in
Volume 6-Part 2
ROCEEDINGS «F
SY PiSI IN
URE THE
A ERICAN MATHE ATiCAL SOCIETY

PROCEEDINGS OF SYMPOSIA
IN PURE MATHEMATICS
Volume 46, Part 2
Algebraic Geometry
Bowdoin1985
Spencer J. Bloch, Editor
with the collaboration of H. Clemens,
D. Eisenbud, W. Fulton, D. Gieseker, J. Harris,
R. Hartshorne, and S. Mori
American Mathematical Society
Providence, Rhode Island

PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE
ON ALGEBRAIC GEOMETRY
HELD AT BOWDOIN COLLEGE
BRUNSWICK, MAINE
JULY 8-26, 1985
with support from the National Science Foundation, Grant DMS-8415200
1980 Mathematics Subject Classification (1985 Revision):
Primary 14-06, 14-XX, 11G25, 11L40, 11Q25, 11R39, 13CXX, 13D05, 13E05, 13H15,
32JXX, 53C55.
Secondary 11G40, 13DXX, 13E15, 13F20, 13H10, 20C99, 22E55, 30F30, 32G13,
43A32, 55P62, 57D35.
Library of Congress Cataloging-in-Publication Data
Summer Research Institute on Algebraic Geometry (1985: Bowdoin College)
Algebraic geometry: Bowdoin 1985/Spencer J. Bloch, editor; with the collaboration
of H. Clemens. . . [et al.].
(Proceedings of symposia in pure mathematics, 0082- 0717; v. 46)
Proceedings of Summer Research Institute on Algebraic Geometry, Bowdoin College,
Brunswick, Maine, July 8-26, 1985.
Includes bibliographies.
1. Geometry, Algebraic-Congresses. I. Bloch, Spencer. II. Clemens, C. Herbert (Charles
Herbert), 1939- . III. American Mathematical Society. IV. Title. V. Series.
QA564.S86 1985 516.3'5 87-12306
ISBN 0-8218-1476-1 (part 1)
ISBN 0-8218-1480-X (part 2)
ISBN 0-8218-1481-8 (set) (alk. paper)
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Contents
PARTI
Algebraic Varieties; General Results
On varieties of minimal degree
David Eisenbud and Joe Harris 3
On the topology of algebraic varieties
William Fulton 15
Curves
Fay's trisecant formula and a characterization of Jacobian varieties
Enrico Arbarello
Deformations and smoothing of complete linear systems on reducible curves
Mei-Chu Chang and Ziv Ran
Complete subvarieties of the moduli space of smooth curves
Steven Diaz
The irreducibility of the Hilbert scheme of smooth space curves
Lawrence Ein
On theta functions for Jacobi varieties
R. C. Gunning
Curves and their moduli
Joe Harris
On the classification of algebraic space curves, II
Robin Hartshorne
The rationality of certain spaces associated to trigonal curves
N. I. Shepherd-Barron
Surfaces
Canonical rings and "special" surfaces of general type
F. CATANESE
An introduction to the geography of surfaces of general type
ULF PERSSON
Threefolds
Contributions to Riemann-Roch on projective 3-folds with only canonical
singularities and applications
A. R. Fletcher 221
iii
49
63
77
83
89
99
145
165
175
195

IV
CONTENTS
Vanishing theorems for cohomology groups
JÂNOS KOLLÂR 233
Deformation of a morphism along a foliation
YOICHI MlYAOKA 245
Classification of higher-dimensional varieties
Shigefumi Mori 269
Tendencious survey of 3-folds
Miles Reid 333
Young person's guide to canonical singularities
Miles Reid 345
Affine Algebraic Geometry
Classification of noncomplete algebraic varieties
TAKAO FUJITA 417
The Zariski decomposition of log-canonical divisors
YUJIRO KAWAMATA 425
Open algebraic surfaces with Kodaira dimension -oo
M. MlYANISHI AND S. TSUNODA 435
Classification of normal surfaces
FUMIO SAKAI 451
Divisors with finite local fundamental group on a surface
A. R. SHASTRI 467
PART 2
Groups in Algebraic Geometry
Rationality of fields of invariants
Igor V. Dolgachev 3
Automorphisms of K3-like rational surfaces
Brian Harbourne 17
Fundamental groups of the complements to plane singular curves
A. LlBGOBER 29
Galois coverings in the theory of algebraic surfaces
B. MOISHEZON AND M. TEICHER 47
Vector Bundles
Geometry of the Horrocks-Mumford bundle
Klaus Hulek 69
Variétés de modules de faisceaux semi-stables de rang élevé sur P2
J. Le Potier 87
Vector bundles and submanifolds of projective space: Nine open problems
Michael Schneider 101

CONTENTS v
Geometry in Characteristic p
F-isocrystals and p-adic representations
Richard Crew ill
Foliations and inseparable morphisms
TORSTEN EKEDAHL 139
Deligne's ^-adic Fourier transform
LUC ILLUSIE 151
Lifting algebraic curves, abelian varieties, and their endomorphisms to
characteristic zero
FRANS OORT 165
Hodge Theory
The geometry of the extension class of a mixed Hodge structure
James A. Carlson 199
The local geometry of the Abel-Jacobi mapping
Herbert Clemens 223
Generic Torelli and infinitesimal variation of Hodge structure
DAVID A. COX 235
The geometry of the mixed Hodge structure on the fundamental group
Richard M. Hain 247
Degeneration of mixed Hodge structures
Steven Zucker 283
Enumerative Geometry
Hilbert scheme of points: Overview of last ten years
A. IARROBINO 297
Intersection theory and enumerative geometry: A decade in review
Steven L. Kleiman with Anders Thorup 321
Completed quadrics and linear maps
DAN LAKSOV 371
Local Chern characters and intersection multiplicities
Paul C. Roberts 389
Enumerating contacts
Robert Speiser 401
Algebraic Cycles
Cycles on arithmetic schemes and Euler characteristics of curves
Spencer Bloch 421
Zero-cycles and Zf-theory on singular varieties
MARC LEVINE 451
Zero cycles modulo rational equivalence for some varieties over fields of
transcendence degree one
Chad Schoen 463

VI
CONTENTS
Rational equivalence of 0-cycles on normal varieties over C
V. SRINIVAS
Commutative Algebra
Syzygies: The codimension of zeros of a nonzero section
E. Graham Evans, Jr. and Phillip Griffith
Intersection problems and Cohen-Macaulay modules
MELVIN HOCHSTER
Decompositions of torsionfree modules over afBne curves
Roger Wiegand and Sylvia Wiegand

Groups in Algebraic Geometry

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Rationality of Fields of Invariants
IGOR V. DOLGACHEV
1. Introduction. Let G be an affine algebraic group acting linearly on a
complex vector space V of finite dimension. The G-invariant rational functions
on the affine space V form a field C(V)G. The problem which is considered here
is:
Whether C(V)G is a purely transcendental extension of C.
Apparently this is a very old problem (cf. [Bu, p. 360]). In geometrical terms
it can be expressed as follows. Let V be the open G-invariant subset of V
such that the geometric quotient space X = V'/G exists in the category of
algebraic varieties [RI, M]. The problem is: whether X is a rational variety.
Passing to the associated projective representation of G in P(V), we may ask
a similar question about the rationality of the orbit space P(V)'/G. It follows
from a result of Rosenlicht [R2] that the rationality of P(V)'/G implies the
rationality of V'/G. Since both orbit spaces are obviously unirational, they are
rational if dimP(F)7G < 2 by Castelnuovo's theorem. Note also that P(V)'/G
is birationally isomorphic to {V ® C)'/G, where G acts trivially on C.
By a result of E. Fischer [F] the answer to our problem is positive if G is an
abelian group. However, this is not the case in general. An example was recently
provided by D. Saltman [Sa]. In this example G is a finite solvable group which
acts on V by means of its regular representation. The rationality of k(V)G in
this case was conjectured by E. Noether. Earlier counterexamples to the Noether
conjectures were known in the case where the ground field is not algebraically
closed (see [Ke]). It is still hoped that the answer is positive in the case where
G is a connected complex algebraic group.
One of the main interests of algebraic geometers in this problem is explained
by the fact that many moduli spaces are birationally equivalent to the orbit
spaces P(Vy/G. The purpose of this talk is to give a survey of some known
general methods of attacking the problem of rationality of k(V)G and to illustrate
them in examples of some rational moduli spaces.
1980 Mathematics Subject Classification (1985 Revision). Primary 14M20; Secondary 14D99,
20C99.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
3

4
I. V. DOLGACHEV
In the future we will denote by X/G any variety birationally isomorphic to the
orbit space X'/G. We also let the symbol S denote the birational equivalence.
The author is very grateful to F. Bogomolov, H. Kraft, and N. Shepherd-
Barron for all their help and advice during the preparation of this article.
2. Explicit computation. This is the most straightforward approach to
the problem. Here one computes explicitly the algebra A(V, G) = S(V*)G of G-
invariant polynomial functions on V. The algebra A(V,G) has a natural
structure of a graded C-algebra; the nth graded piece A(V, G)n consists of invariant
polynomials of degree n. If the character group of G is finite, the field C(V)G
(resp. C(P(V))G) is isomorphic to the field of fractions of A(V, G) (resp.
homogeneous fractions). One searches for a nice system of homogeneous generators
of A(V, G) satisfying the simplest possible relations.
One can be especially lucky if it turns out that A(V, G) is isomorphic to the
polynomial algebra. If G is finite, this happens if and only if the action of G in V
is generated by complex pseudoreflections. The complete list of representations
(V, G) for which this can happen was given by G. Shephard and J. Todd [ST]. In
the case where G is a connected simple group, a list of irreducible (resp. reducible)
representations with A(V, G) isomorphic to the polynomial algebra was provided
by V. Kac, V. Popov, and E. Vinberg [KPV] (resp. O. Adamovich-E. Golovina
[AG] and G. Schwarz [Se]). A partial list for a semisimple group was given by
V. Popov [PI]. He proved also that the number of isomorphism classes of such
representations is finite [P2].
As a very special example, let us mention a standard representation of the
Weyl group W of a simple Lie algebra of rank n on the space V = Q(R)®C, where
Q(R) is the root lattice of rank n. For example, W can be the symmetric group
Sn+i which acts in Cn by permuting the coordinates of points (xi,..., xn+i) £
Cn+1 lying in the hyperplane x\ H h zn+i = 0.
We refer to [Sta] for more information about A(V, G) when G is finite.
EXAMPLE 1. Let V = 56(C2)* be the space of binary sextic forms and
G = SL(2,C). The algebra A(G,V) was computed by A. Clebsch [CI] (see a
modern proof in [I]). It is generated by homogeneous polynomials I2, I4, Iq,
J10, and J15 of the degree indicated in the subscript. They satisfy a single basic
relation:
hb = ^30(^2,^4,^6,^10),
where P30 is a quasi-homogeneous polynomial of degree 30 with weights 2, 4,
6, and 10 (expressed as a certain determinant). Let X = I4/I2, Y = h/I^,
Z = Iio/tf. Then Il^/I^ is expressed as a polynomial in X, y, Z. Let
W = hs/ll e C(V)G. Then I2 = (^V^isKW^)2 is a rational function in
X, y, Z, W. Hence J4, 76, -^10, and J15 are rational functions in X, Y, Z, W
and C{V)G S C(X,Y,Z,W). Also C(P(F))G = C{X,Y,Z). This shows that
V/G and P(V)/G are rational varieties [I].
EXAMPLE 2. Let V = S3(C4)* be the space of ternary cubic forms on C4 and
G = SL(4, C) which acts naturally on V. The algebra A(V,G) was computed

RATIONALITY OF FIELDS OF INVARIANTS
5
explicitly by Clebsch and Salmon (see [P] and a rigorous proof in [Be]). It is
generated by the invariants Ig, lie, J24, I32, ho, and J100, where the subscript
denotes the degree. They satisfy one relation
hoo = Psoih, I161 h*, h2, ho),
where P50 is a quasi-homogeneous polynomial of degree 50 with weights 2, 4, 6,
8, and 10.
The projective space P(V) is isomorphic to the space of cubic surfaces in P3.
The orbit space P(V)/G is birationally isomorphic to the moduli space of cubic
surfaces. Its field of rational functions is isomorphic to the field of G-invariant
rational functions on V which are homogeneous of degree 0. Each such function
can be written as a ratio of two homogeneous polynomials of the same degree
d. Multiplying the numerator and the denominator by J100, if needed, we may
assume that d is divisible by 8. Then the ratio can be written as a rational
function in he/Ig, ht/ll, Im/I*, ho/ll- Thus P(V)/G is a rational variety.
3. The slice method. A subspace E of V is called a slice if the following
two conditions are satisfied:
(i) the orbit of E is an open Zariski subset of V;
(ii) there exists an open subset E' of E such that g(x) G E for x G E' and
g eG implies g(E) c E.
It can be proven that for such a slice E,
C{Vf = C{E)N^E\
where
N(E) = {geG:g(E)cE}.
This method allows us to reduce the problem to the case of smaller space V and
smaller group G.
EXAMPLE 3 [Ba]. We consider again the problem of Example 2. By a theorem
of Sylvester [Se], there exists an open Zariski subset of V such that every F G V
can be written uniquely in the form:
F = A0/io + Aj/i? + A2/12 + A3/13 + Xihl,
where hi G (C4)* are linear forms which are four by four linearly independent
and add up to zero, and Ai are nonzero constants. Let g be a projective
transformation which sends the forms hi (0 < i < 3) to the coordinate functions Z{.
Then
g{F) = A0sg + A12? + A2*f + A3^| + A4*i
where 24 = —(^0+^1+^2+^3)- The 5-dimensional subspace W of V consisting of
cubic forms Ao^o+^1^1+^2^|+^3;2r3+^4;2r4 is a s^ce in V witn N(W) = S5 which
acts on W by permuting the coefficients At. Obviously W/S5 is rational, and the
algebra A(W, S5) is freely generated by the elementary symmetric functions.
REMARK. One should not confuse the existence of a slice with the existence
of "canonical forms" as in the above example. In general, a canonical form is a
nonlinear subvariety in V whose orbit is a Zariski open subset.

6
I. V. DOLGACHEV
4. The no-name method (reducible representations). I have learnt this
method from F. Bogomolov. I believe it was known to many other people (e.g.,
to Lenstra in the case of finite groups (see [Ke]) or to Shepherd-Barron [S-B2]).
We say that a reductive algebraic group G acts almost freely on an irreducible
algebraic variety X if there exists a Zariski open subset U of X such that the
stabilizer subgroup Gx = {1} for any x EU. In this case G acts freely on an open
subset of X (i.e., a canonical morphism GxX—>XxXis& closed immersion
[M]).
An example of an almost free linear representation is a natural irreducible
representation of GL(n) on the space 5d(Cn)* of homogeneous forms of degree
d in n variables where d > 2, (n,d) # (2,3), (2,4)(3,3). This follows from
the nonexistence of global vector fields on nonsingular hypersurfaces in Pn_1
of degree > 2 [KS, Lemma 14.2], which implies that the stabilizer subgroup of
the general point of V is finite. All irreducible representations of semisimple Lie
groups with nontrivial finite stabilizer subgroups of general points are known
[Po].
MAIN LEMMA. Let G be a reductive algebraic group which acts almost freely
on an irreducible variety X, and let E be a G-linearized vector bundle on X [M].
Then E/G is birationally isomorphic to the total space of a vector bundle on
X/G.
PROOF. By the theorem of Rosenlicht [Rl] and Luna's étale slice theorem
[Lu] there exists an open subset U of X such that the quotient U —► U/G exists
and is a principal bundle in the étale topology. In particular, it is an absolutely
flat morphism. The restriction Eu of E to U is a vector bundle satisfying a
descent data which comes from the structure of a G-linearized bundle. According
to Grothendieck, this data is effective, i.e., there exists a vector bundle E' on U/G
whose pull back to U is isomorphic to Eu as a G-linearized bundle ([SGA], cf.
also [M, p. 32]). In particular, this implies that Eu/G is birationally isomorphic
to £", hence the assertion of the lemma.
REMARK. Instead of assuming that G acts freely on [/, one may assume that
for every x EU the stabilizer Gx acts trivially on the fibre Ex [Kr].
COROLLARY 1. Let V = V\ ©Vb be a reducible representation ofG. Assume
that the action of G on Vb is almost free and the orbit space V2/G is rational.
Then V/G is rational.
PROOF. View V as a G-linearized vector bundle over V2 and apply the lemma.
COROLLARY 2. Let W be a linear almost free representation of G with
rational W/G. For any two linear representations V\ and V% of G such that V\ is
almost free and dimV2 > dim IV, the quotient (V\ © Vï)/G is rational.
PROOF. By Corollary 1, X = Vi®W/G is rational. Projecting to Vi/G, we see
that X is birationally isomorphic to (Vi/G) x CdimW. Projecting (Vi © V2)/G

RATIONALITY OF FIELDS OF INVARIANTS
7
to V\/G, we obtain that it is birationally isomorphic to (V\/G) x CdimV2 =
X x Cdimy2-dimVy; hence it is rational.
Recall that a variety X is said to be stably rational if V x An is rational for
some n. A recent example of a nonrational but stably rational variety was given
in [BCSS].
COROLLARY 3. Assume that G admits an almost free linear representation
V with a rational quotient V/G. Then for every almost free linear representation
W the quotient W/G is stably rational.
PROOF. Let G act diagonally on V x W. By the Lemma V x W/G is
birationally isomorphic to a vector bundle over a rational variety V/G. Hence
it is rational. On the other hand, V x W is birationally isomorphic to a vector
bundle over W/G. Hence W/G is stably rational.
THEOREM [Bl]. Let G be a connected semisimple simply connected algebraic
group with no simple factor of type Eg. Then there exists an almost free linear
representation of G with rational quotient.
EXAMPLE 4 [S-B2]. Let C C P5 be a canonical nonsingular curve of genus
6. It is known that C lies on a unique Del Pezzo surface S of degree 5 and is
cut there by a unique quadric Q. Since any two Del Pezzo surfaces of degree 5
are projectively equivalent, the component of the Hilbert scheme parametrizing
canonical curves of genus 6 is birationally isomorphic to the projective space
pi5 ^ |05(2)| of dimension 15. It follows easily from this that the moduli space
Mq of curves of genus 6 is birationally isomorphic to the orbit space P15/ Aut(S),
where Aut(S) = S5 is the automorphism group of S. In the presentation of S
as the blowing up of 4 points pi,...,p4 in P2, the automorphisms of S
correspond to Cremona transformations of the plane with the fundamental points
among the points p\,..., P4 which can be taken as the standard points of
reference (1,0,0)(0,1,0), (0,0,1), and (1,1,1). The elements of the space |05(2)|
correspond to plane curves of degree 6 with at least double points at the p^'s. It
is very easy to analyze explicitly the action of S5 on this space. In particular,
we find that |Os(2)| contains exactly two Ss-invariant curves corresponding to
two So-invariant quadrics in P15. The first corresponds to the curve L G |Os(2)|
which is the union of all lines on S ; it is represented by the sextic which is the
union of 6 lines joining the pairs of the points p^'s. The second one is a smooth
curve T G |Os(2)| which is the proper transform of the sextic:
2Y,*U* ~2Y1 xîxiXk ~2Y1 xix3J " Y,x^x2JXk + 6*0*1*2 = 0-
Here the summation is taken over the set of distinct indices i,j, k G {0,1,2}.
In the corresponding linear representation, V = T(S, Os(2)), S5 has two
invariant one-dimensional subspaces V\ and V2 corresponding to L and T
respectively. On V\ the group S5 acts by the sign character; on the second one it acts
trivially. Next we observe that V\ and V2 are the only one-dimensional invariant
subspaces of V. In fact, the existence of an S^-invariant curve from |Os(2)| not

8
I. V. DOLGACHEV
belonging to the pencil spanned by L and T would imply the existence of S5-
invariant g20 on T. This is obviously absurd. Looking into the list of irreducible
representations of S5, we see that V must contain an irreducible representation
W as an irreducible factor. This representation is isomorphic to the standard
representation of the Weyl group of type A4 (see §2). Since W is an almost free
representation with rational quotient, we deduce from Corollary 2 that V0C/S5
is rational. Hence Me is rational.
5. The method of covariants (irreducible representations). Recall
that a covariant is a polynomial G-equivariant map <p:V\ —► V2 between two
representations of a group G. It induces a rational map ïp:Vi/G —► V2/G. The
following idea is due to N. Shepherd-Barron [S-B2]. Assume that we can prove
that Tp makes V\/G birationally isomorphic to V2/G x An and the projection
V2 —► V2/G makes V2 birationally isomorphic to (V2/G) x Ar for some n and r.
Then
Vi/G S V2/G x An S (Va/G x Ar) x An~r S V2 x An~r 3 AN.
Shepherd-Barron applies this idea to some cases G = SL(n), V\ = Sd(Cn)*,
V2 = Sr(Cn)*, where A: is chosen minimal with the property that G acts almost
freely on V2 (e.g., k = 4 for n = 3). The covariant <p is given by using the
symbolic method [GY].
Another idea belongs to P. Katsylo [Kl]. One finds a good slice H in V2,
then considers its inverse transform R = ^_1(iif) in Vi, and proves that R
is birationally isomorphic to a 7V(if)-slice M in Vi. Finally one checks that
V\/G S M/N(H) is rational. Katsylo successfully applies this method to the
cases Vi = S2n(C2)*, V2 = S4(C2)*, n ? 4 [K2].
Finally in [BK] one more approach is suggested. A bilinear G-equivariant
map <p: V x U —► W of representations of a group G is said to be a double
fibration if there exists an irreducible component X of ^_1(0) of dimension
equal to dim V + dim U — dim W which is mapped dominantly by the projections
V x U — V and V x U — U. If <p: V x U — W is a double fibration with dim V >
dimC/ = dim W + 1, then the rationality of P(C/)/G implies the rationality of
P(V)/G. Using this method, Bogomolov and Katsylo prove the rationality of
P(S2n(C2)*)/PGL(2) for all n [BK].
6. Solvable groups. By a result due to Miyata [Mi] and Vinberg [V], the
orbit space V/G is rational, if G is a solvable connected algebraic group. One
applies the slice method seeking the slice W such that N(W) is a solvable connected
subgroup.
EXAMPLE 5. Let X2 be the universal curve of genus 2, i.e., the coarse moduli
variety of isomorphism classes of pointed curves of genus 2. Let (X,p) G X2,
where X is a nonsingular projective curve of genus 2 and p G X. The linear
system \KX + 2p\ maps X to P2 and its image is a plane quartic X with a cusp
p. It is easy to see that the map
(X,p) H-+ (X,p)/projective equivalence

RATIONALITY OF FIELDS OF INVARIANTS
9
defines a birational isomorphism between X2 and the orbit space QC/PGL(3),
where Qc is the variety of cuspidal plane quartics. By a projective transformation
we can fix the cusp and its tangent cone of such a quartic. For example, we may
assume that the tangent cone is the line xq = 0 and the cusp is the point (0,0,1).
This shows that Qc/ PGL(3) is birationally equivalent to the orbit space P(V)/G,
where V is the subspace of 54(C4)* consisting of quartic ternary forms
X2Xq+X2{ciiXq +a,2xlxi+asXoxl +aiXQ)+(a5XQ+a6XQXi+a7xlxl+agx0xl+a9xj)
and G consists of triangular projective transformations
xo »—► £0, x\ »—► axo + /?xi, X2 »—► 7x0^^1 + ^x2-
Since G is solvable, the orbit space is rational (§6). Thus, we see that X2 is
rational.
7. Special groups [Kr]. This was communicated to me by H. Kraft. Recall
[Gr] that an algebraic group G is said to be special if every principal G-bundle
is locally trivial in Zariski topology.
Let G be a reductive algebraic group which acts on an affine variety X with
an orbit O isomorphic to G. We say that the orbit O has a local G-retraction if
there is an open G-stable neighborhood U of O and a G-equivariant retraction
r: U —► O such that r\0 = ido-
PROPOSITION (LUNA). The following are equivalent:
(i) G is special.
(ii) There exists a linear representation V of G containing an orbit O = G.
(iii) For any affine variety X on which G acts and a closed orbit O isomorphic
to G there is a local G-retraction of O.
COROLLARY l. The groups GL(2), SL(2), Sp(n) are special.
PROOF. Look at the representations of these groups on the space of matrices
Mn by left multiplication.
COROLLARY 2. Let G be a special algebraic group and V be its almost free
linear representation. There exists an open subset V C V such that
V' = G x (V'/G)
as G-varieties.
COROLLARY 3. Let G be a special algebraic group and V be its almost free
linear representation. For every Borel subgroup B of G there exists an open
V C V such that
[V'/G) x {G/B)
is rational. In particular V'/G is stably rational.
PROOF. By the previous section, V'/B is rational for some open V C V.

10
I. V. DOLGACHEV
COROLLARY 4. Let G be a special algebraic group and V be its linear
representation. Assume that G acts almost freely on T*{V). Then P(V)' jG x {G/B)
is rational for some open P(V)' Ç P(V).
EXAMPLE 6. Let V be an almost free linear representation of G = SL(2).
Then V/G is rational. In fact, since G/B = P1, Corollary 4 implies that
P{V)'/G x P1
is rational. But the latter is birationally equivalent to V'/G for some open
V C V. Indeed, the projection V'/G —► P(V)'/G is a principal C*-bundle and
C* is special.
This gives another proof of the Bogomolov-Katsylo theorem.
EXAMPLE 7. For every reducible representation of SL(3) with at least two
nontrivial factors, the orbit space V/ SL(3) is rational. In fact, dim SL(S)/B = 3
is less or equal to the dimension of any nontrivial irreducible representation of
G. Thus, we may apply the no-name method (Corollary 2).
8. Moduli problems. Here we give some examples of moduli spaces in
algebraic geometry which are birationally equivalent to the orbit spaces P(V)/G.
We will call the birational type of this form the linear type.
(a) Mg, the coarse moduli space of curves of genus g. It is known that Mg is
of linear type if g < 6. We have seen this already for g = 2 (Example 2) and
g = 6 (Example 4). The cases g = 0 and 1 are trivial.
The cases g = 3 and 4 are easy. The canonical model of a nonsingular nonhy-
perelliptic curve of genus 3 is isomorphic to a plane nonsingular quartic curve.
In this way, we find that M3 is birationally isomorphic to 54(C4)*/GL(3).
The canonical model of a nonsingular nonhyperelliptic curve of genus 4 is
isomorphic to the intersection of a quadric Q and a cubic in P3. If the curve
does not have an effective even theta characteristic, the quadric Q is
nonsingular. Fixing this quadric by a projective transformation, we obtain that M4 is
birationally equivalent to the orbit space |Oq(3)|/Aut(Q).
The canonical model of a nonsingular nonhyperelliptic curve of genus 5 is
isomorphic to the intersection of 3 quadrics in P4. This can be used to show that the
moduli space M5 is birationally isomorphic to the orbit space G(3,15)/SL(5),
where G(3,15) is the Grassmann variety of 3-dimensional linear systems of
quadrics in P4. It is known that G(3,15) = Ms,id/ GL(3), where the group
GL(3) acts on the space of 3 x 15-matrices by left multiplication. Thus,
M5SM3ii5/GL(3) xSL(5)
is of linear type.
We have seen already the rationality of Mg for g < 2 and 6. The only other
known case of rational Mg is the case M4 [S-Bl]. We will discuss the case M3
in the next section.
Note that it is known that Mg is of Kodaira dimension —00 for g < 15 and of
nonnegative Kodaira dimension for g > 19.

RATIONALITY OF FIELDS OF INVARIANTS
11
(b) 0ig, the coarse moduli space of hyperelliptic curves of genus g. These
spaces are all of linear type. The set of the branch points of the unique g\ on a
hyperelliptic curve of genus g is an unordered set of 2g + 2 points in P1. It is easy
to see that 0ig is birationally isomorphic to the orbit space (P1)^2g+2VPGL(2).
The latter is of linear type 52g+2(C2)VGL(2). The rationality of this orbit
space was first proven by P. Katsylo for g ^ 4 [K2] and later by F. Bogomolov
and P. Katsylo for all n [B2, BK]. The proof uses the method of covariants.
(c) Hyp(d, n), the coarse moduli space of hyper surfaces of degree d in Pn.
These spaces are obviously of linear type Sd(Cn+1)*/GL(n+l). The rationality
of these spaces is known in the following cases:
n = 1, d is odd [K2], n is even (see (b));
n = 2, d < 3 (trivial), d = 5 (slice method), and d = 1 (mod9), d = 10
(covariants) [S-B2];
n = 3, d < 2 (trivial), d = 3 (see Examples 1 and 3);
n > 3, d < 2 (trivial).
(d) MKs{d), the coarse moduli space of polarized KZ surfaces of degree d.
Its points are the isomorphism classes of pairs (5,/), where S is a nonsingular
projective surface of type KS and / E Pic(S) is the class of an ample divisor with
I2 = d= 2k. It is known that MKs{d) is of linear type for d = 2,4,6, and 10. In
fact,
Hyp(6,2) if d = 2,
Hyp(4,3) ifd = 4,
|Oq(3)|/PGL(5) ifd = 6,
|Ox(l)|/Aut(X) if d = 10,
where Q is a nonsingular quadric in P4, X is a certain Fano threefold embedded
into P22 by the anticanonical linear system |—Kx\, and Aut(X) = PGL(2). The
first three isomorphisms are rather obvious; the last one is due to S. Mukai. He
also shows that H°(X,Ox(-Kx)) = V(0) © V(8) © V(12) as a representation
of PGL(2), where V(n) is the space of binary forms of degree n. Applying the
no-name method and the Bogomolov-Katsylo result, we obtain the rationality of
Mk3(10). This is the only known case where I know the rationality of MK3{d).
It would be very interesting to know whether Mks {d) is of general type for large
d (by analogy with Mg).
(e) Mjç;(d), the coarse moduli space of polarized Enriques surfaces of degree
d = 2k. The definition of this space is similar to the definition of Mj^d). For
d = 2 and d = 6, this space is of linear type.
If d = 2, a general point of Me{2) is represented as a double cover of a 4-
nodal quartic Del Pezzo surface ScP4 branched along a nonsingular curve W G
|05(2)| and the four nodes. In this way, Me{2) becomes birationally isomorphic
to |05(2)|/Aut(S) (also birationally isomorphic to the coarse moduli space M5
of curves of genus 5 with two vanishing theta constants). The rationality of this
space can be proven by explicit computations of invariants (I. Dolgachev).
The space Mjç;(6) is birationally isomorphic to the orbit space P13/G, where
G is the normalizer of the maximal torus C*3 in SL(4). In fact, the classical
MK3(d)

12
I. V. DOLGACHEV
construction of an Enriques surface as a sextic surface passing twice through
the edges of the coordinate tetrahedron shows that a general point of Me(6) is
isomorphic to a surface in P3 given by the equation:
x0x1X2Xs<p(x0,Xl' X2> Xs>X4^+ coxix2xl + c\x\x\x\ + o^x\x\x\ + c?>x\x\x\ = 0,
where <p is a quadratic form and ct ^ 0. The group G acts naturally by permuting
the coordinates and multiplying them by constants A^ with À0A1À2A3 = 1.
Normalizing the coefficients by requiring that C0C1C2C3 = 1, we represent the space
of the coefficients as the product C10 x C*3, on which G acts diagonally. Since
C*3 is a homogeneous space with respect to G, the rationality of the quotient
C10 x C*3/G follows from the no-name method.
The period map for Enriques surfaces defines a quasi-finite map P:Me(2) —►
Me, where Me(2) is the period variety whose open Zariski subset parametrizes
the isomorphism classes of Enriques surfaces. The degree of this map is 27 •
17 • 31 = 24(25 - 1)23(24 + 1) [BP]. On the other hand there is a natural map
of the same degree from M'l % Me{2) to the coarse moduli space M'5 trig of
trigonal curves of genus 5 with one vanishing theta constant. The latter space is
birationally isomorphic to the moduli space of cuspidal plane quintic curves and
is rational by Vinberg's theorem. This suggests that Me may be rational too.
(f) Coarse moduli spaces of surfaces of general type. The simplest approach
to proving that such a space is of linear type is the following. Let A(F) =
0n>oi/°(F, Of{tiKf)) be the graded canonical ring of a surface F of general
type. Choosing appropriate homogeneous generators of this ring, one embeds the
canonical model F' = Proj(A(F)) into a weighted projective space P(q). It may
happen that F' is a hypersurface of degree d in P(q). In this way, the moduli
space is birationally isomorphic to the orbit space |Op(q)(d)|/ Aut(P(q)). The
automorphism group of P(q) is often solvable. This allows us to apply Vinberg's
theorem. Even when F' is not a hypersurface, one can find a nice resolution for
the ideal of F' in P(q) allowing us in some cases to prove the rationality of the
moduli space [C2].
(g) Moduli spaces of semistable vector bundles. In a few cases where the
rationality of these spaces is known, the proof follows directly from the geometry
and does not use much of the technique described in this article. Nevertheless,
we list the known cases for completeness.
The rationality of the moduli spaces M(g, r, d) of semistable vector bundles E
of rank r and c\ (E) = d on a curve of genus g is not known. (The paper of P.
Newstead [N], where this result was claimed, contains a mistake.)
The rationality of the moduli space M(n, m) of semistable vector bundles E
of rank 2 on P2 with deg(ci(£')) = n and 02(E) = m was proven by W. Barth
[Bar] in case n is even and by K. Hulek [H] in case n is odd. The proof uses a
little of the no-name method (the Main Lemma).
The rationality of the moduli space M(n, m, A:) of semistable reflexive sheaves
E of rank 2 on P3 with deg(ci(£')) = n, deg(c2(£)) = m, and 03(E) = k is

RATIONALITY OF FIELDS OF INVARIANTS
13
known in the following cases (see [Ch]):
n = 0: m = 1 and 2; m = 3, A: = 4 or 8;
n = -l: (m,A:) = (2,2),(3,3).
Note that by normalizing, we can assume that n = 0 or — 1.
9. Del Pezzo surfaces. Here I will give a "philosophical proof" of the
rationality of M3. Recall that M3 is birationally isomorphic to the moduli space
Hyp(4,2) of quartic curves. Let C be a nonsingular plane quartic curve. The
double cover of P2 branched along C is a Del Pezzo surface S of degree 2 [De].
It is isomorphic to the blowing up of 7 points in P2. Conversely, if S is a Del
Pezzo surface of degree 2, the regular map given by the anticanonical linear
system |—Ks\ is a double cover of P2 branched along a quartic curve. In this
way, we obtain a birational isomorphism between the spaces Hyp(4,2) and the
moduli space DCP(2) of Del Pezzo surfaces of degree 2. Now, we recall that the
moduli space DCP(3) of Del Pezzo surfaces of degree 3, being isomorphic to the
moduli space of cubic surfaces, is rational (Example 1). All the moduli spaces
DCP(cf) are rational for d > 3 by the dimension argument. Let us see that 1XP(1)
is rational too. A general Del Pezzo surface S of degree 1 is obtained by blowing
up 8 points pi,... ,p8 in general position in P2. The anticanonical system on
S is represented by the linear pencil of plane cubic curves with base points at
the points pi. Blowing up the ninth base point po of this pencil, we obtain a
relatively minimal rational elliptic surface with a fixed zero section (the
exceptional line blown up from po). In this way we obtain that D9(l) is birationally
isomorphic to the moduli space of relatively minimal rational jacobian surfaces.
By the theory of Weierstrass models of elliptic fibrations [MS], each such surface
is uniquely determined by a pair (a, b) G V(4) 0 V(6) up to the action of GL(2).
Thus, we obtain that
1XP(1)SV(4)©V(6)/GL(2).
The rationality of the latter orbit space immediately follows from the rationality
of V(6)/GL(2) (Example 1) by applying the no-name method.
Note that the anti-bicanonical map of a Del Pezzo surface of degree 1 is a
double cover onto a singular quadric in P4 with the canonical curve of genus 4
as the branch locus [De]. Lying on a singular quadric, the latter curve has a
vanishing theta constant. One can reverse this construction and show that
aXP(l)»Mi,
where M4 is the moduli space of curves of genus 4 with a vanishing theta constant.
This implies the rationality of the latter space.
Thus we can prove the rationality of all the spaces DCP(cf) except DCP(2) S M3.
Is this not a convincing evidence that M3 must be rational?
If it is not enough for you, add another one. Let P% = (P2)n/PGL(3), where
PGL(3) acts diagonally. This is a rational variety [M]. For 6 < n < 8, the Weyl
group W(En) of root system En acts on an open subset of P^ w^h tne quotient

14
I. V. DOLGACHEV
space isomorphic to IXP(9 — n) [Do]. We know that this quotient is rational for
W(E6) and W(Eg), but we do not know this for W(E7).
10. A list of known rational moduli spaces of curves. Here I
summarize all examples of rational moduli spaces which are known to me. No
completeness of this list is claimed.
We will use the following notation (all varieties are defined up to a birational
equivalence):
Mg, the coarse moduli space of nonsingular projective curves of genus g\
Xg, the coarse moduli space of nonsingular projective curves of genus g with
an ordered point;
!Kg, the coarse moduli space of nonsingular hyperelliptic projective curves of
genus g\
Mg , the coarse moduli space of nonsingular projective curves of genus g > 4
with n vanishing theta constants;
ftg, the coarse moduli space of nonsingular projective curves of genus g
together with a nontrivial divisor class of order 2;
6®v, the coarse moduli space of nonsingular projective curves of genus g
together with an even theta characteristic;
Ggdd, the coarse moduli space of nonsingular projective curves of genus g
together with an odd theta characteristic;
Mg(2), the coarse moduli space of nonsingular projective curves of genus g
together with a 2-level structure (i.e., a symplectic isomorphism between F%g
and the group of divisor classes of order 2).
Space
Mi
M2
M4
Me
Xi
%2
X3
X4
'Kg
U^
^
*i
#2
#3
#4
Credit
K. Weierstrass
J. Igusa
N. Shepherd-Barron
N. Shepherd-Barron
everyone
?
N. Shepherd-Barron, ?
F. Catanese [C2]
P. Katsylo, F. Bogomolov
I. Dolgachev
I. Dolgachev
F. Klein
?
F. Catanese
F. Catanese [C2]
Method
explicit computation
explicit computation
slice
no-name
algebraic surfaces
solvable groups
solvable groups
explicit computation
covariants
no-name (see §8)
explicit computation
algebraic curves
solvable groups
no-name
explicit computation

RATIONALITY OF FIELDS OF INVARIANTS
15
Space Credit Method
9£dd ? solvable groups
6§dd F. Bardelli [Ba], E. Looijenga slice
Mi(2) F. Klein algebraic curves
Mg(2), g = 2,3 A. Coble [Co], B. van Geemen Del Pezzo surfaces
Mi1}(2) A. Coble [Co], B. van Geemen Del Pezzo surfaces
References
[AG] O. Adamovich and E. Golovina, Simple groups with free algebra of invariants, Questions
on the theory of groups and homological algebra, Yaroslavl, 2 (1979), 3-41. (Russian)
[B] A. Beauville, J.-L. Colliot-Thélène, J. J. Sansuc, and P. Swinnerton-Dyer, Variétés
stablement rationnelles non rationnelles, Ann. of Math. (2) 121 (1985), 283-318.
[Bl] F. Bogomolov, Stable rationality of factor spaces for simply connected groups, Mat. Sb.
130 (1986), 3-17. (Russian)
[B2] , Rationality of the moduli of hyperelliptic curves of arbitrary genus, (Conf. Algebraic
Geom., Vancouver, 1984), CMS Conf. Proc, vol. 6, Amer. Math. Soc, Providence, R.I., 1986,
pp. 17-37.
[Ba] F. Bardelli and A. del Centina, Nodal cubic surfaces and the rationality of the moduli
space of curves of genus 2, Math. Ann. 270 (1985), 599-602.
[Bar] W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977),
63-91.
[Be] N. Beklemishev, Invariants of cubic forms in four variables, Vestnik Moscov. Univ. Ser.
I Mat. Mekh. 2 (1982), 42-49.
[BK] F. Bogomolov and P. Katsylo, Rationality of certain quotient-varieties, Mat. Sb. (N.S.)
126(168) (1985), 584-589.
[BP] W. Barth and C. Peters, Automorphisms of Enriques surfaces, Invent. Math. 73 (1983),
383-411.
[Bu] W. Burnside, Theory of groups of finite order, Dover, New York, 1911.
[Cl] F. Catanese, On the rationality of certain moduli spaces related to curves of genus 4,
Algebraic Geometry. I, Dolgachev, editor, Lecture Notes in Math., vol. 1008, Springer-Verlag,
Berlin and New York, 1983, p. 30-50.
[C2] , Commutative algebra methods and equations of regular surfaces, Preprint.
[Ch] M.-C. Chang, Stable rank 2 reflexive sheaves on P3 with small C2 and applications, Trans.
Amer. Math. Soc 284 (1984), 57-89.
[Cl] A. Clebsh, Théorie der binaren algebraischen Formen, Teubner, Leipzig, 1872.
[Co] A. Coble, Algebraic geometry and theta functions, Amer. Math. Soc Colloq. Publ., vol.
10, Amer. Math. Soc, Providence, R.I., 1929, 2964.
[De] M. Demazure, Surfaces de del Pezzo. II-V, Lecture Notes in Math., vol. 777, Springer-
Verlag, Berlin and New York, 1983, pp. 23-70.
[Do] I. Dolgachev, Weyl groups and Cremona transformations, Proc. Sympos. Pure Math.,
vol. 40, part 1, Amer. Math. Soc, Providence, R.I., 1983, pp. 283-294.
[F] E. Fischer, Die Isomorphie der Invariantenkôrper der endlicher Abelschen Gruppen linearen
Transformations, Nachr. Kônig. Ges. Wiss. Gottingen (1915), 77-80.
[G] A. Grothendieck, Torsion homologique et sections rationnelles, Sem. Chevalley, vol. 2,
E.N.S., 1958, 5-01-5-29.
[GY] J. Grace and A. Young, The algebra of invariants, Cambridge Univ. Press, Oxford,
1903.
[H] K. Hulek, Stable rank 2 vector bundles on P2 with ci odd, Math. Ann. 242 (1979), 241-
266.
[I] J. Igusa, Arithmetic theory of moduli for genus two, Ann. of Math. (2) 72 (1960), 612-649.
[Kl] P. Katsylo, Rationality of the orbit spaces of irreducible representations of the group SL2,
Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 26-36.

16
I. V. DOLGACHEV
[K2] , Rationality of the moduli spaces of hyerelliptic curves, Izv. Akad. Nauk SSSR Ser.
Mat. 48 (1984), 705-710.
[Ke] M. Kervaire, Fractions rationnelles invariantes, Sem. Bourbaki, vol. 1973/74, no. 445,
Lecture Notes in Math., vol. 431, Springer-Verlag, Berlin and New York, 1975, pp. 170-184.
[KPV] V. Kac, V. Popov, and E. Vinberg, Sur les groupes linéaires algébrique dont
l'algèbre des invariants est libre, C. R. Acad. Sci. Paris Sér. I Math. 283 (1976), 875-878.
[Kr] H. Kraft, a letter to the author.
[KR] J. Kung and G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc.
(N.S.) 10 (1984), 27-85.
[KS] K. Kodaira and D. Spencer, On deformations of complex analytic structures, Ann. of
Math. (2) 67 (1958), 328-466.
[M] D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin and New York, 1965.
[Mi] K. Miyata, Invariants of certain groups. I, Nagoya Math. J. 41 (1971), 69-73.
[MS] D. Mumford and K. Suominen, Introduction to the theory of modules, Algebraic
Geometry, Oslo, 1970, (Proc. Fifth Nordic Summer School in Math.) Wolters-NoordhofF, Groningen,
1972, pp. 171-222.
[N] P. Newstead, Rationality of moduli spaces of vector bundles on algebraic curves, Math. Ann.
215 (1975), 251-268.
[P] C. Pascal, Repertoires, Teubner, Leipzig, 1895.
[PI] V. L. Popov, Representations with free module of covariants, Functional Anal. Appl. 10
(1976), 91-92.
[P2] , A finiteness theorem for representations with free algebra of invariants, Izv. Akad.
Nauk SSSR Ser. Mat. 46 (1982), 347-370.
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position, Functional Anal. Appl. 12 (1978), 91-92.
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489.
[R2] , Tori operating on projective varieties, Rend Mat. e Appl. (5) 25 (1966), 129-138.
[Sa] D. Saltman, Noether's problem over an algebraically closed field, Invent. Math. 77 (1984),
71-84.
[S-Bl] N. Shepherd-Barron, Rationality of spaces associated to trigonal curves, this volume.
[S-B2] , a unpublished manuscript.
[Sc] G. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent.
Math. 50 (1978), 1-12.
[Se] B. Segre, Nonsingular cubic surfaces, Clarendon Press, Oxford, 1942.
[SGA] Revêtements étales et groupe fundamental, Lecture Notes in Math., vol. 224, Springer-
Verlag, Berlin and New York, 1971.
[S ta] R. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer.
Math. Soc. (N.S.) 1 (1979), 475-511.
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University of Michigan

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Automorphisms of i£3-like Rational Surfaces
BRIAN HARBOURNE
Abstract. This paper is an expanded version of the author's talk at the
1985 AMS Summer Institute on Algebraic Geometry at Bowdoin College.
It studies the question of which rational surfaces have infinite groups of
automorphisms (which are nonregular when regarded as Cremona
transformations of the plane). This is poorly understood for rational surfaces of
nonpositive anticanonical dimension k* . But under appropriate
hypotheses, rational surfaces with k* = 0 behave in some ways like K3 surfaces.
We review results regarding automorphism groups of K3 surfaces, and we
define a special class of rational surfaces, the K3-\\ke surfaces, for which all
of these results carry over.
Introduction. After the work of Nikulin [Nl, 2] and Kondo [Ko] on KS
surfaces and Enriques surfaces, it is well understood which smooth complex
projective surfaces have infinite groups of birational transformations. It is now
reasonable to ask, in place of which surfaces have an infinite group of birational
transformations, which ones have an infinite group of (biregular) automorphisms.
If X is a surface (and by surface we will always mean a smooth complete
irreducible algebraic variety of dimension 2 over an algebraically closed ground field
A:, of arbitrary characteristic unless otherwise specified), a standard approach to
studying Aut X is to first study Aut Y where y is a (relatively) minimal model of
X, which means y is a surface without irreducible exceptional curves for which
there is a birational morphism X —► Y. If the projective plane P2 is not a
minimal model for X, then up to finite groups Aut X is the group of automorphisms
which lift from automorphisms of Y, and if X is not ruled, then Aut Y coincides
with the group of birational transformations Autfc K(X) of X. It is only in the
case that X is a blowing-up of Y = P2 that Aut X may be infinite even though
no automorphisms of Y lift to X.
Now, for any surface X, Aut X has both a continuous aspect and a discrete
aspect, and it is the latter that is interesting with respect to the behavior above.
In particular, suppose that X is a rational surface, so its divisor class group Pic X
1980 Mathematics Subject Classification (1985 Revision). Primary 14M20; Secondary 14J25.
Key words and phrases. Algebraic surfaces, rational surfaces, K3 surfaces, automorphism
groups, roots, nonpositive anticanonical dimension, exceptional configurations, Cremona
transformations.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
17

18
BRIAN HARBOURNE
is a finitely generated free abelian group, isomorphic to H2(X, Z) in case A: is the
complex field C. Naturally, AutX acts on PicX. If Y is any minimal model for
X, then Aut Y is an algebraic group, the kernel Aut# X of the representation
of AutX on PicX is an algebraic subgroup of Aut F, and the image Aut* X is
a discrete group. If P2 is a minimal model for X, we will say that X is a basic
surface. For a rational surface X, Aut* X is infinite only if X is basic [HI], and
if X is basic and Aut* X is infinite, then Aut# X is finite [HI]. It is the behavior
of Aut* X for a basic surface X that is our primary interest in this paper.
In analogy with the classification of surfaces by canonical dimension /c, rational
surfaces can be classified by anticanonical dimension k* [S], and k* takes on the
values — oo, 0, 1 or 2. Since Aut* X is finite [S, Proposition 4.4; HI, Lemma 1.1]
for a basic surface X with k*(X) = 2, this case will not concern us, nor will the
case that k* = 1 which, at least for characteristics other than 2 and 3, reduces
by [S, Theorem 3.4] and [G, Theorem 2] to the well-understood one of minimal
elliptic surfaces [G]. And except for the case of Coble surfaces [CD, Remark
4.6], all rational surfaces with k* = 0 about which anything general is known
of the behavior of Aut* X satisfy the additional property of having an effective
anticanonical divisor of a special form. A key result and the one that instigated
the present paper is Looijenga's Torelli theorem [L, 1.5.3] for such surfaces over
C, in analogy with KS surfaces. We therefore begin the paper by recalling in
§1 some results for complex KS surfaces, and in §2 we review some facts for
basic surfaces generally. In §3 we reformulate Looijenga's Torelli theorem in a
characteristic-free way for basic surfaces of anticanonical dimension 0 having a
reduced irreducible anticanonical divisor. Among these we define a special class,
the 7T3-like surfaces, for which all of the results of §1 in some form carry over.
In §4 we close with a few remarks regarding the mysterious question of basic
surfaces of anticanonical dimension — oo with infinite automorphism group (the
occurrence of which was shown in [HI]) and a discussion of open problems.
1. K3 surfaces. Work by, for example, Friedman [F] and especially Looi-
jenga [L] shows that many results for KS surfaces have analogs for certain rational
surfaces. This behavior is remarkable enough that where no analog is known,
the existence of a result for KS surfaces motivates one to look for an analog.
Therefore we detour to review some pertinent results for KS surfaces.
By a KS surface we will always mean an algebraic surface S over the field
C of complex numbers with trivial canonical class and with irregularity q = 0.
The divisor class group Pic S is a free abelian group of rank 1 < p < 20, p being
called the Picard number of 5, and Pic S has an intersection form of signature
(1, p — 1). The set of roots A for S is defined to be the set of all divisor classes
of self-intersection —2. By the formula of Riemann-Roch for surfaces it is easy
to see that for every root r either r is the class of an effective divisor or — r is.
Now, the set A+ of positive roots is the set of effective roots and the set A_
of negative roots is the set of antieffective ones. The set B of nodal roots is the
set of roots which are the classes of irreducible divisors, and the Weyl group

AUTOMORPHISMS OF RATIONAL SURFACES
19
Wb is the group generated by reflections by nodal roots, where a reflection by
b G B is the homomorphism Pic S —► Pic S defined by s(x) = x + (x • b)b;
it is an involution which preserves the intersection form of Pic 5, so Wb also
preserves the intersection form. We continue to denote by Aut* S the image of
the representation of Aut S on Pic S.
Since we are mainly interested in whether or not Aut* S is infinite, we can
simplify our discussion with the notion of equivalence up to finite groups. Groups
G and H are said to be equivalent up to finite groups, written G ~ H, if each has,
respectively, a subgroup G' and H' of finite index such that one has a quotient
by a finite normal subgroup isomorphic to such a quotient of the other.
Denoting by T the subgroup of elements of GL(Pic S) which preserve the
intersection form and by TB the subgroup of T which preserves B set-wise but
not necessarily element-wise, we can state one of the main results for X3 surfaces,
a version of the Torelli theorem.
THEOREM 1.1. For a (complex algebraic) KS surface S, TB is equivalent to
Aut* S up to finite groups.
PROOF. This is originally due to Piateckiï-Shapiro and Shafarevich [PS]. The
statement and notation given here are taken from [St]. □
Another key property of KS surfaces is
PROPOSITION 1.2. The set of orbits of Aut* S in the set B of nodal roots
is finite.
PROOF. This is Proposition 2.5 of [St]. □
REMARK 1.3. We merely remark that the key to the proof given in [St] is
that for a K3 surface S whose set of nodal roots is B, Wb x Tb appears as an
arithmetic subgroup of GL(PicS'). Since Aut* S ~ TB, Aut* S is finite if and
only if Wb is an arithmetic subgroup of GL(PicS). This leads to the following
result, for the proof of which it is convenient to denote by P(E) for any subset
E C A+ the polyhedron
P(£) = {x e Pic s\x • a > 0 V g e £}.
THEOREM 1.4. Let S be a (complex algebraic) K3 surface.
(a) Aut* S is finite if and only ifWs is of finite index in T.
(b) If S has Picard number p at least 3, then Aut* S is finite if and only if
the set B of nodal roots of S is finite and spans a submodule of Pic S of finite
index.
PROOF, (a) Since T is of finite index in GL(Pic 5), this follows from Remark
1.3.
(b) If B generates a full sublattice of Pic 5, the action of any g G TB on Pic S
is determined by its action on B. So if B is finite, TB is a finite group which
implies that Aut* S is also since TB ~ Aut* S by Theorem 1.1.

20
BRIAN HARBOURNE
Conversely, suppose Aut* S is finite. Then B is finite by Proposition 1.2.
Moreover, Wb is then an arithmetic subgroup of GL(PicS), generated by
reflections. This is equivalent in the case that p is at least 3 to P(-B), defined
above, being contained in the closure Cl($) of the positive cone $ C Pic S 0 R
(Corollary 4.2.4 of [D], and p. 415 of [A]). We take a moment to explain: Pic 5,
and hence P{B), lies in Pic S 0 R, which we endow with the ordinary vector
space topology. The positive cone $ is the component of the subset of Pic S 0 R
of elements of positive self-intersection which contains the ample divisor classes.
So suppose that the span N of B has rank less than p. Since the intersection
form of a K3 surface is known to be nondegenerate, the space N1- C Pic S
perpendicular to N contains an element t of nonzero self-intersection. If t ■ t > 0,
then either t or — t lies outside Cl($) but in P(-B), which is a contradiction. On
the other hand, if t • t < 0, then t itself lies outside Cl($) but in P(-B), which is
again a contradiction. □
2. Basic surfaces and exceptional configurations. We establish some
terminology and some facts that are important for the study of basic surfaces.
Let X be a basic surface, i.e., one with a birational morphism X —► P2. It is
well known that any birational morphism factors into a sequence
X = Xn —► Xn-i —►•••—► X\ —► Xq = P
of blowings-up of points pi G -Xt_i, i = 1,..., n. We denote the total transform
of pi to X by Ei and its divisor class in Pic X by e^. We denote the pullback to X
of the class of a line by eo- Then PicX is a free abelian group generated by the
classes e», 0 < i < n. Moreover, PicX has a bilinear form, the intersection form,
with respect to which these generators are orthogonal and — eo • eo = e\ • e\ =
• •. = en ■ en = — 1. A basis of PicX coming in this way from (and conversely
determining) a factorization of a birational morphism X —► P2 is called an
exceptional configuration of X. In terms of any exceptional configuration 5 =
{eo,..., en} of X the canonical class K of X is —3eo + e\ -\ h en.
Associated with any exceptional configuration 5 = {eo,... ,en} of X is the
set of E-simple roots ro = eo — ei — e<2 — e% and ri = ei — e^+i, i = 1,..., n — 1. A
reflection, defined as in §1, by a 5-simple root r preserves the intersection form
of Pic X and fixes K\ it is called a E-simple reflection. Here the Weyl group W is
the group generated by these simple reflections, and the roots are the elements
in the orbits of the simple roots under the action of W on Pic X. It is easy to
check that each root has self-intersection —2. The set of roots A is the disjoint
union of the E-positive roots A+ and the E-negative roots A_, where A+ is the
set of roots which are nonnegative linear combinations of 5-simple roots and A_
is the set of their additive inverses.
The importance of the notion of an exceptional configuration to us is that,
since an exceptional configuration of X is a geometrically significant basis of
PicX, Aut*X must permute the set of exceptional configurations. The next
result, due to Nagata [N] in different terminology, shows that the same group

AUTOMORPHISMS OF RATIONAL SURFACES
21
W is obtained regardless of the exceptional configuration used to define it, and
this leads us to the connection between Aut* X and W.
THEOREM 2.1. Let X be a basic surface with exceptional configurations E =
{eo,... ,en} and E' = {e^, • • • ^nl- Then the homomorphism w.PïcX —► PicX
defined by w(ei) = e'if i > 0, is an element ofW. Moreover, if X is obtained
by blowing-up sufficiently general points 0/P2, then every W-translate of any
exceptional configuration is also an exceptional configuration.
PROOF. For Nagata's original work, see [N]; for a presentation using the
terminology used here, see [H2]. □
COROLLARY 2.2. If X is a basic surface, then Aut* X is a subgroup ofW.
PROOF. This is immediate from the remarks above. □
Since for any exceptional configuration 5 and any element a E Aut* X, o£ is
also an exceptional configuration, it is useful to have a criterion for when a
translate of an exceptional configuration by an element of W is also an exceptional
configuration. The next result gives such a criterion in terms of the nodal roots
and the notion of numerical effectivity. A divisor class is numerically effective
if it meets each effective class nonnegatively, and a nodal root here is any root
which is 5-simple for some exceptional configuration 5 and which, considered as
an element of Pic X, is the class of an effective divisor.
PROPOSITION 2.3. Let E = {eo,...,en} be an exceptional configuration of
a basic surface X, and let w be an element ofW. Then wE is an exceptional
configuration if and only ifw(eo) is numerically effective andw~l{r)) is E-positive
for every nodel root r).
PROOF. This is Corollary 1.2 of [H2]. □
We refer to a basic surface X as an anticanonical surface if the anticanonical
class — K of X contains a reduced irreducible curve. The importance of such a
hypothesis for the utilization of root system theory was made clear by Looijenga's
paper [L]. This is suggested by the facts that in this case the nodal roots are
precisely the classes of irreducible curves of self-intersection —2 perpendicular to
K [L] and that Proposition 2.3 now takes the following form:
PROPOSITION 2.4. Let E be an exceptional configuration of an anticanonical
surface X and let w be an element ofW. Then wE is an exceptional configuration
if and only if w~l(r)) is E-positive for every nodal root r).
PROOF. This is Proposition 2.4(2) of [H2], which is essentially just Corollary
1.4.2 and Theorem 1.4.6 of [L]. □
A hopeful conjecture, motivated by the case of KS surfaces, would be that
Aut* X is closely related to the group WB of elements of W preserving the set B
of nodal roots. Unfortunately, this need not be so. If X is obtained by blowing up
n > 9 sufficiently general points of P2, then B is empty, WB is W (and infinite),
and Aut*X is the identity (see [G] for n = 9). Searching for circumstances

22
BRIAN HARBOURNE
under which there is a close relationship leads us to the K3-\ike anticanonical
surfaces.
3. K3-\\ke anticanonical surfaces. Let us begin by pointing out significant
differences between basic surfaces and K3 surfaces. For the latter, every (—2)-
class (i.e., a divisor class of self-intersection —2 perpendicular to the canonical
class K) is a root and every root is either effective or antieffective. For a basic
surface X it is unusual for a root to be either effective or antieffective and every
(—2)-class is a root if and only if X has Picard number p (i.e., rk(Pic X)) no more
than 11. Also, whereas for any KZ surface 5, TB ~ Aut* S and Wb x Tb is an
arithmetic subgroup of GL(Pic 5), for a basic and even an anticanonical surface
X, Wb x Aut* X may be of infinite index in W. And W is of infinite index
in GL(PicX) as long as p is at least 12 [D, p. 262]. Because of this, even for
anticanonical surfaces there is no direct analog of Theorem 1.4, while Proposition
1.2 is not known to be true. However, all of the results of §1 have analogs for a
special class of anticanonical surfaces; those we dub to be K3-like.
Consider an anticanonical surface X, and let D denote a reduced irreducible
anticanonical curve on X. Having an anticanonical curve implies /c* > 0; if
«* < 1 it follows that D D < 0 [S], or, equivalently, that X is obtained by
blowing up n > 9 smooth points of a reduced irreducible cubic curve D' in P2.
(The proper transform of D' is D.) But we are interested in the case that /c* = 0,
which occurs if and only if either n > 9, or n = 9 and no multiple of D moves
in a pencil [S]. If n = 9 and /c* = 0, then it follows from Theorem 2 of [G]—at
least for char A: ^ 2,3—that Aut*X is finite. Because of this we will hereafter
assume that n > 9 or, equivalently, D • D < 0, or p > 10.
So consider an anticanonical surface X with its reduced irreducible
anticanonical curve D of negative self-intersection. Denote the kernel of the natural ho-
momorphism PicX —► PïcD by L. Of course, since D is anticanonical, L lies
in K±, the submodule of PicX of classes perpendicular to K. It is easy but
important to observe that L contains the set B of nodal roots. Define Q to
be the subgroup of the Weyl group W of X which preserves L and induces the
trivial action on K±/L. As usual, QB denotes the subgroup of H preserving the
set (but not necessarily each element of) 5, and Wb denotes the subgroup of W
generated by reflections by elements of B.
DEFINITION 3.1. An anticanonical surface with p > 11 will be said to be
K3-like if L is of finite index in K^.
The following theorem justifies this terminology.
THEOREM 3.2. Let X be a KS-like surface with canonical class K. Then
the following statements are true:
(a) QB is a subgroup of Aut* X of finite index;
(b) Aut* X has only finitely many orbits in the set B of nodal roots;
(c) Aut* X is finite if and only ifWs is of finite index in W; and
(d) Aut* X is finite if and only if B is finite and its span has finite index in

AUTOMORPHISMS OF RATIONAL SURFACES
23
REMARK 3.3. It is worth remarking that, for a basic surface X with Picard
number p, W is infinite if and only if p > 10 [H2, p. 132].
Proving Theorem 3.2 requires some further development, so we defer it to the
end of this section. To provide evidence that the property of being if 3-like is
interesting, we mention some examples.
EXAMPLE 3.4. The first example given of a #3-like surface X with Aut* X
infinite is the surface V of Example 2.8 of [H2]. This example is over k = C,
has p = 11, and 5, Wb, Hb, and Aut* V are all infinite. Perhaps the simplest
examples occur in positive characteristics. Let p = char/c ^ 0 and let D' be
a cuspidal cubic curve in P2. Let X be the blowing-up of n > 10 smooth
points of D'. Then X is anticanonical and the proper transform D of D' is the
unique anticanonical curve on X. But Pic0 D is pure p-torsion, so K1- jL, being
finitely generated and the image of PicX in Pic0 D, is finite. Thus X is 7T3-like.
Moreover, if the points of D' that are blown-up are general enough, then L is
pK^. Since roots are primitive elements of if-1, L contains no roots; hence
X has no nodal roots and B is empty. Theorem 3.2(c) and Remark 3.3 now
imply Aut*X is infinite. (Indeed, QB equals Q which is just the p-congruence
subgroup of W. This situation is very similar to that of generic Coble surfaces
[Dl, Remark 3.9].) Finally, more examples are obtained by considering the
case that the ground field k is the algebraic closure of a finite field. Then any
anticanonical surface with Picard number at least 11 is #3-like.
In preparation for proving the theorem, we present some results for
anticanonical surfaces that need not be 7T3-like. We begin with an analog of Theorem
1.4(b).
PROPOSITION 3.4. Let X be an anticanonical surface with p > 10. Then
the set of exceptional configurations of X is finite if and only if the following two
conditions hold:
(i) the set B of nodal roots is finite, and
(ii) the span of B is a sublattice of K1- of finite index.
PROOF. This is Theorem 3.1 of [H2]. □
Another ingredient in the proof of Theorem 3.2 is an analog of Theorem 1.1, a
Torelli theorem for anticanonical surfaces. This is a reformulation of Looijenga's
Theorem 1.5.3 of [L].
THEOREM 3.5. Let X be an anticanonical surface with p > 11, and let D
be its anticanonical curve. Then QB is a subgroup of Aut* X, and if D is either
smooth or has at most one simple node, then QB is, moreover, of finite index.
The proof depends on understanding how Aut D is related to Aut* X; we
give it after Lemma 3.6. At this point it is worth mentioning that since p > 11
implies that D • D < 0, D is the only anticanonical curve of X, and hence any
automorphism of X restricts to one of D. The next result is a partial converse.
LEMMA 3.6. Let X and D be as in Theorem 3.5 and let 5 be an exceptional
configuration of X. Suppose that w G W is such that wE is also an exceptional

24
BRIAN HARBOURNE
configuration and that there is an automorphism a G Aut D making diagram (*)
commutative,
K1- —+ Pic0 D
(*) iw i a*
K1- —+ Pic0 D
where the horizontal maps o/(*) are the composition of the inclusion K1- C PicX
with the natural map PicX —► PicD induced by the inclusion D C X. Then w
lies in Aut* X.
PROOF. Consider the exceptional configurations 5 = {eo,..., en} and wE =
{eo,...,e^} and the birational morphisms to P2 that they respectively
determine:
X = Xn —► Xn-\ —►•..—► X\ —► Xq = P ,
V Yf ». V' ». ». Yf ». Y1 Czi T>2
A. — A.n —► ^\n_i —►■•■—► ^ —► ^\0 _ f .
In each of these sequences of blowings-up, for each i let pi and p\ indicate the
point of Xi and X[ blown-up. Denote by {ro,..., rn-i} and {r'0,..., r^j} the S-
simple and E'-simple roots respectively, and by {£o, • • •, tn-\} and {t'Q,..., ^-1}
their restrictions to D.
Composing a with a translation from Pic0 D leaves the induced homomor-
phism a* unchanged, so we may assume a (pi) = p\. If we set J to be
Od(3pi)0(^i)"20(^)"10^o
and J' to be
Od(3p,1)0(^,1)-20(^,2)-10^,
it is easy to check that J = eo <8> Od and a* J = J' = e'0 0 Od- Thus a extends
to an isomorphism ao giving a commutative diagram (').
Pi G £> C X0 = P2
(') |a Ta î ao
pi G D C X'Q = P2
By hypothesis a*r\ = r[ and by construction ao(p'i) = Pi, so, blowing up p\ and
pi, ao lifts to an isomorphism ai'.X^ —► Xi for which ai(p2) = P2- Continuing
in this manner we eventually obtain an automorphism an: X = X'n —► Xn = X
with o£ = w. □
PROOF OF THEOREM 3.5. This theorem is essentially just Lemma 3.6 in
the case that a* is the identity. For suppose that w G Hfî, and consider an
exceptional configuration 5 = {eo,..., en} of X. Since w preserves 5, so does
w_1; hence w_1(r?) is a nodal root for any nodal root r?. But it follows from
Proposition 2.4 by change of coordinates that a nodal root is positive with respect
to every exceptional configuration. So again by Proposition 2.4 it follows that
wE is also an exceptional configuration.

AUTOMORPHISMS OF RATIONAL SURFACES
25
Now we claim, taking a* to be the identity in diagram ("), that (") commutes.
K-L —> P'ic°D
(") I - I «•
if-1 —- Pic0 Z?
Since w preserves the kernel L of the horizontal maps, to see this we only need to
check commutativity for elements of K1- not in L. So let s be such an element.
Since w acts trivially on K±/L, we have w(s) E s + L and it follows that both
s and w(s) restrict to the same element of Pic0 D, so the diagram commutes. It
now follows by Lemma 3.6 that w is induced by an automorphism and therefore
that QB is a subgroup of Aut* X.
For the rest of the theorem, it is enough to consider the case that Aut* X is
infinite. By the remark preceding Lemma 3.6 we have a natural map from Aut X
to the group Aut(Pic° D) of group-variety automorphisms of Pic0 D, the kernel
of which we denote G. Now D is the proper transform of a reduced irreducible
cubic curve D' in P2. In particular, either D' is smooth or it has one simple
node or cusp, and, except for the cuspidal case, Aut (Pic0 D) is a finite group,
so G is of finite index in Aut X. And since Aut# X is finite if Aut* X is not
[HI], we have AutX ~ Aut*X. Since G clearly maps to QB under the map
Aut X -► Aut* X, QB is of finite index in Aut* X. □
The next lemma is perhaps the key fact for 7T3-like surfaces:
LEMMA 3.7. If X is KS-like, then Aut* X has only finitely many orbits in
the set of exceptional configurations.
PROOF. By Lemma 3.6, taking a to be the identity, we see that if S =
{eo,..., en} and S' = {eg,..., e'n} are exceptional configurations of X such that
the restrictions to D of the 5-simple roots agree with the corresponding
restrictions of the E'-simple roots, then the homomorphism w.PicX —► PicX taking
e{ to e'{ for i = 0,..., n is induced by an automorphism of X. Since K1- contains
all roots and the image of K1- in PicD is finite by the hypothesis that X is
i£3-like, it is therefore clear that the set of exceptional configurations comprises
only finitely many orbits of Aut* X. □
We can now prove Theorem 3.2:
PROOF, (a) We see QB c Aut* X by Theorem 3.5 and since QB is the kernel
of the homomorphism
Aut*X-^Aut(jR:-L/L)
and the latter is finite, the result follows.
(b) This follows because Aut* X has only finitely many orbits in the set of
exceptional configurations by Lemma 3.7, while each exceptional configuration 5
has only finitely many 5-simple roots and each nodal root is by definition simple
for some exceptional configuration.
(c) As in §1, any set E C A defines a polyhedron P(E). Let II be the set of S-
simple roots for some exceptional configuration 5. Then P(II) is a fundamental

26
BRIAN HARBOURNE
domain for the action of W on the union W -P(I1) of its VF-translates [L, (1.3.2)].
Likewise, for the set of nodal roots B, P{B) is a fundamental domain for the
action of WB on WB • P(B) [St, cf. Lemma 1.2]. The ^-translates of P(IÏ)
contained in P(B) correspond one-to-one with the exceptional configurations
of X [L, (1.4.6)]. But Aut*X is finite if and only if the set of exceptional
configurations is finite (Lemma 3.7) if and only if P(B) contains only finitely
many VF-translates of P(I1) if and only if WB has finite index in W.
(d) This follows directly from Proposition 3.4 and Lemma 3.7. □
4. Minimal surfaces, /c* = — oo, and open problems. Very little is known
about basic surfaces with k* = — oo, except that there does indeed occur such a
surface X for which Aut* X is infinite [HI]. The example given in [HI] leads to
a class of such examples, but they all are obtained by blowing up (Aut Y)-fixed
points for an anticanonical surface Y for which Aut* Y is infinite and «* = 0,
and no examples other than these are known. In particular, it is not known
whether examples exist which are minimal, in a sense which we now formulate,
modifying somewhat the terminology of [G].
Say that a basic surface X is a G-surface if G is a subgroup of Aut* X. A
G-morphism X —► Y of G-surfaces is a morphism for which the action of G on
Pic y —► PicX is equivariant. Now X is said to be: G-minimal if any birational
G-morphism X —► Y is an isomorphism; weakly minimal if X is Aut* X-minimal;
minimal if, for some infinite cyclic group H C Aut* X, X is G-minimal for every
nonidentity subgroup G C H\ and strongly minimal if, for every infinite cyclic
group H C Aut* X, X is G-minimal for every nonidentity subgroup G C H.
It is easy to see that strongly minimal implies minimal which implies weakly
minimal. The notion of being minimal means essentially being weakly minimal
up to finite groups, while the notion of X being strongly minimal means that up
to finite groups there is no birational morphism X —► Y for which an element of
Aut* X is induced by an element of Aut* Y.
Question 1. Does there exist a minimal basic surface with /c* = —oo? Gizat-
ullin has suggested in correspondence with Dolgachev that the answer is no. The
example of a basic surface X with k* = — oo and infinite Aut* X given in [HI]
is probably not even weakly minimal, although not even this is known.
Question 2. Does there exist a basic surface X with k* = —oo, infinite
Aut* X and Picard number p = 11? The method of construction of surfaces X
with infinite Aut* X and k* = — oo discovered in [HI] seems to require p > 12.
Question 3. Does there exist a strongly minimal 7T3-like basic surface XI
(There do exist minimal 7T3-like surfaces.) Every 7T3-like surface X has elliptic
or quasi-elliptic fibrations, but if X is 7T3-like and Aut* X is infinite, must there
always be one such fibration such that Aut* Y is infinite for the corresponding
elliptic or quasi-elliptic surface F?
Question 4. Is the set of orbits of Aut* X in the set B of nodal roots always
finite if X is an anticanonical surface? Is Aut* X finitely generated? The answer
is yes to both if X is a KZ surface [St].

AUTOMORPHISMS OF RATIONAL SURFACES
27
Question 5. If X is anticanonical with Picard number at least 11, does QB
ever not have finite index Aut* XI By Theorems 3.2 and 3.5 and Example 3.4
we see that any examples occur only in characteristic 0.
Question 6. If X is a rational surface and Aut* X is infinite, then Aut# X is
finite [HI]. Must Aut# X actually be trivial?
Question 7. The similarities between Theorems 1.1, 1.2, 1.4 and Theorem 3.2
justify our definition of K3-like surfaces, but it is essentially ad hoc and must be
considered as still tentative. One wonders whether there is a deeper connection
between KS surfaces and K3-\ike rational surfaces. For example, the referee
remarks that Coble surfaces (cf. Introduction) appear as the degenerations of
Enriques surfaces, which accounts for their behaving very much like Enriques
surfaces.
We finish with an example showing that there do exist strongly minimal
anticanonical surfaces.
EXAMPLE. Let X be a smooth anticanonical surface of Picard number p = 11
with a smooth anticanonical curve D. Moreover, let the kernel L of PicX —►
PicD be generated by elements 2s and 2£, where s and t are roots of X with
s -t = 3. This is easy to achieve, for example, if the ground field A: is C, since
K±/L^Z8®Z2®Z2
embeds in Pic0 D in this case, and if 5 = {eo,..., eio} is an exceptional
configuration of X we can, for example, take
s = 9eo — 2ei — 4e2 — Ses ~ %e± — • • • — 3eg and t = eg — e\o.
Since L has no (—2)-classes, there are no nodal roots on X. Thus every VF-
translation of 5 is an exceptional configuration (Proposition 2.4). Denoting by a
the composition of the reflections by s and £, a has infinite order [K, p. 63], and
it follows by Lemma 3.6 (taking a there to be the identity) that some positive
power om must lie in Aut* X.
If X were not strongly minimal, there must at the very least be a nonisomor-
phic birational morphism X —► Y such that Aut* Y is infinite. But X has Picard
number 11, so the Picard number of Y is no more than 10 and in this case Y
must have an elliptic fibration and Picard number 10 if Aut* Y is to be infinite
[G, p. 130]. Since this leads to the occurrence of an isotropic element in L and
since it is easy to check that r • r ^ 0 for every element r ^ 0 of L, it follows that
X is strongly minimal.
ACKNOWLEDGMENTS. I offer my thanks to the University of Texas
Mathematics Department and the American Mathematical Society for supporting my
trip to the Bowdoin conference, to the organizers for bringing it together and
Igor Dolgachev for having me talk, and to the University of North Carolina for
its hospitality and the University of Nebraska for its giving me leave during the
preparation of this paper. I also wish to recall the stimulating conversations I
had with J. Wahl, I. Dolgachev, and E. Looijenga and the helpful remarks of the
referee.

28
BRIAN HARBOURNE
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University of Nebraska, Lincoln

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Fundamental Groups of the Complements
to Plane Singular Curves
A. LIBGOBER
1. Introduction. The purpose of these notes is to give an exposition of some
recent developments in the study of the fundamental groups of the complements
to plane algebraic curves. Historically the subject was very active in the first
third of the century due to its role in viewing algebraic surfaces as multiple
planes and in the study of the families of plane curves with fixed number of
nodes and cusps. The results of this period are presented in Chapter VIII of
Zariski's Algebraic Surfaces [Zl]. 35 years later, in an appendix to the new
edition of this book, D. Mumford remarked,
"The classification of plane curves C with d nodes and A: cusps
and the computation of tti (P2 — C) has unfortunately not been
pursued. Zariski's techniques in §3, theorem 1 are closely
connected to certain techniques in knot theory. For instance the
polynomial /(x), which is determinant appearing in the proof
of the theorem 1, §3, is analogous to the Alexander polynomial.
If T = tti(P2 -C), F = [r,T], r" = [r',r'] it would be
interesting to investigate the structure of T'/T" as Z[T/T'] module... "
[Mu].
Here we shall describe the progress made in the directions pointed out in
this quotation. The background describing the results known at the time when
Zariski's book was written and which weren't developed further, examples, and
some generalities are outlined in the rest of this introduction. §2 presents the
basics of the techniques of B. Moishezon of braid monodromies, which are closely
related to the study of the fundamental groups of the complements of plane curves,
especially those which appear in the study of algebraic surfaces via generic
projections. §3 surveys the results on Alexander invariants of plane curves. In
particular we give examples of the direct computation of Alexander modules for
1980 Mathematics Subject Classification (1985 Revision). Primary 14H30; Secondary 14E20.
Supported by a National Science Foundation grant.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
29

30
A. LIBGOBER
certain curves avoiding calculations of the fundamental groups. In §4 we describe
M. Nori's theory allowing establishment of the commutativity of 7Ti(P2 — C) in
many cases. Note that we concentrate primarily on the fundamental group of
the complements and omit many important developments in the study of plane
singular curves since Mumford's appendix, e.g., J. Harris's proof of irreducibility
of the families of plane curves with a fixed number of nodes and no cusps (
"Seven's conjecture"), deformation theory for the curve with nodes and cusps [Tan,
No], possible incompleteness of the characteristic series of complete continuous
families of plane curves with cusps and nodes [Wahl], complements to unions of
lines [OS], and others.
The problem of finding for which values of n, d, A: there exists an irreducible
curve of degree n with d nodes and A: cusps is largely untouched since Zariski's
survey [Zl]. Recall that some restrictions, going back to Lefschetz, come from
the fact that if a curve with such n, d, A; should exist then the numbers obtained
from the Pliicker formulas for the degree and the numbers of nodes and cusps of
dual curve should be nonnegative. Zariski [Z2, Z3] showed that these restrictions
are insufficient to guarantee the existence of a curve. For example, he showed
that there is no curve of degree 7 (resp. 8) with 11 (resp. 16) cusps and no
nodes. Recently, however, new restrictions on possible values of d and A: were
obtained based on Yau-Miyaoka inequalities. For example, for a curve of even
degree one obtains §d + 8A: < n(5n — 6)/2 by applying Yau-Miyaoka to the
desingularization of the double cover of P2 branched over this curve. This implies
the following asymptotic for the maximal number of cusps on a curve of degree n:
lim(A:(n)/n2) < 5/16. Along different lines Varchenko obtains lim(A:(n)/n2) <
23/72.
Methods of construction of singular curves are rather limited. They amount
to:
(a) considering branching curves of generic projections of surfaces;
(b) taking generic plane sections of discriminants of linear systems on algebraic
manifolds (this includes dual curves);
(c) writing explicit equations;
(d) using deformation theory to prove the existence of curves which can be
degenerated into curves of types (a), (b), (c).
The curves of type (a) are discussed in §2. Curves Ca,b,c given by equation
fac + fbc = 0 where // is a generic form of degree / are an example of curves
of type (c). The singularities of Ca,b,c are given locally by xa + yb = 0 and are
located at the intersection of the curves fac = 0 and fbc = 0. The fundamental
groups of the complement to such curves Ca,b,c were computed in the following
cases:
(1) a = 2, b = 3, c arbitrary. ni(P2 — C2,s,c) has a presentation on two
generators a and /? with relations a2 = /?3 and a2c = 1 ([Z4] for c = 1, [Tu]
any c).
(2) c = 1, a and b are relatively prime. tti(P2 — Ca,b,i) is the group on two
generators a and /? with the relation aa = (3b = 1 [Oka].

THE COMPLEMENTS TO PLANE SINGULAR CURVES 31
The curves of type (b) are obtained as follows. Let V be an algebraic manifold
and L a line bundle. In many cases the set of elements in the linear system
P(H°(V, L)) (projective space associated to H°(V, L)) that have a singular set
of zeros form a hypersurface, say D(V, L). A generic section of D(V, L) by a
plane H is a curve such that
7Ti(tf -Hf) D(V,L)) = 7n(P(i/0(F,L)) - D(V,L)).
An important special case of this construction is the case when V is a subvariety
of PN and L = Opn(1)\v. Then D(V,L) is just the dual variety of V, i.e., the
set of hyperplanes in PN tangent to V.
IfV = P1,L = 0(n) then the elements in P(H°(V, L)) can be viewed just as
unordered collections of n points in a two-dimensional sphere and
P{H0{P\O{h)))-D{P\O{n))
is the set of collections in which all points are distinct. Therefore
*1{P{lP(P\0[n))) - D{P\0[n)))
is just the braid group of a sphere on n strings (cf. [Bir] and §2). This group
is isomorphic to Artin's braid group with generators cr^o^,... ,crn_i factored
by the normal subgroup generated by the element o\ • • -crn_2 cr^_1crn_2 • • -o\.
The curve H fl D(P1,0(n)) has degree 2(n — 1), and it has 3(n — 2) cusps and
2(n — 2)(n — 3) nodes. It is the curve which is dual to the generic rational nodal
curve of degree n [Z6, DL].
If V is an elliptic curve and D is any divisor of degree n, then this construction
applied to O(D) leads to the curve of degree 2n with Sn cusps and 2n(n — 3)
nodes. The fundamental group of the complement to such a curve is isomorphic
to the kernel of the homomorphism of the braid group of the torus (underlying
the elliptic curve V) on Hi(V, Z) obtained by sending any braid to the homology
class of the sum of the loops traced by the points of D in the motion denned this
braid [Z5, DL].
Another case in which the computation of the fundamental group was carried
out to the end is V = P2, L = Op2 (3) and V = P1 xP1, L = p\Ov> (l)0p$OPi (1)
[DL, L3].
As an example of the use of method (d) above of constructing singular curves,
we shall mention Zariski's argument [Zl, p. 223] showing the existence of a curve
of degree 6 with 6 cusps not on a conic (an example of a sextic with 6 cusps on
a conic is the curve C2,s,i considered above). The curve which is dual to a
nonsingular cubic is a sextic with 9 cusps. All these cusps clearly cannot belong
to the same conic. The idea is to pick up six cusps not on a conic and to deform
the curve to eliminate the remaining 3 cusps. Such a deformation does exist
because in a complete family of curves of degree n with A: cusps and d nodes the
cusps impose independent conditions on the family provided A: < 3n. (This in
turn follows from the fact that in this case the degree of the characteristic series
is greater than 2p — 2, where p is the genus of a curve in the system, and hence
this series is not special.)

32
A. LIBGOBER
The link between the plane curves with singularities and algebraic surfaces
is provided by the existence theorem of Grauert-Remmert-Enriques-Riemann
[GR], which implies that given any finite unbranched covering F : P2 — C —►
P2 — C (which in turn is uniquely determined by a homomorphism of tti (P2 —
C) into a symmetric group) there exists a normal surface V with a projection
F' : V —► P2 that is a branched covering of P2 and such that P'If'-^ps-c)
coincides with F. On the other hand, any projective nonsingular surface can be
obtained in such a way for example by considering its generic projection on a
plane.
Many interesting surfaces can be obtained in such a way (i.e., as multiple
planes) even for rather simple branching curves and Galois groups, e.g., to name
a few: Enriques surfaces, Campedelli surfaces, and more lately surfaces with
c\ = 3c2 [BPV, H]. It would be interesting, however, to understand better this
relationship between surfaces and singular curves, for example, to find invariants
of surfaces such as Betti and Chern numbers or tti in terms of the numerical data
of the singular curve such as its degree, the number of singularities of fixed type,
their position, and inevitably the fundamental group. Some results of such a
type do not require tti [H]; others such as the computation of the Betti numbers
of cyclic covers do ([Z2, L3] and §3). As pointed out in [M2] (attributed to
K. Chakiris) the fundamental group of the complement to a ramification locus
may provide a discrete invariant distinguishing different connected components
of the moduli spaces of the surfaces of general type. Namely, one can define
a map from the set of connected components of the moduli space of surfaces
of general type with fixed c\ and c2 into the set of connected components of a
family of plane curves with fixed degree and numbers of nodes and cusps. One
can do this by relating to a surface the branching curve of the generic projection
of the image of the surface in some PN under some pluricanonical embedding,
say 5K. It can be verified easily that this map separates components of the
moduli space.
2. Braid monodromies and branching loci of generic projections.
Recall first the various definitions of Artin's braid group which are useful in the
sequel. We shall mention four of them. Let Hi and H2 be two parallel planes in
i23, Pi,..., Pn be a set of n distinct points in Hi, and Qi,..., Qn be projections
of Pi,..., Pn on Hi along the direction perpendicular to the planes Hi and
#2- A braid is a union of nonintersecting segments connecting Pi,..., Pn and
Qi? • • • ? Qn or rather the class of isotopy of such a union leaving Pi,..., Pn and
Qi? • • • ,Qn fixed (Figure 1). If Hz is a plane under the lower plane among Hi
and #2, say i/2, parallel to Hi and i/2, then any braid between Hi and H2
defines the braid between Hi and Hz by extending paths between Hi and i/2
by segments of straight lines perpendicular to i/2 and i/3. This allows us to
identify braids between different planes. The composition of the braids between
planes Hi and i/2 and braids between i/2 and Hz is defined by attaching together
portions of the space between Hi and i/2 and between i/2 and Hz. In such a way

THE COMPLEMENTS TO PLANE SINGULAR CURVES 33
Figure l Figure 2
Figure 3 Figure 4
one obtains a group Bn which has as generators elements X{ (i = 1,..., n — 1)
corresponding to the braids interchanging Pi and P{+i given in Figure 2. The
complete set of relations among X^s is given by [Ar].
(1) X{Xj = XjX{ for \i — j\ > 2, XiXi+iXi = Xi+iX{Xi+i.
This presentation can be taken as a definition of the braid group Bn.
The braid group Bn is isomorphic also to the group of isotopy classes of
orientation-preserving homeomorphisms of the plane which are identity outside
of a fixed disk and which map into itself a fixed set of n points Pi,..., Pn
inside the disk. The identification of the group of isotopy classes of such
homeomorphisms with the braid group Bn introduced above depend on a choice of
a system of (n — 1) paths ai,... , an-i in the plane such that (a) each of the
points Pi,..., Pn belongs to one of the paths, and (b) any two paths have no
points in common except maybe for one of the points Pi,..., Pn. To each path
from such a system one can associate the following homeomorphism of the plane.
Take a metric in the plane such that the path connecting points, say Ps and Pt,
is a diameter of a circle D\ of radius 1 and such that the concentric circle D2 of
radius 2 contains no points P{ for i ^ s,t (see Figure 3).
Then the homeomorphism associated to a is Dehn's half twist about the
boundary of the circle Z}3/2 of the radius 3/2 concentric with D\ and D2, i.e., the
homeomorphism which is the identity outside of D2, which is the rotation by it
inside D\ and which is the rotation by m on the circle of radius r concentric with

34
A. LIBGOBER
D\ and D<2. One can show that these homeomorphisms generate the group of
isotopy classes of homeomorphisms of the type described above, and the relations
among them are given by (1).
Finally, the braid group can be described as a subgroup of the group of
automorphisms of a free group F which take any generator of F into a conjugate of
a generator and which leave invariant the product of chosen generators of free
group, i.e.,
{n n
<peA\itFn{x1,...,xn)\<pxi = TixiT~1, Y\xi = Yl<p(xi)
z=l i=l
where Fn(xi,..., xn) is the free group on symbols xi,..., xn and T{ are certain
words in Fn(xi,... ,xn). The isomorphism Bn —► AutFn is given by assigning
to a generator Xi from presentation (1) the automorphism of Fn(x\,..., xn).
\Z) Afc : \X\, . . . , X{, 2^-fi, • • • î %n) * V^lî • • • ? %i%i+l%{ %ii • • • ? ^nj«
Geometrically this action is the action of the braid group interpreted as group
of the classes of homeomorphisms as above on the punctured plane with the
generators of ni(H — \Ji=1 Pi) chosen as in Figure 4.
Now we are going to define the braid monodromy of a plane algebraic curve.
We shall denote it by C and d will denote its degree. We'll be concerned primarily
with the affine portion of this curve but we assume that C is transversal to the
line L at infinity. Let II: P2 — L = C2 — C be a linear projection, n = II|c,
and let 5 = {pi,... ,piv} be the subset of C consisting of points the 7r-preimage
of which contains less than d elements. Points of S correspond to the members
of the pencil of lines defined by the projection II which either pass through
singularities of C or are tangent to C. (Lines corresponding to S we call the
singular elements of the pencil.) Let us fix a system of generators </i,..., </jv
(N = card5) of 7Ti(C — S,p) for some p G C. A trivialization of the locally
trivial fibration II: C2 — C — II~1(S') —► C — S over a path representing g\
defines a homeomorphism of n_1(p), which can be chosen to be identity outside
of a sufficiently large disk. A choice of a system of paths in II-1 (p), as in the
aforementioned definition of the braid group as a group of homeomorphisms,
defines therefore the homomorphism 6: 7r'i(C — S) —► B<i. This homomorphism
0 is called the braid monodromy defined by C (and the choices made all along).
Particularly important are the so-called well-ordered systems of generators of
7Ti(C — S) [Ml]. Let us fix a system of paths 7» (i = 1,..., N) connecting p
with the points Pi,..., Pn G S such that any two have only p as a common point.
Let Di be small disks about Pi. Then we define the path 7» corresponding to Pi
as 7» = (7» — 7i n Di) U dDi which runs in a counterclockwise direction. If Dq is
a small disk about p, then the counterclockwise direction defines an ordering of
points of dDi fl 7» (i = 1,..., N) (cf. Figure 4). Such a system 7» (i = 1,..., N)
(depending on a choice of 7») is called a well-ordered system of generators of
7Ti(C — S,p). For any two well-ordered systems of generators 71,... ,7jv and
7i ? • • • ? 7jv? one can find an element 7 G ni (C — 5, p) such that 71,..., 7jv) can

THE COMPLEMENTS TO PLANE SINGULAR CURVES
35
be obtained from 77i7~1,7727_1> • • • ? 77iv7_1 by means of the transformations
(2). Indeed n»=i 7* an(^ Ilz=i li represent homotopic unbased loops in C — S
(each is homotopic to a circle containing the set S). Hence they are conjugate
in 7Ti(C - S,p), i.e., n»=i 7» = YliLillil'1 f°r some 7- Moreover 7» and
77a(t)7_1 are conjugate for some permutation s of indices 1,..., N. Therefore
the automorphism of the free group 7Ti(C - S,p), taking 7» —► 77^) 7"1 is the
action of a braid on 7Ti(C — s,p), and the claim follows.
The collection of elements 0(7») in the braid group constructed using the braid
monodromy 0 for some well-ordered system of generators of 7Ti(C — S,p) is not
arbitrary and it is of great interest to find conditions which #(7») satisfy. One of
them is that #(7») belong to a semigroup in Bd generated by conjugates to the
generators X{ of Bd (cf. [M2, Ru]) (which is called the semigroup of positive
braids). Indeed a small perturbation of a given curve produces a nonsingular
curve Ct such that the lines of the pencil defined by projection have at most
simple tangency points with the curve Ct locally given by x = y2 relative to the
projection (x, y) —► x. Each singular point of C splits into several points in which
the lines of the pencil are tangent to C. Suppose that the ith singular point of C
splits into rii such points. Moreover each element #(7») is replaced by elements
9(li,j) (j = 1, • • • ,Wj), each of which is conjugate to the braid corresponding to
the simple tangency point of a line from the pencil. The latter braid is conjugate
to a standard generator of Bd because in the case when C is given by x = y2
and the projection by (x, y) —► x the corresponding braid is just the standard
generator interchanging a pair of points. On the other hand, for Ct sufficiently
close to C we have 0(7») = YYjLi Hli,j) and the claim follows.
Another restriction on collections of elements 0(7») is given by the following:
PROPOSITION l [Ml, Ch]. n£Li0(7») *5 eQua^to t^ie positive generator of
the center of Bd, i.e., to A2 = H(Xi '-'Xd-i)d. Note that the product in the
statement is well defined because the system of paths 7» is ordered.
PROOF. Let us consider a deformation of a given curve C into a curve U
which is a union of lines passing through a point. This can be done, for example,
by considering C as a section of a cone by a plane and then by moving this plane
into the one passing through the vertex of the cone. The element n»=i Hli)
in the braid group can be viewed as the monodromy along a loop containing
the whole set S of singular points of the chosen projection of C, and clearly
this monodromy is unchanged in the deformation of C into U. Furthermore,
we can assume that U is given the equation xd = yd. Let x = exp(27ri^). If
<p = 0 we obtain in the fibre of the pencil d points located on the unit circle.
Now the increase of <p by 1/d amounts to rotation of this circle by 27r/d, which
corresponds to the braid X{ • • • Xd-\. Therefore the change of between 0 and 1
amounts to one full twist of the unit circle mentioned above, i.e., corresponds to
A2 = U(X1---Xd-i)d. Q.E.D.
It is convenient to consider the collection 0(7») as a factorization of the work
A2, and that is actually what Moishezon calls the braid monodromy. (Chisini

36
A. LIBGOBER
Figure 5. A2 for d = 3.
and his school were calling this situation a helicoidal braid (i.e., A2) separated
into pieces corresponding to the singular points of the pencil by the diaphragms
[Ch].)
Braid monodromy determines the fundamental group of C2 — C. Moreover
one can recover from it even the homotopy type of C2 — C. To describe how to
do this, recall that to any presentation of a group G with generators ei,..., en
and relations R\,..., Rn one can associate a 2-dimensional complex with single
0-cell, l-cells corresponding to the generators ei,..., en, 2-cells corresponding to
N relations of the chosen presentation and such that the attaching map of ith
cell (i = 1,..., N) takes its boundary into the class Ri in -K\ (S1 V • • • V S1). Note
also that the decomposition of 7» as union (7» — 7» flD{) UdDi U (7* — 7* D A)-1
corresponds to the splitting of 9(h) as QifaQ~l, where Qi is a trivialization of
C2 — C\li composed with fixed identification of II-1^ — dDi) with n_1(p) (for
example from a decomposition C = C + C), fa represents the local monodromy
near Pi. Usually one takes a system of generators of tti (II_1 (Pi)) and generators
of the corresponding braid group in such a way that the fa's have simple form
reflecting the type of singularities at the point corresponding to P. For example,
for simple tangency point (resp. node, resp. cusp) for appropriate choices can be
made equal to one (resp. square, resp. cube) of a generator (cf. Figure 6).
THEOREM 2 [L4]. Let C be a curve in C2 with the braid monodromy
Yli=i QifaQ^1 = A2- Let mi denote the multiplicity of the singular point of
C corresponding to Pi. (Notations are as in the text above.) Assume that the
braid fa acts nontrivially on e^,..., ejm. and acts trivially on the rest of the
generators of 7Ti(U~1(;)i fl dDi)). Then C2 — C has the homotopy type of a 2-
dimensional complex corresponding to the following presentation of tti (C2 — C) :
{ei,..., ed\Qifa(ej) = Qifa), j = ju ... ,jmi-i}.
For example, using the computations [Ml] described below one obtains that
the complement to the affine portion of a sextic with six cusps on a conic has
the homotopy type of the wedge of 13 copies of a 2-dimensional sphere with the
complement to the trefoil knot in S3.

THE COMPLEMENTS TO PLANE SINGULAR CURVES
37
X S 8
tangency point node cusp
Figure 6
Computation of braid monodromy at this point is a rather painful process.
In works [Ml, M2, M3] certain techniques are developed for this in the case of
the curves which are branching loci of generic projections of surfaces. Numerical
invariants of such surfaces are easy to find. We have the following:
PROPOSITION 3. LetVc CPN be a nonsingular surface, p: V —► CP2 a
generic projection, and B C CP2 the branching curve of p. Let c\ and C2 be
Chern numbers of V and let H be the class of hyperplane section. Then B has
degree 3H2 — (ci,H), the number of cusps of B is equal to 12H2 — 9(ci,i/) +
2c? - c2, and the number of nodes is \((cuH)2) - 6{cuH) • H2 + 9{H2)2 +
30(ci,tf)-6c? + 2c2-42i72.
PROOF. Clearly Ky = p*(Kp2) + R, where R C V is the ramification
locus. B = p(i2), and R is the normalization of B. Hence degi? = (H,R) =
(Ky + ZH)H and the first formula follows. Next recall that the set of cusps of
B is the image of the Boardman stratum E1,1 and that the homology class of
E1'1 is given by (c\ +c2)(pTP2 - Ty) U [V] (cf. [Ro]). This implies the formula
for the number of cusps. Finally we have
X(R) = x(B) +d = deg£(3 - degB) + 2k + 2d.
Using the adjunction formula x{R) = —R{R-\-Ky) and the formula for the class
of R used above, we arrive at the expression for the number of nodes d.
For example, we obtain, as the branching curve of the generic projection of
the image Fn of P2 under the map defined by the linear system i/0(P2,O(n)),
a curve of degree 3n(n — 1) with 12n2 — 27n + 15 cusps and
§(n-l)(n-2)(3n2+3n-8)
nodes. For the branching curve of the generic projection of the surface Xa,6, which
is the image of the quadric P1 x P1 under the map defined by the linear system
H°(P1xP1,p*10Pi(a)<g>P20pi(b)) (where pi,p2 are projections of P1 x P1 onto
its factors), we have the degree equal to 6ab - 2a - 26, A: = 24ab - 18a - 186 +12,
andcf = 4(a + 6)2-24a6(a + 6)+36a262+60(a + 6)-84a6-40(cf. [MT]).Inthe

38
A. LIBGOBER
case of a nonsingular surface Vn in P3 the branching curve has degree n(n — 1),
k = n{n- l)(n - 2), and d = (l/2)(n - l)(n - 2)(n - 3).
It is of great interest to characterize the branching curves of generic
projections. The following restriction was pointed out in [Ml]: for such a curve one
has the following congruences: A: = 0 (mod 3) and d = 0 (mod 4). (This follows
immediately from Proposition 2 using c\ + c<i = 0 (mod 12).)
The braid monodromies of the branching curves of surfaces Vn,Fn,Xa,b &re
computed in [Ml, M2, M3]. For example, for V3 one has as the branching curve
a sextic with 6 cusps on the conic, and the corresponding braid monodromy is
(3) A = [^24^13^56^35^12^34^56] ^24^13^50^35,
where Z{j = Xj+i • • • Xi+\Xi{Xj+\ • • • Xi+i)~x. The corresponding
fundamental group (of the complement in the affine plane to the branching curve of the
generic projection of V) is the braid group Bn. The results on the braid
monodromies of the branching curves of projections of Fn and Xa^, or even on tt\
of their complements, are too cumbersome to quote here. Note, however, that
this computation led to the proof in [MT] that the Galois covering of P2,
corresponding to the full symmetric group, branched over the branching curve of the
generic projection of Xa,b, is simply connected for a and b relatively prime, and
has positive index for a > 4, b > 4.
3. Alexander modules. An Alexander module of a plane algebraic curve is
an invariant of the fundamental group of its complement. It turns out that the
first homology group of the cyclic covering of P2 depends only on the Alexander
module of the branching curve. On the other hand, the Alexander module can
be found in a simple manner in terms of the degree, the local type, and the
position of the singularities of the curve. In particular, we shall show below how
this allows us to derive information on the fundamental group of the complement
to a curve just from the geometry of the set in the plane of singular points of
the curve.
The idea of the definition of the Alexander module essentially suggested by
Mumford [Mu] (cf. Introduction). However, for a number of technical reasons,
it is more convenient to work with the affine portion of the curve. One of the
reasons is to allow coverings of P2 of arbitrary degree. If C is an irreducible
curve of degree n and L is a line in infinity, then Hi (P2 — C,Z) = Z/nZ
while Hi(P2 — C — L,Z) = Z. The compactified cyclic covering of degree m of
P2— C — L has canonical map onto P2 branched over C and for certain m over L.
Let T = 7Ti (P2 — C — L) (it depends in general on a choice of L), T' = [r, T] is the
commutator subgroup, and T" = [T',T']. For irreducible C we have T/T' = Z.
DEFINITION [LI, L3]. The Alexander module A(C, Z) of C relative to L is
T'/r" considered as module over Z\T/T'], where the action of T/T' is obtained
from the exact sequence
o — r'/r" — r/r" — r/r — o.

THE COMPLEMENTS TO PLANE SINGULAR CURVES 39
Z[T/T'] is actually just the ring Z[£,£_1] of the Laurent polynomials with
the integral coefficients. We will be concerned with the rationalized Alexander
module A(C, Q) = A(C, Z) (g> Q, though a number of results have been obtained
over Z/pZ [L2]. For an irreducible curve C, A(C, Q) is a Q[t, £_1]-torsion module
(cf. [LI]) and hence is isomorphic to 0Z=1 Q[M_1]/(^*) for some integer /,
where (A^) denotes the principal ideal generated by the Laurent polynomial A^.
The order of A(C,Q) as a Q[t,t~l] module, i.e., A(C) = nUi An is called the
Alexander polynomial of C.
Another description of the Alexander module comes from the fact that it can
be identified with the fundamental group of the infinite cyclic cover of P2—C — L
and r'/r" is just Hx of this cover. The action of T/T' on T'/T" is the usual
action of the group of deck transformations on the homology of the covering.
The relation between -K\(P2 —CuL) and the homology of a desingularization
of a cyclic m-fold cover Xm of P2 branched along C and L can be described as
follows. The rank of H\ (Xm, C) is just the sum over i of the number of common
roots of X{ and tm — 1 [LI]. The proof can be obtained by showing that the first
Betti number of the m-fold unbranched cover of P2 — C U L is greater by 1 than
the first Betti number of a desingularization of Xm and by deriving the formula
for the homology of an m-fold cyclic unbranched cover in a way similar to [Mi].
Next we shall describe the dependence of the Alexander module on the degree
of the curve and on the local structure of the singularities. For each singular paint
Pi (i = 1,..., N) of C let Si be the boundary of a small ball about the point Pi
and let Sqq denote the boundary of a sufficiently small tubular neighborhood of
Lin P2. For each branch Cij of the curve C at pi let ~ denote the infinite cyclic
cover of the corresponding space relative to the map iri(Si — Sij) —► Z defined
by appropriate linking number. Then the local Alexander module corresponding
to Cij (resp. the Alexander module at infinity) is Hi (Si — Si H Cij,Z) (resp.
Hi(Soo — Sqo H C, Z)) considered as a module over the ring of Laurent
polynomials. We'll denote these Alexander modules by Ac{j(C,Z) and Aoo(C,Z)
respectively. The order Actj(C) of Aci3(C,Q) is just the characteristic
polynomial of the local monodromy of the singularity Cij. For example, if Cij near pi
is given locally by xp +yq = 0, then ACij(C) = (t™ -l)(t- l)/(t? -1)(*« -1). If
C is transversal to L, then the order Aoc(C) of A^C, Z) is (tn - l)n~2(t - 1).
The analysis of the proof of the theorem from [LI] leads to the following
THEOREM 4. The maps 0^ ACij(C,Z) — A(C,Z) and A^C.Z) —
j4(C, Z) induced by inclusions are surjective.
This implies that the Alexander polynomial of a curve divides the product of
the local Alexander polynomials of all branches of all singularities of C as well
as the Alexander polynomial of C at infinity. For example, if all singularities
of C are either nodes or locally given by xp + yq = 0, then all roots of the
Alexander polynomial are roots of unity of degree pq. On the other hand, roots
of the Alexander polynomial of a curve of degree n are roots of unity of degree
n. This, combined with the fact that the Alexander polynomial (defined up

40
A. LIBGOBER
to a unit in the ring of Laurent polynomials) can be normalized in such a way
that A(l) = 1, implies that if pq \ n then A(C, Q) = 0. Using this with
aforementioned formula for the first Betti number of cyclic covers and specializing
this to the case p = 2, q = 3 we obtain Zariski's theorem that the cyclic branched
cover of P2 is regular (i.e. irregularity h1,0 is zero) unless both the degree of the
cover and the degree of the curve are divisible by 6 [Z2].
Another corollary from Theorem 4 is the semisimplicity of A(C, Q). Indeed in
this case each Actj (C, Z) is semisimple because of semisimplicity of monodromy
in the unibranched case [Le, A'C].
Next we shall describe the relation between the Alexander module and the
geometry of the set of singularities of C. Assume that all singularities locally
are given by xp + yq = 0 or are the nodes (the more general case is discussed
in [L3]). Recall that for a finite set N of points in P2 the superabundance
sjv(fc) of N relative to the curves of degree A: is the difference betweeen the
actual and expected dimensions of the space of curves of degree A: containing
N (the latter one is just (A: + 1)(A: + 2)/2 - card(TV)). In other words s/v(fc) =
dimi/1(P2,/iv(A:)), where InW is the ideal sheaf of N. We can assume that
pq\n because otherwise, as noted earlier, A(C, Q) = 0.
THEOREM 5. If C is an irreducible curve of degree n, all singularities which
are either nodes or locally given by xp + yq = 0 for a fixed pair of integers such
that pq\n, then
MC, R)= 0 R[t, t-l\/{{t™ -i)(t- i)/(t? - i)(t« - l))
Ssing(n-3-(l-l/p-l/<2)n)
where Sing is the set of singularities of C.
According to the Cayley-Bacharach theorem [GH] the superabundance of a
complete intersection of two curves of degrees a and b relative to the curves of
degree a + b — 3 is equal to 1. For Turpin's and Oka's (cf. Introduction) curves
one obtains that the Alexander modules over R are isomorphic respectively to
R[t, t-l]/{t2 -t + 1) and R[t, r1]/^9 - 1)(* - l)/((*p - l){tq - 1)).
Somewhat different approaches to this subject were taken in works [Ran, Esn,
and Ko]. R. Randell showed that the Alexander polynomial relative to a generic
line in infinity of a curve given by the equation f(x,y,z) = 0 can be found as the
characteristic polynomial of the monodromy of the two-dimensional singularity
at the origin of f(x,y,z). As a corollary he obtained the cyclotomicity of the
Alexander polynomial, the normalization A(l) = 1, and the fact that the degree
of the Alexander polynomial is equal to the first Betti number of the covering
of P2 of degree equal to the degree of the branching curve which it contains
in the above discussion. Kohno [Ko] described the Alexander polynomial in
purely algebraic fashion from the cohomology groups of the de Rham complex
constructed using a flat connection defined in terms of f(x,y,z). H. Esnault
[Es] computed, purely algebraically using logarithmic complex, the mixed Hodge
structure on the Milnor fibre of the two-dimensional singularity at the origin of

THE COMPLEMENTS TO PLANE SINGULAR CURVES 41
f(x,y,z). Her approach allows her to find, among other things, the first Betti
number of the covering of P2 of degree equal to the degree of the branching
curve, thus recovering the phenomenon of dependence of this Betti number on
the position of singularities.
4. Abelian fundamental groups. The first systematic work on finding
plane curves for which the fundamental group of the complement is abelian was
done by S. Abhyankar in the late 1950s [Ab; Zl, Appendix 1 to Chapter VIII].
His approach included the treatment of the complements to the divisors on a
nonsingular simply connected surface (actually on any simply connected non-
singular algebraic variety, which is the same as far as the fundamental group is
concerned) rather than only the case of curves on P2. Abhyankar also worked
with the algebraic fundamental group and introduced geometric methods in the
subject using intersection theory, Bertini's theorem, etc. He showed that if a
plane curve has a small number of nodes and cusps relative to its degree then the
fundamental group of its complement in P2 is abelian. His inequalities weren't
sharp however. Further improvements were made in [AM, Ed, Pr]. The next
step was made by W. Fulton and P. Deligne, who found the long-sought proof of
the statement (known as Zariski's conjecture) that the complement to a plane
curve with nodes as the only singularities has abelian fundamental group (the
case not always covered by Abhyankar inequalities). Their proof (for algebraic
and topological fundamental groups respectively) is based on the connectedness
theorem and couldn't be extended to arbitrary surfaces. Shortly after that,
M. Nori [N] brought a new circle of ideas which we are going to describe. Like
Abhyankar he works with an arbitrary nonsingular surface, but does not assume
that they are simply connected or projective.
THEOREM 6. Let D and E be (possibly reducible) curves on a nonsingular
projective surface X. Assume that D has only nodes as singularities and that
C2 > 2r(C) for any irreducible component C of the curve D, where r(C) is the
number of nodes on C. Then N = Ker7Ti(X — D U E) —► tti (X — E) is abelian.
In particular, this implies that -K\ (P2 — D) is abelian for any nodal curve
D because for any irreducible curve C the maximal number of nodes is
±(degC - l)(degC - 2) and C2 = (degC)2 > (degC - l)(degC - 2) > 2r(C).
To illustrate the ideas involved we shall outline Nori's proof of the commuta-
tivity of 7Ti(P2 — C) for nodal C and then we'll sketch the general case. First
note that any loop in a connected topological space defines a conjugacy class
in the fundamental group by connecting any point of this loop with the base
point. Now if H is a nonsingular curve and T is a tubular neighborhood of H in
a nonsingular surface, then the class of the fibre of the restriction of the normal
bundle T — i/-*i/is a circle whose class in tti(T — H) belongs to its center. Let
C{ be an irreducible component of C. Because C is nodal, we can assume that
C{ is the image of a generic projection n: Ci —► Ci of a nonsingular curve Ci. We
can assume that Ci belongs to a surface in which we consider a sufficiently small

42
A. LIBGOBER
tubular neighborhood U{. Genericity of projection means that the projecting
cone is transversal to £/», which implies that the projection p: Ui —► P2 is a local
homeomorphism.
The main step is to show that the map p* : ni(Ui — p~l (IJ Ci)) —► -k\(P2 — C)
is surjective for any i. If so then the conjugacy class in tti (P2 — C) of a fibre of the
normal bundle of Ci contains a central element in ni(P2 — C). But -K\(P2 — C)
is generated by the conjugates to the fibres of normal bundles and hence the
commutativity of ni (P2 - C) follows.
To show the surjectivity of p* let us consider the action 0: SL(3) x P2 —► P2
of the linear group SL(3) on P2. Let P: C{ x SL(3) -► P2 be given by P(c, g) =
0(g)c (c ECi, g G SL(3)). The center of the proof is the following diagram:
Ui-P-l(\JCi) CixSL(S)-P-1([jCi)
ci 7 P2-\JCi
Here 6 is the restriction of P on P~l(P2 — C), /? is the restriction of the
projection p: Ui —► P2 on p~1(P2 — \JCi). The curve C^ is a generic fibre of the
map Ci x SL(3) - P"1 (U Ci) -► SL(3) over a point £ G SL(3) sufficiently close to
identity and 7 is the embedding as a fibre. Note that for a dominant morphism
E —► B of algebraic varieties with connected fibres for generic fibre one has
^(F)-+iniE)-+miB)-+1.
Because SL(3) is simply connected, we can assume that
*i(CÏ) - *i{Ci x SL(3) - P-'iJCi))
is surjective. Construction of a uses the genericity of p: Ui —► P2. So 7(Ct-) is
the part, contained in P2 — (J C^, of the image of Ci under linear transformation
g close to identity. This linear transformation induces a transformation of the
projecting cone which produces the map of Ci onto the intersection of this cone
with Ui and which makes the diagram above commutative. The map <5* : 7Ti (Ci x
SL(3) - P~l(P2 -{J Ci)) — 7Ti(P2 - \Jd) is surjective because the fibres of 6
are connected. Surjectivity of /3 hence follows.
In the proof of the general case Nori reconstructs a diagram somewhat similar
to the one used above. For any irreducible component C C D C X and its
normalization C, Nori considers an open set U D C with a local homeomorphism
h: U —► X extending the normalization map C —► X. Let X' = X - D - E and
U' = h~l(X'). The inequality C2 > 2r(C) implies that C has positive self-
intersection index on U. If C were lying on X, then by a Lefschetz-type theorem
this would imply that 7Ti(C//) —► ni{X') is surjective. In general, however, C —► X
is only a local homeomorphism but Nori shows that nevertheless the image of
7Ti(C//) in 7Ti(X') has finite index. He calls this fact the weak Lefschetz theorem,
which is very interesting for its own sake. Nori also has an estimation of the index
[ImTr1 ([/'): 7Ti(X/)]. Now let us consider a covering s: Y' —► X' corresponding

THE COMPLEMENTS TO PLANE SINGULAR CURVES 43
to Im7Ti([/'). One can show that there exist a finite morphism of a normal
projective variety <p : Y —► X and a lifting s : U —► Y such that the composition
<p o s is unbranched over D. This implies that N = Ker7Ti(X — D — E) —►
tti(X — E) belongs to the image of ni(Y') in tti(X') because this kernel is
generated by the loops which are unions of paths and the boundaries of small
normal disks to D. Moreover ni(U,)ni(Yt) is surjective. Hence each of these
loops belongs to the center of ^\{Y') and therefore the kernel N is abelian.
It is interesting to compare Nori's weak Lefschetz theorem with the classical
one. In the latter case one assumes that C is a curve with C2 > 0 and concludes
that 7Ti(C) —► tti{X) is surjective. 7Ti(C) is however much bigger then 7r1(Cr)
where C is a normalization of C (tti (C) is a free product of ni (C) and a free
group). Nori assumes that C2 > 2r(C) and obtains that the fundamental group
of normalization is very close to (has finite index in) tti(X). In fact, he poses
the question of whether this inequality can be weakened; i.e., does C2 > 0 imply
that the image of the fundamental group of the normalization of C has finite
index in tti(X)? In [GS] a positive answer to this question is given in the case
when S is an elliptic surface with at least one fibre not of the type ml®.
As a corollary Nori obtained the strongest theorem on the commutativity of
the fundamental group of the complement to a plane curve with cusps and nodes.
THEOREM 7. Let C be an irreducible curve of degree n which has a cusps
and b nodes as the only singularities. Ifn>6a + 2b, then m (P2 — C) is abelian.
PROOF. Blow up P several times to obtain a surface P2 on which the proper
preimage C of C is transversal to the exceptional set E. Then C2 = C2 — 6a.
Theorem 6 applied to P2 implies the claim.
References
[Ab] S. Abhyankar, Tame coverings and fundamental groups of algebraic varieties. I, II, III,
Amer. J. Math. 81 (1959), 46-94; 82 (1960), 120-178; 82 (1960), 179-190.
[A'C] N. A'Campo, Sur la monodromie des singularités d'hyper surf aces complex, Invent. Math.
20 (1973), 147-169.
[AM] D. Alibert and G. Maltsiniotis, Groupe fondamentale du complémentaire d'une courbe
a point double ordinaires, Bull. Soc. Math. France 102 (1874), 335-351.
[AR] E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101-126.
[BPV] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Springer-Verlag,
1984.
[Bir] J. Birman, Braids, links and mapping class group, Ann. of Math. Stud. 82, Princeton
Univ. Press, Princeton, N.J., 1975.
[Ch] O. Chisini, Courbes de diramation des planes multiples et tresses algébriques,
Deuxième Colloque de Géométrie Algébrique (Liège, 1952), Masson, Paris, 1952, pp. 11-27.
[De] P. Deligne, Le groupe fondamental du complement d'une courbe plane n'ayant que des
points doubles ordinaires est abelien, Sem. Bourbaki, 1979/80, Lecture Notes in Math., vol. 842,
Springer-Verlag, 1981, pp. 1-10.
[DL] I. Dolgachev and A. Libgober, On the fundamental group of the complement to a
discriminant variety, Lecture Notes in Math., vol. 862, Springer-Verlag, 1981, pp. 1-25.
[Ed] G. Edmunds, Coverings of node curves, J. London Math. Soc. 1 (1969), 473-479.
[Es] H. Esnault, Fibre de Milnor d'un cone sur une courbe plane singulière, Invent. Math. 68
(1982), 477-496.

44
A. LIBGOBER
[F] W. Fulton, On the fundamental group of the complement to a node curve, Ann. of Math.
(2) 111 (1980), 407-409.
[GH] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, 1978.
[GR] H. Grauert and R. Remmert, Komplexe Raume, Math. Ann. 136 (1958), 245-318.
[GS] R. Gurjar and A. Shastri, Covering spaces of elliptic surfaces, Compositio Math. 54
(1985), 95-104.
[H] F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry,
Papers Dedicated to I. R. Shafarevich, Birkhauser, 1983, pp. 113-140.
[H2] , Singularities of algebraic surfaces and characteristic numbers, Contemp. Math.,
vol. 58, Amer. Math. Soc, Providence, R.I., 1986, pp. 141-152.
[Ko] T. Kohno, An algebraic computation of the Alexander polynomial of plane algebraic curve,
Proc. Japan Acad. Ser. A. Math. 59 (1983), 94-97.
[Le] Dung Trang Le, Sur les nodes algébriques, Compositio Math. 25 (1972), 281-321.
[LI] A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple planes,
Duke Math. J. 49 (1982), 833-851.
[L2] , Alexander modules of plane algebraic curves, Contemp. Math., vol. 20, Amer.
Math. Soc, Providence, R.I., 1983, pp. 231-247.
[L3] , Alexander invariants of plane algebraic curves, Proc Sympos. Pure. Math., vol.
40, Amer. Math. Soc, Providence, R.I., part 2, pp. 135-144.
[L4] , On the homotopy type of the complement to plane algebraic curves, Crelles J. (to
appear).
[Mi] J. Milnor, Infinite cyclic covers, Topology of Manifolds, Prindle, Weber & Schmidt,
Boston, 1967, pp. 115-133.
[Ml] B. Moishezon, Stable branch curves and braid monodromies, Lecture Notes in Math.,
vol 862, Springer-Verlag, 1981, pp. 107-192.
[M2] , Algebraic surfaces and arithmetic of braids. I, II, Arithmetic and Geometry, Papers
Dedicated to I. R. Shafarevich, Birkhauser, 1983; Contemp. Math, (to appear).
[MT] B. Moishezon and M. Teicher, Simply connected algebraic surfaces of positive index,
Preprint, 1985.
[Mu] D. Mumford, "Appendix 2 to chapter 8," in Algebraic surfaces by O. Zariski, Springer-
Verlag, 1971.
[No] A. Nobile, Families of curves on surfaces, Math. Z. 187 (1984), 453-470.
[N] M. Nori, Zariski's conjecture and related problems, Ann. Sci. École. Norm. Sup. (4), 16
(1983), 305-344.
[Oka] M. Oka, Some plane curves whose complement have non abelian fundamental group,
Math. Ann. 218 (1978), 55-65.
[OS] P. Orlik and L. Solomon, Combinatorics and the topology of complements to hyperplanes,
Invent. Math. 56 (1980), 167-189.
[Pr] D. Prill, The fundamental group of the complement of an algebraic curve,
Manuscript a Math. 14 (1974), 163-172.
[Ran] R. Randell, Milnor fibres and the Alexander polynomials of plane algebraic
curves, Proc. Sympos. Pure. Math., vol. 40, part 2, Amer. Math. Soc, Providence, R.I., 1983,
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Comment. Math. Helv. 47 (1972), 15-35.
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150-170.

THE COMPLEMENTS TO PLANE SINGULAR CURVES 45
[Z4] , On the problem of existence of algebraic functions of two variables possessing a given
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University of Illinois at Chicago

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Galois Coverings in the Theory of Algebraic Surfaces
B. MOISHEZON AND M. TEICHER
1. Branch curves and their fundamental groups. In the modern
development of the theory of complex algebraic surfaces some deep and challenging
problems are related to the classification of surfaces of general type. Very often
such surfaces are simply connected. So with the present state of our knowledge
it is natural to restrict ourselves to surfaces of general type which are simply
connected. U. Persson [1] showed how to construct families of such surfaces
covering a big domain in the space of basic pairs of discrete invariants (ci,x)>
or equivalently (cf,C2), or (62,^") (H = 2-dimensional Betti number, r = index
of corresponding 4-manifolds). Each of these pairs together with the type (even
or odd) of the corresponding intersection form defines a unique homotopy type
(and even unique homeomorphism type by M. Freedman). Thus already for
U. Persson surfaces there are interesting questions about global moduli spaces
related to given topological types of 4-manifolds.
Using divisibility properties of the canonical class it is not difficult to construct
infinite series of examples of the simplest U. Persson surfaces (so-called double-
double coverings of CP1 x CP1) for which moduli spaces have more than one
connected component. A more general approach is due to F. Catanese [2], who
computed explicitly dimensions of local (Kuranishi) moduli spaces for many
double-double coverings of CP1 x CP1. In his work F. Catanese obtained a series
of examples in which the number of irreducible components of moduli spaces goes
to infinity. It seems that for algebraic surfaces of general type disconnectedness,
or not irreducibility of moduli space is a quite common phenomenon, which
is in striking disagreement with our knowledge of moduli spaces of algebraic
curves. It is possible that different connected components of moduli spaces of
simply connected surfaces of general type represent main classes of different
C°°-structures existing on corresponding 4-manifolds.
To distinguish connected components of moduli spaces we have to look for new
discrete invariants. The following approach was suggested a few years ago by
1980 Mathematics Subject Classification (1985 Revision). Primary 14E20.
Partially supported by the National Science Foundation Grant number DMS 85-03743.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
47

48
B. MOISHEZON AND M. TEICHER
K. Chakiris: let V be any surface of general type, Ky the canonical class of V, m
an integer > 5, <pm : V —► CPN a pluricanonical embedding of V to a projective
space corresponding to mKy, CPN —► CP2 a generic projection to CP2, and
/ : V —► CP2 the corresponding map. Denote by Sy C CP2 the branch curve of
/, Sy C V the corresponding ramification curve, /' = f\Si : Sy —► CP2. Define
Sv (the modified branch curve) as follows: if £>m(V) is nonsingular, Sy = Sy. If
^m(V') is singular (it can have only rational double points), replace /' by a close
C°°-map /': Sv —► CP2 so that f'{Sv) will correspond to local regeneration
(smoothing) of singular points of Sy corresponding to singularities of <pm{V).
Then Sy = f'(Sv). It is easy to see that the topological type of (Sy *-► CP2)
is not changing under deformation of the complex structure of V. K. Chakiris's
suggestion was to study the topological type of (Sy <-► CP2) and in particular
7Ti(CP2—Sv, *) as the main invariants of the connected components of the global
moduli space of V (that is, the space of complex structures on the topological
4-manifold corresponding to V).
Let n = deg /, and let Sn be the symmetric group of order n. There is an
evident surjection
^:7n(CP2-Sv,*)-^Sn
(monodromy map of finite morphism /). To study 7Ti(CP2 — Sv, *) and ^, we
can use the so-called braid monodromy technique developed in [3, 4, 5]. The
problem is similar to problems in classical knot theory and could be considered
as a complex-analytic analog of it. A certain advantage of the complex-analytic
case is that algebraic curves in CP2 are "minimal surfaces." Thus a variational
principle is involved, and geometrical phenomena related to "complex-analytic
knots" in CP2 have some properties of rigidity and beauty unknown in the more
free and artificial world of classical :nots.
7Ti(CP2 — Sv, *) is an infinite grc up with finite presentation, and to get some
finite invariants from it we have tc descend to different levels of depth in its
structure. We can start as follows. Let £ be a generic line in CP2 containing
(*), Q = ^flSv- Q is a finite set, say (6i,..., bm). Consider on £ small circles c^,
i = 1,..., m, around points b{, i - 1,..., m, and let 7'^..., 7^ be a system
of simple paths in ^, each 7^ com ecting (*) with a point on c^, Diii l[ =
(*). Denote by 7» the element of '\(£ — Q, *) corresponding to 7^ U C{ (take
Ci with positive orientation). It is \ ell known that the natural homomorphism
tti(^ — Q, *) —► tti(CP2 — Sv, *) (corresponding to £ <-► CP2) is a surjection.
Let Ti,... ,Tm be images of 71,... ,7m in 7Ti(CP2 — Sv,*)- We call Ti,... ,Tm
geometric generators of 7Ti(CP2 - Sv, *). If we use a different choice of £ and
7i,..., 7^, then the given geometric generators are replaced by their conjugates.
For any positive integer v denote by Gv the normal subgroup of 7Ti(CP2 — Sv, *)
(normally) generated by ((r^, i = 1,..., m). It is clear that the definition of
Gv is independent of the choice of £ and 7i, • • •, 7^ •
Denote by A — Ker tp. For all i = 1,..., m, ip(Ti) is a transposition in Sn
(because / corresponds to a generic projection to CP2). Thus, for any even 1/,

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 49
Gv C A. So we can start to consider A/G2,G2/G4, etc. and corresponding
abelianizations. It is easy to see that A/G2 is isomorphic to tti(V), where V is
the Galois covering of CP2 corresponding to /.
We mentioned above that the simplest U. Persson surfaces are double-double
coverings of CP1 x CP1. It happens that to study generic CP2-projections
corresponding to their pluricanonical embeddings we have to understand
similar questions for generic CP2-projections of different projective embeddings of
CP^CP1. Denote by £x = CP1xpt,£2 = ptxCP1 (generators ofCP^CP1),
by i£ab = a£i + b£2 (a, b positive integers) and by Xa^ the image of CP1 x CP1
under the projective embedding CP1 x CP1 C CPN corresponding to the
complete linear system \Ea,b\- Taking a generic projection CPN —► CP2 we get a
finite map fa,b'> Xa^ —► CP2. Let Sa,b C CP2 be the corresponding branch
curve, n = degfa,b (^ = 2a6), tpa,b'- tti(CP2 — Sa,b,*) -* Sn the corresponding
monodromy homomorphism (surjection in our case), Aa,b = Ker^a^.
Introducing as above geometric generators Ti,..., Tm of 7Ti(CP2 — Sa,b, *) we define
normal subgroups G^-a,b of 7Ti(CP2 — Sa,bi*) (normally) generated by {(Ti)",
i = 1,... , ra}.
Let fa,b- Xa,b —► CP2 be the Galois covering corresponding to fa,b-
If generally g : X —► CP2 is a finite morphism of a nonsingular algebraic
surface corresponding to a generic projection ( "stable morphism" ) and g : X —► CP2
the corresponding Galois covering, then it can be shown that X is
nonsingular. Moreover, if S C CP2 is the branch curve of g, S C X the ramification
curve of gf, and £ a line on CP2, then the canonical class K^ of X is equal to
g*(—3£) + S and because as a divisor 5 = ^g*{S) = \mg*(£), m = deg5, we
have Kx = (y — 3)<7*^. Thus when m > 6, K^ is ample and X is a minimal
surface of general type. If <p: 7Ti(CP2 — 5, *) —► Sn is the monodromy surjection
corresponding to /, A = Ker^, and the G^'s are introduced as above, we have
in general: ni(X) = AjG<i.
For Xa,b introduced above we have m = deg Sa,b = 6a6 — 2a — 26, so m > 6
when a > 2, b > 2.
In our recent study of fundamental groups 7Ti(CP2 — Sa,b, *) we succeeded in
proving that for a > 3, b > 2 all groups Aajb/G2-,a,b are finite and commutative
and actually trivial when a, b are relatively prime (see [6]). Using arguments
similar to those used in §3 below, we can show now that also for a = b = 2,
Aa,b/G2;a,b is a finite commutative group.
Because 7Ti(Xa>b) = Aa,b/G2-,a,b we obtained the following
THEOREM 1. For a>2,b>2all Galois coverings Xa^ are minimal surfaces
of general type with finite commutative fundamental groups. For a, b relatively
prime these surfaces are simply connected.
Our analysis of 7Ti(CP2 — Saib, *) could be applied with some modifications
(see §3 below) to a class of surfaces similar to Xa,b, namely, to images Y^, k > 1,
of Veronese embeddings of CP2 in CPN corresponding to \k£\ (monomials of
degree k).

50
B. MOISHEZON AND M. TEICHER
Denote by fk : Yk —► CP2 a generic projection, by fk : Yk —► CP2 the
corresponding Galois covering, by Sk the branch curve of fk in CP2, and by
ipk'- tti(CP2 — Sk,*) —► Sn (n = deg/fc = A:2) the monodromy homomorphism
of fk. Let A*; = Ker tpk and let G^;^ be the normal subgroup of 7Ti(CP2 — S&, *)
(normally) generated by {(r^}, where {I\} is a system of geometric generators
Of7Ti(CP2-5fc,*).
It is interesting that in the case of the Yjt's only for A: > 3 are the groups
Ak/G2,k finite commutative. The degree of Sk is equal to 3A:(A: — 1) and it is
> 6 also for A: > 3. We proved an analog of Theorem 1.
THEOREM 2. For k > 3 all Galois coverings Yk (corresponding to generic
CP2-projections of kth Veronese surfaces Yk) are minimal surfaces of general
type with finite fundamental groups.
The classical Veronese surface Y2 is an exceptional case here (as in some other
problems of algebraic geometry). The corresponding Galois covering Y^ has an
infinite fundamental group which is commutative. We can construct Y2 explicitly
using the following
PROPOSITION 1. Let f\Y2 —► CP2 be a generic projection, f: Y2 —► CP2
the corresponding Galois covering, S C CP2 the branch curve of f, S' the
ramification curve of f in Y2, and Si = f*(S) — 2S'. Identifying Y2 with a projective
2 2* 2 2*
plane CP , denote by CP the dual projective plane to CP and by C C CP
the dual projective curve to Si.
2* ~
Then C is a nonsingular cubic curve in CP and Y2 is isomorphic to C xC.
In particular, ^1(^2) = Z©Z©Z©Z.
PROOF. S is a sextic in CP2 with nine cusps. Let P^, i = 1,..., 9, be cusps
of S, Pi = f~l(Pi) H S'. In each P/ three sheets of the covering f:Y2-+ CP2
are coming together. Because deg/ = 4 we get that f~x(P{) — Pi is a single
point. Denote it by Q\. It is clear that each Q\ is a cusp on Si (= f*(S) — 2S')
and that Si fl f~1(S — \Ji=1 Pi) is nonsingular.
Direct verification shows that the branch of Si passing through P/ is
nonsingular. Thus the only singularities of Si are the nine cusps Qi,..., Qg. Let I be
a straight line on Y2 {= CP2), I a straight line on CP2. Then f~l(t) = 2Î,
-3£ = KY2 = f*{KcP*) + S' = f*(-M) + S' = -Û+ S'.
Thus S' is linearly equivalent to Si From 2S' + Si = /_1(S) ~ /_1(^) = 12^
_ 2
we get: Si ~ 6£. Thus Si is a curve of degree 6 on y2 (considered as CP ) with
9 cusps. In particular, the geometrical genus of Si is equal to one, and the
2* ~
dual curve C to Si is a nonsingular cubic curve on CP . Let g : y2 —► ^2 be
a canonical map corresponding to /: F2 —► CP2. There exist a degree-three
covering gi : Z —► Y2 and a degree-two covering g2 : Y2 —► Z, g = gi o g2 (and
/ = f ° 9i ° 92)' The branch curve of gi is Si and the branch curve of 02 is
#2 = 9Ï1{Si) — 2SJ, where S[ is the ramification curve of gi in Z.

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 51
Denote by E = g^ 1(£). It is clear that the canonical class Kz of Z is equal
to gl(-Û) + S[ and
2g(E) -2={KZ + Ë)Ë = [gl(-Sê) + S[ + g{{£M{£)
= (-3) ■ 3 + Si • £ + 3 = -6 + 6 = 0.
Denote by /i the number of singular elements in a generic pencil of \E\. We
have:
c2{Z) + deg gi = 2(2 - 2g(E)) + /i
or
c2(Z)+3 = /i.
But /i is equal to the degree of the projective dual curve C to Si, that is, /i = 3.
Thus c2(Z) = 0 and 6i(Z) ^ 0. From 0(£) = 1 we get that bx(Z) = 2 and
that singular elements of generic pencils of \E\ are reducible. These singular
elements correspond to the family {^a, a G Si} of lines in Y2 (= CP ) tangent
to Si at a. Let a be any nonsingular point of Si. From g(E) = 1, b\(Z) =
2, Z nonsingular, it follows actually that #f 1{£a) consists of two nonsingular
irreducible components, one of which is isomorphic to a projective line and the
second has genus one. Denote the first by £'a. From deg^i = 3 it follows that
9i\i' • t'a ~~+ ta is an isomorphism. We see that Z has a 1-dimensional family
{£'a} of nonsingular projective lines parametrized by an elliptic curve (Si or C).
From the fact that 0i |^ : t!a —► £a is an isomorphism it follows that f!a (1 S[ is the
single point gïl(à) DS[. We have
-2 = 2g(Q -2 = Kz.ta + (Q2 = (-3Ë + S[) • t!a + (Q2
= -3 + S[ • t!a + (Q2 > >3 + 1 + (l'a)2 = -2 + (Q2.
Thus (£fa)2 < 0 and because £'a is a member of the algebraic family {£fa} we see
that (£'a)2 = 0. Because Z is nonsingular there exists a regular map <p: Z —► C
with generic fibers equal to elements of {£'a}. Let <p: Z —► C be the corresponding
minimal CP1-bundle. Then c<i(Z) — 0 (because g(C) = 1). Since c2(Z) = 0 we
see that already <p: Z —► C is a CP1-bundle with base C. Let ifr: T —> C be the
graph of the algebraic family of projective lines tangent to Si. (Remember that
C is dual to Si.) There exists a regular map a: Z —► T with ^ o a = <p. On
fibers of <p and ip the map a induces an isomorphism. Thus h is an isomorphism.
Thus we can identify (Z,<p) with (r,^).
We can identify Y2 with the closure Z Xy2 Z — A, where A is the diagonal in
Zx Z.
Let (A, B) G C x C, and let ^U,b be the straight line in CP2* defined by
(A, B) (ïî A = B we take for £a,a the tangent to C at A). Define two maps
hi : C x C —► Z, i = 1,2, as follows:
M(i4,fl)) = ((/a,b)*,A*), /i2((A,5)) = ((^lfl)*,B*).
Here ( )* means projective dual. So for example A* is a tangent £a to Si at some
point a G Si. We use identification (Z, <p) with (r,t/;), where T = {(c,^), c EY2

52
B. MOISHEZON AND M. TEICHER
(= CP ), £ is a straight line in Y2 (= CP ), £ 3 c, £ is a tangent to Si},
tp(c,£) = £* (e C, since ^ is a tangent to S\).
It is clear that after identification (Z, <p) = (r, ^) £i is defined by #i((c, ^)) = c.
We see that gih\ = g\h2 and there exists a map h: C x C —► Z Xy2 Z
canonically corresponding to (hi, /i2)-
Let (x,y) G Z Xy2 Z — A be a general point. We have x E £'ai, y e £'a2 for
some ai,a2 G Si. From i^y, ^i(x) = ^i(y) G ^ H^a2, it follows that ai ^ a2.
Let A = (4J*, 5 = (42)*. Then £ai D £a2 = (£a,b)*- We can write (using
Z = T):
x = {£ai n£a2,£ai) = {{£a,b)*,A*),
y = {£ai n£a2Ja2) = {£a,b)*,b*).
We see that
x = hx{A,B), y = h2{A,B).
So (x,y) = h((A,B)).
If (x, y) = h((Ai,Bi)) we must have
x = (£fll n42,4J = fci((i4i,Bi)) = (^1|fll,AÎ),
y = {tai n£a2,£a2) = h2((A1,B1)) = (^lifll,Bj).
Thus
-Ai = £aii A\ = (£a\) = A,
B\ — ta2, B\ — (£a2Y — B.
We see that h(C x C) = Z Xy2 Z — A = Y2 and that h is a map of degree one.
Because C x C is a minimal surface we get that /i is an isomorphism. Q.E.D.
2. Simply connected surfaces of general type with positive and zero
indices. Miyaoka [7] was the first to observe that Galois coverings of generic
CP2-projections very often have index r > 0, or equivalently c\ > 2c2. If
/: X —► CP2 is a generic CP2-projection and /: X —► CP2 the corresponding
Galois covering, then the precise formula for the index r(X) is the following:
t(X) = in! [§p - 3ra - 3/i + 12]
where n = deg/, m = deg5 (branch curve of / in CP2), p = the number
of cusps of 5, and /i = degree of projective dual curve of S. (Use 3r(X) =
c?(X) - 2c2{X) and formulae from [6]: c\{X) = \n\(m - 6)2; c2{X) =
n!(|ra2 — §ra + 3 — \d — |p), m2 — 2d = m + /i + 3p (d = number of nodes
of 5).) For Xa>b, 5a>6 introduced above we have
n = 2ab; m = 6ab — 2a — 26;
p = 24a6 - 18a - 186 + 12; /i = 6a6 - 4a - 46 + 4.
Thus
T(Xatb) = |n!(a6 - 3a - 36 + 5).
From this we see that r(Xa,b) > 0 for a > 7, 6 = 4 or a > 6, 6 > 5 and
t(X7A) = t(X^) = 0.

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 53
The Bogomolov-Miyaoka-Yau inequality states that for surfaces of general
type c\ < 3c2- After this inequality was discovered, empirical evidence suggested
that for simply connected such surfaces a stronger inequality must hold:
c\ < 2c2, or equivalently r < 0.
The following conjecture was formulated (see [8]):
CONJECTURE 1. Any algebraic surface X of general type with r(X) > 0 has
nontrivial fundamental group (or equivalently, infinite fundamental group).
By a theorem of Yau, surfaces of general type with c\ = 3c2 are exactly those
which are uniformizable in the unit ball of C2. These surfaces can be considered
as an extremal case in the classification of surfaces of general type.
All homogeneous bounded domains in C2 are isomorphic either to the unit
ball or to the unit polydisk (= D x D, D = unit disk in C1). For surfaces
uniformizable in the polydisk it is known that c\ = 2c2. If Conjecture 1 were
true, that is, if simply connected surfaces of general type would have c\ < 2c2,
then we could expect that on their "classification boundary" lie very special
surfaces. So the following analog of Yau's theorem was conjectured.
CONJECTURE 2. Minimal surfaces of general type with c\ — 2c<2 (or
equivalently t = 0) are exactly those which are uniformizable in the polydisk.
Our results show that both Conjectures 1 and 2 are not true. Namely,
Theorem 1 (see §1 above) states that all Xa,b, a > 2, b > 2, are minimal surfaces of
general type with finite commutative fundamental groups. Because r(Xa^) > 0
for a > 7, b = 4 or a > 6, b > 5 we see that all surfaces Xa,4, a > 7, and Xa,b,
a > 6, b > 5, are counterexamples to Conjecture 1.
Surfaces X4J and X^^ are counterexamples to Conjecture 2 (t(X4j) =
T(*5,5)=0).
Another series of counterexamples we obtain by considering "Veronese
surfaces" Yfc and their Galois coverings Y^ (see §1). Let Sk C CP2 be the branch
curve of the generic projection fk : Yk —► CP2, rrik = deg Sk, Vk = degree of the
projective dual curve of S^, and p = number of cusps of SV It is easy to show
that
mk = Sk(k - 1); /ifc = 3(fc - l)2; pk = S(k - l)(4fc - 5).
From r(Yk) = ^n![|pfc - 3rafc - Sfik + 12] we get now:
r(Yk) = ^n! [§ • S(k - l)(4fc - 5) - 3 • Sk(k - 1) - 3 • 3(fc - l)2 + 12] ,
(n = deg/fc = A:2),
or
r(Yk) = \n\(k-2)(k-l).
In particular t{Yt) = 0 and r(Yk) > 0 when A: > 8. Theorem 2 (see §1) states that
for A: > 3 each Yk is a minimal surface of general type with finite commutative

54
B. MOISHEZON AND M. TEICHER
fundamental group. We see that all Yit, A: > 8, are counterexamples to Conjecture
1 and Y? is a counterexample to Conjecture 2.
3. Veronese surfaces of order three. In this section we illustrate the
proofs of Theorems 1 and 2 (§1) giving an outline of the proof of the simplest
case of Theorem 2, namely the case A: = 3. That is, we show below how to prove
the following
STATEMENT. Let X be a Veronese surface of order three (Y3 in terms of
Theorem 2). Let Y be its Galois cover with respect to a generic projection f : X —►
CP2. Then ni(Y) is a finite commutative group.
OUTLINE OF THE PROOF. Let FAff be the part of Y lying over a generic
affine part of CP2. 7ri(FAff) —► tti(Y) is evidently a surjection, so first we
consider 7Ti(yAff).
1. Reidemeister-Schreier method (RMS method). (See [9, p. 89].) Consider
the following data:
(1) an exact sequence of groups 1—> A —> G —> S —► 1, where S is finite and
G has finite presentation;
(2) a finite set of generators for G: {gi}iei\
(3) a finite complete set of relations for G (corresponding to {gi}iei)- {R}',
(4) a splitting p: S —► G for rj) (rj) o p = Id).
The RMS method gives a finite presentation of A (corresponding to (l)-(4))
as follows: Denote x = pip(x) (x G G). Then a set of generators of A is
{p{<j)9ig-\p{<j)rl\<jes, is I}.
A complete set of relations for A is obtained from {R} as follows:
Each Ra — p(a)R(p(a))~1 = g^ • • ^, a G S, Sj = ±1, induces the following
relation in A:
k
A(R(t) = HA3(R(t),
3 = 1
where
A3{Ra) = gtl---g\l:l9eil{gtl---0-\
We shall apply the RMS method for a description of 7Ti(YAf£).
2. A set of generators for 7Ti(C2 — 5, uo). f: x —► GP2 can be degenerated
into /o: Xq —► CP2 (see [5]), where Xq is a union of nine projective planes
Pi,..., Pq defined over R forming the configuration shown in Figure 1.
(Here each 2-simplex corresponds to a projective plane, each 1-simplex to a
projective line, and each 0-simplex to an intersection point.) We shall numerate
the intersection lines £i,..., £9 between the planes Pi,..., P9 as shown in Figure
2.
The union U»=i /bC^t) we denote by So and consider it as the "branch curve"
of /o which is a degeneration of the branch curve S of the map /.
Let 7T : C2 —► C be the projection on the first coordinate (which we can assume
to be generic). Let M' C S C C2 be the set of all points of S where ^15 is not

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 55
Figure l Figure 2
étale, M = 7r(M') (M is finite). Take a real positive u G C - M which is far
enough from the points of M. Let uq be a point in C1 = 7r-1(u), u £ S. Denote
qi = f0(£i) fl C1, i' = 1,..., 9. Regeneration of S0 to S involves a "doubling" of
each of fo{£i). Thus, after regeneration, to each & will correspond a pair çz, #/,
in S C\ C1 and S n C1 = {qi,qv,.. • , <79><79'}-
Let ci, ci/,..., eg, eg/ be small circles (in C1) around points <?i, q\>,..., çg, <?g/
and let 71,7i',..., 7g, 7g>, be a system of simple paths in C1, each 7» (resp. 7»/)
connecting uq with point on C{ (resp. c»/), 71 fl 71/ fl ■ • • fl 79 fl 7g' = ^o- Denote
by 7z (resp. 7^) the element of ^(C1 - C1 fl S,u0) corresponding to 7» U c^
(resp. 7»/ UCt).
Denote by Ti, Ti/,..., Tg, Tg/ a geometric system of generators of
7ri(C2 - S,ii0) corresponding to 7i»7i'»---»79»79' (see §1).
2 2 2
Let (Tjj,) be the normal subgroup of tti(C2-S, u0) generated by r^T!,,...,
Tl,Tl„ Denote by G = tti(C2 - 5,u0)/(r^).
It is easy to see that we get the same G when using different systems of
geometric generators of 7Ti(C2 — 5, uo).
3. Zariski-Van Kampen method for obtaining complete sets of relations for
7Ti(C2 — 5, uq) and G. A classical theorem of Zariski and Van Kampen says
how to get all relations between geometric generators of 7Ti(C2 - S,u$) (see
[10, Chapter VIII; 11]). This theorem can be reformulated in the terms of the
so-called braid monodromy (see [3, pp. 127-130]).
Let 5i8 = 5i8[C1, C1 (IS] be the braid group corresponding to the Euclidean
plane C1 and to the finite set C1 fl S (c C1). There exists a naturally defined
homomorphism 6: 7Ti(C — M, u) —► Bi8 which is called the braid monodromy
corresponding to S C C2, tt: C2 —► C, and u G C. Because / is a generic
projection we see that, for each a^ G M, n~1(ai) D M' is a single point, say a'^
and a[ is either a nonsingular (stable) ramification point of S (corresponding
to 7r) or a node or a cusp of S. Let {6i} be a good ("geometric") system of
generators for 7Ti(C — M, u) ordered naturally by going in positive direction
around u.

56
B. MOISHEZON AND M. TEICHER
By the Zariski-Van Kampen theorem each 0(6i) induces a relation between
geometric generators ri,IY,... ,rg,rg/ of 7Ti(C2 — 5, uo) and such relations
generate all the relations.
In our case (S is a curve with singularities only nodes and cusps) each of these
relations has the following form:
(1) A = B (corresponds to a nonsingular ramification point of S);
(2) AB = BA (corresponds to a node of S);
(3) ABA = BAB (corresponds to a cusp of S); where A and B are some
conjugates of ri,IY,... ,rg,rg/.
To define explicitly the relations prescribed by the Zariski-Van Kampen
theorem we need explicit formulae for the braid monodromy 6. Such formulae for
all Veronese embeddings Yk of CP2 were obtained in [5].
4. A set of relations for G. In [6] it was shown in particular how results of [5]
must be translated in Zariski-Van Kampen relations for 7ri(C2 — Sa,b, uo), where
Sa,b is the branch curve in CP2 corresponding to a generic CP2-projection of
Xa,b (defined in §1 above).
For surfaces Yk we get a similar translation which in our case (F3, Veronese
surface of order three) provides a set of relations in G (for a special choice of
ri,IY,... ,rg,Tg/) which we can describe as follows.
First introduce some notations. Denote by ri,IV,... ,rg,Tg/ the images of
Ti,Ti/,...,Tg,Tg/ in G. For any i G (1,..., 9) and any k G Z denote by
^2fc+i(ri) = (riT0fcr^(r^ri)-fc,
and by Tq an arbitrary element of the set {<pn(Ti), n G Z}.
Denote by
1^3' = ^^3^2', P3 = r2'T3r3/r3r2';
r4/=r4/, T4 = r4, r5/=r5/, r5 = r5;
Tv = rgr7/r7r7/r8, r? = TsT^tTs.
Define Tq similarly to Tq.
Let (A, B) = ABA(BAB)-1 and [A, B] = AB(BA)~l. A set of relations
which we obtain almost directly using [5, Theorem 8.2 (p. 342), Proposition 7.1
(p. 338)] and the last Remark of [5, p. 343] is the following (remember that
r2 = 1, r2 = 1, i' = 1,..., 9, and compare with [6, Chapters 0 and 1]):
(A)
(A4)
(A5)
(A6)
(A7)
(A8)
(A9)
(A10)
Commutativity relations:
lr®>r®] = i;
lr©>r®] = !»
[r©,r©] = i,
[r@,r©] = l,
[r©,r©] = l,
[r®,r©] = l,
[f3,f7] = l;
[f3',f7-] = i.
i=l,2;
i= 1,2,3,5;
i = l,2;
1 = 1,2,3,4,6;
t = 1,2,3,4,6;

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 57
(B) Relations of type (A, B) = 1 :
(B2) (r@,r0) = ]
(B3) <r@,r@) = ]
(B4) <r@,r@) = ]
(B6) <r®,r®> = ]
(B7) <r©,r®> = ]
(B8) <r®,r©> = ]
(B9) <r®,r©> = ]
(BIO) <f®,f@) = ]
(Bll) (f®,f®) = .
^
L, i=l,2;
-Î
-Î
-Î
L, t = 5,7;
L, i = 5,8.
-Î
(C) "Equality" relations:
(Cl)
(C2)
(C3)
(C4)
(C5)
(C6)
(C7)
(C8)
(C9)
(CIO)
(Cll)
(C12)
^'Ti^' = l^IVl^;
Ts'TiTs' = r3lYr3;
T6'r4r6' = rg^rg
Tg/r5rg' = rgr5'Tg;
r7'T6r7' = r7r6'r7;
Tg'Tgrg/ = TgrsTg;
^'•T4^3' = *J*15^Z;
r3/r4'r3' = TrT^TT,
r3r4r3 = r^TTTyT^^TT-r;
r3r4'T3 = ^iv^^i^^i^;
r2 = r4'r4r3'r3r2'r3r3'r4r4';
IV = rsrs^^rg^'^rs/rs.
We shall now prove several claims.
claim a. [r®,r7/r7r7/] = 1.
PROOF. Because of (AlO) [f3/,f7/] = 0 and using the definitions of f; we
get
Now
Thus
[r2'r3r2',r8(r7/r7r7/)r8] = 1.
[r2T3r^r8(r7*r7iY)r8] = i
# (because of (A8), (A7))
[r3,rrr7rr] = i
# (because of (C2) and (B3))
[ri/rir3/riri/,r7/r7r7/] = 1
# (because of (A7))
[T3,,TrT7Tv} = l.
[r®,r7T7rr] = i. q.e.d.

58 B. MOISHEZON AND M. TEICHER
Claim /?. [r®,r©] = 1 and [f®,f©] = 1.
PROOF. Arguing as above we get:
[r2'r3r3/r3r2',r8r7/r8] = i
# (because of (A8), (A7))
[T3r3T3,IV] = l.
With [r3r3/r3,IVr7r7/] = l (see Claim a) we get [r3r3/r3,r7] = 1 and thus
[r3r3,r3,r©] = i.
Using (C2) and (B3) we can write:
r3(r3,)r3 = r3(r!ivr3ivri)r3 = r3rir3ivr3rir3
= rir3ririTir3ri =riririTir3rirrriri =rrrir3rirr.
So we have
[rrrir3rirr,r(Z)] = i
$ (because of (A7))
(r3,r©] = i.
We see that [T®,r©] = 1. By of (A8), (A7) this is equivalent to [f®,f@] =
1. Q.E.D.
claim 7. <f®,f©) = i, <f®,f®) = i, and <r®,r©) = i, <r®,r®) = i.
PROOF. We have „ „
(r5,r7) = 1 (see (Bll) above)
# (because of (C7))
<f7f3,f4f3,f7,f7) = l
<f3,f4f3*,f7) = l
$ (because of Claim j3 above)
(f4,f7) = i
$ (because of (A8))
<r4,r7) = i.
Using Proposition 7.1 from [5] ("Invariance property") we get (r@,T©) = 1 and
because of (A8), (f@,f@) = 1.
Similarly we can write:
(f3/,f4) = 1 (see (BIO) above)
$ (because of Claim (3 above)
(r^irvrvi^ivirv) = i
# (because of (C7))
<f3,,f5) = l
$ (because of (A5))
(r3,r6) = i.
Using again the "Invariance property" [5, Proposition 7.1] we get (r@,r@) = 1
and because of (A5)
(f®,f®) = l. Q.E.D.

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 59
Claim 6. [r@,r@] = 1.
PROOF. Consider S4, the symmetric group of degree 4. Denote by X\ = (12),
x2 = (23), x3 = (34), x4 = (14).
A finite presentation of S4 is given by generators Xi,X2,X3,X4 and
relations: xf = 1, i = 1,...,4; (xi,x2) = 1; (x2,x3) = 1; [xi,x3] = 1 and
£4 = X3X2X1X2X3.
Using this presentation define two homomorphisms 1/1,1/2 of 5 in G by:
^i(zi) =f3, 1/2(^1) =T3,
vifa) =f4', ^2(^2) =f4,
i/i(x3) = f7frf7, i/2(x3) = f 7f 7'f 7,
i/i(x4) = f5/ 1/2(^4) = r5.
It follows from Claims 7 and /? above and from (C9), (CIO), (BIO) that 1/1,1/2
are well-defined homomorphisms. Because [x2,X4] = 1 in S4 we get, using v\
and 1/2,
[f4',f5'] = l, [f4,f5] = l
or equivalently
[r4,,r5^] = i, [r4,r5] = i.
From (C3) we get T4 = r6/r6r4/r6r6/, r4/ = r6r6/r4r6/r6. So we have
[r6r6T4r6/r6,r5'] = 1 and [r6/r6r4T6r6/,r5] = 1
(because of (A6)) $ $ (because of (A6))
[r4,r5*] = i [iv,r5] = i.
Thus [TqT®] = 1. Q.E.D.
Claim e. [r@,r0] = 1, [r®,r©] = 1.
PROOF. We can write:
[r2,,r5r8rrr8r5] = 1 (see (A5), (A7), (as))
[r3r2'r3,r3r5r8r7/r8r5r3] = 1
# (because of (B3))
[r2'r3r2',r3r5r8r7/r8r5r3] = 1
[r3,r2'r3r5r8r7/r8r5r3r2'] = 1.
From (C7) we see that T4 = f4 = f3/f7f5f7f3/. Using the definition of the
fys (p. 538) and (A5), (A7), (A8) we get: T4 = r2'r3r5r8r7,r8r5r3r2' and
so [r3,r4] = 1. Using the "invariance property" [5, Proposition 7.1] we get

60
B. MOISHEZON AND M. TEICHER
[T®,r®] = l. Similarly:
[r4,r8] = l (see(A8))
$ (because of (A8))
[r3rvr4rvr3,r8] = i
[r6r3ivr4ivr3r5,r5r8r5] = 1
# (because of (B8))
[r5r3r2,r4rvr3r5,r8r5r8] = i
[r8r5r3iYr4ivr3r5r8,r5] = 1
$ (because r4 = rvr3r5r8ry r8r5r3rv
(see above))
[rr,r6] = i.
Here again by the "invariance property" [5, Proposition 7.1] we get that
[r©,r®] = i. q.e.d.
Adding the relations given by Claims a,/?,7,6,e to relations (A4)-(A10),
(B2)-(B11), (C1)-(C12) and using the definitions of the f z's (p. 538) we obtain
the following list of relations in G:
(Rl) I^ = r), = l0- = 1 9);
(R2) [r0,T0] = 1 for i and j such that the corresponding l-simplices of
Figure 2 (p. 537) are not on the same triangle (2-simplex);
(R3) (r©,T0) = 1 for i and j such that the corresponding l-simplices of
Figure 2 are on the same triangle;
(R4) ri = r3T3riT3r3';
(R5) Tô^rgTgrsTgrg/;
(R6) r2,=riivr2iYri;
(R7) Tq' = r^^Ter^r^
(R8) Tq = ^^^^7^;
(R9) rg^rgTgrgTgrg/;
(Rio) r2 = r4'rjYr3iYr3r3>rjV;
(Rii) r8, = r5r5T7r7T8r7'r7r5T5;
(R12) r4r2T3r2T4 = r5rrr8r7T5;
(R13) r4/r2/r3r2/r4/ = r5/r7/rgr7/r5/;
(Ri4) r4r2T3r3T3r2T4 = r5r7r8r7r5;
(R15) r4/r2/r3r3/r3r2/r4/ = r5/r7rgr7r5/.
REMARK 1. Actually it is possible to check that (R1)-(R15) is a complete
list of relations between Ti, Ti/,..., Tg, Tg/.
5. Splitting of a "covering monodromy homomorphism" Consider the
covering monodromy homomorphism rj): 7Ti(C2 — S, uq) —► Sg corresponding to /
(deg/ = 9, Sg = symmetric group of degree 9). Because / is a generic
projection, V> is a surjection and all ^(T^^ÇTji) are transpositions in Sg. Thus
(Tj,j') Q Ker ip. By the standard isomorphism theorem we get an exact sequence
1 -> Ker^/fl^-,) - 7n(C2 - S,uo)/Flj.) ^ S9 ^ 1

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 61
~ o
(where xj) corresponds to xp). Recall that G = 7Ti(C2 - S,uo)/(TJ3,). Denote by
~ 2
A = Kerxp/(Tjj,). So we get:
It is easy to see that
w1(YAB) = Kerrp/(Tl}l) = A.
Using the degeneration of / to /o we can show that for our choice of Ti,...,Tg,
we have, V/ = 1,..., 9, xp(Tj) = xp(Tjf) = (A:/), where the pair (A:, /) corresponds
to the pair of planes Pk,Pi ( in Xq) with Pkf)Pi = £j. Denoting by ai = xp(Ti)
we see from Figures 1 and 2 that
<*i = (12); a2 = (23); a3 = (24); a4 = (35); a5 = (47);
a6 = (56); a7 = (58); a8 = (78); a9 = (79).
{a», z = 1,..., 9} is a set of generators of 5g and all relations between them
follow from:
(51) a? = l,i=l,...,9;
(52) [a», o^] = 1 if the transpositions a», o^ have no common index;
(53) (otiOtj) = 1 if a», o^ have exactly one common index;
(54) ag = a7a4a2asa^a3a20t4(^7•
Using (S1)-(S4) we define a homomorphism p: Sg —► G (a splitting for ^) by:
ri> fory = 2,7,
p(a?) = <
PV " IT,- for;/#2,7.
We have to check that (S1)-(S4) are satisfied when we replace a3-'s by p(a3-ys.
For (SI) it is evident. For (S2) and (S3) it follows from (R2), (R3) above and
for (S4) it follows from (R12) above. (From (R12) we have
Then by (R2)
Tg = r7/r4r2'F5r3r5r2'r4r7/
and by (R3)
Ts = r7/r4r2'F3r5r3r2'r4r7/.
6. Applying the RMS method for A. Using the splitting homomorphism p
and the RMS method (see §3.1) we get the following set of generators for A:
k e p(S9),
<?eS9,
o e Sg, /ce p(Sq).
Applying the RMS method to (R1)-(R15) we can get the following relations
between Aa.j (a G Sg, j = 1,..., 9) (compare with [6, Chapters 3-6]).
p{<T)TjTj,{p(<T))-1
We denote by
,<reSg.
j\.fç"j —— rvJ. i% J. tji rv
Aa;j = Ap(ayj,
A-aK'j = ^p{a)K\ji

62
B. MOISHEZON AND M. TEICHER
(Dl) Aaaj;j = (Aa;j)-\j = l,...,9;
(D2) Aaai;j = Aa]j • AaocjOCi.j = Aa0Cj0ti^A^j if i and j are such that
<r<;r0) = i;
(D3) 4^ = ^;i if [k^-IY] = l;
(D4) Aa.2 =ACrr1r2/;i;
(D5) Aa;3 = A<TTlr3;i;
(D6) Aa.4 = Aar2,r4;3 ' A<Tr2,r4;2{A<Tr2,r4r2,;3)~1''>
(D7) Aa.5 = ACTr8r7/r3r2/;4;
(D8) Aa;e = Avr^eA'i
(D9) A<r;7 = Aar6r7,&
(D10) Aa.8 = ACTr5r7,r3r4;2;
(Dll) Aa;9 =Acrr8r9;8;
(D12) Aa,g = AaT5T9;5.
REMARK 2. Using Remark 1 (see p. 542) it is possible to show that (Dl)-
(D12) is a complete list of relations between Aa.Js (a G Sg, j = 1, •.., 9).
7. Obtaining a smaller set of generators for A. We can see from relations
(D4)-(D11) that each Aa;j is a product of some A^i's and their inverses (with
different r G Sg). So already the set {Aa.i,a E Sg} generates the group A.
Consider all elements r G Sg which stabilize supp(ai) = {1,2}.
Such t are permutations of indices (3,4,..., 9) and form a subgroup % of Sg
which is isomorphic to 57. Clearly % is generated by {c*4, as, aç, #7, ag, ag}. It
follows from (R2) that each p(a^), j = 4,5,6, 7,8,9, commutes with Ti and IV.
Thus Vr G JC we have: p(r) commutes with Ti and IV. From (D3) we get:
Aar-,1 =Aa.1 VaeSg, T E%.
This means that Aa;i is determined by the values of cr-1 on supp(ai) = {1,2}.
Introduce some new notations. Namely, let Am = Aa-i for a E Sg s.t.
a_1(l) = A:, (T~1{2) = /. Thus A is generated by {Aw|Jfe,/ = 1,... ,9, A; ^/}.
8. Commutativity and finiteness of A.
PROPOSITION. Generators {j4fcj|fc,Z = 1,... ,9, A: ^ 1} satisfy the following
relations:
(El) Am = (An,)-1 VAm;
(E2) Akm = AkiAim = A/mAfc/ VA:, /, m G (1,..., 9), A: ^ /, A: ^ ra, / ^ m;
(E3) ,4// Am commute-,
(E4) (Afc/)3 = lVAfc/.
7n particular A is a finite commutative group.
Proof.
(1) Proo/ of (El). Take any A^ and then a E Sg s.t. jV.i = AM. By (Dl)
>W;i = OVi)"1. Evidently Aaai;i = Alk. Thus A/fc;i = (Afc/)_1. Q.E.D.
(2) Proof of (E2). Take jfe, /, m G (1,..., 9), k ^ /, k ^ ra, / ^ m. There exists
cr G Sg s.t. (l)cr-1 = A:, (2)cr_1 = /, (4) a~l = ra (we write permutations on the

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 63
right side of the symbols). Because (r3;r0) = 1 (see (B3)) we have from (D2):
Aaa3 ;i = Aa^\AaOL xot^\\ = -A<7c*ia3;l-A<7;l'
(Recall that ol\ = (12), a3 = (24).) But Aa-i = Am, Aaa3-i = Akm (because
(1)^3 V"1 = [l)e~l = A:, (2)a^1a~1 = (4)a_1 = m) and AGOClOC3-i = AXm
(because (l)c*3 ^ V"1 = (l)a^ V"1 = (2)a~l = I, {2)a^1a^1a~1 = {A)a^lG~l
= (4)cr_1 = m). So we get
Akm = AkiAim = AimAki. Q.E.D.
(3) Proof of (E3). Take k, m G (2,..., 9) with k ^ m. From (E2) we have
(take / = 1):
Akm = ^fcl^lm = A\mAki
# (because of (El))
Afcm = {An,)'1 Aim = ^lm(^lfc)"1.
We see that actually the elements Aik, k G (2,..., 9), generate the group A
and that they all commute.
(4) Proof o/(E4). It follows from (E3) that A is a commutative group. Using
that we can rewrite the relation (D6) as follows:
Ar;4 = Arr2,r4;3(Arr2,r4r2/;3)~ Arr2/r4;2-
Using (B3) and (D2) we obtain that
Arr2/r4r2/;3 = Arr2/r4r3r2/;3 • Arr2/r4;3
or
^crr2/r4;3(^crr2,r4r2/;3)~ = (^4<rr2/r4r3r2/;3)~ •
Thus we get the following relation:
(D6) Aa;4 = (ACTr2,r4r3r2,;3)~ •^ar2,r4;2-
Express Aa;g as a product of Af.^, using relations (1)12), (D7), (D6)', (D5),
(D4). We get
Aa;9 = (AaTi;i)~ o AaT2-i
where
t\ = a^agasa7asa4asa2aia3,
It is easy to check that (l)rf1 = 9; (2)rf1 = 7 and (l)^"1 = 7; (2)r2_1 = 9.
Denote by A: = (9)cr_1, / = (7)cr_1. We see that Aau;i = Am, AaT2.\ = Aik =
A^i and so Aa;9 = (AM)~2.
Another expression of Aa;g in generators {Ar;i} we obtain from relations
(Dn), (Dio), (D4): Aa.9 = AaT3;1 where r3 = asaga5a7a3a4aia2^ It is easy
to check that (l)r^-1 = 9, (2)r^_1 = 7. This shows that Arr3;i = Am and
Aa;g = Am- Comparing it with Aa.g = {Am)~2 we get Akl = (Afc/)"2 or
(Aw)3 = l. Q.E.D.
REMARK 3. Using Remark 2 we can show that A is a free abelian group over
Z/3Z with eight generators {A\k, k G (2,..., 9)}.

64
B. MOISHEZON AND M. TEICHER
9. Explicit description of tti(Y). From the proposition above we see that
tti{Yas) = A is a finite abelian group generated by elements Au, l G (2,..., 9).
Because tti(Yas) —► tti{Y) is a surjection, we get that tti(Y) is also
commutative. In particular, we see that 7Ti(yAff) = #i(yAff; Z) and 7n(F) = #i(F; Z).
Denote by efci the image of AM in #i(yAff; Z), by a: #i(yAff; Z) — #i(y; Z)
the natural homomorphism, and by ëfcj = a(ew).
To stress the general character of arguments we replace in our notations below
the number three by r and 9 = 32 by n = r2. There is a "Galois" action of Sn
on yAff and Y which induces an action also on Hi{YAfi\Z) and Hi(Y;Z).
We can describe this action as follows: take r E Sn and Am = Ar,i (that is,
(1)<t_1 = fc,(2)a"1 = /). Let r(Aw) = ATa;1. Because (l^V"1 = (fc)r-1
and (2)cr~1r~1 = (/)r_1 we have: t(j4m) = A(fc)T-1(i)T-1 • Thus a "Galois"
action of Sn on H\(YAn\Z) and Hi(Y;Z) is defined by r(ejfcj) = e^k)T-i^)T-i
and r(gfcf) = ê(fc)T-i(j)T-i.
Denote by /: y —► CP2, g: Y -^ X the maps canonically corresponding to
f: X -+ CP2, by 2? = y - yAff, and by Z, the image of È in CP2. Because L
is generic, E = f*{L) is nonsingular and irreducible (Bertini's theorem). Take a
point p in E and a small 2-dimensional disk D in Y with D H E = p and such
that D is transversal to E. Let c = dD and let c be the corresponding element
in i/i(yAff; Z). Assume that rac = 0 and let Z be a 2-dimensional chain in YAS
with <9Z = rac, Z\ = Z — raD. It is clear that dZi = 0 (that is, Z\ is a 2-cycle
in Y) and that Zi./*L = Zi.£ = ra.
Rewriting this as Z\.g*f*L = ra and applying g: Y —► X we get g*Z\,f*L =
ra. Because /*L ~ rL (L is a projective line inl^ CP2) we conclude that
ra = 0 (mod r).
From Zariski-Van Kampen theorem it follows that to get 7Ti(CP2 — S,uo)
from 7Ti(C2 — 5, uo) we have to add one relation:
nïvïw.
all i
Translating it to geometrical language we get that Ker a is generated by {o~(c),
aeSn}.
Because E is nonsingular and irreducible, all cr(c), a G Sn, are equal to c.
Thus Ker a is generated by c. Clearly {eut, A: G (2,..., n)} is a set of generators
of i/i(yAff; Z), ejfcj = eu — eut (where we assume en = 0), and for a E Sn
o-(eifc) = 6(1)^-1(^-1 = ei^fcj^-i - ex^ij^-i.
We can write c = S"=2 &fceifc? ^ € Z. Take /i, /2 G (2,..., n), /i ^ /2, and let
o — {h,h)' It is clear that cr(ei,i1) = ei,j2, cr(ei5j2) = ei,^ and so
( n \
\kïh,i2 J
Because a(c) = c we get: 0 = cr(c) —c = (6/x — 6j2)ei,j2 + (&z2 — fyjei,^ and since
ei,«2 ~ «i,ii = eJi,J2» (b*i ~ bh)eh,h = 0- Applying appropriate r G Sn we see

GALOIS COVERINGS IN THE THEORY OF ALGEBRAIC SURFACES 65
that (6/j - bi2)eki = 0 VA:,/ G (1,... ,n), A: ^ /, and in particular (bix - bi2)c = 0.
From our remark above it follows that bix = b\2 (mod r).
Actually the proposition above was an illustration of the following general fact
which we can prove for any Yr (r > 3): 7ri(yrAff) is a finite abelian group and
Va G 7ri(yrAff), ar = 1 (compare with [6, Chapter 6]). Thus r#i(yAff;Z) = 0
and from bix = b\2 (mod r) it follows that we can choose coefficients bk all equal
to an integer b. So
n n
c = ^6eifc = fry^eifc.
fc=2 fc=2
Let a1 =g.c.d. (6,r), n = r/d. Because r\b = 0 (mod r) we have r\c —
ri^Z)fc=2ei^ = 0. From our remark above it follows that r\ = 0 (mod r) and
so d = 1. This means that 3a G Z, (a,r) = 1 with a6 = 1 (mod r), and so
ac = YHz=2 eifc- We see that Ker a is generated by YHz=2 eifc-
Remark 3 above was an illustration of a general fact which we can prove for
any Yr (r > 3) : n\ (YAS) is a free abelian group over Z/rZ with n — 1 generators:
{Ai*, fc€(2,...,n)}.
Because Ker a is generated by X)fc=2 ei^ (m ^i (^Aff ) this element corresponds to
11^=2 -^lfc) we conclude that ni(Y) is a free abelian group over Z/nZ generated
by {Au, fce(3,...,n)}.
It is easy to describe the "Galois" action of Sn on 7Ti(F).
References
1. U. Persson, C/iern invariants of surfaces of general type, Compositio Math. 43 (1981),
3-58.
2. F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19
(1984), 483-515.
3. B. Moishezon, Stable branch curves and branch monodromies, Lecture Notes in Math.,
vol. 862, Springer-Verlag, 1981, pp. 107-192.
4. , Algebraic surfaces and the arithmetic of braids. I, Progr. Math., vol. 36, Birkhàuser,
1983, pp. 199-269.
5. , Algebraic surfaces and the arithmetic of braids. II, Contemp. Math., vol. 44, Amer.
Math. Soc, Providence, R.I., 1985, pp. 311-344.
6. B. Moishezon and M. Teicher, Simply-connected algebraic surfaces with positive index,
Invent. Math, (to appear).
7. Y. Miyaoka, Algebraic surfaces with positive indices, Classification of Algebraic and
Analytic Manifolds, Birkhàuser, 1983, pp. 281-301.
8. J. Feustel and R. Holzapfel, Symmetry points and Chern invariants of Picard modular
surfaces, Math. Nachr. Ill (1983), 7-40.
9. W. Magnus, A. Karras, and D. Solitar, Combinatorial group theory, Interscience, New
York, 1966.
10. O. Zariski, Algebraic surfaces, Springer-Verlag, 1971.
11. E. R. Van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55
(1933), 255-260.
Columbia University
Bar Ilan University, Israel

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Vector Bundles

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Geometry of the Horrocks-Mumford Bundle
KLAUS HULEK
0. Introduction. The Horrocks-Mumford bundle F is a rank 2 vector bundle
on P4 which was first constructed in [9]. Since then it has attracted considerable
interest (e.g., [1, 2, 5, 6, 11]). There are at least two reasons for this:
(1) So far it is essentially the only known indecomposable rank 2 vector
bundle on P4. All other known examples are derived from the Horrocks-Mumford
bundle by taking twists or pull-backs.
(2) There is an amazing wealth of geometry attached to the Horrocks-Mumford
bundle. In particular it is very closely related to the icosahedron. This makes it
an object worth studying in its own right.
The aim of this article is to describe some of the relations between the
Horrocks-Mumford bundle, the icosahedron, and Shioda's modular surface 5(5).
Most of these results were originally published elsewhere (see [2, 3]). Hence
many of the proofs are omitted. The only new result in this paper is a proof
that the "general translation scrolls" (see §111 for a definition) are zero-sets of
suitable sections of the bundle F(3).
I should like to emphasize that the results mentioned are joint work with
W. Barth and R. Moore. We have worked together on this very closely for the
last two years, and in most cases it would be futile to-try to attribute specific
results to any one of us in particular.
1. The monad construction. In [9] the bundle F was constructed by means
of a monad. To describe this, let V := C5 and let {ei}i^z5 be the standard basis
of V. By P4 := P(V) we denote the projective space of lines through the origin
of V. The key tools are the maps:
/+: V - A2 V; /+ f J2 v^ ) = J2 "&+* A c*+3
and
f~\V -> A2V; /" f^^et] =531/^+1 A et-+4.
1980 Mathematics Subject Classification (1985 Revision). Primary 14F05.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
69

70
KLAUS HULEK
The middle part of the Koszul complex on P4 reads
V®0(-2) ^ A2V®0{-1) ^ A3V^O ^ A4F®0(1)
(A2T)(-3)
/ \
0 0
where s: 0(—1) —► V <8> 0 is the tautological map. Together with /+ and /" this
gives rise to maps
p: V ® 0(-l) ^ 2A2F 0 0(-l) ^ 2(A2T)(-3),
q: 2{A2T){-3) ^ 2A3F <g) 0 -^ F* <g) 0,
where / = (/+,/") and g = ('/", -</+).
Proposition 1. The complex
(M) F ® 0(-l) *+ 2(A2T)(-3) -2> F* ® 0
zs a monad, i.e.,
(i) p zs injective as a map of vector bundles,
(ii) q is surjective, and
(iii) q o p = 0.
/n particular the cohomology
F = ker ç/im p
of M is a rank 2 vector bundle on P4.
PROOF. (See [9, p. 67fF.] and [1, p. 35fF.].) From the above description one
concludes immediately that the Chern classes of F are given by
Ci(F) = -l, c2(F) = 4.
Since the polynomial
1 - h + Ah2
is irreducible over Z, it follows that the bundle F is indecomposable. Also from
the monad one concludes without difficulties that h°(F) = 0, which in view of
Ci(F) = — 1 is equivalent to the fact that F is stable.
II. Elliptic curves. At this point it is necessary to make a slight detour
and recall some facts about elliptic normal curves. Although many of the results
below generalize readily to higher degree, we shall restrict ourselves, for the sake
of brevity, to degree 5. Let C be an elliptic curve with fixed origin 0. The
line bundle Oc(l) = 0^(50) embeds C as a normal curve C Ç P4. The group
G5 = Z5 x Z5 of 5-torsion points acts on C and on the linear system |0o(l)|.
The same holds for the involution P i-+ —P. In particular the action of G5 and
the involution of C extend to automorphisms of P4. Whereas the involution lifts

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE 71
to an automorphism of V, the situation with regard to G5 is more difficult. It
is, however, still true that the projective representation of G5 lifts to a linear
representation of a finite extension of G5.
To make this precise, let i/5 be the subgroup of SL(5, C) generated by
a:e{ »—► e^_i,
nei^e'ei {e = e2^5).
i/5 is called the Heisenberg group of level 5. Its order is 125 and it is a central
extension
1 -+ M5 -> H$ -► Z5 x Z5 -► 1,
e »-► e ■ lay = [cr, r], a »-+ (1,0),
rt-(0,l).
Finally we consider the involution
Then one has
PROPOSITION 2. After a suitable change of coordinates the following hold:
(i) The Heisenberg group H§ leaves C invariant (as a curve) and operates on
C as the group G5 ofh-torsion points.
(ii) The involution t operates on C as the involution P i-+ —P.
PROOF. (See [10, Theorem 1.2.3].)
From now on we shall always assume that the coordinates are as in Proposition
2. Using the exact sequence
0 - Jc{2) - Op4(2) - Oc(2) - 0
and the fact that C is projectively normal, one finds
h°(Jc(2)) = 5.
Moreover, C is the scheme-theoretic intersection of the quadrics containing it
(see [10, Theorem IV. 1.3]). Since C is invariant under i/5 the same holds for
the space H°(Jc{2)). It is then easy to see that this space is spanned by the
quadrics
Qi = QiW = x* + Azi+2Zi+3 - (l/\)xi+iXi+4 (i G Z5),
where À G Pi is a suitable parameter depending on C. For details see [10,
Chapter IV].
One can now reverse this process and ask about the intersection
GA:=Qo(A)n---ng4(A)
for any given À G Pi. In this context it is useful to define the set
A = {0,OO,^(£ + £4),^(£2 + £3), ^ = 0,...,4}.

72
KLAUS HULEK
A Ç Px consists of 12 points. Under the identification
Pi = CU{oo} = S2
which is given by stereographic projection the set A can be identified with the
vertices of an icosahedron sitting inside the unit sphere S2. This means that A
enjoys the symmetries of an icosahedron, i.e., the icosahedral group A§ acts on
Pi as a group of projective transformations in such a way that A is the unique
orbit consisting of 12 points.
PROPOSITION 3. (i) If X £ A, then C\ is a smooth elliptic normal curve of
degree 5.
(ii) If X G A, then C\ is a pentagon, i.e., consists of 5 lines.
(iii) Two curves C\ and C\> are disjoint, unless X and X' belong to opposite
vertices of the icosahedron. In this case the two pentagons come together to form
a complete pentagon:
Figure l
PROOF. (See [2, Propositions 3 and 6].) It is now natural to consider the
union
Si5= U C*
AGPi
of the curves C\. Si 5 is an irreducible surface.
PROPOSITION 4. The surface S15 has degree 15. It is smooth outside the 30
vertices of the 6 complete pentagons formed by the singular curves C\, X G A.
There two smooth branches meet transversally.
PROOF. (See [2, Propositions 9 and 10].) By the above results the
normalization 5i5 of Si5 has a natural structure as an elliptic surface
7r:Si5-+Pi
over Pi. Its fibers are precisely the curves C\. In particular it has exactly 12
singular fibres of type J5, namely, the pentagons Ca, À G A.
In order to give an interpretation of the surface Si5 we recall that the modular
group T = SL(2, Z) acts on the upper half-plane
<K={zeC, lm*>0}

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE 73
by
az + b , ( a b\ _,T /rt _
2 »-* -, where , G SL(2,Z).
cz + cf Ve V
The principal congruence subgroup of level 5 is defined as the group
r(5):={7er;7 = lmod5}.
As a subgroup of T it also operates on !K, and the quotient
is an open Riemann surface parametrizing the isomorphism classes of elliptic
curves with level 5 structure. Adding the 12 cusps one gets the compact modular
curve X(5) = X'(5). X(5) is rational. The icosahedral group
A5 = PSL(2,z5) = r/±r(5)
acts on Pi = X(5) and defines a 60 : 1 branched covering
p:Pi -Pi-
The 12 cusps are the ramification points of order 5. In particular they form
the unique minimal orbit of A5 in Pi and can, therefore, be identified with the
icosahedron A.
It is well known that a universal elliptic curve with level 5 structure exists. It
can be constructed as a quotient
S'(5)=:KxC/(ZxZ)xir(5).
It was shown by Shioda that S"(5) has a natural compactification
tt:S(5)-+X(5) = Pi
which is an elliptic surface with singular fibres of type J5 precisely over the 12
cusps of X(5). Moreover 5(5) admits exactly 25 sections L^\ i,k G Z5 x Z5
which form a group Z5 x Z5.
THEOREM 1. (i) There exists a unique divisor class I G Pic 5(5) such that
i,k
(ii) The complete linear system |J + 2F|, where F denotes the class of a fibre,
is base point free and defines an immersion
£>|J+2F|:S(5) —► P4
whose image can be identified with S15. In particular 5(5) is isomorphic to the
normalization Si^ 0/S15.
PROOF. See [2, Proposition 9].
REMARK. Part (i) of Theorem 1 was first proved by Inoue and Livné, who
used this fact to construct a surface of general type with c\ = 3c2-

74
KLAUS HULEK
The group Z5 x Z5 x SL(2, Z5) operates on 5(5) as the group of
automorphisms. Note that this group is closely related to the Horrocks-Mumford bundle.
Indeed, if N is the normalizer of the Heisenberg group in SL(5, C) then
JV/ff5 = SL(2,Z5)
and the group
7V = tf5xSL(2,Z5)
is the famous group of order 15,000 which acts on F as its symmetry group.
III. Sections. We already saw that h°(F) = 0. The first twist of F which
admits nonzero global sections is F(3), and one has
h°(F(3)) = 4.
Since h°(F(k)) = 0 for all A: < 2 it follows that for every section 0/ sG
H°(F(3)) its zero-set X8 = {s = 0} is a surface of degree c2(F(3)) = 10.
THEOREM 2. For a general section s G H°(F(3)) the zero-set X3 = {s = 0}
is a (smooth) abelian surface of degree 10.
PROOF. See [9, Theorem 5.1].
REMARKS, (i) Every abelian surface X CP4 has necessarily degree 10 (see,
e.g., [7, p. 60]). Its polarization H is of type (1,5).
(ii) Every abelian surface X Ç P4 arises (modulo PGL(5, C)) as the zero-set
of a suitable section s G H°(F(S)).
(iii) For a large class of polarized abelian surfaces (X,H) of type (1,5) Ra-
manan [15] showed directly that the linear system \H\ is very ample. For the
other cases see [12].
(iv) I was informed by H. Lange that A. Comessatti [4] was (probably) the
first who showed that certain abelian surfaces can be embedded in P4. For a
discussion of Comessatti's results see Lange's paper [13].
(v) The vector bundle F(3) can be reconstructed from the zero-sets X9 by the
Serre-construction.
It is also interesting to study the possible degenerations of abelian surfaces in
P4. For this let C = C\ Ç P4 be an elliptic normal quintic as in §11. Let Po be
a fixed point which is not a 2-torsion point, i.e., 2P0 7^ O. For each point P EC
we consider the line L(P, P + Po) spanned by the points P and P + Po (here +
denotes addition on C); see Figure 2.
We consider the scroll
X= (J L(P,P + P0).
Pec
We shall see later (Proposition 5) that X has degree 10. It is singular along C,
which it contains as a double curve. We shall call such an X a general translation
scroll In Theorem 3 we shall show that every such X is the zero-set of a suitable
section s G H°(F(3)).

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE
75
Figure 2
The general translation scrolls have further degenerations. If Po = 0, then
X becomes the tangent scroll TanC of C. Its connection with the Horrocks-
Mumford bundle was first mentioned in [9]. For a proof see [10, Theorem
VII.3.1]. If Pq becomes a 2-torsion point different from 0, then X becomes
a smooth scroll of degree 5, the well-known quintic elliptic scroll. It was shown
by Van de Ven and the author that these elliptic scrolls admit a multiplicity-2
structure X (a double quintic) such that X = {s = 0} for a suitable section
s e H°(F(3)).
Further degenerations occur if the smooth elliptic curve C — C\ degenerates
to a pentagon, e.g., Coo'.
In this case X degenerates to a union of 5 quadric surfaces
X= [j {x{ = aXi+2Xi+3 + Xi+iXi+4} (a^O, oo).
For this see [9, p. 80]. Finally if a = 0 or oo, then X becomes the union
of 5 double planes. Summarizing this we have found the following hierarchy of
degenerations:
, tang, scrolls
ab. surfaces — gen. transi, scrolls double quintics 5 double planes
dim: 3
' 5 quadrics '
1

76
KLAUS HULEK
In a forthcoming joint paper with W. Barth and R. Moore we shall show that
these are all possible cases.
Let us now return to the general translation scroll
X= (J L(P,P + P0).
Pec
We consider the map
<p: C - Gr(l, 4), P ^ L(P, P + P0).
Let U be the universal subbundle on Gr(l,4). We set
X:=P(<p*(U)).
X is a Pi-bundle over C. We denote the projection map by
p:X-+C.
Projection to P4 gives a map
tt:X-+X
which is a desingularization of X. Let H be the hyperplane section of X —► X Ç
P4.
PROPOSITION 5. (i) X = P{0C © £) for some line bundle JC e Pic0 C.
(ii) H = Co + 5/, where Co is a section with Cq = 0 and f denotes the class
of a fibre.
(iii) degX= 10.
PROOF. In order to show that X splits we have to construct two sections
which do not intersect. This can be done as follows:
C0: <p0:C-+X, P^PGL(P,P + P0),
Ci: (px-.C^X, P^PeL(P- P0,P).
We can write
H = Co + bf
and find that
5 = H • Co = Cl + 6, 5 = tf • Ci = b.
Hence 6 = 5 and Cq = 0. In the same way one shows C\ = 0. This proves (i)
and (ii). Since
H2 = {Co + 5/)2 = 10
and since tt is birational—two secants of C do not meet outside C—we get (iii).
We are, therefore, in the situation of Figure 3.
Note that we have identified the sections Co and C\ via the maps <po (resp.
<Pi) with the curve C (with origin 0). After possibly replacing £ by £-1 we can
assume that the section Co (resp. C\ ) is given by the quotient
Oc®£-+>C (resp. Oc 0 £ — Oc).

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE
77
0
1C
■
-Po
f
Po
0
'f—
\
Ci
Co
Then (see [8, p. 373]) one has
(1) o;^=O^(-2-Co)0p^,
(2)
0^(C0)|Co = £.
PROPOSITION 6. (i) 0$ = O^(C0 + 5/0)®p*£_1, where f0 denotes the
fibre over the origin 0 G C.
(ii)£ = Oc(5P0-5O).
PROOF, (i) It follows from Proposition 5 (ii) that
Ox(H) S 0* (C0 + 5/o) 0 P*M
for a suitable line bundle M G Pic0 C. Using (2) this gives
0Co(50) = Ox{H)\C0 = 0Co(50) 0 JC ® M,
and the assertion follows immediately,
(ii) Similarly one has
0Cl(50) = 0*(ff)|Ci S (Ox(Co + 5/o)0p^"1)|C1
sOc^SPo)®?**"1^!.
Since £ is a line bundle of degree 0, it is invariant under translations; hence
p*£|Ci = £
and we find
OCl(5O) = OCl(5Po)0^-1,
from which the assertion is obvious.
We can now formulate the main result of this section.

78 KLAUS HULEK
THEOREM 3. Let X be a general translation scroll. Then a section s G
H°(F(S)) exists such that
X={s = 0}.
PROOF. Let r#(Op4(5)) be the space of quintic forms which are invariant
under the Heisenberg group i/5. It is well known (cf. [9, Theorem 3.5]) that
dimrH(Op4(5))=6.
The crucial step is to show that
dimrtf(Jx(5))>3,
i.e., that there are at least three independent invariant quintics containing X.
Once one has seen this, the proof can be completed exactly as in [10, VII.3.3],
where the case of the tangent scrolls was treated similarly (cf. also [9, p. 77]).
We shall, therefore, omit these details.
To show that dimTH{Jx{5)) > 3 we fix a point P eC with -P = P + P0,
i.e.,
(3) 2P~O + *(P0).
There are four possible choices for P. The line
L0:=L{P,P + P0)
is fixed under 1 which operates nontrivially on L0. Let Pi, P2, P3 G Lq be general
points, i.e., points with i(Pi) ^ Pj.
Let
W:={QerH{Op<&)), Q(Pi) = o> « = 1,2,3}.
Since dimT^(Op4(5)) = 6, it follows that
dim^>3.
We shall show that
(4) WCTH{Jx{5)),
which will conclude the proof. In order to prove (4) we consider for each Q £W
its pull-back
Q:=7T*QeT{Ox{5H)).
Every quintic form Q G r#(0p4(5)) is not only invariant under i/5 but also
under 1. Hence Q vanishes at the 6 points Pt-, i{P%)\ i = 1,2,3 on L0. Since Q is
a quintic, this implies that Q contains the line Lq, i.e., Q vanishes on the fibre
/p. By i/5-symmetry this implies that
Qer (0* Uh- Yl f°(p)))=T(°x(w-25/p))•
Clearly it will be sufficient to show that
h°(Ox(5H - 25/p)) = h°(Ox(5C0 + 50/o - 25/Po - 25/») = 0.

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE 79
For this purpose we set
Ox(Dn) := Ox(nC0 + 50/o - 25/Po - 25/P).
We have exact sequences
0 - 0^(Dn_x) - 0*(Dn) - O^(Dn)|C0 - 0.
It is enough to show that
h°{Ox(Dn)\C0) = 0 for 0 < n < 5.
Using Proposition 6 and formula (2) it follows that
0^{Dn)\C0 = Oc((5n - 25)P0 + (50 - 5n)0 - 25P).
Hence we have to see that none of the equations
(An) 25P - (50 - 5n)0 + (5n - 25)P0
holds. First assume that Pq is a 5-torsion point. Then for all n the equation
(An) is equivalent to 25P ~ 250. But we can always avoid this by replacing
P if necessary by one of the other three solutions of the equation 2P = —Pq
which differ from P by a 2-torsion point. Hence we can from now on assume
that 5Po 7^ O. Using (3) we find that (An) is equivalent to
(Bn) P ~ (14 - 5n)0 + (5n - 13P0).
Multiplying (Bn) by 2 and using (3) once more, one sees that (Bn) implies
(Cn) (25 - 10n)O - (25 - 10n)P0.
For 0 < n < 5 these equations read explicitly:
±25O~±25P0,
± 150 - ±15P0,
±5O-±5P0-
Since we may assume at this stage that Po is n°t a 5-torsion point, we see that
at most one of these pairs of equations can hold. This implies that at most 2 of
the equations (Bn) (or equivalently (An)) can hold for 0 < n < 5—independent
of the choice of P. But this shows that we can again replace P if necessary
by another solution of the equation 2P = —Po in such a way that none of the
equations (An) hold for 0 < n < 5. This completes the proof.
IV. Jumping planes. A rather successful method in the theory of vector
bundles is to study the restriction of a given vector bundle F to linear subspaces
Pfc Çl Pn- Since the restriction of the Horrocks-Mumford bundle F to every
hyperplane H Ç P4 is stable, we shall turn immediately to planes E Ç P4. It
follows from general theory that for a general plane E the restriction F\E is
stable. This is, however, not the case for all planes.
DEFINITION. A plane E Ç P4 is called unstable or a jumping plane if F\E is
unstable, i.e., if h°(F\E) ^ 0.

80
KLAUS HULEK
To the bundle F we can associate the variety
S{F) := {E G Gr(2,4); E is unstable}.
This variety of jumping planes reflects many properties of the bundle itself (see
in particular [6]). In order to describe S(F) more closely we need the following
characterization of jumping planes.
LEMMA 1. A plane E Ç P4 is a jumping plane if and only if the two linear
forms
X±:V -* AbV S C, v ^ /±(t;) A E
are linearly dependent.
PROOF. The proof follows closely [1, Proposition 1]. The display of the
monad (M) of Proposition 1 shows that
H°(F\E) = kev{q: 2H°{{A2T{-S))\E) -* V*).
The map q can be factored in the following way:
2H°((A2T)(-S)\E) -i V*
2ao \ /(7",-7+)
2A3F
Here <?o comes from the Koszul complex and
im(<7o) = E Ç A3V.
The assertion is then obvious.
In order to analyze the variety S(F) of jumping planes more closely we
consider the Plucker embedding Gr(2,4) Ç P(A3V) = Pg which identifies the Grass-
mannian Gr(2,4) with the variety of decomposable vectors in P(A3V). By the
above lemma and the definition of the maps /+ and /~ a plane
E = ^2 *»ifce» A ej A efc G Gr(2,4)
i<j<k
is a jumping plane if and only if there exists some (A : /i) G Pi such that
A£t,t+l,t+4 + A^m+2,1+3 = 0 (i G Z5).
For fixed (A : /i) G Pi these five linear equations define a linear subspace P4 '^ Ç
Pg. In order to describe the union of these subspaces we consider the Segre
embedding
s: P4 x Pi -+ P9, ((x0 : ... : rr4), (A : /i)) -+ ^ tijkei A e3- A efc,
where
^t,t+l,t+4 = V>xi-> U,i+2,i+3 = — Ax^.
Then
P<frt = s(P4 x {(A : /i)}).

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE
81
Identifying P4xPi with its image under the Segre map s we have in this way
shown that
S(F) = Gr(2,4)n(P4xP!).
In order to exhibit the relation between the variety S(F) and Shioda's modular
surface 5(5), we have to recall the Plucker relations:
^,1+2,1+3^,1+3,1+4 — ^,z+2,i+4^z,z+l,z+3 + ^,z+2,z+3^,i+l,i+4 =0 (i G Z5).
They give quadratic equations for the Grassmannian Gr(2,4) CP9. If we restrict
these equations to the linear subspace P4 = P4 : , we get the following five
quadrics:
Qi(X : fi) = XfiXi + A2^+2Zz+3 - fi2Xi+iXi+4 (i G Z5).
The important point is that this is just the homogeneous form of the quadrics
which appeared in §11. By the results of this section the intersection
C(xm) = £o(A : /i) H • • • H Q4{\ : /i)
in either an elliptic normal curve of degree 5 or a pentagon if (A : /i) G A. Hence
we have seen that
S(F)= (J C(A:M)=5(5).
(A:/z)GPi
We can summarize our results as follows:
THEOREM 4. (i) The variety S(F) of jumping planes is a smooth surface of
degree 25 in P9. It is the transversal intersection of the Grassmannian Gr(2,4)
with the Segre variety P4 x Pi Ç P9.
(ii) The surface S(F) is isomorphic to Shioda's modular surface 5(5). The
embedding 5(5) —► P9 is given by the complete linear system \I + 3F|.
PROOF. For more details see [2] and [3, Proposition 18].
V. Jumping lines. Another method to study a vector bundle on some
projective space Pn is to restrict it to the lines L CPn. For the Horrocks-
Mumford bundle F this was done in [1] and [2]. Since ci(F) = —1 one finds
from Grothendieck's theorem that
F\L = 0L{-k - 1) © 0L{k) {k > 0).
Here A: depends on the line L. The Grauert-Mulich theorem says that A: = 0 for
the general line LCP4.
This motivates the following
DEFINITON. A line L is called a jumping line if
F\L = 0L{-k-l)®0L{k)
with A: > 1. The integer A: is called the order of the jumping line L. Similarly to
the case of jumping planes we define the following varieties of jumping lines:
J(F) := {L G Gr(l, 4); L is a jumping line},
J2{F) := {L G J(F); order of L > 2}.

82
KLAUS HULEK
THEOREM 5. J (F) is an irreducible rational variety of dimension 4. It is
smooth outside the surface J2(F).
IDEA OF PROOF. Let S15 Ç P4 be the surface constructed in §11. Let J :=
blow up of P4 in the ideal of Si 5. Then there is a natural embedding
JCP4 xGr(l,4).
Let q be the projection onto Gr(l, 4). Then one finds that
J(F) = q(J).
From this description of J(F) one can verify the assertions of the theorem.
For more details see [2].
The projection map q: J —► J is geometrically very interesting. It is closely
related to the sexti-secants of the surface S15. These are separated in J after
blowing up Si5. The map J —► J(F) essentially consists of blowing down the
proper transforms of these sexti-secants.
Before we can give more details we first have to recall a few facts about Bring's
curve. For this we recall the involution
t:V-+V, en-+e-i.
It defines a decomposition of V into eigenspaces V = V+ © V~, where
y- = (ei - e4, e2 - e3), V+ = (e0, ex + e4, e2 + e3).
Let
L := {x0 = xi + X4 = x2 + %3 = 0} = P(V~),
E := {xi - x4 = x2 - x3 = 0} = P(F+).
By construction the line L intersects every smooth curve C(^:/i) Ç S15 in its
origin 0. Hence L can be identified with the zero-section of the modular surface
S(5). Similarly the plane E intersects every smooth curve C(\:fA) in the three
2-torsion points different from 0. Hence the intersection
D := S15 H E
is a curve. It is not difficult to see that this is a sextic curve with equation
XqXiX1} XqX-^X<2 1 ZX-^X2 — XqX-^ — XqX2 == U.
This sextic is called Bring's curve. It has the property that it is invariant under
the (natural) action of the icosahedral group A$ on the plane E. Moreover, it
has precisely 6 nodes which form the minimal A5-orbit. By construction the
normalization D of D is the modular curve which parametrizes elliptic curves
with a level-5 structure and a distinguished 2-torsion point different from 0.
Hence
where

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE 83
The genus of D is 4. There is a natural 3 : 1 map
tt: D = M/r0(2,5) - Pi = X/T(5),
which consists of forgetting the distinguished 2-torsion point. As in the case of
S (5) one can also associate a fine moduli scheme to To (2,5), namely, the modular
surface So(2,5). The map tt induces a rational map
7r':S0(2,5)34s(5).
We now return to the surface Si 5. It has two types of sexti-secants which arise
in the following way:
(1) Let C(A:/z) Q Si5 be a smooth fibre. Fix a 2-torsion point P0 ¥" 0- Then
the union of the secants joining two points P and P + Po (here P € C(a./z) is an
arbitrary point), i.e.,
X= (J L(P,P + P0)
pec
is the well-known quintic elliptic scroll. As an abstract surface it is the unique
indecomposable Pi-bundle over the elliptic curve C = C/(Po) with invariant
e = — 1 (cf. [8, Theorem 2.15]). The scroll X contains altogether three smooth
fibres of S15 which are 2-sections on X (see [11]). Hence every ruling of X is
a sexti-secant of Si 5. It follows from this construction that the variety of pairs
consisting of sexti-secants of the above type and one of their points of intersection
with S15 is birationally equivalent to So (2,5). The variety of the sexti-secants
themselves is the image of So(2,5) under the 6 : 1 map
Sb(2,5)^Sb(2,5)^S(5).
Here u is the quotient map with respect to the distinguished 2-torsion point in
every smooth fibre.
Since the sexti-secants described above are mapped to jumping lines of order
2, the above discussion also shows that the surface J2{F) is birationally equivalent
to S(5).
(2) The other type of sexti-secants comes from Bring's curve D = S15 D E.
Since D is a sextic curve, every line L Ç E is a sexti-secant of S15. By Z5 x Z5-
symmetry we get 25 planes
Eik = {Xi+i — 6 Xi+4 = Xi+2 — ^ Zz+3 = 0}
which all have this property. All the sexti-secants in one plane are mapped under
q to a single point in J2(F). In this way one recovers the 25 jumping lines of
order 3 of F. In the original P4 these are just the 25 Horrocks-Mumford lines
Lik = {xi = Zj+i + e3kXi+4 = Xi+2 + ekXi+s = 0}.
VI. Further results. There are two more results about the Horrocks-
Mumford bundle which I should like to mention. The first is the following
uniqueness result.

84
KLAUS HULEK
THEOREM 6 (Decker/Schreyer) . For every stable rank 2 vector bundle
F on P4 with c\{F) = —1 and 02(F) = 4 there exists a projective transformation
<p G PGL(5, C) such that
F = <p*F.
In particular
MP4(-1,4) = PGL(5,C)/Z5 x Z5 x SL(2,Z5).
In [6] Decker and Schreyer give a proof for this result under the additional
hypothesis h2(F(—1)) = 0. A key point in their argument is a close analysis of
the variety S(F) of jumping planes of F. They can show that S(F) is a smooth
surface isomorphic to 5(5). From this their result follows. In the meantime they
have been able to show that the hypothesis h2(F(—1)) = 0 is always fulfilled.
For the latter result see also [14].
THEOREM 7 (DECKER). There exists no stable rank 2 bundle on P5 with
ci = — 1 and C2 = 4. In particular, the Horrocks-Mumford bundle cannot be
extended to P5.
This theorem was first proved by Decker [5]. The following beautiful proof
which uses Theorem 6 is due to Ellingsrud and Str0mme.
PROOF. Let F be a stable rank 2 bundle on P5 with a = — 1 and c2 = 4. Its
restriction F|P4 to a general hyperplane is stable. By Theorem 6 the restricted
bundle FIP4 is projectively equivalent to the Horrocks-Mumford bundle F. Since
F restricted to every hyperplane P3 Ç P4 is stable, it follows that FIP4 is stable
for every P4 Ç P5.
Now consider the product
P4XP1
P / \ Q
P4 Pi
with
a:=p*0P4(l), /?:=tf*0Pl(l).
Let
s:P4 xPi-^Pg
be the Segre embedding. Let P3 Ç P9 be a linear subspace which does not meet
P4 x Pi. Projection from P3 gives a 5 : 1 map
tt:P4xPi -+P5.
Let J:= tt*F. By construction
c(^ = l-(a + /?) + 4(a + /?)2.
J is a vector bundle on P4 x Pi and can hence be viewed as a family of stable
vector bundles over P4 with base space Pi. The associated map
<p: Pi - MP4(-1,4) = PGL(5, C)/Z5 x Z5 x SL(2, Z5),
t K-.y|P4 x {t}

GEOMETRY OF THE HORROCKS-MUMFORD BUNDLE 85
must be constant. Hence
3*=p*F®g*0Pl(k)
for some A: G Z. But this implies
which contradicts the expression for c(3) which we have found above.
REFERENCES
1. W. Bart h, Kummer surfaces associated with the Horrocks- Mumford bundle,
Journées de Géométrie Algébrique d'Angers, (Angers, 1979), A. Beauville, editor, Sijthofî and
Noordhoff, Alphen aan den Rijn, 1980, pp. 29-48.
2. W. Barth, K. Hulek, and R. Moore, Shioda's modular surface 5(5) and the Horrocks-
Mumford bundle, Proc. Tata Conf. on Algebraic Vector Bundles over Algebraic Varieties
(Bombay, 1984) (to appear).
3. W. Barth and K. Hulek, Projective models of Shioda modular surfaces, Manuscripta
Math. 50 (1985), 73-132.
4. A. Comessatti, Sulle superficie di Jacobi, Tipografia délia R. Accademia dei Lincei,
Rome, 1919.
5. W. Decker, Uber stabile 2-Vektorraumbùndel mit Chern-Klassen c\ = — 1, C2 = 4,
Preprint No. 84, Kaiserslautern, 1984.
6. W. Decker and F. Schreyer, On the uniqueness of the Horrocks-Mumford bundle, Math.
Ann. 273 (1986), 415-443.
7. W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), Band 2, Springer-Verlag,
Berlin, Heidelberg, New York, Tokyo, 1984.
8. R. Hartshorne, Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag,
Berlin, Heidelberg, New York, Tokyo, 1977.
9. G. Horrocks and D. Mumford, A rank 2 vector bundle on P4 with 15,000 symmetries,
Topology 12 (1973), 63-81.
10. K. Hulek, Projective geometry of elliptic curves, Astérisque, no. 137, Soc. Math. France,
Paris, 1986.
11. K. Hulek and A. Van de Ven, The Horrocks-Mumford bundle and the Ferrand construction,
Manuscripta Math. 50 (1985), 313-335.
12. K. Hulek and H. Lange, Examples of abelian surfaces in P4, J. Reine Angew. Math.
363 (1985), 201-216.
13. H. Lange, Embeddings of Jacobian surfaces in P4, J. Reine Angew. Math. 372 (1986),
71-86.
14. Lipshutz, Doctoral Thesis, Berkeley.
15. S. Ramanan, Ample divisors on abelian surfaces, Proc. London Math. Soc. 51 (1985),
231-245.
Mathematisches Institut, Universitat Bayreuth, Federal Republic of
Germany

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Variétés de Modules de Faisceaux Semi-stables
de Rang Elevé sur P2
J. LE POTIER
Soient r un entier > 2, c\ et c2 G Z. On désigne par M — M(r,ci,c2)
l'espace de modules de Gieseker et Maruyama des faisceaux semi-stables de rang
r, de classes de Chern c\ et c2 sur P2 = P2(C). C'est une variété projective
irréductible, de dimension 1 + r2(2A — 1), où A désigne le nombre rationnel
On se propose de décrire le groupe de Picard de M. Ce calcul avait été
entrepris quand r = 2 pour la variété de modules M0 des fibres stables par
EUingrud, Str0mme, et moi-même en 1979 [6, 9]; il a été étendu par Str0mme
en 1984 à la variété M des faisceaux semi-stables de rang 2 satisfaisant à la
condition supplémentaire 4c2 — c\ ^ 0 mod. 8 [13]: dans ce cas la variété M est
alors lisse, et c'est un espace de modules fin. C'est J. M. Drezet qui a traité le
cas général [4]. Afin d'énoncer son résultat, nous commençons par rappeler les
conditions d'existence des faisceaux semi-stables obtenues dans [5].
1. Formulaire. Soit E un faisceau algébrique cohérent de rang r > 0, de
classes de Chern c\ et c2 sur P2. Les nombres rationnels,
ci A 1 / r-1 2\
sont appelés respectivement pente et discriminant de E. La formule de Riemann-
Roch prend alors la forme suivante
X = Ç(-l)W(£)=r(P00-A)
i
où P(/i) = 1 + 3/i/2 + /i2/2 est le polynôme de Hilbert du fibre trivial de
rang un sur P2. Plus généralement, si Ef est un autre faisceau algébrique
cohérent de rang r' > 0, de pente /i', de discriminant A', on pose x{E,E') =
XV(-l)Mim Ext*(£,£"); on a alors
X(£, E') = rrf(P(!i' - /i) - A - A').
1980 Mathematics Subject Classification (1985 Revision). Primary 14J05; Secondary 14C22.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
87

88
J. LE POTIER
Dualité de Serre. Soit Kp2 ~ 0(—3) le fibre canonique sur P2; il existe un
accouplement canonique
Ext*(£,£") x Ext2-\E',E® Kj>2) -* C
qui fait d'un de ces espaces vectoriels le dual de l'autre.
2. Faisceaux semi-stables. La notion de stabilité et semi-stabilité dont
nous aurons besoin est celle de Gieseker et Maruyama [7, 11].
DÉFINITION. Un faisceau algébrique cohérent E sur P2 est dit semi-stable
(resp. stable) s'il est sans torsion, et si pour tout sous-faisceau cohérent 0 ^ E' ^
E on a
H{E') < n{E)
et, en cas d'égalité, A(£") > A(E) (resp. A(£") > A(E)).
Soit S une variété algébrique; on considère le foncteur M_(r, ci, c2) qui associe
à S l'ensemble des classes d'isomorphisme de faisceaux algébriques cohérents E
sur SxP2, 5-plats, et tels que pour tout s G S, E(s) soit semi-stable de rang
r, de classe de Chern c\ et c2. Alors Gieseker et Maruyama ont construit une
variété projective M = M(r, ci, c2) et un morphisme fonctoriel
M(r, ci, c2) —► Mor( , M)
qui fait de M un espace de modules grossier, c'est-à-dire qu'il satisfait à la
propriété suivante: pour toute variété algébrique M' et tout morphisme fonctoriel
M(r< ci, c2) —► Mor(, M'), il existe un morphisme de variétés et un seul M —► M'
rendant commutatif le diagramme
Mor( ,M)
M(r,ci,c2)
Mor( ,M')
Ceci caractérise la variété M. Dans M, les points qui proviennent de
faisceaux stables forment un ouvert lisse Ms qui s'identifie à l'ensemble de classes
d'isomorphisme de faisceaux stables de rang r, de classes de Chern ci et c2.
Lorsque, r, ci, et x sont premiers entr'eux, on a M = Ma, et le morphisme
fonctoriel M(r, ci,c2) —► Mor( ,M) est surjectif: autrement dit, il existe un faisceau
universel sur M x P2.
Filtration de Harder-Narasimhan. Tout faisceau algébrique cohérent sans
torsion E possède une filtration par des sous-modules cohérents
{0} C Fi C F2 • • • C Fi = E
tels que (1) le gradué gr^ = Fi+i/Fi soit semi-stable,
(2) /i(grj > /i(gri+1), et en cas d'égalité
A(gr<) < A(gr<+1).

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES 89
Une telle filtration est déterminée de manière unique par ces conditions; on
l'appelle la filtration de Harder-Narasimhan [8].
3. Fibres exceptionnels.
PROPOSITION (3.1). Soit E un faisceau stable sur P2, de rang r, de pente
fi, de discriminant A. Les assertions suivantes sont équivalentes:
(1) Ext1(£', E) = 0; autrement dit, E est rigide;
(2) A < 1/2;
(3)X(£,£)>0.
DÉMONSTRATION. Par dualité de Serre, l'espace Ext2 (25,25) est le dual de
Hom(2£, E(—3)), et donc nul par stabilité. D'autre part, on a Hom(2£, E) = C.
L'énoncé résulte donc de la formule de Riemann-Roch
X{E,E) = 1 - dim Ext1^,^) = r2(l - 2A).
DÉFINITION. Un faisceau stable sur P2 satisfaisant à l'une des conditions (1),
(2), (3) ci-dessus est dit exceptionnel Un faisceau semi-stable de discriminant
A < 1/2 est dit semi-exceptionnel
EXEMPLES. (1) Tous les fibres de rang un sont exceptionnels.
(2) Le fibre tangent T = T(P2) est un fibre exceptionnel de pente 3/2.
(3) Si E est un fibre exceptionnel, il en est de même des fibres E(i) pour
ieZ, et du fibre dual E*.
Nous allons montrer comment on peut déterminer tous les faisceaux
exceptionnels.
PROPOSITION (3.2). Pour tout a G Q, il existe au plus, à isomorphisme
près, un faisceau exceptionnel de pente a. Un tel faisceau est localement libre;
son rang ra est le dénominateur de a, et son discriminant est donné par la
formule
DÉMONSTRATION. Soit E un fibre exceptionnel de rang r, de pente a, de
discriminant A. La formule x(E,E) = 1 = r2(l — 2A) s'écrit, en revenant aux
classes de Chern c\ et c2:
l = r2 -2rc2 + (r- l)c\.
Il en résulte que r et c\ sont premiers entr'eux, donc déterminés par la pente
a = ci/r. La même formule donne le discriminant. Si E' est un autre faisceau
exceptionnel de pente a, il aura même rang r et même discriminant A. D'après
la formule de Riemann-Roch
X(^,^) = r2(l-2A)
et par suite x{E, E') > 0. D'autre part, Ext2(i£, E') = 0 d'après le théorème de
dualité de Serre. Par suite, Hom(i£, E') ^ 0; par stabilité, on obtient E = E'.

90
J. LE POTIER
Il reste à voir que E est localement libre. Mais si g G AutP2, g*{E) est
encore un faisceau exceptionnel de pente a, donc isomorphe à E. Il en résulte
que l'ensemble des points de P2 au voisinage desquels E est localement libre est
invariant par le groupe projectif A ut P2; c'est donc le plan P2 tout entier.
L'ensemble <£. On désigne par <£ l'ensemble des nombres rationnels qui sont
pentes de fibres exceptionnels.
Soient /i G Q, rM le plus petit dénominateur > 0 de /i (on dira aussi son rang).
Le nombre rationnel,
A, = i (1 - 1/r»),
est appelé le discriminant de /i.
PROPOSITION (3.3). Soit p, un nombre rationnel de rang r, de discriminant
A. Les assertions suivantes sont équivalentes:
(1) M e <8;
(2) On a r(P(fi) — A) G Z, et pour tout a E (E tel que ra < rM, \a — \i\ < 3,
on a A > P(-|/i - a\) - Aa.
L'implication (1) => (2) est facile: si Ea (resp. I?,,) est un fibre exceptionnel
de pente a (resp. /i), et si par exemple a > /i, on a
Hom(^,^) = 0, Ext2 {E^E^) = 0
et par suite xi^a^E^) < 0. On raisonne de manière analogue si a < p. Le cas
a = p est exclu par la condition ra < rM. Le fait que r(P(p) — A) soit entier
résulte de la formule de Riemann-Roch.
L'implication (2) => (1) peut se démontrer en suivant le même méthode que
pour le théorème général d'existence des fibres stables (cf. §4).
Construction de <£. On considère pour a et 0 G Q tels que 3 + a — 0 ^ 0,
l'équation en t
P{a - t) - Aa = P{t - 0) - Ap.
Cette équation a une seule solution, notée t = a • 0, donnée par
a + 0 A^-Ag
2 3 + a-/3*
Soit D l'ensemble des nombres rationnels de la forme p/2a, où p G Z, et où ç
est un entier > 0. On définit une application e: S —► <£ par récurrence sur ç, en
posant
e:(n) = n si n G Z,
«(*£)-'(£) •'(*£)
PROPOSITION (3.4). (1) L'application e est bien définie et strictement
croissante.
(2) Pour p impair, e(p/2q) est de rang > 2q.
(3) Pour a G e(î>), ra{P(a) - Aa) G Z.
(4) Pour p G 35 e(neZ^(/? + n) = e(p) + n, e£ e{-p) = -e(p).

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES 91
Il résulte de l'assertion (4) que la fonction e est connue dès que l'on connaît
sa restriction à S H [0,1/2]. Le calcul donne par exemple
p/2*
e(p/2q)
0 I I 1 1
u 8 4 8 2
n 5 2 12 1
u 13 5 29 2
THEOREME (3.5). On a e(D) = <£.
L'inclusion e(T)) C <£ peut se démontrer en utilisant la proposition (3.3) [5].
On peut aussi comme dans [3] donner une construction plus explicite, qui utilise
la caractérisation suivante des fibres exceptionnels:
LEMMA (3.6). Soit E un fibre simple et rigide sur P2. Alors E est un fibre
exceptionnel.
La démonstration de ce lemme repose sur l'examen de la filtration de Harder-
Narasimhan de E: on vérifie qu'elle est réduite à un seul terme, et on utilise
le fait qu'un faisceau semi-exceptionnel de pente /i est somme directe de fibres
exceptionnels de même pente /i [5].
Soit Q le fibre quotient canonique de rang 2 sur P2. On peut construire un
fibre exceptionnel de pente 2/5 de la manière suivante: considérons le morphisme
canonique
Hom(Q,0(l))®g3 0(l).
rang 3
Ce morphisme est évidemment surjectif; son noyau Z = Keret; est alors un fibre
de rang 5 à la fois simple et rigide, donc exceptionnel, de pente 2/5. De même,
le morphisme canonique
Hom(Z,Q)<g)Z™ Q
V v '
rang 3
est surjectif: s'il ne l'était pas, il serait de rang constant et égal à un, et donc se
factoriserait à travers un sous-fibré de rang un de Q. Ceci est absurde, car Q n'a
pas de sous-fibré de rang un. Le noyau de ev est alors un fibre de rang 13 à la
fois simple et rigide (donc exceptionnel) de pente 5/13. Par le même argument,
on voit que le noyau du morphisme canonique
Hom(Q,0(3))®Q™ 0(3)
> v '
rang 3
est un fibre exceptionnel de pente 12/29.
Plus généralement, on peut en suivant ces modèles donner une
construction d'un fibre exceptionnel E de pente e(p/2q) comme noyau d'un morphisme
évaluation
Hom(F, G) ® F ™ G
où F et G sont des fibres exceptionnels de pentes respectives e(p'/2q~1) et
e(p"/2q~1) bien choisis. Ceci permet donc d'obtenir l'inclusion £(£>) C (H par
récurrence sur q.
Pour obtenir l'égalité e{D) = (H, il suffit de vérifier le lemme suivant,
conséquence immédiate de l'assertion (2) de la proposition (3.3).

92
J. LE POTIER
LEMME (3.7). Soient a = e{p/2q), 0 = e((p + l)/2«), fi = a- /3. Pour tout
a' G <£ tel que a < a' < {3, on a ra< > rM.
Etant donné v G <£, on choisit un entier q tel que 2q > rv, et un entier p tel
que
e{p/2q)<v<e{{p + l)/2q).
D'après la proposition (3.4), /i = a • /? est de rang > 2a+1 > r^. Le lemme (3.7)
impose v = e(p/2q). Par suite <£ C e(T>).
4. Conditions d'existence [5]. Soient r un entier > 1, c\ et c2 G Z, /i =
ci/r, et
A = -r{C*-r^)-
On suppose A > 1/2. La même démonstration que dans la proposition (3.3)
montre que s'il existe un faisceau semi-stable de rang r, de classes de Chern c\
et C2, pour tout a G <£ tel que |/i — a| < 3, on a
A>P(-|/i-a|)-Aa.
On pose
<5(/i)= Sup P{-\fi-a\)-Aa.
|/i-a|<3
La fonction 6 : Q —► Q ainsi définie est périodique de période un. Les points
/i G Q tels que P(—\fi — a\) — Aa > 1/2 constituent un intervalle ouvert Ia de
Q, centré en a; les intervalles Ia sont disjoints et recouvrent Q, et on a en fait
6\ia=P(-\n-a\)-Aa.
Ceci donne Figure 1 pour le graphe de 6.
Il en résulte en particulier que 6 > 1/2 (donc il n'existe pas de faisceau semi-
stable de discriminant 1/2).
THEOREME (4.1). On suppose A > <5(/i). Alors il existe un fibre stable de
rang r, de classes de Chern c\ et c<i sur P2.
Soit d un droite fixée de P2. La démonstration de ce théorème repose sur la
construction d'une famille E —► S x P2 de fibres vectoriels algébriques de rang
r, de classes de Chern c\ et c2, paramétrée par une variété algébrique lisse 5, et
satisfaisant aux conditions suivantes:
(L) Pour tout seS, Ext2{E(s), E{s)) = 0.
(KS) Le morphisme de déformation infinitésimale de Kodaira et Spencer
TSS-+Ext1 {E{s),E{s))
est surjectif.
(R) Pour tout s G 5, E(s)\ a est rigide, c'est-à-dire
Ext1d(E(s)\d,E(s)\d) = 0.

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES
93
Figure 1
Pour construire une telle famille, on peut supposer, quitte à changer /i en
/i + 2, que /i > 0, et que si l'on pose x{i) = r{P{l*> + i) — A), on a
N2 = -X(-2) > 0, Nx = -x(-l) > 0, N0 = X > 0.
On considère alors l'ouvert Q des morphismes injectifs (comme morphismes de
fibres vectoriels)
s: {0{-l))N> ® {Q*)Nl -+0No.
Du fait que TVo — 2N\ + 7V2 = r, il résulte que cet ouvert est non vide si r > 2.
Pour s G H, le conoyau E(s) est un fibre vectoriel de rang r, de classes de Chern
ci et C2, et on obtient ainsi un fibre E —► Q x P2 qui satisfait aux conditions (L)
et (KS). Plus précisément, on a même la condition suivante, plus forte que (L)
(L,)Ext2(^(s),^(s)(-l))=0
d'où il résulte que l'ouvert S Cfl des points s tels que E(s)\d soit rigide est non
vide.

94
J. LE POTIER
Stratification de Shatz. Soit S une variété algébrique lisse, E un faisceau
algébrique cohérent sur SxP2, S-plat, et tel que pour tout s G S, le faisceau
induit E(s) soit sans torsion. On suppose que la famille E satisfait aux conditions
(L) et (KS) ci-dessus. Soient H\,..., H\ des polynômes à coefficients rationnels;
on dira qu'un point s G S est de poids Hi,..., H\ (relativement à E) si la
filtration de Harder-Narasimhan de E(s)
{0} C Fi C F2 C •••cFj = E{s)
est de longueur /, et si le gradué associé gri(E(s)) a Hi pour polynôme de Hilbert.
Le rang r^, la pente //*, le discriminant A» de gri(E(s)) sont déterminés à partir
de Hi par la formule
Hi{m) = ri(P(fii + m) - Ai).
L'énoncé suivant étend à P2 l'énoncé obtenu par Atiyah et Bott dans le cadre
des courbes algébriques [1]:
PROPOSITION (4.2). Sous le conditions (L) et (KS), l'ensemble Y(Hi,..., H{)
des points s E S de poids Hi,..., H\ est une sous-variété localement fermée lisse
de codimension
Y^nr0{^i + A, - p{fij - m)).
Ces sous-variétés sont en nombre fini.
L'énoncé s'applique en particulier à la famille de fibres construite ci-dessus.
Pour obtenir l'existence de fibres semi-stables, il suffit donc de vérifier que si
/ > 0, y(i/i,... ,i//) est de codimension > 0. Ceci est une conséquence de
l'assertion (1) dans la proposition suivante:
PROPOSITION (4.3). Soient d une droite de P2, E un faisceau algébrique
cohérent sans torsion sur P2, de rang r, de pente fi, de discriminant A,
satisfaisant à la condition suivante:
(R7) Le faisceau E\ d est localement libre, et
dimExtJ(£|d,£|d) <1.
Soit {0} C F\ C - - C Fi = E la filtration de Harder-Narasimhan de E, de
gradué gr^E). Alors si l > 0,
(1) Si A > S(jt), EiKjXteMVjiE)) < 0;
(2) Si A > 6(n), Zt<j X(grt(E), gr,.(£)) < -1.
L'assertion (2) est un raffinement de (1); c'est un des arguments essentiels
utilisés dans le calcul du groupe de Picard (cf. §6). Démontrons par exemple
l'assertion (1): désignons par r^, /i^, A» le rang, la pente et le discriminant de
gri{E). L'hypothèse (R7) entraîne 0 < fi\ — {i\ < 2. De la semi-stabilité de
gTi{E), il découle x(gri{E),gTj{E)) < 0 si i < j. Il s'agit de vérifier qu'un au
moins de ces termes est non nul si / > 0. Supposons que pour tout (i, j) tel que

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES 95
i < j, on ait x{8Ti{E)i£rj{E)) = 0. On aurait en particulier, d'après la formule
de Riemann-Roch
P{fH-fi1) = Ai + Af.
Si /ii - fii > 0, ceci implique Ai + A/ < 1, donc soit Ax < 1/2, soit A/ < 1/2.
Si /ii = /i/, Ai < A/ par définition de la filtration de Harder-Narasimhan; par
suite, Ai < 1/2. Dans les deux cas, l'un des faisceaux gri(E'), grt(E) serait
semi-exceptionnel. Supposons par exemple que gri(i£) soit semi-exceptionnel; il
est donc somme directe de fibres exceptionnels de même pente /ii. On a alors
X(gri(^),^)=x(gri(^),gri(^)) + ^x(gri(^),grz(^))
Z>1
= x(gri(£),gr1(£))>0.
Ceci entraîne A < P(/i — /ii) — Ai; comme |/i —/ii| < 3, ceci contredit l'hypothèse
A > <5(/i). Dans le cas où c'est grt(E) qui est semi-exceptionnel, on obtient à
nouveau une contradiction en étudiant x{F^grl{E)).
Construction de fibres stables. Dans la famille de fibres vectoriels
construite ci-dessus, on obtient ainsi un ouvert non vide 9JI C S correspondant aux
fibres semi-stables. Considérons une suite (r^) d'entiers > 0 telle que Ylri = r-
L'ensemble des points s G 9Jt tels que E(s) ait une filtration (dite de Jordan-
Hôlder)
{0} C Fi C F2 C • • • C Fi = E{s)
dont le gradué gr^ soit stable de rang r^, de pente /i, de discriminant A est un
fermé Y (ri,..., r/) dont on peut minorer la codimension, de manière tout à fait
semblable à ce qui a été vu dans la proposition (4.2), bien que ces fermés n'aient
ici aucune raison d'être lisses:
Codimy(n,...,n) > X>^(2A -1)-
La réunion Y = \J F(rx,..., r/) est un fermé de 9JI de codimension > 0, dont
le complémentaire est l'ouvert des points s tels que E(s) soit stable, ouvert qui
est donc non vide. Ceci conduit au théorème (4.1).
5. Groupe de Picard: Le résultat de Drezet [4]. Si r, ci, et c2 sont
choisis de sorte que A > <5(/i), la variété de modules M = M(r,ci,C2) est non
vide, projective, irréductible de dimension 1 + r2(2A — 1). C'est une variété
normale, qui n'est pas lisse en général. Plus précisément, J. M. Drezet démontre
le résultat suivant:
THEOREME (5.1). (1) La variété de modules M = M(r, ci, C2) est localement
factorielle.
(2) Si A > <5(/i), Pic M = Z © Z;
(3) Si A = é(/i), PicM = Z.
On peut en fait donner une base de Pic M, en procédant de la manière
suivante. Soient S une variété algébrique lisse, irréductible, E un faisceau algébrique

96
J. LE POTIER
cohérent sur SxP2, S-plat, et tel que pour tout s G S, le faisceau E(s) soit
semi-stable de rang r, de classes de Chern c\ et c<2. Une telle famille détermine
un morphisme de variétés /#: S —► M qui ne change pas quand on remplace
E par E ® prJ(A), où A est un fibre de rang un sur S. Désignons par E(i) le
faisceau -E® pi2(Op2(i)). Dans le groupe de Grothendieck K(S) de S, on peut
considérer les éléments pr1!(£'(2)), et pour toute suite v — (i/o, ^i, ^2) d'entiers,
le fibre de rang un sur S défini par
£*,.,= ® (detpr^H)))®"'.
z=0,l,2
Posons x(«) = r(P(fi + 0 - A); c'est le rang de prx[(E(i)). Si A est un fibre
vectoriel de rang un sur 5, on a
pourvu que v satisfasse à la condition suivante:
Désignons par f) le sous-groupe de Z3 des suites 1/ = (^0,^1,^2) satisfaisant
à cette condition.
PROPOSITION (5.2). Soit v G S). Il existe un élément et un seul Lv G Pic M
satisfaisant à la propriété suivante: pour toute famille plate E de faisceaux
semi-stables de rang r, de classes de Chern c\ et C2, paramétrée par un variété
algébrique lisse S, on a
fE\Lv) — Le,v>
Ainsi, on obtient un homomorphisme de groupes
À:#-+PicM,
La proposition suivante précise le théorème (5.1):
PROPOSITION (5.3). Vhomomorphisme À est un isomorphisme si A > 6(fi),
et un épimorphisme si A = 6(fi).
Il reste à préciser le noyau de À dans le cas où A = 6(fi). Pour ceci, on
remarque d'abord que par définition de l'application <5, il existe un élément a G (H
et un seul tel que a — S < fi < a et
ô(fi)=P(a-fi)-Aa.
Désignons par Ea un fibre exceptionnel de pente a, et considérons une famille
plate E de faisceaux semi-stables de rang r, de classes de Chern c\ et C2,
paramétrée par une variété lisse et irréductible S. Dans la catégorie des faisceaux
algébriques cohérents sur S x P2, il existe une suite spectrale, dite de Beilinson
[14], d'aboutissement £*, dont le terme E\ est donné par
^•« = JR«pr1.(£?(-p))HA"Q*.

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES 97
Dans le groupe de Grothendieck K(S x P2), on obtient donc
V
Supposons A = <5(/i). D'après le choix de a, on obtient alors pour tout point
s G S et tout ç, Extq{Ea,E(s)) = 0. Il en résulte
et par suite prx; (Homfpro (E„), E) = 0 dans le groupe de Grothendieck K(S).
On obtient donc dans K(S)
0 = £(-l)px(£a, A?Q*)ptv(E(-p)).
V
En prenant le rang, on voit que la suite ((—l)%x(Ea, A*Q*))t=o,i,2 appartient au
sous-groupe #. En prenant le déterminant, on voit qu'elle appartient en fait au
noyau de À. On vérifie en fait que cette suite définit un élément non divisible
de #; le fait que Pic M n'ait pas de torsion entraîne que cet élément engendre
KerA.
Dans le cas particulier où r, c\ et x sont premiers entr'eux, la situation est
plus simple: la variété M est lisse, et il existe un faisceau universel U sur M xP2.
On a alors pour v G $)
i
Ces fibres sont indépendants du choix du faisceau universel.
6. Un exemple. Considérons l'espace de modules M = M(2,0,3) des
faisceaux semi-stables de rang 2, de classes de Chern (0,3); l'ouvert correspondant
aux fibres stables a été étudié en détail par Barth [2], Nous identifierons M à
M' = M(2,2,4). Si E est un fibre semi-stable de rang 2, de classes de Chern
ci = 2, C2 = 4, il est en fait stable, et hq(E(i)) est donné par le tableau suivant:
i
q
2
i
0
-2
0
3
0
-1
0
1
0
0
0
0
3
L'égalité hl(E) = 0 s'obtient en constatant comme dans [10] que la suite i •—►
hl(E(ï)) est strictement décroissante pour i > —2, jusqu'à ce qu'elle s'annule.
De la suite spectrale de Beilinson, il résulte que E est le conoyau d'un morphisme
inject if de faisceaux
H = {0{-1))3 -^03®Q* =K.
Considérons une droite d C P2 et l'ouvert f^ des morphismes s qui satisfont
aux conditions suivantes.

98
J. LE POTIER
(1) la restriction s\ d est injective (comme morphisme de fibres);
(2) i/1(cokers|d(-l)) = 0.
La condition (2) signifie que
coker s\ d(-l) = O(fc) © O(-fc)
où A: est un entier tel que 0 < A: < 1. Alors Qd est un ouvert dont le
complémentaire est de codimension > 2 dans l'espace vectoriel de tous les morphismes, et
pour s G Hd, coker s est sans torsion. Soit Qd,s l'ouvert de Qd des morphismes s
tels que coker s soit semi-stable (ou stable, ce qui revient au même ici). On peut
appliquer à la famille E(s) = coker s paramétrée par f^, les propositions (4.2)
et (4.3): en effet, les hypothèses (L) et (KS) sont trivialement satisfaites, ainsi
que la condition
dim Ext^Eis)] dlE{s)\ d) <1.
Puisque A > <5(/i), il en résulte que le complémentaire de Qd,s dans Qd est un
fermé de codimension > 2. Considérons l'ouvert
m = \Jnd,a.
d
La variété M' = M(2,2,4) est le quotient de 9)1 par l'action propre et libre du
groupe algébrique G = (AutH x AutK)/C*. Bien que le groupe G ne soit
pas réductif, le quotient a quand même un sens, c'est-à-dire que les fonctions
régulières G-invariantes sur un ouvert V de 9)1 invariant par G proviennent de
fonctions régulières sur un ouvert de M'; de plus, on a
PicM' = PicG(9Jt)
le membre de droite désignant le groupe des classes d'isomorphisme de fibres
vectoriels de rang un, munis d'une action de G compatible avec l'action de G sur
9)1. D'autre part, le complémentaire de 9)1 étant de codimension 2 dans l'espace
vectoriel de tous les morphismes, on a
0*(9)l) = C*, Pic(0R)=0.
Il en résulte que le groupe PicG (9)1) s'identifie au groupe des caractères Char G
de G. Ce groupe s'identifie au sous-groupe de Z3 des suites v = (^0,^1,^2)
satisfaisant à l'équation 3i^o + ^1 + ^2 = 0. Soit U un faisceau universal sur
M' x P2. Les faisceaux sur M'
A = prlt(U), B = R1pru(U(-l)), C = R1 pru(U (-2))
sont localement libres de rangs respectifs 3, 1, 3. L'isomorphisme Char G -^
Pic M' obtenu ci-dessus associe à v le fibre de rang un sur M' (indépendant du
choix de U)
(det A)®"Q 0 B®»1 0 (det C)®"2
qui n'est autre, avec les notations du paragraphe 5, que le fibre de rang un Lv<
associé à l'élément v' = (i/0, —^i, —^2) de S). On constate donc que l'homomor-
phisme À : S) —► Pic M' est bien un isomorphisms

VARIÉTÉS DE MODULES DE FAISCEAUX SEMI-STABLES 99
La démonstration que nous avons adoptée dans ce cas particulier diffère de
celle de Str0mme et de celle de Drezet [13, 4]. J'ignore si elle peut s'étendre au
cas général A > <5(/i). On peut utiliser le résultat pour donner une description
plus précise de M = M(2,0,3):
THEOREME (6.1). Soient Gr = Grass3 T(0(2)) la grassmannienne des
réseaux de coniques de P2, P2 —► Gr le plongement du plan projectif dual dans
Gr obtenu en associant à la droite l d'équation z = 0 le réseau de coniques
dégénérées zT(0(l)) C T(0(2)). Alors la variété de modules M = M(2,0,3) est
isomorphe à l'éclaté Gr de P^ dans Gr.
Voici l'idée de la démonstration: on considère à nouveau un faisceau universel
E sur M x P2, et les fibres vectoriels de rangs respectifs 3, 1, 3 associés sur M:
A = pvlt(E(l)), B = R1pvu(E), C = R1pvu(E(-l)).
La suite spectrale de Beilinson sur M x P2, d'aboutissement U — E(l), fournit
des morphismes de faisceaux [10]
CBA2Q* ABBQ*,
qui ont même noyau et même conoyau. En fait, le morphisme v est surjectif en
dehors d'un fermé de codimension 2, d'où il résulte que
Ker v S L El 0(-2), avec L = det C ® .B®"2.
On peut supposer que le fibre universel est choisi de sorte que L soit trivial.
On obtient alors un morphisme inject if 0(—2) —► A, et par suite un morphisme
de fibres vectoriels sur M
A* -* M x r(0(2))
où A* désigne le dual de A.
PROPOSITION (6.2). Ce morphisme est injectif comme morphisme de fibres
vectoriels.
Ainsi, A* est un sous-fibre vectoriel de rang 3 du fibre trivial de rang 6 sur
M de fibre T(0(2)). Il définit donc un morphisme de variétés <p: M —► Gr. On
vérifie alors les points suivants:
• au-dessus de Gr — P2 , <p est un isomorphisme;
• l'image réciproque ^_1(P^) (au sens des schémas) est une hypersurface
lisse de M: c'est l'hypersurface correspondant aux points s de M pour lesquels
il existe une droite / C P2 telle que /i1(£,(s)| /) ^ 0.
Il résulte de la propriété universelle de l'éclatement d'une sous-variété que le
morphisme <p se factorise à travers Gr en un morphisme
<p: M — Gr
lui aussi birationnel. Puisque ces deux variétés ont même groupe de Picard, fi
est un isomorphisme, ce qui conduit au théorème (6.1).

100
J. LE POTIER
REFERENCES
1. M. F. Atiyah et R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans.
Roy. Soc. London Ser. A 308 (1983), 523-615.
2. W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977),
63-91.
3. J. M. Drezet, Fibres exceptionnels et suite spectrale de Beilinson généralisée sur P2,
Preprint, Université Paris 7, 1985.
4. , Groupe de Picard des variétés de modules de faisceaux semi-stables sur P2, Preprint,
Université Paris 7, 1985.
5. J. M. Drezet et J. Le Potier, Fibres stables et fibres exceptionnels sur P2, Ann. Sci.
École Norm. Sup. (4) 18 (1985), 193-244.
6. G. Ellingsrud et S. A. Str0mme, The Picard group of the moduli for stable rank 2 vector
bundles on P2 with odd Chern class, Preprint, Oslo, 1979.
7. D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math (2)
106 (1977), 45-60.
8. G. Harder et M. S. Narasimhan, On the cohomology groups of moduli spaces of vector
bundles on curves, Math. Ann. 212 (1975), 215-248.
9. J. Le Potier, Sur le groupe de Picard de l'espace de modules des fibres stables sur P2, Ann.
Sci. École Norm. Sup. 14 (1981), 141-155.
10. , Stabilité et amplitude sur P2(C), Vector Bundles and Differential Equations
(Proc. Conf., Nice, 1979), Progr. Math., vol. 7, Birkhàuser, 1980, pp. 146-181.
11. M. Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978), 557-614.
12. S. Shatz, The decomposition and specialisation of algebraic families of vector bundles,
Compositio Math. 35 (1977), 163-187.
13. S. A. Str0mme, Ample divisors on fine moduli spaces on the projective plane, Math. Z.
187 (1984), 405-423.
14. J. L. Verdier, Instantons, Les Équations de Yang-Mills, Astérisque, no. 71-72, Soc.
Math. France, Paris, 1980, pp. 105-134.
Université Paris 7, France

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Vector Bundles and Submanifolds of Projective Space:
Nine Open Problems
MICHAEL SCHNEIDER
Introduction. In the last 10 years there has been an enormous activity in the
theory of algebraic vector bundles on Pn = Pn(C) and other model manifolds.
There were essentially two reasons for this boom:
(i) the connection of algebraic vector bundles on P3 with solutions of the
Yang-Mills equations (via the Penrose transform) on 54;
(ii) the connection between algebraic vector bundles on Pn and subvarieties
X C Pn of projective space.
In the last few years there has been a slight depression in the vector bundle
industry. This has recently been overcome through the pioneering work of
Donaldson relating Yang-Mills fields to algebraic geometry and differential topology
of certain 4-dimensional manifolds.
I believe that when one formulates problems at this moment it is more
important to concentrate on a few central questions rather than give an exhaustive list
of open problems (including many minor ones). I will formulate nine unsolved
problems. Some of these problems, conjectures and questions are rather "old"
and due to various people. I'll try to trace their origin wherever I am aware of
it. The nine problems fall into three different classes:
(1) geometric problems, submanifolds of Pn;
(2) moduli and existence of stable vector bundles;
(3) vector bundles and differential geometry.
For survey-type papers on vector bundles I refer to [36, 17, 37, 18, 6] and
for a more complete (but somewhat outdated) account to the book [31].
1. Geometric problems. The outstanding open problem here is Harts-
linebreak home's conjecture [16].
CONJECTURE (HARTSHORNE). Let X C Pn be a smooth subvariety of
dimension d > |n. Then X is a complete intersection.
1980 Mathematics Subject Classification (1985 Revision). Primary 14F05, 14M07, 53C55.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
101

102
MICHAEL SCHNEIDER
There are some results supporting this conjecture. If the degree of X is small
compared to n then the conjecture is true (cf. [3, 22]). Moreover Zak [39] has
proved a weak form of this conjecture (also due to Hartshorne), namely, that X
is linearly normal for d > |(n — 1), i.e.,
H°(Pn,QPn(l))-+H°(X,Ox(l))
is surjective.
A next step in proving Hartshorne's conjecture would be a solution of the
following
PROBLEM. Let X C Pn be a smooth subvariety of dimension d with d > |n.
Prove that X is quadratically normal, i.e.,
H°(Pn,0Pn(2) -+ H°(X,Ox(2))
is surjective.
In a recent paper [34] this has been proved for codimX = 2 and n > 12 using
Faltings's approach [11] to Zak's theorem mentioned above. The conjectured
bound in this case would be n > 7. Note also that for codimX = 2 Hartshorne's
conjecture is for n > 6 in fact equivalent to the surjectivity of
H°(Pn,QPn(k)) -+ H°(X,Ox(k))
for all A:, which means that X is projectively normal.
Let X C Pn be a smooth subvariety. If X is a complete intersection, then the
normal bundle Nx/pn of X C Pn is of the form
Nx/rn ^Ox(ai)©---©Ox(ar),
r = codimX. By a nice result of Faltings [12] the converse is true for d =
dimX > ^n. In the range d > |n all line bundles on X are of the form Ox (a)
by [2].
There is a relation between vector bundles on Pn and subvarieties of Pn
which is well understood only in codimension 2 (Serre's construction). A variant
of Hartshorne's conjecture for vector bundles would be the following problem: do
vector bundles of rank r on Pn split into a direct sum of line bundles if n > r?
For rank 2 Hartshorne's conjecture is in fact equivalent to the splitting of all
2-bundles on Pn, n > 7.
PROBLEM. Let X C Pn be a smooth subvariety of codimension r.
(a) When does there exist a vector bundle E of rank r on Pn such that
E\X~NX/PJ
(b) When does there exist a vector bundle E of rank r on Pn and a section
a G H°(Pn,E) such that X = {a = 0} as schemes?
Note that in the case (b) one has automatically E\X ~ NX/pn- For r = 2
and n > 3 by Serre's construction (b) is true precisely if X is subcanonical, i.e.,
wx = Ox {a) for some a G Z. By [2] X is subcanonical for n > 6.
The relation between subvarieties of Pn of codimension two and vector
bundles (or sheaves) of rank two has been exploited very often. For the case P3 see

VECTOR BUNDLES AND SUBMANIFOLDS OF PROJECTIVE SPACE 103
the paper [19] of Hartshorne in this volume. For the case of surfaces in P4 and
3-folds in P5 there is a survey paper of Okonek [33].
2. Moduli of stable vector bundles. For a smooth projective variety X
over C we denote by Mx(r, C\, c2,..., cn) the coarse moduli scheme of semistable
sheaves of rank r with Chern classes ci,..., cn. Here semistability is in the sense
of Gieseker [14] and Maruyama [28]. These moduli schemes are known to be
quasiprojective [13].
For X = P2 one has a rather good understanding of these moduli spaces. For
a survey of recent results in this case we refer to le Potier's paper [35] in this
volume. The main open problem here is the following
PROBLEM. Are the moduli spaces Mp2(2, ci, c2) rational (or stably rational)?
They are always unirational [28] and rational for C\ odd [23] and for C\ even
and C2 — cï/4 odd [4]. (In [4] the rationality is asserted also for c<2 — c\/A even,
but there is a gap in the proof as noticed by Maruyama.) The corresponding
problem for higher rank is of course also very interesting.
The properties of the moduli spaces Mp2 have been proved important in
the study of the solutions of the Yang-Mills equations on S4. For instance
Donaldson [8] related the space of real A:-instantons on S4 via the moment map
to Mp2(2; 0, A:). Using the irreducibility of Mp2, he obtained the connectedness
of the space of A:-instantons.
Another problem for the spaces Mp2 is the analogue of a question for the
moduli spaces of curves of genus g.
PROBLEM (HirsCHOWITZ, HULEK [20], [24]). Determine the maximal
dimension of a complete subvariety of Mp2ab(2,ci,c2).
Here Mp2ab C Mp2 denotes the open part of stable bundles. It is shown
that except in some obvious cases there always exist complete curves. What one
would expect is a function of c<2 (once c\ normalized to 0 or —1). Of course one
could ask the stronger question to determine the ç-completeness of Mp2ab in the
sense of Andreotti and Grauert.
As was mentioned at the beginning, one of the driving forces in vector bundle
theory has been the connection to real instantons. For the case of "complex
instantons" the main unsolved question is the following
PROBLEM. Are the spaces of complex A:-instantons Mp^(k) nonsingular,
irreducible, rational?
Here Mp^(k) denotes the moduli space of stable rank 2 vector bundles E on
P3 with d(E) = 0, c2(E) = fc, and the instanton condition H1^, E(-2)) = 0.
The problem has a positive answer for A: < 4 (cf. [5], rationality is known only
for k < 3).
As a last problem in this section we shall consider rank 2 bundles on P4. As
is well known there is essentially only one known indecomposable rank 2 bundle
on P4—the Horrocks-Mumford bundle. For a survey on this beautiful bundle
and its geometry see the article of Hulek [25] in this volume.

104
MICHAEL SCHNEIDER
There have been attempts [15] to prove that rank 2 bundles on P4 which are
not stable have to split. Possibly this is true for Pn, n > 5.
PROBLEM. Do there exist indecomposable unstable 2-bundles on P4? If not,
this would imply the existence of topological rank-2 bundles on Pn which do not
admit an algebraic structure.
3. Vector bundles and differential geometry. Kobayashi [26] (see also
[27]) observed that holomorphic vector bundles on compact Kàhler manifolds
admitting a Hermitian-Einstein metric are direct sums of /i-stable bundles (all
with the same /i). Here a vector bundle E on a compact Kâhler manifold X
(with fixed Kâhler class u) is /i-stable if /i(F) < fi(E) for all coherent subsheaves
F C E with 0 < rank F < rank E. By definition,
/i(F) = (ci(F) Ucjn-1)/rank(F),n = dimX
The converse of this has been conjectured by Kobayashi [26] and Hitchin. A
proof when X is a projective surface has been given by Donaldson [9]. Several
years ago Uhlenbeck and Yau [38] announced a proof also for higher-dimensional
base spaces.
Conjecture (Kobayashi, Hitchin). Let X be a projective variety with
fixed polarization and let E be a fi-stable vector bundle on X. Then X admits a
Hermitian-Einstein metric.
Recall that a holomorphic vector bundle E on a compact Kâhler manifold
(X, g) admits a Hermitian-Einstein metric if there is a Hermitian metric h on E
such that RiCg(h) is proportional to /i, i.e.,
/] ^Oaiijthp = Chap,
where 6aiij are the components of the curvature matrix 6ai.
The Yang-Mills fields giving the connection between differential and algebraic
geometry just mentioned have been used by Donaldson [7] in a spectacular way
to show that the intersection forms of simply connected, compact, oriented dif-
ferentiable 4-manifolds are of a very special form.
Recently Donaldson [10] has used Yang-Mills fields in order to introduce a
new invariant T which distinguishes difFeomorphism classes of compact, oriented,
simply connected smooth 4-manifolds X with intersection form equivalent (over
R) to
T2 2 2
In case X is a complex projective surface, T can be computed via the moduli
space of stable 2-bundles on X with c\ = 0, C2 = 1 (with respect to a suitable
polarization of X). In this way Donaldson has shown that P2 blown up in 9
points has two different differentiate structures. This has renewed the interest
in determining certain moduli spaces of stable 2-bundles on algebraic surfaces.

VECTOR BUNDLES AND SUBMANIFOLDS OF PROJECTIVE SPACE 105
Okonek and Van de Ven [32] and Friedman and Morgan have used
Donaldson's T-invariant to show that P2 blown up in 9 points admits infinitely many
inequivalent differentiable structures.
In this context many interesting problems arise, but I will just recall one old
problem [21].
PROBLEM. Determine all complex structures on S2 x S2.
Concluding remarks. I hope that it has become apparent that the
determination of moduli spaces of stable vector bundles on certain base manifolds
is not just interesting by itself but has various important applications. The
most important base spaces are complex projective surfaces (e.g., P2, ruled
surfaces, elliptic surfaces, abelian surfaces, if3-surfaces, ...). In this case the
connection between algebraic and differential geometry and topology is extremely
tight.
Moduli spaces of stable bundles seem more and more to serve the purpose of
giving insight into the structure of the base space itself. For instance Mukai [29]
has studied the moduli spaces of stable bundles on an algebraic if 3-surface. As
an application he proves in certain cases the Hodge conjecture for products of
two if3-surfaces [30].
Another interesting direction is the case of nonalgebraic base manifolds. For
complex surfaces there is a rather complete answer [1] to the question of which
topological vector bundles admit a holomorphic structure. For Kàhler manifolds
moduli spaces of bundles can be constructed by differential geometric methods.
In most cases it remains to be shown that these enjoy the natural universal
properties.
Added in proof. The conjecture of Kobayashi and Hitchin has recently
been established by Donaldson ("Infinite determinants, stable bundles, and
curvature" ) for projective manifolds and by Uhlenbeck and Yau ( "The existence of
Hermitian Yang-Mills connections on stable bundles over Kàhler manifolds" ) for
compact Kàhler manifolds.
References
1. C. Banica and J. le Potier, Sur Vexistence des fibres vectoriels holomorphes sur les surfaces
non-algébriques, Crelle's J. (to appear).
2. W. Barth, Transplanting cohomology classes in complex-projective space, Amer. J. Math.
92 (1970), 951-967.
3. , Submanifolds of low codimension in projective space, Proc. Internat. Congr. Math.
(Vancouver, 1974), Canad. Math. Congress, Montreal, 1975, pp. 409-413.
4. , Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63-91.
5. , Irreducibility of the space of mathematical instanton bundles with rank 2 and C2 = 4,
Math. Ann. 258 (1981), 81-106.
6. , Report on vector bundles, Proc. Internat. Congr. Math. (Warsaw, 1983), North-
Holland, 1984, pp. 783-789.
7. S. K. Donaldson, An application of gauge theory to the topology of 4-manifolds, J.
Differential Geom. 18 (1983), 279-315.
8. , Instantons and geometric invariant theory, Commun. Math. Phys. 93 (1984), 453-
460.

106
MICHAEL SCHNEIDER
9. , Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable
vector bundles, Proc. London Math. Soc. 50 (1985), 1-26.
10. , The differential topology of complex surfaces, C. R. Acad. Sci. Paris Sér. I Math.
301 (1985).
11. G. Faltings, Verschwindungssatze und Untermannigfaltigkeiten kleiner Kodimension des
komplex-projektiven Raumes, Crelle's J. 326 (1981), 136-151.
12. , Ein Kriterium fur vollstdndige Durchschnitte, Invent. Math. 62 (1981), 393-
401.
13. O. Forster, A. Hirschowitz, and M. Schneider, Type de scindage généralisé pour les fibres
stables, Vector Bundles and Differential Equations, Nice 1979, Progr. Math., vol. 7, Birkhauser,
1980, pp. 65-81.
14. D. Gieseker, On the moduli space of vector bundles on an algebraic surface, Ann. of Math.
(2) 106 (1977), 45-60.
15. H. Grauert and M. Schneider, Komplexe Unterraume und holomorphe Vektorraumbundel
vom Rang zwei, Math. Ann. 230 (1977), 75-90.
16. R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math.
Soc. 80 (1974), 1017-1032.
17. , Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979),
117-128.
18. , Four years of algebraic vector bundles: 1975-1979, Géométrie Algébrique, Angers,
1979, A. Beauville, editor, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, pp. 21-27.
19. , On the classification of algebraic space curves. II, this volume.
20. A. Hirschowitz and K. Hulek, Complete families of stable vector bundles over P2,
Complex Analysis and Algebraic Geometry (Proc. Gottingen SFB Conf., 1985), Lecture Notes in
Math., vol. 1194, Springer-Ver lag, 1986, pp. 19-33.
21. F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math.
(2) 60 (1954), 213-235.
22. A. Holme and M. Schneider, A computer aided approach to codimension 2 sub-
varieties ofPn, n > 6, Crelle's J. 357 (1985), 205-220.
23. K. Hulek, Stable rank-2 bundles on P2 with a odd, Math. Ann. 242 (1979), 241-
266.
24. K. Hulek and S. A. Str0mme, Appendix to the paper Complete families of stable vector
bundles on P2, Complex Analysis and Algebraic Geometry (Proc. Gottingen SFB Conf., 1985),
Lecture Notes in Math., vol. 1194, Springer-Verlag, 1986, pp. 34-40.
25. K. Hulek, Geometry of the Horrocks-Mumford bundle, this volume.
26. S. Kobayashi, Curvature and stability of vector bundles, Proc. Japan. Acad. Ser. A Math.
Sci. 58 (1982), 158-162.
27. M. Lubke, Stability of Einstein-Hermitian vector bundles, Manuscripta Math. 42 (1983),
245-257.
28. M. Maruyama, Moduli of stable sheaves. I and II, J. Math. Kyoto Univ. 17 (1977),
91-126 and 18 (1978), 557-614.
29. S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or KS surface,
Invent. Math. 77 (1984), 101-116.
30. , On the moduli space of bundles on KS surfaces. I, Proc. Tata Sympos. 1984 (to
appear).
31. C. Okonek, M. Schneider, and H. Spindler, Vector bundles over complex projective spaces,
Progr. Math., vol. 3, Birkhauser, Boston, 1980.
32. C. Okonek and A. Van de Ven, Stable bundles and differentiable structures on certain
elliptic surfaces, Invent. Math. 86 (1986), 357-370.
33. C. Okonek, On codimension-2 submanifolds in P4 and P5, Mathematica Gôttingensis
50 (1986).
34. T. Peternell, J. le Potier, and M. Schneider, Vanishing theorems, linear and
quadratic normality, Invent. Math, (to appear).
35. J. le Potier, Variétés de modules de faisceaux semi-stables de rang élevé sur P2, this
volume.
36. M. Schneider, Holomorphic vector bundles on Pn, Sém. Bourbaki 530 (1978/79).

VECTOR BUNDLES AND SUBMANIFOLDS OF PROJECTIVE SPACE 107
37. A. Van de Ven, Twenty years of classifying algebraic vector bundles, Géométrie
Algébrique, Angers 1979, A. Beauville, editor, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980,
pp. 3-20.
38. S. T. Yau, A survey on Kàhler-Einstein metrics, Proc. Sympos. Pure Math., vol. 41,
Amer. Math. Soc, Providence, R.I., 1984, pp. 285-289.
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transi, in Math. USSR-Sb. 44 (1983).
Mathematisches Institut der Universitat Bayreuth, Federal Republic
of Germany

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Geometry in Characteristic p

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
F-Isocrystals and p-adic Representations
RICHARD CREW
Introduction. On a complex manifold X, the category of representations
of 7Ti (X) in finite-dimensional C-vector spaces is equivalent to the category of
holomorphic vector bundles endowed with an integrable connection. This paper
is concerned with the various analogues of this statement for varieties defined
over a perfect field of characteristic p > 0. The basic result, due to Katz, asserts
that on a smooth scheme X/k over a perfect field of characteristic p, the category
of representations of tti(X) in free VF(Fg)-modules of finite rank is equivalent to
the category of unit root F^-crystals on X; this can be thought of as a nonabelian
generalization of Kummer-Artin-Schreier theory.
The term "isocrystal" is meant to suggest "crystal up to isogeny," but it
seems that this is the right notion only when the base is a perfect field. On a
more general base the category of crystals up to isogeny seems not to have very
many useful properties, and the need for a stronger notion first surfaced in the
work of Berthelot and Ogus [6, 20] on the variation of de Rham cohomology.
They called these objects "convergent isocrystals" because, roughly speaking,
they have a full set of horizontal sections in any open disk of radius one (cf. 1.4
below), and their properties are investigated at length in [20]. It is, however,
well known that a unit-root F-crystal has a full set of horizontal sections in any
open disk of radius one, and one is led to ask whether a unit-root convergent
F-crystal is indeed nothing more than a unit-root F-crystal up to isogeny. In
§2 we show that this is true when the base is smooth, and as a result we see
that the category of unit-root convergent F-isocrystals on a smooth A:-scheme X
is equivalent to the category of representations of -K\ (X) on finite-dimensional
p-adic vector spaces.
Berthelot introduced yet another refinement in [1, 3]. Motivated by the
problem of giving an intrinsic formulation to the "overconvergent" cohomology
theories of Dwork and Washnitzer-Monsky, he constructed a category of
"overconvergent isocrystals" and a cohomology theory for them. Our concern here is to
find out what sort of representations of -K\ give rise to overconvergent unit-root
1980 Mathematics Subject Classification (1985 Revision). Primary 11Q25.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
111

112
RICHARD CREW
F-isocrystals. So far we only have results when the base is a smooth curve, and
we show in §3 that if a p-adic representation of -K\ has "finite local monodromy
at infinity" then the corresponding unit-root F-isocrystal is overconvergent. It
is tempting to believe that all overconvergent unit-root F-isocrystals on a curve
arise in this way, but so far we can only prove this for isocrystals of rank one.
This is one of the main results of §4; the rest of §4 is concerned with other basic
questions for which we have complete results only for isocrystals of rank one on
a curve. For example, is the functor of restriction to a dense open subset fully
faithful? A result of Ogus guarantees that this is true for isocrystals of rank one
on a curve, but beyond this case very little is known. The question is related, via
the Dieudonné theory of Berthelot-Breen-Messing [4], to the equicharacteristic
case of Tate's theorem on p-divisible groups.
From the results mentioned above, one can deduce (Corollary 4.13) that if
(M, $) is an overconvergent unit-root F-isocrystal of rank one on a smooth curve
over a finite field, then some tensor power of (M, $) is geometrically constant.
This is an analogue of the well-known fact that a suitably high tensor power of a
rank one lisse /-adic sheaf on a curve over a finite field is geometrically constant.
In a subsequent paper I will show how to define "geometric" monodromy groups
for overconvergent F-isocrystals (not necessarily unit-root) and prove for them
an analogue of Grothendieck's "Global Monodromy Theorem" [10, 1.3.8]. These
and other facts justify the hope that the category of overconvergent F-isocrystals
is the correct p-adic analogue of the category of lisse /-adic sheaves. It would
be wrong however to assume that nonoverconvergent objects are uninteresting,
for they too occur quite often in nature, even as subquotients (in the convergent
category) of overconvergent F-isocrystals. Such is the example in 4.14, which is
important in many arithmetic questions [16, §4; 9, 11, 12].
Probably not a single result in this paper is definitive. In addition to the
questions already touched upon, there are other points that could be cleared up.
For example, one would think that the equivalence of Theorem 2.1 should be
valid for any separated fc-scheme of finite type. There is also the question of
whether an analogue of the local monodromy theorem holds for overconvergent
F-isocrystals (see 4.16.2).
I am indebted to a number of people for helpful suggestions during the
preparation of this paper, notably Dwork, Berthelot, Ogus, Robba, and Ekedahl. I
am also indebted to the referee for pointing out Corollary 4.14. I would like to
thank them all heartily.
Notation. Throughout the paper R is a complete discrete valuation ring
with perfect residue field fc of characteristic p > 0 and field of fractions K of
characteristic 0, W = W(k) is the ring of Witt vectors of fc, tt is a uniformizer
of i2, and v is the valuation, normalized so that v(tt) = 1.
We fix a power q of p, which is assumed to be such that Fq Ç fc. The qih
power endomorphism of any scheme of characteristic p will be denoted F or Fq.
We also fix a lifting a : R —► R of the qth power map fc —► fc which is assumed to
satisfy <j(-k) = tt. Let R°, Ka be the subsets of R, K fixed by a.

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
113
Formal schemes are assumed to carry the p-adic topology.
If U is an affinoid space with affinoid algebra A, we denote by | \u the spectral
seminorm on A, and by A0 the set of elements of seminorm < 1.
If C is any Z-linear category, we denote by C <g) Q the category whose objects
are those of C, and whose morphism groups are those of C tensored with Q.
1. Isocrystals.
1.0. In this section we will review and discuss Berthelot's construction of
the categories of convergent and overconvergent isocrystals. For more details,
the reader should consult the papers of Berthelot [1, 2] and Ogus [20]. In what
follows A: is a perfect field of characteristic p > 0, R a complete discrete valuation
ring with residue field A: and field of fractions K of characteristic 0. We begin
with some preliminaries on rigid-analytic geometry.
If X/R is any p-adic formal i2-scheme flat over R with closed fiber X = X <8> k
and "generic fiber" the rigid-analytic space X^/K (in the sense of Raynaud),
then the points of Xan are in a bijection with the set of closed subschemes of X
integral, finite, and flat over R. The specialization map
sp:Xan-+X
is defined by taking a point of Xan, viewed as a subscheme Z C X, and assigning
to it the support of Z <g) A:, which is a closed point of X. If now Z Ç X is any
subscheme of X (or X), we define the tube ]Z[% of Z in X to be the inverse image
For example, if Z Ç X is open and Z Ç X is a flat lifting over R, then ]Z[%= £an;
in particular ]X[x= Xan. If Z Ç X is closed and defined by the vanishing of
sections /i,..., fn G T(Ox), then
]Z[x={zeXan| |/,(*)|<lVt}.
We will write ]Z[ for ]Z[% when there is no chance of confusion.
1.1. Now let X/k be a separated A:-scheme of finite type and I ^ ^ a
closed immersion into a flat p-adic formal i2-scheme that is formally smooth
in a neighborhood of the image of X. The tube ]X[^xy of X for the diagonal
embedding is endowed with two natural projection maps
(1.1.1) PuP2:]X\w-*]X[y.
A convergent isocrystal on (X/if, y) is a locally free sheaf of Ojx[y -modules M
endowed with an isomorphism
(1.1.2) p\M^p\M
restricting to the identity on the image of the diagonal embedding ]X[y—^X^x^
and satisfying the usual sort of cocycle condition (cf. [5, 2.10.4] for example).
A morphism of convergent isocrystals on {X/K, y) is just a morphism of locally
free sheaves compatible with the isomorphisms (1.1.2). Berthelot shows in [3]
that the category of convergent isocrystals on (X/if,y) is independent, up to

114
RICHARD CREW
canonical equivalence, of the choice of X ^ y, and is furthermore functorial in
X/K and of local nature on X. Since any separated X/k of finite type always
has such embeddings locally on X, we can define the category of convergent
isocrystals on a general X/k by glueing.
1.2. Since ]X[yxyC]X[yx]X[y, a convergent isocrystal M on X/K gives rise
to a locally free sheaf of Ojx[y -modules endowed with an integrable connection
V': M —► M <g) fipq; recall that the data of an integrable connection on M is
equivalent to specifying an isomorphism
(1.2.1) q\M^q*2M, Ql,q2: Ai -]*[*
of the two pullbacks of M to the first infinitesimal neighborhood Ai of the
diagonal ]X[c]X[x]X[, the isomorphism being required to restrict to the identity
on the diagonal, and satisfy a cocycle condition. Again, we shall say that a
connection V on a locally free 0]x[-module M is convergent if the associated
isomorphism (1.2.1) extends to an isomorphism (1.1.1) (if so, it extends uniquely).
If V is convergent, t = (£i,..., tn) are local coordinates on y, 6 = (<$i,..., 6n)
are derivations dual to (£i,..., £n), and m G M, then the "Taylor series"
(1.2.2) P(*,r,ro)= £ ^(V((5)r(m)) 0 rr G M 0 Ov[[r]]
rGNn
converges in the open unit polydisk \t{\ < 1, i.e., in every open unit disk on y,
the connection has a full set of horizontal sections given by convergent power
series (whence the name "convergent connection"), and the isomorphism (1.1.1)
is given explicitly by
m \-+ P(t ® 1,1 ® t - t ® 1, m).
1.3. We record for later use some results of Ogus:
1.3.1. PROPOSITION [20, Theorem 2.16]. Let M be a locally free sheaf
of 0}x[y-modules endowed with an integrable connection V, and suppose that X
is smooth and that there is an open dense subset Z Ç X such that V induces a
convergent connection on M\]Z[. Then V is convergent.
1.3.2. PROPOSITION [20, Corollary 2.9]. Suppose that X = Spf(A)/R
is a noetherian formally smooth lifting of a smooth affine k-scheme X, and M
is a locally free sheaf on Xan underlying a convergent isocrystal on X/K. If
M = r(Xan,M), then M is a locally free A <g) Q-module.
Actually [20, Corollary 2.9] merely asserts that M is flat over A <g) Q, but in
fact it is of finite type over A <g) Q ([20, 1.2.1]; this is really a tautology in the
setup of [20]), hence locally free since A <g) Q is noetherian.
1.4. The reader should compare the definitions and facts we have just reviewed
with the corresponding ones for a crystal of Ox/w-m°dules: if X/W is formally
smooth, the latter can be realized as locally free Ox-modules endowed with an
integrable connection, and the corresponding Taylor series (1.2.2) will converge
on the closed disk of radius IpI1/^-1) [15, §1]. Thus a convergent isocrystal is a

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
115
much more rigid object than just a "crystal up to isogeny." This is particularly
important if one wants to work over ramified R when the absolute index of
ramification is > p — 1; then crystals do not possess the necessary rigidity.
Since we will have no use for "nonconvergent" isocrystals, we will drop the
modifer "convergent" when it helps to lighten the terminology.
1.5. Just as crystals on a formally smooth X/W can be viewed as a category
of sheaves on the crystalline site Cris(X/W), so also can isocrystals on X/K be
viewed as a category of sheaves on a suitable site (in Ogus's terminology, the
category of enlargements of X/K). This point of view has been developed in
detail by Ogus [20].
1.6. To define the notion of overconvergent isocrystal, we need some more
analytic constructions. Suppose j: X C X is an open immersion, X <-+ y is a
closed immersion with y/i2 smooth in a neighborhood of X, and let Z = X — X.
If Z is defined by the vanishing of sections /i,..., fn G r(0y), we set, for an
A < 1, _ _
zx = {x e]x[\ \fi(x)\ < x v<}, xA =}x[-zx,
and let
jx:Xx^]X[
be the natural inclusion. As defined, the Xa depend on the choice of fa, but the
pro-object {Xa}a—l does not, so that we may unambiguously define, for any
coherent sheaf M on a Xa,
yfM = limjx*JxM-
The sheaf i^jxf c 0]x[ is the ring of germs of functions on ]X[ extending
into the tube ]Z[, i.e., "overconvergent" functions on ]X[. As always, there
are two functors p\,P2 fr°m the category of j^0\x\ -modules to the category of
■7*®ix r-modules. An overconvergent isocrystal on (X/K, y, Z) is a locally free
sheaf of y ^q^ -modules M endowed with an isomorphism
(1.6.1) p\M^p\M
satisfying all the usual conditions. Morphisms of overconvergent isocrystals on
(X/if,y, Z) are of course morphisms of ^Oi^r-modules compatible with (1.6.1).
Berthelot shows that the category of overconvergent isocrystals on (X/K, y, Z) is
independent of y, up to canonical equivalence of categories, is of local nature on
X, and is functorial in the pair X C X. Therefore we may, just as before, define
a category of overconvergent isocrystals on (X/K, Z) for any X Ç X with X/k
separated of finite type by choosing local embeddings of X into formally smooth
y 's and glueing. Objects of this category will also be referred to as "isocrystals on
X/K, overconvergent around Z." Finally, if X/k is separated and of finite type,
and X C X is a compactification of X, then Berthelot shows that the category of
overconvergent isocrystals on (X/K, X) depends up to canonical equivalence on
X/K only, and is of local nature on X and functorial in X/K. It is the category
of overconvergent isocrystals on X/K.

116
RICHARD CREW
Heuristically, the extension of (1.1.2) to an isomorphism (1.6.1) means that
the radius of convergence of the Taylor series can be made as close as we want
to one at points of ]X[ sufficiently close to ]X[. To be more precise, we say that
an admissible open set U in ]X[ (in the sense of [8, 9.1.4, Proposition 2]) is a
strict neighborhood of ]X[ in ]X[ if
(1.6.2) ]X[C U.
1.6.3. For any affinoid W C]X[, there is a A < 1 such that Xx D W Ç U fl W.
Then the category of overconvergent isocrystals on X/K is equivalent to the
category of locally free modules on some strict neighborhood of ]X[y in ]X[y, with
an integrable connection whose associated Taylor series defines an isomorphism
as in (1.6.1) on some strict neighborhood of ]X[^xy m l^lvxv-
Suppose now that X c X,y,X\ are as in 1.6 (we give ourselves a set of
fi 6 r(Oy) defining Z = X-X). In §3 we will need
1.7. THEOREM (BOSCH-DWORK-ROBBA [7]). Let A C]X[ be an affinoid
subspace and set A\ = A fl X\, A\ = Afi]X[. If f G I^OaJ is integral over
T(Oax) for some A < 1, then there is a A' with A < A' < 1 such that f G T(Oax,)-
1.7.1. COROLLARY. ^Oi^r is integrally closed in 0]X[.
1.8. Now let F = Fq be a fixed power of the absolute Frobenius of fc, with
fixed field Fq. We choose once and for all a homomorphism a : K —► K extending
the natural action of F on W(k) C if, and fixing a uniformizer -k of R—this can
always be achieved after making a finite extension of R.
Suppose X/k is separated and of finite type. Since the category of isocrystals
on X/K is functorial in X/K, the pair {Fq,a) gives rise to a (semilinear) functor
F* from the category of isocrystals on X/K to itself. An F-isocrystal on X/K
is an isocrystal M endowed with an isomorphism
(1.8.1) $: F^M^M.
Similarly an overconvergent F-isocrystal on X/K is an overconvergent
isocrystal M endowed with an isomorphism (1.8.1) in the category of overconvergent
isocrystals. The map $ is called the Frobenius structure of the F-isocrystal. Note
that even if M is overconvergent, a map (1.8.1) is not known to be an
overconvergent map, so such a (M, $) might not be an overconvergent i2-isocrystal. We
should also point out that the definition we have adopted is that of Berthelot; in
[20] Ogus gives another definition which is different and much more restrictive,
as his Proposition 2.21 shows.
Suppose now that X <^-> y is a closed immersion as in 1.1 with y/R formally
smooth, and 0 is a lifting of F to y. If (M, $) is an F-isocrystal on X/R,
realize M as a locally free 0]x[-module with integrable connection V: M —►
M 0 Qpq. The condition that (1.8.1) be a morphism of isocrystals means that
V "commutes" with $; i.e., for any local section m of M such that
V(m) = 2^ mi ® rç*î w^h rji sections of H?xr,

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
117
we have
(1.8.2) V(*(m)) = ^2 *K) ® ^V
1.9. When X = Spec(fc), an F-isocrystal on X/K is simply a
finite-dimensional K-vector space V endowed with a cr-linear automorphism: $: a*V -^ V.
Since we have assumed that <j(-k) = 7r, the theory of Dieudonné-Manin [19]
applies: one can decompose V as a direct sum of F-isocrystals
(1.9.1) V = 0VA
A
indexed by a finite set of A G Q, such that if A = a/6, then Va ®w W(k) has
a basis whose elements satisfy $b(v) = 7rat;; the rational numbers A occurring
in (1.8.2) are the slopes of $ in V, and dim Va is the multiplicity of A. The
Newton polygon of (V, $) is the convex polygon in the plane whose sides are line
segments, one for each A in (1.8.2), each with slope A and horizontal projection
dim Va . It is a standard fact that dim Va is a multiple of the denominator of A
(in lowest terms) so that the break-points of the Newton polygon have integral
coordinates (cf. [17, §1] for more details). If V = Va for some A, we say that
(V, $) is purely of slope A, and if it is purely of slope zero, we also say that it is
a unit-root isocrystal. Suppose now that (M, $) is an F-isocrystal on X/R and
x —► X is a point with values in a perfect field. Let Spf(iZ') —► Spf(iZ) cover
W(x) -+ Spf(W(fc)) and let & \ R' -+ R' be a lifting of F compatible with o\
then the pullback of (M, $) to xjR! is an F-isocrystal on x/R' which we will
call simply the fiber of (M, $) at x, and the Newton polygon of (M, $) at x is
the Newton polygon of this fiber (it is independent of the choice of R'). Again,
we say that an F-isocrystal (M, $) is a unit-root F-isocrystal if its fiber at every
x —► X is.
1.10. For technical reasons we shall want to use a less rigid kind of structure
that that of an F-isocrystal. Let i2, X, cr, be as before and suppose that X/W
is a formally smooth lifting of a smooth fc-scheme X, endowed with a lifting
(j>: X —► X of the absolute Frobenius of X. An F-lattice on X/(i2, </>) is a locally
free R <g) Ox-module M endowed with a map
(1.10.1) $: <j)*M-+M
such that $ <g) Q is an isomorphism. If $ itself is an isomorphism, we say that
(M, $) is a unit-root F-lattice (note how this differs from the definition of a
unit-root F-isocrystal, but cf. nonetheless 2.4 below).
In the case X = Spf(VF), an F-lattice is just a finite free i2-module V endowed
with a (7-linear isogeny $, and its Newton polygon is by definition that of the
F-isocrystal (V<8>Q, $). The Hodge polygon is defined as follows: by elementary
divisors, there is a direct sum decomposition of i2-modules V = ®^ V% such that

118
RICHARD CREW
The Hodge numbers of (V, $) are the integers {rank^ V*}, and the Hodge polygon
is drawn in the same way as the Newton polygon, with one side of slope i and
horizontal projection rank# V% for each i. According to a basic result of Mazur,
the Newton polygon lies on or above the Hodge polygon, and both polygons
have the same endpoint (cf. [17, 1.4.1] for the case R = W, but the proof in the
general is no different).
In the case when X/W is an arbitrary noetherian formally smooth lifting of
a smooth fc-scheme X and (M, $) is an F-lattice on X/(i2,</>), one can define
the Newton and Hodge polygons of M at a perfect-field-valued point x —► X
by means of the "0-Teichmuller lifting" W(x) -+ X of x -► X (cf. [15, §1]); if
R'^a' are as in 1.9 then the pullback of (M, $) by W(x) —► X is an F-lattice
on W(x)/(R'^a')^ whose Newton and Hodge polygons will be called simply the
Newton and Hodge polygons of (M, $) at x, as these are independent of the
choice of R'^a'. The relation between the Hodge and Newton polygons and the
divisibility properties of $ is given by the following result, due, essentially, to
Katz:
1.11. PROPOSITION. Suppose (M,$) is an F-lattice on X/(R, (/)).
1.11.1. The Hodge slopes of (M, $) are > n (resp. < n) at every point of X
if and only if
$(M) Ç 7rnM (resp. $(M) D 7rnM).
1.11.2. (Slope estimate). The Newton slopes of (M, </>) are > X at every point
of X if and only if
^>n+r-1(M)Ç7r^nA}M,
where r = rank M and {s} = —[—s] (cf. [15, 1.4.3])
PROOF (SKETCH). When X = Spf(VF), the argument of [17, §2] applies with
trivial modifications (replace p by it). From the punctual case, one immediately
deduces a variant of 1.11 in which X is replaced by W(XpeTÎ). To deduce the
corresponding results for X, one can adopt the arguments of [17, 2.4, 2.4.1].
1.12. If (n) has divided powers and M can be given an integrable nilpotent
connection V commuting with F, then (M, V, $) is just an F-isocrystal on X/R.
Such a V need not exist of course, unless, as is well known, (M, $) is a unit-root
F-lattice; then (M, V, $) is a unit-root F-crystal and the connection extends to
a stratification (cf. the proof of Proposition 2.3 below).
1.13. From the definitions we have given, it is clear that for any X/k and any
dense open f/cl, there are natural forgetful functors:
(1.13.1) (isocrystals on X) —► (isocrystals on £/),
(1.13.2) (overconvergent isocrystals on X) —► (isocrystals on X).
It is reasonable to ask whether these are fully faithful. For (1.13.1) this is the
analogue of the assertion that a (classical, /-adic) local system is determined by
its restriction to a dense open subset; there is of course no classical analogue of
(1.13.2). Evidence for the full faithfulness of these functors is quite fragmentary

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
119
and we will only obtain results in the case when X is a curve, in which case the
restrictions of (1.13.1) and (1.13.2) to isocrystals of rank one are fully faithful
(4.10.2 and 4.10.3). For isocrystals of higher rank, Ogus has obtained partial
results (cf. [22, 4.3] and 4.4 below). If the fully faithful functor (3.1.2) is an
equivalence, it would then follow from Theorem 2.1 that the functor
overconvergent unit-root \ / unit-root \
F-isocrystals on X J I F-isocrystals on XJ
is fully faithful (X a curve).
2. p-adic representations and F-isocrystals.
2.0. Throughout this section X/k will be a smooth A:-scheme. We shall fix
a geometric point x —► X and let tti(X) = 7Ti(X,x). Let R,K be as in 1.0,
and let R°\Ka be the subrings of R,K fixed by a. Note that RG is finite over
VF(Fg),and so Ka is locally compact. Denote by RepR<T(7Ti(x)) the category of
representations of -k\ (X) in finite free i2CT-modules. We define
(2.0.1) RepK.(7ri(X)) = Rep^(7Ti(X)) ® Q.
Of course, since n(X) is compact and any two maximal compact subgroups of
GL(n, KG) are conjugate, RepK<T(7Ti(X)) is equivalent to the category of
continuous representations of tti (X) in finite-dimensional if ^-vector spaces, but we
will find the above description more convenient for our purposes.
The main result of this section is
2.1. THEOREM. Let X/k be a smooth k-scheme, and suppose Fq Ç A:. There
is natural equivalence of categories
(2.1.1) G: RepKa(7ri(X)) -^ (Unit-root F-isocrystals on X/K).
We shall deduce Theorem 2.1 from the following well-known theorem of Katz
[16, 4.1.1]:
2.2. THEOREM. Let X/k, X be as in 1.10. There is a natural equivalence
of categories
(2.2.1) G: Rep^TripO ^ {Unit-root F-lattices on X/R).
Before we proceed, let us recall the construction of the functor (2.2.1). Let
p: 7Ti(X) —► GL(V) be a representation of ni(X) on a finite free i2CT-module V,
and let X, <\> be as in 1.10. For n > 1 let Xn = X <8> Wn and let Gn be the image
of tti(X) in GL(V/pnV). The homomorphism tti(X) —► Gn classifies an étale
cover Yn —► X which has a unique étale lifting 7rn : yn —► Xn; the action of Gn
on Yn extends uniquely to yn, as does the action of <\> on Xn. The "opposite"
action makes Oyn into a right Gn-module and we let
(2.2.2) Mn = 7rn,0Vn ®Wn[Gn] V, M = UmMn.
n
By uniqueness, the action of Gn on Oyn commutes with 0, so that the map-
ing $ = 0 0 id gives compatible isomorphisms $: <t>*Mn -^ Mn, whence

120
RICHARD CREW
$: (j)*M -^ M. This (M, $) is G(p). We must now construct G(f) for any mor-
phism / in RepK„ (tti (X)), and we can easily reduce to the case when f:V-+V
is morphism in Rep#a(7Ti(X)). Let 7rn: yn —► Xn, Gn,Mn be as before and let
-k'u : y'n —► Xn, G^, Mn be the corresponding objects for V. Furthermore let Hn
be the image of ni(X) in GL((V ® V;)/pn) and let rn: 2^ —► Xn correspond to
7Ti(X) —► i/n. There are natural maps Hn —► Gn, i/n —► Gn inducing morphisms
Zn -* ^n? 2^ ~* yn? and it is easily checked that these give rise to isomorphisms
(2.2.3) Mn ~ rn*0Zn ®w[Hn] V, M'n ~ rn*0*n ®W[Hn] Ve'.
Via (2.2.3), the map id 0 / gives a system of maps Mn —► Mn, compatible in n,
whose inverse limit is G(/).
One concludes the proof of Theorem 2.2 by finding an inverse functor to G.
Roughly speaking, it is Ker(l - $).
2.2.4. REMARK. The construction we have given for G(f) can be used to
show that the functor (2.4.1) is compatible with the tensor product and internal
Hom in the source and target categories. Since the functor (2.2.1) is obtained
from (2.2.1) by, essentially, tensoring with Q, the same is true for (2.2.1).
2.3. PROPOSITION. Let (M, $) be a unit-root F-lattice on X/R. Then Man
can be endowed with a natural convergent connection commuting with $.
PROOF. As is well known, a stronger statement is actually true: Let A be
the formal completion of X x X along the image of the diagonal X <--+ X x X
and let pi,P2* A —► X be the natural projection maps; then there is a natural
isomorphism pi M -^ p\M verifying the cocycle condition and restricting to
the identity on the diagonal (i.e., in the terminology of Grothendieck [14], the
connection extends to a stratification), and commuting with $. In other words
the Taylor series (1.2.2) belongs to M ®Ox[[r]], and hence is not only convergent
in the open unit polydisk but bounded as well. Let An = A <g) Wn; in view
of the preceding discussion, it will be enough to produce a compatible system
of isomorphisms Pi{nn*Oyn) -^ P2(7rn*Oyn) commuting with the Gn-action and
with the (unique) lifting of <j> to yn. Define p*$>n to be the fiber product
PVèn ► Vn
z\n ► xn
The map yn —► Xn -^g An gives rise to formal thickenings, yn —► p*yn for which
the diagram
1 1
An < p£yn

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
121
is commutative. Since p*yn —► An is étale, there is a unique isomorphism pjyn
p^èn of Xn-schemes making
pVèn < Vn
(2.3.1) I ^^
An < p^n
commutative. By uniqueness, the isomorphism is compatible with the Gn-action
and with the lifting of <f> to yn, and satisfies the cocycle condition; the condition
on the diagonal follows by construction. Passing to structure sheaves gives an
isomorphism
whence PÎ(7rn*Oyn) -^ P2(fl"n*Oyn), since p{ is flat.
2.3.2. REMARK. There are many other ways of proving Proposition 2.3.
For example, the "constant" connection on Oyn induces a connection on Mn by
(2.2.2), and one then passes to the inverse limit. Convergence of this connection
is then proven by a well-known argument using the Frobenius structure, cf. Katz
[15].
2.4. It follows from Proposition 2.3 that there is a fully faithful functor
/ Unit-root F-lattices\ / Unit-root F-isocrystals\
2.4.1) , ®Q-+ , ■
V ; ^ on X/R J \ onX/K J
On the other hand, we may tensor (2.2.1) with Q to obtain an equivalence
(2.4.2) RepKt,(7Ti(X)) -^ (Unit-root F-lattices on X/R) <g> Q.
Then Theorem 2.1 will follow once we show that (2.4.1) is an equivalence
whenever X is affine (as the case of arbitrary smooth X follows by glueing). This
amounts to showing
2.5. PROPOSITION. Let X/k be a smooth affine k-scheme and (M, $) a
unit-root F-isocrystal on X/K, and let X/W,(j) be as in 1.10. Then there is a
unit-root F-lattice (M0,$o) on X/R such that (M, $) = (Mo,$o)an-
We will construct (M0, $o) in steps. Let X = Spf(A); then the first step is
2.5.1. LEMMA. There is an open ^-stable subscheme U Ç Spec A and
locally free sheaf M\ on U such that Spec A — U has codimension > 3 in Spec A,
and Mfn ~ (M|C/an), where Û is the p-adic completion ofU.
PROOF. We will identify M with its A-module of global sections. By
Proposition 1.3.2 M is a locally free A <g) Q-module, and since A is regular M can be
extended to a locally free sheaf M across a neighborhood V of the generic point
of the divisor X C Spec A. Writing j for the inclusion j : V —► Spec A we must
now show that j*M is locally free ona[/C Spec A satisfying the conditions of
Lemma 2.5.1. In fact it is enough to show that j*M is free on the local rings
of points of Spec A of codimension 2, but this is clear since A satisfies Serre's
condition 52. Since Spec A — U Ç X the ^stability of U is clear, as is the
isomorphism (Mi|C/)an ~ M|C/an.

122
RICHARD CREW
2.5.2. LEMMA. Let Z be a locally noetherian integral scheme satisfying
Serre's condition S2, and let K,L,M be sheaves on Z such that K,L Ç M and
L is locally free. Suppose that for each point x G Z of height < 1 we have
Kx Ç Lx, where KX,LX are the localizations of K,L at x. Then K Ç L.
PROOF. Since the question is local on X, we may assume that X = Spec A is
afBne, A noetherian, and that G is free. Identify if, L, M with their A-modules of
global sections and choose a basis {e^} of G. Since Kn C L^, where rj = Spec K is
the generic point of X, we can write any m G K as m = J2i ^i^i with rrii G if,
uniquely. Since Kx Ç Lx at every point of height one, we have rrii £ Ay for
every height one prime 9 of A. By condition 52 we then have rrii E A since A is
noetherian, whence m G K.
2.5.3. LEMMA. If Mi is as in Lemma 2.5.1, there are integers r, s such
that
?rrMi Ç $(Mi) Ç ttsMi for all n > 0.
PROOF. By the previous lemma, it is enough to check Lemma 2.5.3 generi-
cally and after localizing around every divisor; of course since $<g)Q is an
isomorphism, the special fiber of U is the only divisor capable of causing trouble. By
the faithful flatness of completions we are reduced to checking the containment
above U. Since Mi is coherent, there is an integer m such that Mi is
7rm«Instable, so (Mi|t/,7rm$) is an F-lattice on U of pure slope m. The containment
$n(Mi) Ç 7rsMi follows from the slope estimate 1.11.2 after dividing by a
suitable power of p. Then (Mx |C/, 7r~s$n) is an Fn-lattice of slope -s for all n > 0.
Since the Hodge polygon has integral breakpoints and lies beneath the Newton
polygon we see that the greatest Hodge slope of (Mi |C/, 7r~s$n) is bounded
independently of n, so by 1.11.1 there is an integer t such that 7r*Mi Ç 7r~s$n(Mi)
for all n. Then r = s + t.
Denote by j the inclusion j: U ^> Spec A. Then j*M\ is a coherent sheaf
on Spec A\ in fact A is Cohen-Macauley, Mi is locally free, and Spec A — U
has codimension > 2 in Spec A, so the assertion follows from SGA2 VIII 3.3.
In particular N = T(C/, Mi) = r(Spec A,j*Mi) is an A-module of finite type.
Furthermore iV(g)Q = r(Xan, M), so N <g) Q inherits a 0-linear isomorphism $
from M. The A-module
N'= ^Im$n: N -+N®Q
n>0
is clearly ^-stable, so we can define
N"= p| Im$: N'-+N',
n>l
M0 = the quasicoherent sheaf on Spec A such that r(M0) = N.
2.5.4. LEMMA. Mo is coherent and $ induces an isomorphism $ : </>*Mo ^
M0. Furthermore N ® Q = N" ® Q.

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
123
PROOF. Applying T(U, ) to the inclusions in Lemma 2.5.3 yields
7rrN ç $n(N) C ttsN
for all n > 0. This shows that N ® Q = N" ® Q and also that N" is an A-module
of finite type, since A is noetherian. In other words M0 is coherent, and the fact
that $ induces an isomorphism follows from the construction.
To conclude the proof of Proposition 2.5, it remains only to show that M0 is
locally free. This follows from the next lemma:
2.6. LEMMA. Let y be a reduced noetherian flat W-scheme, <j> a lifting of
Frobenius to y, and L a coherent sheaf of Oy-modules. If (j)*L ~ L, then L is
locally free.
PROOF. By the local criterion, it is enough to check that Lq = L®w k is flat
over F = y ®w A:, and by the valuative criterion, it suffices to check that for each
morphism f:S—+Y with S the spectrum of a complete discrete valuation ring,
the pullback f*Lo is flat over S. Since L0 is coherent we may write f*Lo = N(&T
where N is free and T is torsion. Then
F*f*L02iF*N®F*T,
while on the other hand
F*f*L0 ^ f*F*L0 c± f*L0 ^ N © T.
Equating torsion submodules, we find T ~ F*T, which is impossible unless
T = 0, since length F*T = q length T.
2.7. REMARK. A coherent module M on X endowed with a (convergent)
connection and a Frobenius structure $: (j)*M —► M need not be locally free
if $ is not an isomorphism, even if (M, F)an is a unit-root F-isocrystal. For
example, let X = Spf A, A = W{T}, M = (p,TP) C A, and let </>(T) = T*\ The
constant connection V(l) = 0 and "trivial" $ = <\> stabilize M, and (M, $)an is
the "trivial" unit-root F-isocrystal on Xan, but M is not locally free.
3. p-adic representations with finite local monodromy.
3.0. From now on we restrict our attention to the case when X/k is a smooth,
geometrically connected curve. If X <^-> X is an open immersion into a smooth
geometrically connected curve and D = X — X, we will say that a representation
P:tti(X)-GL(V)
has finite local monodromy around D if for each x G D, the image under p of
the inertia group at x is finite. The full subcategory of RepKa(7Ti(X))
consisting of such representations will be denoted RepKa (tti(X))d. If X is complete,
we will say simply that p has finite local monodromy, as this condition is
independent of the compactification, and the category of such p will be denoted
Rep^(7n(X))fin.

124
RICHARD CREW
The object of this section is to prove
3.1. THEOREM. Let X/k,X,D be as in 3.0 and suppose Fq C k. The
restriction of the equivalence (2.1.1) induces fully faithful functors
Unit-root F-isocrystals \
(3.1.1) GD: RepKa(7ri(X))D -+ I onX/K, overconvergent
(3.1.2) Gt:RepK.(7ri(X))
fin
around D J
(overconvergent unit-root]
F-isocrystals on X/K J
I do not know if (3.1.2) is an equivalence, although we shall prove in §4 that
the restriction of (3.1.2) to rank one objects (on both sides) is an equivalence
(Theorem 4.11).
3.1.3. REMARKS. The best known examples of overconvergent unit-root F-
isocrystals are of the form G(p), with p: tti(X) —► GL(t;) a finite order character
of tti(X) (not suprisingly, cf. Corollary 4.13). If Y —► X is the principal G-
cover corresponding to p and tt: y —► X is a lifting (y,X formally smooth over
R, of course), then we have G(p) ~ {n*Oy ®w[G] V)3,11. Overconvergence of
F-isocrystals of this type is often not difficult to check directly; this is done in
[2] for Artin-Schreier and Kummer covers. Of course it follows from Theorem
3.1 that 7r*y itself "extends" to an overconvergent F-isocrystal—this is a very
special case of a theorem of Berthelot on the overconvergence of higher direct
images [1, Theorem 5].
3.2. Before beginning the proof of Theorem 3.1, let us consider the situation:
y — y
(3.2.1) I [_n
X *-> X
Here X is an affine formal i2-scheme whose special fiber is a smooth affine curve,
X is dense and open in X, tt is finite and flat, and y = 7r_1(X). We also give
ourselves an action of a finite group G on y/X, and a lifting </>: X —► X of the
Frobenius of the special fiber of X. We ask to what extent do <\> and the G-action
extend to y. The answer is the following:
3.2.2. LEMMA. There is a diagram
y > y > y
«A/ ' «A/ ' «A/
whose squares are cartesian, such that Xan is a strict neighbrohood o/Xan in X
and such that the action of G on y/X extends to y/X, and <\>\ X —► X is covered
by a map <j>: y —► y. The composites of the horizontal maps are the horizontal
maps in (3.2.1).

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
125
an jan -an
—an
, y respec-
PROOF. Let A,B,A,B be the affinoid algebras of Xan,y
tively, so that (3.2.1) is obtained from the diagrams
B° ^ B°
î _î
A0 *-> A°
by taking formal spectra. It is enough to find an affinoid strict neighborhood
SpA of Xan in X such that, in the diagram
yan ► spê > gan
xa
SpA
xan
(in which Sp£ = (7ran)-1(Spi)) the actions of </>an on Xan and G on yan/Xan
extend to SpB. In fact the diagram in Lemma 3.2.2 is obtained from the one
above by passing to affinoid algebras, applying the functor ()•—►( )°, and taking
formal spectra.
To show the existence of such an A, we choose a set {fi} of affinoid generators
of B over A. Since each fi is integral over A, the elements (i>{fi),g{fi) € B
(for any g G G) are also integral over A. The existence of A then follows from
Theorem 1.7, and </>(/i), g(fi) define morphisms (j> : Sp B —► y , g : Sp i? —► y of
the sort we are looking for, except that we must show that g : Sp I? —► y factors
through a map SpB —► Sp 5 over Sp A. This follows from the commutativity of
the following diagram, in which the inside square is cartesian:
SpÈ
3.2.3. REMARK. Of course if 7T is étale, we may taxe JL — JL.
3.2.4. Suppose that we are now given an affine smooth curve X, a dense open
subset X C X, a lifting X C X of X C X of Frobenius, and, finally, an étale
cover Y —► X acted on by a finite group G. The cover Y —► X lifts uniquely to
a cover y —► X of formally smooth formal i2-schemes, on which G acts, and we
claim that the diagram
y
i
x ^-> x
can be completed to a diagram (3.2.1). In fact, Y has a smooth "partial" com-
pactification, i.e., there is a diagram
Y ^ Y
1 i
X *-> X

126
RICHARD CREW
and since X, Y are smooth curves, Y —► X is a finite flat local complete
intersection morphism. We can therefore lift Y —► X to a finite flat map n: y —► X,
and, by the uniqueness of liftings of Y —► X, we have 7r_1(X) ~ y.
3.3. We can now begin the proof of Theorem 3.1, and we obviously need only
to treat the case of GD. The method is a refinement of the proofs of Theorem
2.2 and Proposition 2.3, from which we shall borrow some notation: p is now
an object of RepK<T (ni{X))D, and V, Xn,yn, Gn are as in the proof of Theorem
2.2. Since Theorem 3.1 is a local assertion around D, we can assume that X
is affine, and then choose a formally smooth lifting X/R of X/k and a lifting
(j) of the Frobenius of X. Let Yn = yn <g) A: and let Yn <--+ Yn be a partial
compactification, so that we have a commutative diagram
Yn — Yn
I I
X *-> X
for any n > 0. The projection maps Yn —► ym extend to Yn —► ym; since p has
finite local monodromy around D, there is an integer N such that Yn -+YN is
étale for any n> N.
We can now apply the considerations of 3.2.4 with Y = Yjv, y = Fjv, G =
Gjv, to obtain y,y as in (3.2.1), with Gn acting on y. Since for any n > N
the morphism yn —► y^v is étale, it has a unique étale lifting yn —► y 0vy VFn.
Lemma 3.2.2 is now applicable, yielding a y/X on which Gn acts and to which <\>
extends. Let yn be the pullback of yn to y 0 VFn; since yn —► y 0 VFn is étale, so
is yn —► y 0 VFn, and hence the action of Gn on y 0 Wn is covered by an action
of Gn on yn. Denoting by 7rn the projection yn —► Xn, we can, as in Theorem
2.2, set
(3.3.1) Mn = 7rn*0Sn %[G] F
and
M = limMn.
n
Note that Man is a locally free sheaf on Xan; this is evident for the pullback of
Man to yan, and the assertion follows by descent. We now endow Man with a
Frobenius structure: let X, (j) be defined by the Cartesian diagram
ry 9 ry
(3-3.2) | |
ry 9 ry
and let M be the pullback of M to X by the left-hand vertical map in the above
diagram. Then Ô = <f> 0 id defines an isomorphism
Ô: 4>*M -^M.

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
127
By restriction we obtain a locally free sheaf j^M*n of ^O^r-modules endowed
with a Frobenius structure
*:F;(yt^raa)-*yt^raa-
3.4. If / is a morphism in Rep^a (tti (X))d, one can construct GD(f) along the
same lines as above, and we shall just sketch the construction. Let /: V —► V,
Hn, Gn, yn, Zn, Mn, M'n be as in the proof of Theorem 2.2, and set Y„ = yn ® A:,
Zn Yn, Zn are partial compactifications of Y^, Y^, ^n above X
and TV is sufficiently large, then Zn —► Zjv, Zn -+ Yn, Zn -+ Yn are all étale
for any n > N. As in 3.2.4, we can lift
Zn <"-+ Zn
1 1
X *-> X
to a diagram
Z -+ Z
1 1
X <--+ X
Again, by Lemma 3.2.2 there are X, Z =? Z x^ X such that the actions of i//v
and 0 extend to Z. For n > N, the étale map Zn —► Z;v lifts uniquely to a map
Zn —► Z <g) Wn, and if we let rn : Zn —► X <g) VFn be the puUback of the natural
projection Zn —► X(8)VFn, then Hn and <f> act on Zn/Xn. If Mn is as in (3.3.1) and
Mn is defined analogously, then one can check easily that we have isomorphisms
(3.4.1) Mn ~ rn*0in ®W[Hn] V, M'n ~ rn*0in ®W{Hn] V
extending (2.2.3). (The main point is that one can find a X as above which also
has the properties of Lemma 3.2.2.) Then /: V —► V induces a map Mn —► Mn,
whence a map M —► M' of O^-modules. The pullback of this to Xan is the
desired GD(f).
3.5. Let us now return to the situation of 3.3, and show that M can be
endowed with an overconvergent connection. Let Sp C be an affinoid subspace
of the tube of the diagonal ]^[xxx (n°tation of 3.3) and let Z = Spf C . The
intersection SpCC\]X[xxx is an affinoid SpC and we set Z = Spf C°. There
are natural projections pi,P2- Z —► X restricting to pi,p2- Z —► X, and define
PÎ%9*y by the cartesian diagrams
pjy ► y ry > y
1 1 1 lplXp2

128
RICHARD CREW
and similarly for p*y,g*y. The same arguments which gave (2.3.1) yield a
commutative diagram
PVé < <?*y
(3.5.1) I X^ I
z < P$
3.5.2. LEMMA. There is an affinoid strict neighborhood SpC of SpC in
SpC such that (3.5.1) extends to a commutative diagram
PVè —► <rv
i\i
z > PVè
in which Z = Spf C° and pi, q are the pullbacks of pi,q byZ-+Z.
PROOF. If, in (3.5.1), we pass to analytic spaces and then to amnoid algebras,
then the oblique isomorphism in (3.5.1) become the horizontal isomorphism in
o:B®aC ~ ) C®aB
B®âC C®â~B
(here A, A, B are as in the proof of Lemma 3.2.2, so that y = SpB, etc; and
we are representing pî,p£ by tensor products on the left and right). Let {fi} be
a set of generators of B/A. Each fi is integral over A, so the /j(g)l are integral
over C. Since 6 is C-linear, and therefore C-linear, the <5(/j(g)l) are integral
over C, so that each <5(/i<8>1) is an element of C®-^B integral over C®-^B.
Then Theorem 1.7 yields an amnoid strict neighborhood Sp C\ of Sp C such that
<H/i®l) € C\ ®-fcB. A symmetrical argument with <5_1, 1<8>fi yields a C<2 such
that 6~l(l<è>fi) G B0jC2. Then C = C\ (gJ^-C^ is the amnoid algebra of a strict
neighborhood of SpC and 6 induces an isomorphism 6: B®-^C -^ C®-^B.
This is the oblique arrow in Lemma 3.5.2; commutativity of the triangles is
straightforward and is left to the reader.
3.5.3. At the cost of shrinking Sp C, one can also arrange to have Sp C Ç
Xan x Xan, where X is the formal scheme whose exsistence is guaranteed by
Lemma 3.2.2. In this case we may replace the diagram in Lemma 3.5.2 by
PVè <— <ry
IXJ
Z < PVè
and one can check that the oblique isomorphism is compatible with the action
of G = Gn- There is also a compatibility with 0 whose formulation and proof
we leave to the reader (cf. (3.3.2)).

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
129
We can now construct an isomorphism p\M -^ p\M. From (3.3.1) and the
remarks in 3.5.3, we see, as in the proof of Proposition 2.3, that it is enough to
produce a compatible family of isomorphisms
(3.5.4) PÎ(*n*0Sn)^p5(7rn*0Sn),
where 7rn : yn —► X <g) Wn is as in 3.3. In fact, since yn —► y <g) Wn is étale, the
existence and compatibility of these isomorphisms follows from a brief study of
the commutative diagram
p2on
One must check that (3.5.4) is also compatible with the Gn-action and the
lifting of Frobenius, which is immediate, given 3.5.3 and the fact that yn —►
9 <8> Wn is étale. Note also that the induced isomorphism
p^Man^p*Man
satisfies the necessary cocycle condition, since it already does on the open subset
Zan of Zan.
It remains to check that the connection and the Frobenius structure are
compatible with the maps GD(f). This uses the construction (3.4.1) and is tedious
but straightforward. Finally, the functor GD is necessarily fully faithful, since
its composition with the forgetful functor
(overconvergent F-isocrystals on X/K) —► (F-isocrystals on X/K)
is fully faithful, by (2.1.1).
3.6. REMARK. For the same reasons as in 2.2.4, the functor GD is
compatible with tensor products and internal Horn.
4. Isocrystals of rank one.
4.1. The main results in this section, Theorems 4.10 and 4.12, give partial
answers to the questions raised in 1.13. We will first prove local versions of
Theorem 4.10 (Propositions 4.4 and 4.5), and then explain how to globalize in
4.6-4.9.
We begin with a review of formal Laurent series. For the time being, R is
any complete discrete valuation ring with residue field A: of characteristic p and
quotient field K.
PVèn +
Z®Wn < pm^Wn) +

130
RICHARD CREW
With T a formal indeterminate, we define
K[[T]]b = R[[T}\ ®R K,
K((T)) = the ring of formal Laurent series ^ anTn
nez
with an G jPC, an bounded, and an —► 0 as n —► — oo,
(4.1.1) K((T))i = the subring of K((T)) consisting of series ^ anTn
nez
such that v(an) > a\n\ + /3 for some positive constants
a, /3 and all n < 0.
Since the valuation v on i2 is discrete, K((T)) and if ((T))"*" are actually fields.
We can of course think of if [[T]]6 as the ring of bounded functions on the open
unit disk, and of K((T)Y as the field of Laurent series which converge and are
bounded on some annulus Ur = {T \ r < \T\ < 1} (depending on the series). For
u G K((T)), the Gauss norm |t*|Gauss is defined as
(4.1.2) MGauss = max|an|, where u = Y^anTn.
n ' ^
n
If u e K((T)Y then we have
(4.1.3) Mcauss = hm |u|^r
r—>1
and the function log|u|[/r is a decreasing piecewise-linear function of logr.
Finally if u e K((T)) (resp. u e K((T)Y) then u' e K{(T)) (resp. u' e K({T)Y)
and li^Gauss ^ l^lGauss, and if u G K((T)Y has only finitely many positive
powers of T in its series expansion, and no term in T_1, then there is a v G K((T)Y
such that v' = u.
We now prove a formal version of Corollary 1.7.1, using a method of Dwork-
Robba [7, 13]:
4.2. PROPOSITION. K((T)Y is algebraically closed in K((T)).
PROOF. Suppose u G K((T)) is a root of F(u) = 0, where F is a monic
irreducible polynomial with coefficients in K((T)Y. Given any uq sufficiently
close to u, Newton's method (see, for example, [7, Lemma 1]) will produce a
sequence {un}n>o converging to u in the Gauss norm; more precisely, if
(4-2.D \F(u0)/F'(u0)\anaa < 1
and un is defined by
un+i = un - F(un)/F'(un),
then un —► u as n —► oo. As K((T)Y is dense in K((T)) we can choose uq G
K((T)Y such that (4.2.1) holds; then since
\F(u0)/Ff(uo)2\ur < 1
for r sufficiently close to one, we see that {un} converges to a Laurent series
convergent on f/r, whence u G K((T)Y.

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS 131
For any u G if ((T)), we define the residue Res(u) of u to be
Res(u) = a_i if u = }^anTn.
n
4.3. LEMMA. If u = Y,nanTn e K((T)), then Res(uf/u) is the smallest
integer n such that \an\ = \u\g&-
PROOF. If u has only finitely many terms of negative degree, then this follows
immediately from the Weierstrass factorization theorem. If u is sufficiently close
to 1, then u = exp(v) for some v G K((T)) and so Res(u'/u) = Res(t/) = 0. In
general, any u can be written u = Uiu2, where Ui has finitely many negative
degree terms and u2 is close to 1. Since Res(u'/u) = Res(u/1/^i) + Res^/i^),
the lemma follows.
The next theorem is the rank one case of a theorem of Ogus (cf. [22, 4.3]).
The method of proof is due to Robba.
4.4. Proposition. Ifu eK{{T)) andu'/ue K[[T)]b, then ue K[[T]]b.
PROOF. Let Z be a new formal variable; we will work in K((Z))((T)). Let
g = g(T) = u'(T)/u(T) e K[[T}}b and for n > 0 define gn{T) by
gn = u^/n\u.
We have \gn\ = \u^/nl\/\u\ < 1 for n > 0 and
1 / /
whence gn E K[[T]]b for all n. Now since
UIZ) ~ 2^ 9n{Z)T '
v J n>0
we see that v(T) = J2n>o 9n{0)Tn e K[[T]]b is a solution of
V1 — gv = 0.
Since u satisfies the same equation, u = (const) x v, whence u G if [[T]]6.
More generally, Ogus has shown (unpublished, but cf. [21]) that if a system of
linear differential equations with coefficients in if [[T]]6 has a full set of solutions
in K((T)), then it has a full set of solutions in K\[T\\b.
The next proposition is a formal version of a result of Dwork-Robba ([21], cf.
the remark after Corollary 4.10.1).
4.5. Proposition. Ifu eK((T)) andu'/ueK{{T)Y, then u e K((T))î.
PROOF. By Proposition 4.2 it is enough to show that uN G K((T)Y for some
N. Write
u'/u = g = ^anTneK((T)y.
n
Now (cf. the last paragraph before Proposition 4.2)
Tn+l
£a"^e*((T))t'
n< —1

132
RICHARD CREW
so that for m sufficiently large we have
h = exp
and
Ki>"^Hm)'
(u" h)'
= Pm J2 a"T"-
Now by Lemma 4.3, r = pma_i = Res((upmh)'/upmh)) G Z, so we may write
V ' n>0
whence uprnhT~r G K[[T}]b by Proposition 4.4. Thus upm G K{{T))i.
4.6. To apply these theorems in a geometric situation we will use a simple
localization construction. Let X/k be a smooth curve over A:, where A: is a perfect
field of characteristic p and R, K are as in 1.0. We let a: be a closed point of X
and suppose that there is a formally smooth lifting X/R of X/k satisfying the
following conditions:
4.6.1. There is a section T G T(Ox) which reduces to a local parameter of Ox
at x and is invertible on U = X — x.
4.6.2. There is a derivation 6 of Ox such that 6{T) = 1.
(Of course, for any x G X there is an open neighborhood of x for which such
lifting exists.) Since X/R is formally smooth, the completion of Ox at x is
isomorphic to #[[7]], whence an injective homomorphism
(4.6.3) i:T(Ox*n)-+K[[T]}b.
Let U = X — x,U = X — x, and denote by j any of the natural inclusions U —► X,
U-+X, ]£/[-+Xan. Since
T(0]u{)° ^ T(j*Ou) ~ limOXOxHT-1]) ® Wn,
n
it is clear that (4.6.3) extends to an injective homomorphism
(4.6.4) i:T{Q]u{)^K{{T)).
By construction, we have for any u G r(0][/[),
(4.6.5) *(*(")) = *W>
(4.6.6) K^lcauss = |ti||tf|.
4.6.7. REMARK. Suppose that £nanrn € i^((^))f has the property
that on -4 0 as n h +oo. Then if we let T be the section T G T(Ox), then
X)nanTn defines an analytic function g on a strict neighborhood of ]£/[, such
that 2(0) = J2n anTn (in fact, a strict neighborhood of the form \T\ > r for some
r <1).

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
133
4.7. LEMMA. The homomorphism i satisfies
4.7.1. i(r(0]x[)) = (lmi)nK[[T]]b,
4.7.2. t(r(yto]x[)) = (imt) n #((r))t.
PROOF. 4.7.1 is an immediate consequence of the isomorphism r(0]j/[)° ~
Furthermore 4.7.2 is a consequence of 4.7.1. In fact, suppose u G r(0]j/[) and
i(a) = J2*nTnGK((T))l
n
Then the series Xm<o °"^n actually defines an analytic function on some strict
neighborhood of ]U[ in }X[. Since, by 4.7.1, the function
n<0
extends to a function on all of ]X[, we conclude that u extends to the strict
neighborhood |T| > r of ]£/[, sowE T(pO]u{).
4.8. LEMMA. IfX/R,X,U,x,T are as in 4.6.1, 4.6.2, set B~ = {z e Xan|
\T(z)\ < 1} and Ur = {z e Xan|r < \T(z)\ < 1}. If N is any affinoid supspace
o/Xan such that N fl B~ is not contained in any affinoid subspace of B~, then
Ur Ç N n -B~ /or r sufficiently close to one.
Note that the hypothesis on TV can be rephrased as "TV fl B is not contained
in any closed disk {z\ \T(z)\ < r} = Dr of radius less than one."
PROOF OF LEMMA 4.8. Write X = Spf(A); then by a theorem of Grauert
and Gerritzen [8, 7.3.5, Corollary 3] N has a finite cover by "rational" affinoid
subspaces, i.e., affinoids of the form W = Sp((A<g) Q)(foc/g)aei)i f°r some set of
fa, 9 € A ® Q. Since N fl B~ £ Dr for any r < 1, we must have W fl B~ C Dr
for some W of the above sort, which we fix henceforth. As g G A ® Q, i(g)
is a power series with bounded coefficients, and so the Laurent series i{fa/g)
all necessarily converge on some annulus Ur. Thus all the fa/g define analytic
functions on f/r, and N fl B~ is the locus of \fa/g\ < 1. But by the remarks
following (4.1.3), we have that for r close to one this locus is either empty (i.e.,
NDB- CDr) or all of Ur.
4.8.1. REMARKS. The argument also shows that the map i: A ® Q —►
K[[T}}b extends uniquely to a map i: T(N,0N) -+ K((T)Y (if Nn]U[^]U[ this
case is not covered by (4.6.4)). In fact the above argument and 4.1.3 show that
K(/a/0)|Gauss < 1? which guarantees the existence of the extension. Uniqueness
is clear: i(fa/g) "calculates" fa/g on Ur.
4.9. LEMMA. Let X/R,X,U,x be as in 4.6.1 and 4.6.2 and suppose M is a
vector bundle on a strict neighborhood V of]U[ in Xan. Then there is a strict
neighborhood V Ç V of]U[ such that M\V can be extended to a vector bundle
M' on all o/Xan, and for a sufficiently small neighborhood W of x in X, M'\]W[
is free. IfV = Xan, we can take V = V = Xan, M' = M.

134
RICHARD CREW
PROOF. Since X is a curve we can, after shrinking V, assume that V is
affinoid. Then V is covered by finitely many affinoids W{ on which M is free,
and after further shrinking we can, by Lemma 4.8, assume that W{ n B~ = Ur
for some r. It is then easy to extend M to all of Xan; for example, if r < r' < 1
then a transition matrix for W{0 fl Dr< can be chosen at will, and this determines
the transition matrices for the other W% fl Dri. Suppose now that M is a vector
bundle on Xan; then by Lemma 2.5.1 there is a locally free Ox-module M such
that Man ~ M. But M will be free on a sufficiently small open neighborhood W
of x in X, and if W = W <g) A:, then M is free on }W[.
4.10. THEOREM. Let X/k be a smooth curve, U Ç X a dense open subset,
and D = X — U. The restriction functors
(Rank one isocrystals on U\
—► {Rank one isocrystals on U),
overconvergent around D J
(4.10.2) {Rank one isocrystals on X) —► {Rank one isocrystals on U)
are fully faithful
PROOF. We shall prove only that (4.10.1) is fully faithful, as the argument
for (4.10.2) is similar. By the local nature of these categories we may assume
that D is a single closed point. All categories involved have an internal Horn, so
we must show that a horizontal section on U of an overconvergent isocrystal of
rank one is overconvergent. By Lemma 4.9 we can assume that M is represented
by a trivial vector bundle on a strict neighborhood V of ]U[ in Xan. We can
also assume that the hypotheses 4.6.1-4.6.2 hold, in which case the horizontal
section can be identified with a section u G r(]f/[,0%**) such that 8{u)/u G
r(V,Ox»")- By (4.6.5), Lemma 4.7, and Proposition 4.5 we must then have
u G r(V, Ox»n). Similarly, in the case at (4.10.2) one appeals to Proposition 4.4
in place of Proposition 4.5.
4.10.3. C OROLLARY. The restriction functor
(Rank one overconvergent\ (Rank one isocrystals\
isocrystals on U J \ on U I
is fully faithful.
When U = A1 this was first proven by Dwork and Robba [21, 4.5] using
entirely different methods.
The key point in our last theorem is
4.11. PROPOSITION. Let X/k,U, and D be as in Theorem 4.10, and let
X/R be a formally smooth lifting of X/k, and <\>\ X —► X a lifting of Fq.
Suppose that M is a locally free sheaf of rank one on a strict neighborhood V of
]U[ in Xan endowed with a connection V: M —► M ® &v/k ana* ^at M|]f/[
has a Frobenius structure $ commuting with V {cf. (1.8.2)). Then some
tensor power (M, V,^)®^ can be extended to all o/Xan. // (M, V,$)|]£/[ defines

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS 135
a (convergent) F-isocrystal on U/K, then (M, V,^)®^ defines a (convergent)
F-isocrystal on X/K.
PROOF. The assertion being local around the points of D, we can assume
D is a single point and shrink X at will. In particular, we can assume 4.6 and
Lemma 4.9 are applicable, and that M is free on V with basis e. Write
Ve = e<g)r?, r? e YtfO]x[)dT,
$e = fe, feT(0]u{).
We proceed as in the proof of Proposition 4.5. Write
t'(»?) = 5>»TnÇe*((r))tdr.
n
By Remark 4.6.7 there is a g G r^Oixi) such that
i(g) = exp(pmJ2rin^-)^K((T^
V n<0 /
for m sufficiently large. Thus (M, $)®pm has a basis with respect to which the
induced connection has no terms in TndT/T for n < 0 (i.e., has a "regular
singular point"). Writing anew the connection matrix on (M,$)®pm as
Ve = r?e, i(V) = £ r)nTn —,
n>0
we must now show that r?o £ Q ( of course, if there were a horizontal section in
K((T)), we would have rj0 G Z). From (1.8.2) we get
Since |0(T) - T9|Ga < 1, we have Res(d(j)(T)/(j)(T)) = q, by Lemma 4.3.
Comparing residues and applying Lemma 4.3 yields
which has the unique solution
ft ~
Vo = 7 € Q
q-1
Then M®^^-1) has a basis for which the induced connection extends across
]X[ (by Remark 4.8.1). The induced Frobenius structure extends across ]X[
by Proposition 4.4 and 4.7.1. The last part of the proposition follows from
Proposition 1.3.1.
We can now resume our study of representations with finite local monodromy:
4.12. THEOREM. Let X/k be a smooth curve and suppose that Fq Ç A:. The
functor
G: RepKairi(X) —► (unit-root F-isocrystals on X/K)

136
RICHARD CREW
induces, when restricted to the rank one representations of-K\(X) with finite local
monodromy, an equivalence of categories
(Rank one objects of\ ^ ( overconvergent rank one
RepK*(7Ti(X)) n J y unit-root F-isocrystals on X/K
PROOF. By Theorem 3.1, we only need to show that (4.12.1) is essentially
surjective. Let (M, $) be an overconvergent unit-root F-crystal of rank one on
X and let X D X be a smooth compactification of X. Then by Proposition
4.11, some tensor power (M, $)®N extends to an F-crystal on X. Since G is
compatible with tensor products, we have that if G(p) = (M, $), then G(p®N) =
(M,$)®N; we thus see that p®N extends to a representation of ni(X). In
particular, the images under p of the inertia groups of the points of X — X are
killed by TV, and are therefore finite.
4.13. COROLLARY. Suppose k is a finite field and that X/k is a smooth
curve. If (M, $) is a unit-root overconvergent F-isocrystal on X/K of rank one,
then some tensor power of (M, $) is geometrically constant.
PROOF. Extending k if necessary, we may assume X has a smooth
compactification X defined over k. Let p: tti(X) —► (Ka)x be such that G(p) = (M, $).
Then some power of p extends to tt\ (X). But the hypothesis on A: implies that the
image of 7Ti(X<g) fcalg) in 7Ti(X)ab is finite (this is just the well-known finiteness
theorem of class field theory). Thus a further power of p is trivial on tt\ (X<g)fcalg),
i.e., factors through 7Ti(Spec(A:)). The corresponding powers of (M,p$) are then
constant.
One can show, using results of Katz and Lang [18] that Corollary 4.13 is
still true if one merely supposes that A: is the perfection of an absolutely finitely
generated field.
4.14. COROLLARY. Let X/k, £/,X/i2,F, (M,V,$) be as in Proposition 4.11
and suppose that (M, V, $)|]t/[ defines a convergent unit-root F-isocrystal on
U/K. Then (M, V,$) defines a overconvergent F-isocrystal on (X/K,D).
In other words, a unit-root F-isocrystal of rank one on a smooth curve is
overconvergent if and only if its connection matrix is.
PROOF. By Proposition 4.11, (M, V,$)|]£/[~ G(p) for some p with finite
local monodromy around D, and so the conclusion follows from Theorem 4.12.
4.15. REMARK. Let X/k be an elliptic modular curve of level three or
more with its supersingular points removed, and let / : E —► X be the universal
ordinary elliptic curve on X. Then the étale quotient of the p-adic Tate module
is a twisted form of Zp over X, whence a representation p: ni(X) —► Z£. It
is known that p does not have finite local monodromy; it is in fact a theorem
of Igusa (cf. [16, 4.3]) that the inertia group at each supersingular point on
the complete modular curve surjects onto Z£. Hence the rank one unit-root
F-crystal G(p) = (M, $) corresponding to p is not overconvergent. When E —►
X is the Legendre family of elliptic curves, this had been shown by Dwork

F-ISOCRYSTALS AND p-ADIC REPRESENTATIONS
137
(unpublished). On the other hand, it follows from work of Berthelot [3; 1,
Theorem 5] that the rank two F-crystal R1 fT[g*(E/X) is overconvergent, and it
is well known that (M,F) is in fact the unit-root subcrystal of R1 fTig*{X/E).
We see, then, that subquotient of an overconvergent F-crystal is not in general
overconvergent.
4.16. We conclude with some questions and speculation.
4.16.1. Let p: ni(X) —► (Ka)x be a character with finite local monodromy
and let G^(p) ~ (M, $) be the corresponding F-crystal. Fix a smooth compact-
ification X C X. As p has finite local monodromy, the higher inertia groups
at each point of X — X are finite, and one defines in the usual way an Artin
conductor ax for each x G X — X. Is it possible to calculate ax in terms of the
behavior of (M, $) at xl One obvious possibility is that ax is equal to the local
index of M at x\ this is suggested by the resemblance between the Hurwitz genus
formula and the formula for the index of M as a sum of local terms (cf. [23]).
4.16.2. Suppose now that, in the notation of 4.1, M is a finite-dimensional
K((T))^-vector space endowed with a convergent connection V (we leave it to
the reader to determine the sense of this) and a Frobenius structure $. Does
(M, V, $) satisfy any kind of analogue of the local monodromy theorem? To
formulate one such analogue, note first that any finite extension k[[T}} <--+ k[[X]]
induces in a natural way a finite extension K((T)Y <--+ K((X)Y. The analogue
of the local monodromy theorem would then say that for any (M, V, $) as above
(i.e., for any "overconvergent F-isocrystal on fc((T))") there is a finite extension
k[[T]] ^ k[[X]] such that the pullback F-isocrystal (M, V,$) <g> K((X)Y has a
filtration by sub-F-isocrystals whose quotients extend across if [[X]]6. That this
is true for M of rank one is the essential content of Theorem 4.12, but at the
moment there is no serious evidence for such an assertion for M of higher rank.
On the other hand it is not too difficult to prove that if this local monodromy
theorem is true, then the functor of Theorem 3.1 is an equivalence of categories.
References
1. P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p,
Journées d'Analyse p-adique (Luminy, 1982), Mém. Soc. Math. France, no. 23, suppl. to Bull.
Soc. Math France 114 (1986), 7-32.
2. , Cohomologie rigide et théorie de Dwork la cas des sommes exponentielles,
Cohomologie p-adique, Astérisque, no. 119-120, Soc. Math. France, Paris, 1984, pp. 17-49.
3. , Cohomologie rigide et cohomologie rigide à support propere, (to appear).
4. P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné cristallin, Lecture Notes
in Math, vol. 930, Springer-Verlag, 1982.
5. P. Berthelot and A. Ogus, Notes on crystalline cohomology, Math. Notes, no. 21,
Princeton Univ. Press, Princeton, N.J., 1978.
6. , F-isocrystals on de Rham cohomology, Invent. Math. 72 (1983), 159-199.
7. S. Bosch, B. Dwork, and P. Robba, Un théorème de prolongement pour des fonctions
analytiques, Math. Ann. 252 (1980), 165-173.
8. S. Bosch, U. Gûntzer, and R. Remmert, Non-Archimedean analysis, Springer-Verlag,
1984.
9. R. Crew, L-functions of p-adic characters and geometric Iwasawa theory, Invent. Math,
(to appear).

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RICHARD CREW
10. P. Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math 52 (1980),
137-252.
11. B. Dwork, The Up-operator of Atkin on modular functions of level 2 with growth conditions,
Modular Functions of One Variable. Ill, Lecture Notes in Math., vol. 350, Springer-Verlag,
1973, pp. 57-67.
12. , On Hecke polynomials, Invent. Math. 12 (1971), 249-256.
13. B. Dwork and P. Robba, On ordinary linear p-adic differential equations, Trans. Amer.
Math. Soc. 231 (1977), 1-46.
14. A. Grothendieck, Crystals on the de Rham cohomology of schemes (notes by J. Coates
and O. Jussila), Dix Exposées sur la Cohomologie des Schémas, North Holland, 1968, pp. 306-
358.
15. N. Katz, Travaux de Dwork, Sém. Bourbaki 409, Lecture Notes in Math, vol. 317,
Springer-Verlag, 1973.
16. , P-adic properties of modular schemes and modular forms, Modular Functions of
One Variable. Ill, Lecture Notes in Math, vol. 350, Springer-Verlag, 1973, pp. 69-190.
17. , Slope filtration of F-crystals, Journées de Géométrie Algébrique de Rennes,
Astérisque, no. 63, Soc. Math. France, Paris, 1979, pp. 113-164.
18. N. Katz and S. Lang, Finiteness theorems in geometric classified theory, Enseign. Math.
(2) 27 (1981), 285-319.
19.Yu. Manin, The theory of commutative formal groups over fields of finite characteristic,
Russian Math. Surveys 18 (1963), 1-83.
20. A. Ogus, F-isocrystals and de Rham cohomology. II: Convergent isocrystals, Duke Math
J. 51 (1984), 765-850.
21. P. Robba, Caractérisation des dérivées logarithmiques, Groupe d'Étude d'Analyse
Ultramétrique, 2nd année (1974/75), no. 12, Secrétariat Mathématique, Paris, 1975.
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393-418.
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119-120, Soc. Math. France, Paris, 1984, pp. 191-266.
Harvard University

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Foliations and Inseparable Morphisms
TORSTEN EKEDAHL
The major new phenomenon in positive characteristic algebraic geometry, as
opposed to characteristic zero algebraic geometry, is the appearance of
inseparable morphisms. A geometer who insists on considering positive characteristic
will often, in the course of an argument, be faced with the possibility of a
mapping being inseparable and will then have to cook up a nonclassical argument
to take care of this possibility. Experience shows us that there is a good chance
of succeeding in such an attempt, the best example of this being the work of
Bombieri and Mumford which extends the Enriques classification of surfaces to
positive characteristic.
There seem, in fact, to be two lessons experience has to teach us. First
one should try to understand exactly where inseparability makes a classical
argument break down so as to be able to exploit the existence of one specific
inseparable morphism. A good example of this technique is in the proof by
Rudakov-Shafarevich of the nonexistence of global vector fields on a if 3-surface,
where from a nonzero global vector field on a if 3-surface a purely inseparable
rational map from the projective plane to the given surface is constructed. Second,
the goal towards which one should strive should be to obtain results in positive
characteristic good enough to enable one to obtain the corresponding result in
characteristic zero by specializing to almost all finite characteristics. There are
sometimes a priori reasons why this should be possible. Consider, for instance,
minimal surfaces of general type with fixed K2. It is a fact, and will be proved
elsewhere, that these form a bounded family. Hence, if one considers any finite
set of numerical invariants of such surfaces which are constructible as functions
on parameter spaces and one considers a set of values of these invariants which
do not occur in characteristic zero, it follows that it can only occur in a finite
number of finite characteristics. We will later give a result in any characteristic
which will imply, in characteristic zero, the result of Castelnuovo that c<i > 0,
thus giving an example of this philosophy. It is a challenging problem to obtain
similar results pertaining to the Miyaoka-Yau inequality.
1980 Mathematics Subject Classification (1985 Revision). Primary 14J99; Secondary 14L30.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
139

140
TORSTEN EKEDAHL
The purpose of this article is to give a few examples of how inseparable mor-
phisms can be analyzed. The point of view that the existence of inseparable
morphisms should be traced back to the fact that, contrary to the case of
characteristic zero, a foliation is not determined by its first-order part will be all-
pervasive.
1. Let us recall (cf. [Ga 1]) that a groupoid scheme consists of two schemes X
and y and morphisms Si,s2:X —► F, t:Y —► X, p:XX(Sl)S2) X —► X audi: X —►
X such that for any scheme Z, Hom(Z, X), resp. Hom(Z, F), becomes the set of
arrows, resp. the set of objects, of a groupoid with $i*, s2*, and £*, resp. p*, the
source, sink, and identity arrow, resp. product mapping. (Thus one requires the
usual relations between the different morphisms.) When (s\,S2):XxY -+YxY
is a monomorphism we get, by definition, a schematic equivalence relation. This
is equivalent to Im(Hom(Z, X) l-^2 Hom(Z, Y)) being an equivalence relation
for all Z.
We will be interested in infinitesimal equivalence relations only. By
definition this means that S\ (and so 82) is an affine homeomorphism and that
Ker(£:si*<^x —► &y) is nilpotent. It is then easy to see that an infinitesimal
groupoid scheme is an equivalence relation precisely when (si,S2):X —► Y x Y
is a closed immersion. We will at two isolated occasions consider also formal
infinitesimal groupoid schemes. This will mean that X is a formal scheme rather
than a scheme, S{ is still an affine homeomorphism, and Ker(t: S\*@x —► &y) is
an ideal of definition for X.
Let us further note that if Y —► Z is a faithfully flat morphism, then there is
a schematic equivalence relation with Y as the scheme of objects and Y xz Y —►
y x y as the scheme of arrows (cf. [Ga 1]) such that Z is the coequalizer of
y Xz Y =t y or, as we will say, the quotient of the groupoid scheme. Conversely,
if (X, y, 81, 82, t,p) is a groupoid scheme such that s\ is finite and flat, then the
quotient Z exists [Ga 1]. If (X, F, etc.) is also an equivalence relation, then
X = Y xz Y [Ga 1].
2. We will now suppose that Y is smooth over a perfect field A and all our
schemes and morphisms will be over 4'. When char// = 0 the situation is
very simple. Recall [111, VIII:1.1.5] that a Lie algebra over Y// consists of an
<^x-module L with a //-Lie algebra structure together with an ^x-morphism
0: L —► Ty,j (the tangent sheaf) such that
[/,a0] = 0(/)(a)0 + a[/,0], oG^x, f,geL.
PROPOSITION 2.1. (i) [111, VII : 1.3.6]. There is a 1-1 correspondence
between locally free Lie algebras over Yj4 and formal infinitesimal groupoid
schemes with Y as a scheme of objects. Under this correspondence equivalence
relations correspond to Lie algebras over Yj4 for which 6 embeds L as a sub-
bundle OfTy,^.
(ii) [Mi; Corollary 6.4]. For any formal infinitesimal equivalence relation and
any closed point of Y there are formal coordinates t\,... ,£p at y (i.e., elements

FOLIATIONS AND INSEPARABLE MORPHISMS
141
of the maximal ideal of the completion of Y at the closed point y and forming a
basis modulo the square of the maximal ideal) such that the infinitesimal
equivalence relation in question is formally isomorphic at y to the equivalence relation
associated to the formal morphism
(h,... ,£n) —► (h,.. • ,^r)
for some r <n.
From now on we will suppose that char// = p > 0. The largest formal
equivalence relation on Y is clearly (y, X,...), where X is the formal completion
oiYx/Y along the diagonal. Unlike in characteristic zero this equivalence
relation is the union of finite infinitesimal equivalence relations. In fact, X =
\JnXn, where Xn := Y xY(n) Y and where Y^ is the pullback of Y along
4 -+/:ih xv~n and Y —► Y^ is the pnth power map. We will say that an
infinitesimal groupoid scheme (F,-X",...) is of height n of X —► Y x^ Y factors
through Xn. Recall that a groupoid scheme is said to be finite or flat if si (and
hence 82) is finite or flat. Descent theory now gives
PROPOSITION 2.2 [Gal, Theorem 4.1]. The functor (Y -+ Z) -+
(Y,Yxz y,...) gives a bijective correspondence (on isomorphism classes) between
finite and flat morphisms Y —► Z for which there are morphisms Z —► y(n) such
that the composite Y —► Z —► Y^ is the pnth power map and closed flat sub-
groupoid schemes of (F, Xn,...), i.e., finite, flat equivalence relations of height
n on y.
It is often convenient to describe infinitesimal, finite, flat groupoid schemes
in terms of rings of differential operators. This is done as follows. Let T =
(y, X,...) be a finite, flat, infinitesimal groupoid scheme. As all morphisms
involved are affine, we can describe T by a sheaf of rings s/ on Y together
with ring homomorphisms Si,s2:<^y ~* ^-> t\S/ —> @y, p:S& —► s/ ®&Y stf
and i'.stf -+ stf fulfilling certain relations. As T is infinitesimal the existence
of i is automatic (cf. [Ga, 2.7]). As sf considered as an <^y-module through
s 1 is locally free of finite rank, we lose nothing by taking the dual and hence
considering j/*:= %cwi&Y(srf\@y). Then J/* has a coproduct a dual to the
product of a/. It also has a product; if fa: stf —► @Y and fa: s/ —► <fy, then
fa<h:Sf-*S*®*y S/'lA£PS/®*y &y=rf -+&y.
We also have a ring homomorphism from s/ to ^^fc(^y), the ring of differential
operators (cf. [DG, 16.8]) on F, given by <\> \-> (@y ^ srf -t ffY). Under this
morphism the coproduct corresponds to a(fa(ai <8>a2) = <t>{o>\ 802)? Q>\,Q>2 € $y.
It is easy to reconstruct from these data on srf* the original data on sf.
In the case of height 1 groupoid schemes we can as with group schemes go
even further and get the most obvious analogue of 2.1 (i).
PROPOSITION 2.3. There is a functorial 1-1 correspondence between locally
free coherent p-Lie algebras over Y'j4 (with the obvious definition of that notion)

142
TORSTEN EKEDAHL
and height 1 finite, flat groupoid schemes (X, F,...) with Y as a scheme of objects
such that J^/J^2, where S := Ker(t: Su<fx —>$y), is locally free.
The proof is just a slight modification of the proof of the corresponding
statement for group schemes (cf. [Mu, §14]).
In case the groupoid scheme is an equivalence relation which is equivalent to
the Lie algebra being a subbundle of Der^(^fy), we can take the quotient Y —► Z
and then X = Y xz Y so that the J? JJ7"1 of the proposition is simply fiyyz.
As Y —► Z is flat, Z is smooth, and conversely, if Y —► Z is a finite height 1
morphism between smooth //-schemes, then by a result of [K-N], Qy ,z is locally
free, so we get
PROPOSITION 2.4. There is a 1-1 correspondence between finite, flat height
1 morphisms Y —► Z and subbundles L Ç Ty,k stable under Lie brackets and pth
powers.
As an illustration of the notions introduced, we will give a proof of the theorem
of [K-N] in the case when Y —► Z has degree p: It is clear that Tyiz Ç Ty,^ is
a saturated submodule, i.e., Ty,^/Ty,z is torsion free. As rkTy ,z — 1, the fact
that it is a saturated submodule of a free module implies that Ty ,z is a line-
bundle. Therefore Proposition 2.3 provides us with a groupoid scheme (VF, F,...)
which maps to (F Xz F, F,...). At the points where Ty ,z is a subbundle of Ty,^
this is an isomorphism again by (2.3). Hence it is an isomorphism outside of a
closed set of codimension at least 2, and as W and F Xz Y are both finite and
flat over F, it is an isomorphism everywhere. As the J^/J^2 of (VF, F,...) is
locally free, Qy ,z, which equals the J7j^f2 of (F Xz F, F,...), is locally free.
As a somewhat deeper illustration let us consider the following problem, posed
by P. Blass and solved by S. Bloch (unpublished) in the case n = 2: Classify all
f\Y —► Pn, n > 2, where F is smooth and / is a finite, height 1 morphism of
degree p and for simplicity k = k.
If p = 2 and n is odd, we can consider a quadric
F := {y2 = xix2 + x3x4 + ... xnxn+i} Ç Pn+1
and the projection onto the x-coordinates F —► Pn. This morphism clearly has
the required properties and it turns out that it is the only one.
PROPOSITION 2.5. Let f:Y —► Pn be a finite height 1 morphism of degree
p with Y smooth. Then p = 2, n is odd, and f is isomorphic to the projection
{y2 = XXX2 + • ■ ' + XnXn+x} -+ Pn.
PROOF. We consider instead Pn —► Y^ which is of degree pn~l. Hence we
have an exact sequence
0 —► Tpn/y(l) —► Tpn// —► 3* —► 0,
where Sf is a linebundle by Proposition 2.4. Taking Chern classes and denoting
by h the canonical generator of CH^P71), we get (1 + /i)n+1 = (1 + ch)f(h) in

FOLIATIONS AND INSEPARABLE MORPHISMS
143
CH*(Pn), where ci(Jî?) = ch, c G Z, and f(h) is a polynomial of degree n - 1.
As /in+1 = 0 is the only relation in CH*(Pn), we get an equality of polynomials
(1 + h)n+1 - hn+x = (1 + ch)f(h). From this we get c ^ 0 and by substitution
x = ft-1, (x + l)n+1 — 1 = (x + c)xn/(x_1), and this implies that n is odd and
c = 2, i.e., «S* = <^(2). Recall now that we have an exact sequence
0 - ^pn - ^(l)**1 -, r£n/, - 0,
in homogenous notation the ith <^(l)-factor has Xjd/dxi, 0 < j < n, as basis
for its global sections and 1 •—► Xid/J2xid- We thus get a surjective morphism
^>n(l)n+1 —► d?(2) described by n + 1 linear forms lo,h,...,ln or m other words
an (n + 1) x (n + l)-matrix L. The surjectivity amounts precisely to det L ^ 0
and the composite <fpn —► ^Pn(l)n+1 —► ^(2) sends 1 to xZ,x* where x =
(xq, ..., xn), whence xLx* = 0. Finally, if we make a linear coordinate change
in Pn by the matrix A, then L is changed to ALA1. Any L with xZ,x* = 0 and
det L^O can, through changes L —► ALA1 with det A ^ 0, be put in one single
standard form. As Y is determined by Tpn ,yl, and so by ^pn(l)n+1 —► <^Pn(2),
we get at most one possibility, up to isomorphism, for Y. This proves the result
for p = 2. We may therefore assume that p is odd.
When L then is put in the standard form for a nondegenerate alternating
matrix we have /o = xi, h = — xo,..., ln — — £n-i- This implies that xod/dxo~\-
x\djdx\ and x^d/dx^ + xid/dx^ give global sections of Tpn ,yl. However,
[xod/dxo + x\dldx\,x<idjdx§ + Xi<9/dx3] = —x^d/dx® + Xi<9/3x3,
which is mapped to —2xix2 in <^(2) and Tpn/y(1) is not closed under Lie brackets
as p ^ 2.
As a contrast to the computations in the proof of Proposition 2.6 we will prove
PROPOSITION 2.6. Let f:Y-^Pnbea radical finite morphism of degree
pm where Y is smooth. If n is even, then m is odd.
Indeed, as Pn is defined over the prime field, Pn is isomorphic to pn(m), and
we can consider the pmth power morphism as a //-endomorphism of Pn,Fm.
Hence we have a morphism g:Pn —► Y s.t. fg = Fm. We will use the
following properties of crystalline cohomology (cf. [Be, VII]): If X is a purely n-
dimensional connected smooth and proper A:-variety, then H*(x) := H*ris(X/W)/
p-tors is a graded ring finitely generated and free as a VF-module, where W is
the ring of Witt vectors of / (still assumed to be algebraically closed).
Furthermore, H2n(X) = W, the multiplication induces a perfect pairing Hl(X) ®w
H2n~i{X) —► H2n(X), and a surjective morphism h:Xi —► X2, where Xi
and X2 both fulfill the conditions above, induces multiplication by deg h: W =
H2n(X2) -* tf2n(Xi) = W. Finally, H*(Pn) = W[h\/(hn+l), where h =
ci(^(l)) G H2{Pn) so that hn = leW = H2n(Pn). From the perfect pairing
property and the fact that h*: i/2n(X2) —► H2n(X1) is nonzero, it follows that,
for h as above, /i*:i/*(X2) —► H*(Xi) is injective.
Returning to the situation g:Pn —► F, f:Y —► Pn we see that /* and g* are
injective and so Hn(Y) -^ VF, as n is even. Let s G Hn(Y) be a generator so that

144
TORSTEN EKEDAHL
there is a A G W s.t. f*(hn/2) = Xs. By the perfectness of Hn(Y)®w Hn(Y) -+
H2n(Y), we see that s2 is a unit in H2n{W) = W. Now, pm = /*(/in) =
if*(hn/2))2 = AV e H2n{Y) = W. This shows that m is even.
3. We will now pick out certain infinitesimal equivalence relations which
should as faithfully as possible mirror the equivalence relations obtained by
intersecting the equivalence relation obtained from a smooth morphism X —► Z and
that obtained from X —► X^n\ Recall [DG, 16.8] that the algebra of differential
operators on a smooth variety X has a filtration 0 = 3ffl~l Ç 3<ffl® Ç • • • by
the order of differential operators such that 9fffli • 3$j Ç 3$?*+* and such
that the associated graded algebra g?3ffl = r(TjL^), the free divided power
algebra on T^^. If we, locally, choose coordinates t\,..., tn on X, then
w/i= y ^J-AmiJ_ATO2...J_Am".
, , , mildti m2!3^2 mn\ dtn
m\ +m2i rfrinS^i
In this basis the divided power operations take
d/dnegrl3^ to {l/m\){d/dti)megvn3^.
Note however that these operations are not well defined in 3^.
DEFINITION 3.1 (cf. [Mi]). Let X be a smooth A:-variety. A foliation (resp.
of height n) is a subsheaf s/ Ç 3c/^{fx) such that % := s/HT^/jf is a subbun-
dle of Tjtyfc, stf is closed under product and coproduct and the graded algebra
associated to the filtration induced by {3tffl1} equals r(lf ) (resp. J2i<pn rz(2")).
In particular, if J/ is a foliation of height n it defines as above a schematic
equivalence relation Z —► X of height n. By a slight abuse of language we will
call the quotient of this equivalence relation the quotient of X by the foliation
of height n.
PROPOSITION 3.2. Let X be a smooth /-variety, and letX-+Y be the
quotient by a height n foliation. At each closed point <z G X there are formal
coordinates £i, $2î • • • ? *m su°h that X -+Y is formally isomorphic to (£i, t^ ..., £m) •—►
(£i,..., £fc, ^+1,..., tvm ). In particular, around &
*fc+H Hm<Pn
Indeed, we see that if rk8" = r then the degree of f:X —► Y equals rkj/ =
prn. We also have an exact sequence
(3.3) / ÏÏY// ~~* ^x// ~* ^x/y ~* 0,
where VLlx,Y — %&n&x{E,&x) and so has rank r. If we put y = f{<z) we get an
exact sequence from (3.3)
xxiy/m2y -+ n^/mj -+ n^/y/m^/y -+ 0.

FOLIATIONS AND INSEPARABLE MORPHISMS
145
Hence we can find for A: = m — r, A: elements t\,..., tk G m^ whose images
in m^/rn^ form part of a system of coordinates £1,..., £&, £fc+i, • • •, tm. If we
complete around a and y and put L := ^x/^, we get morphisms
L[[*l, . . . , tk, tPkl,, . . . , *£]] - ^y,y - ^ ^ £[[*, . . . ,*m]].
Now <^x,a; has rank prn over ^y,y, and it has visibly rank prn over
REMARK: (i) From this and a linear compactness argument one gets that if
A/ is a foliation then there are, at each point of X, formal coordinates £1,..., tm
such that s/ consists of the differential operators tangent to £1,..., £*. See [Mi]
for a uniform proof of this fact in any characteristic.
(ii) We never used the full hypothesis grsf = J2i<pn 1^(2"), only that sf is a
subbundle of 3tff p""1 and rkj/ = pnr.
COROLLARY 3.4. Let f:X -+ Y be as in Proposition 3.2 and X -► Y -^
X^ a factorization of the pnth power map Fn. Ifan: X^ —► X is the canonical
(non-k-linear) isomorphism there is an exact sequence
(3.5) 0 -, Tk/Y - TV - rTYM - Fn*(an*Tk/Y) - 0.
In particular,
(3.6) ojx = f*uY ®&x (detTi/y)pn-1.
PROOF. 0 —► T^,Y —► TjL^ —► /*7y is of course always exact. Applying
this to X^ —► y(n) we get an exact sequence 0 —► T^(n)/y(n) —► T^(n)// —►
/(n)*^y(n)//r, which equals 0 -+ <rn*ï£/y -+ <rn*T^//f -+ <rn*f*T*//f. We also
have a morphism Ty,^ —► 9*T^(n)/^^ and as Fn induces zero on tangent vectors,
we get an induced morphism Ty,^ —► g*<rn*Tx/Y- Applying / to it we get the
zero sequence (3.5), the exactness of which can be verified locally with the aid
of Proposition 3.2. Finally, (3.6) follows by taking determinants.
From Proposition 3.2 it follows that a foliation of height n can, locally, be
extended to a foliation. This extension is not unique, however, and the following
example shows that a foliation of height 1 may not be extendable to a foliation
of height 2. Let X be a smooth, proper, and connected surface (we assume for
simplicity >/ = //) for which there is no dominant rational map Y —► X with Y
ruled, i.e., X is not uniruled. That b2 ^ p(X) (:= rkNS(X)) or that X -+ AlbX
has a 2-dimensional image are, for instance, conditions ensuring this. Let S* be
a linebundle such that S?v is very ample, there is a Lefschetz pencil in |^p|, and
S"9 P <8>&x ojx1 is ample. For any global section s of S"9 there is an ^-linear
morphism ds:2p —► fiL,^ defined locally by the property that if u G 2 is a
local generator of «S5*, then ds: uv •—► d(s/up) G HL^. By the assumption on <2fp
there is a global section s such that 2V —► fiL,^ has only isolated (in fact simple)
zeros. Consider such a section s and let a: Y —► X be obtained by taking a pth

146
TORSTEN EKEDAHL
root of s, i.e., Y = Spec(0P=o^-*) with multiplication &-*&&-' -^ S?'i~:j
by identification when i + j<p and 2"^ ® 2^ -* &-*-* id4s ^-»-i+p if
not. Locally, Y is of the form Spec(^x[£]/(£p - /)) with / = s/up as above.
If F is the set of zeros of ds, then Y' := Y^-^F) ^ X\F =: X' is a height
1 foliation, and it is easy to see that over Y\ T^/y = tt**S? defined locally by
d/dt <-► u. From (3.6) we then get ojy = n'*(ux' ®^p~1). Suppose now that
the height 1 foliation T^, iY, extends to a height 2 foliation and let Y' —► Z' be
the quotient. Again by (3.6) a*ojz> = ^(ux1 <8>i?p~p2). This gives for r > 0,
p-i
r(z>y - r(y,7T,*((o;x' ®^p"p2)r)) = r(x',0(wx, ®^p"p2)r o-S*-*)),
1=0
which equals 0 by assumption. Hence any smooth compactification of Z' is ruled,
by the classification of surfaces, and the same is true for Z'(~2) which dominates
X contrary to the original assumption.
4. We will now concentrate on foliations of height 1. In fact we will also allow
foliations with singularities: If X =£ U is an infinitesimal finite flat equivalence
relation on an open dense subset U of a smooth variety F, then we can take
the schematic closure X1 of X in Y x Y. Then X1 =t Y will be a finite, but
not necessarily flat equivalence relation on Y. It corresponds to a subalgebra srf
of «S^(^y) closed under coproducts which is saturated as a submodule. If sf
also has the property that there is an open dense subset U' of Y such that srfjji
is a height 1 foliation we will say that srf is a 1-foliation. From (2.4) we get
LEMMA 4.1. There is a 1-1 correspondence between 1-foliations on Y, a
smooth variety, and saturated subsheaves ê? Ç Ty .^ closed under Lie brackets
and pth powers.
The following lemma simplifies the checking that a subsheaf is closed under
Lie brackets and pth powers.
LEMMA 4.2. Let A be an Ôy -submodule ojTy,^.
(i) The Lie bracket induces an @y-linear morphism
Aê?2 —► Tyi£i'%', a A 6 •—► [a, &].
(ii) Suppose ê? is closed under Lie brackets. Then the pth power morphism
induces an <fy -linear morphism
where F:Y -+Y is the absolute Frobenius endomorphism.
Indeed, for (i) we just compute;
[DufD2] = f[DuD2] +£>i(/)£>2 = f[DuD2] modgT.
For (ii) the Jacobson identity [Ja, p. 187]
(£>i +D2)P = D* + L% + y%2sl{D1,D2) = r%+r% modgr,

FOLIATIONS AND INSEPARABLE MORPHISMS
147
the last congruence being true as the s» are Lie polynomials, gives additivity and
the identity [Ka, 5.3.0]
(fD)p = fpDp - fDp-l(fp-l)D = fpDp modr
gives p-linearity.
We will now also restrict ourselves to surfaces. The Mumford-Bombieri proof
of the Enriques classification of surfaces differs very much from the classical
proof in the case which leads to abelian surfaces. If one knew that the Albanese
mapping were separable, an altogether classical proof would be possible in that
case (cf. [Bea, p. 126]).
PROPOSITION 4.3. Let Y be a connected, smooth, and proper surface of
Kodaira dimension < 0 over a field 4 of characteristic p > 0. If the image Y'
of Y in Alb F is 2-dimensional, Y -+Y' is separable.
PROOF. Assume that Y —► Y' is inseparable. Consider f:Y —► Alb y and
the induced morphism f*^1AihY// ~~* ^y//' As F —► F' is inseparable this
morphism has rank < 1, and by a theorem of Igusa (see [Se] for a proof in
the vein of this article) it is injective on global sections. Hence the saturation
of the image is a line bundle & C Q\r with h°(£?) > 2 as dim Alb y > 2.
Let J? Ç Tyjf be its annihilator. As J£ is the kernel of Ty,^ —► f*Ty,^, it
corresponds to a 1-foliation, and restricting to the complement of the finite set of
points outside of which */# corresponds to a height 1-foliation, i.e., the open set
where J£ is a subbundle, one sees that M -^ ojyx ®Sf. If Y" is the quotient of Y
by the corresponding schematic equivalence relation, we can factor Y —► Alb Y
as y —► Y" —► Alb y, and from (3.6) we get that, outside of the singularities of
the 1-foliation, ojy = h*ojY" 0 (cjy1 <g) J?)?"1, i.e., uvY = h*ojY" <Si<2fp~1. As
h°(*S?) > 2 we see that if some positive power of uy^ had a nonzero global
section, then some power of ujy would have two linearly independent sections,
thereby contradicting that k(Y) < 0. Hence a resolution Y of Y" has Kodaira
dimension — oo. We then finish by
LEMMA 4.4. Let Y be a connected, smooth, and proper surface with k.(Y) =
f
—oo. Then Y —► Alb y does not have a 2-dimensional image.
Indeed, assume not and let Y' be the image. We argue by induction on the
inseparability degree of Y —► Y'. If Y —► Y' is separable, /*H^lby —► QY
has rank 2, so /*A2fi^lby/^ —► ojy is nonzero and h0(ujy) > 1. If Y —► Y1 is
inseparable we argue as above to get a diagram
7 -* Aib F
1 1
Y -+ Y" -+ Alb y
Hence Y —► Alb Y has 2-dimensional image Y , Y -+Y has lower inseparability
degree than Y —► y', and by the argument above k(Y) = —oo. By the induction
assumption we get a contradiction.

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TORSTEN EKEDAHL
Raynaud has conjectured that if a surface of general type has c2 < 0, then it
is uniruled. The following argument, which is heavily inspired by an attempt by
Miyaoka to prove Raynaud's conjecture, gives a result in the desired direction.
PROPOSITION 4.5. Let X be a minimal surface of general type. Then either
X is uniruled or
c*(X) < P
c\{x)-{P-ir
PROOF. If c2 > 0 there is nothing to prove. Hence we may assume c2 < 0
and then AlbX ^ 0. Then, by the theorem of Igusa mentioned above, the
image of /*^Aibx// ""* ^x// nas a nonzero global section. Let 2? be the
saturation of such a section. Then we get c2 — d = 2f2 — (ux,&), where d is
the number of zeros of S* Ç Çlxx counted with multiplicities. Hence —c2/cj <
-(ojx,°S?)/ci + S?2jc\. (Note that c\ > 0 as X is minimal.) Furthermore,
Hodge's index theorem gives £f2/c\ < ((ojx,<2f)/c2)2. Let Jt Ç T*, be the
annihilator of S?. Then j# -^ S? <g) oj^1. By (4.2(i)) ^ is closed under Lie
brackets and by Lemma 4.2 (ii) the pth power map induces an ^x-morphism
J£v —► J£~x <g) oj^1. If this is nonzero, then as ojx is numerically positive
we get p(u>x,^) < —c\ - (ujx,^), i.e., (p + 1)(((jJx,°S?)/c2) < p so that
(ux,&)lc\ < 1, and so by S?2/c\ < ((ux^)/c\)\ S?2/c2 < (^x,^)/c?,
which contradicts 0 < -c2/c\ < J*?2/cl - {ux,-&)lc\-
We therefore see that J£ defines a 1-foliation. Let F be the set zeros of
j# —► Tx/j and f:X —► Y the quotient corresponding to J£. By (3.6) /*cjy =
ux®>^l~v outside of F. As ux is numerically positive, if (ux, f*uy) < 0 (where
f*ojy denotes the unique linebundle extension of f*ojy\XF to X), then any
resolution of Y is (birationally) ruled. Hence we may assume that (cjx, /*^y ) >
0, i.e., pel + (p - l)(wx,-S^) > 0 or {u)X^)/c\ < p/{p - 1). Now -c2/c? <
^2/cï-(^x,^)/cï < {{ux,^)/cî)2-{ux,^)/cî and as^ by construction
has a nonzero global section, (ujx,^) > 0- Finally, the function x2 — x takes,
in the interval 0 < x < p/(p — 1), its maximum for x = p/(p — 1) so we get
-C2/C\ < (P/(p ~ l))2 - P/(p - 1) = p/(p - l)2.
REMARK. One may ask if there similarly exists a restricted class of minimal
surfaces of general type and a function /(p), so that /(p) —► 1/3 when p —► oo
and that any minimal surface of general type either belongs to this class or fulfills
/(p) < c2/ci- Examples of Szpiro [Sz, 3.4] show that the infimum of c2/cj over
minimal nonuniruled surfaces of general type is < 0, but all his examples have
the property that they map onto curves of genus > 2.
5. A more substantial application of the ideas presented in this paper will be
found in a forthcoming paper by the author where the following result is proved.
THEOREM 5.1. Let X be a minimal surface of general type over an
algebraically closed field of characteristic p. Then Hl(X, oj^1) = 0 for i > 0 except,
possibly, when 2 = 1, p = 2, x(^x) = 1 and there exists a dominant inseparable

FOLIATIONS AND INSEPARABLE MORPHISMS 149
rational map of degree 2, Y —► X where Y is rational or a KS-surface.
Furthermore, \mK\ considered as a linear system on the canonical model of X is very
ample for m > 5.
References
[Bea] A. Beauville, Surfaces algébriques complexes, Astérisque, no. 54, Soc. Math. France,
Paris, 1978.
[Be] P. Berthelot, Cohomologie cristalline des schémas de carctéristique p > 0, Lecture Notes
in Math., vol. 407, Springer-Verlag, 1974.
[DG] J. Dieudonné and A. Grothendieck, Éléments de géométrie algébrique. IV, Inst. Hautes
Études Sci. Publ. Math. 32 (1967).
[Ga] P. Gabriel, ExpVII#, SGA 3, Lecture Notes in Math., vol. 151, Springer-Verlag, pp.
474-548.
[Ga 1] , Exp V, SGA 3, Lecture Notes in Math., vol. 151, Springer-Verlag, pp. 250-
283.
[Ill] L. Illusie, Complexe cotangent et déformations. II, Lecture Notes in Math., vol. 283,
Springer-Verlag, 1971.
[Ja] N. Jacobsson, Lie algebras, Interscience, New York, 1962.
[Ka] N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of
Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175-232.
[K-N] T. Kimura and H. Niitsuma, On Kunz's conjecture, J. Math. Soc. Japan 34 (1982),
371-378.
[Mi] Y. Miyaoka, Deformations of a morphism along a foliation and applications, this volume.
[Mu] D. Mumford, Abelian varieties, Oxford Univ. Press, Oxford, 1970.
[Se] J.-P. Serre, Quelques propriétés des variétés abéliennes en caractéristique p, Amer. J.
Math. 80 (1958), 715-739.
[Sz] L. Szpiro, Sur le théorème de rigidité de Parsin et Arakelov, Astérisque, no. 64, Soc.
Math. France, Paris, 1979, pp. 169-202.
University of Stockholm, Sweden

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Deligne's £-adic Fourier Transform
LUC ILLUSIE
While applying his results in [5, 6] to the study of exponential sums, Deligne
discovered a geometric transformation on ^-adic sheaves on the affine space over
a finite field lifting in some sense the usual Fourier transform on functions on its
set of rational points. Some hidden properties of this transformation were found
later, and in the past few years many applications have been obtained, due to
Brylinski, Katz, Laumon, and Lusztig. It should be mentioned, too, that during
the same period other geometric Fourier transforms have been investigated, such
as the Mukai Fourier transform for coherent sheaves on an abelian variety [16]
and the Brylinski-Malgrange-Verdier Fourier transform on D-modules on a vector
space over C [3].
1. Functions and sheaves. Fix a prime p and a nontrivial additive
character
^0: Fp-C*.
Let q be a power of p. Denote by tp the (nontrivial) additive character deduced
from ipo by composition with TrFq/Fp.
1.1. Let V be a vector space of finite dimension d over Fg, let V be the dual
vector space, and denote by
(,):VxV'-*Fq
the canonical pairing. As discrete groups, V and V are isomorphic to (Z/p)ad
(if q = pa), and tp(( , )) identifies V with the Pontryagin dual of V. We can
therefore consider the usual Fourier transform, which sends a C-valued function
/ on V to the function J/ on V defined by
(1.1.1) (J/)(y) = £/(*M(z,y».
xev
As is well known (and rather easy to show), we have
(1.1.2) y(«o) = lv,
1980 Mathematics Subject Classification (1985 Revision). Primary 11L40, 11R39, 14G10,
14G15; Secondary 11F66, 11G40, 14F20, 22E55, 43A32.
©1987 American Mathematical Society
0082-0717/87 $1.00+ $.25 per page
151

152
LUC ILLUSIE
(1.1.3) 7{lv) = QdSo
(where 6X denotes the Dirac function with value 1 at a: and zero elsewhere), and
the Fourier inversion formula
(1.1.4) (W)(f)(x) = qdf(-x).
We also have
(1.1.5) *(/* 9) = WW),
where / * g is the usual convolution product (/ * g)(x) = J2Xl+x2=x f{xi)o(x2)-
The motivation for Deligne's construction was to try to understand better the
effect of J on certain functions coming from geometry. For example, if we view
V as an F^-scheme, and if we have a morphism of finite type X —► V, we can
consider the function x •—► #Xx(Fq) defined on V(Fq). Such functions have a
cohomological interpretation, which I will now recall.
1.2. Fix a prime I ^ p and an algebraic closure Qe of Q^. If X/Fq is a
scheme of finite type, we have the notion of a constructible Q^-sheaf on X (see
[20, V, VI; 19, Rapport §2]), which is analogous to that of a constructible sheaf
of C-vector spaces on a complex algebraic variety. Among the constructible
Qrsheaves, those which are unramified, also called lisse, correspond (for X
connected, pointed by x) to continuous representations -Ki(X,x) —► GL(n, A:),
where A: is a finite extension of Q^. Examples of these are the Tate sheaves
Qi(r) = (Hm/4r) ®z< Q* (r G Z),
which are lisse of rank 1. In general, given a constructible Q^-sheaf on X, there
exists a stratification of X such that its restriction to each stratum is lisse. The
fibers of a constructible Q^-sheaf are finite-dimensional Q^-vector spaces.
We also have a good derived category D%(X, Qe) (where b, c means: bounded,
constructible cohomology (see [6, 1.1.2 and 1, 2.2.14])), which is stable under
the usual operations (<g), RKom, Rf*, Rf\, /*, Rf{).
If E is a constructible Q^-sheaf on X, we can consider the Frobenius
correspondence, consisting of the çth-power map F on X and the isomorphism
F: F*E ^ E (see, e.g., [19, Rapport 1.2]). Let Fq be an algebraic closure of
Fq. If £ G X(Fq), then x, viewed as an F^-point, is fixed under F; hence, the
Frobenius correspondence defines an automorphism F of the fiber Ex, and we
can consider its trace Tr(F, Ex) G Q^. We thus get a function
tE: X{Fq) - Q€, tE{x) = TV(F, Ex).
More generally, for E G D}ï(X, Qe) we define
ïe = /A-iytxHE)-
Examples 1.3. We have
*(Q£x(r» = q " ' 1X'

DELIGNE'S £-ADIC FOURIER TRANSFORM
153
t(K®L)=tK-tL (K,L€D*(X,Qe)),
t(f.L) = ftL (/: X - Y of finite type,L € Dhc{Y,Q,)),
tRflK = MtK) (KGD*(X,Qt)),
where, for u a Q^-valued function on X(Fg), f\u is the function on Y(Fq) defined
by {f\u)(y) = J2f(x)=yu(x)' This ^ast identity is Grothendieck's trace formula
(see [20, 19, Rapport]). In particular, for K = Q€, we get a cohomological
interpretation for the function considered at the end of 1.1:
(y » #Xy(Fq)) = tRfi^t = (jh ^(-l)'lV(F,^/!(Q£)y)) •
So we see that natural operations on functions reflect—for functions of the
type %e—natural operations on sheaves. Working with Q^-valued functions
rather that C-valued functions, we would like to interpret the Fourier
transform (1.1.1) in this style.
1.4. To do this, we just need to find a sheaf on the affine line A = A\ such
that its associated t function is the character ^, viewed as a function from A(Fq)
to Q{Çp)* C Q^. For this, consider the Artin-Schreier exact sequence
0 -+ Fq -+ A*—? A -+ 0.
7T
It makes A into an étale torsor over A of Galois group Fq. Associated to this
torsor and the character tp~1:Fq —► Q£, we have a lisse Q^-sheaf of rank 1 L^
on A (denoted £F(^) in [19, Sommes trig. 1.8]). It is easy to show (loc. cit.) that
tL^ = ^. Sometimes it is convenient to use the following alternate description
of L^: consider the decomposition of 7r*Q^ under the action of the Galois group
Fq and take the piece on which Fq acts through \j)~l.
Here are some basic properties of L^ :
(a) We have a canonical isomorphism
s*Lip —► pIL^p <g) y^Lty,
where s,pi,p2- Ax A —► A are the addition map and the two projections,
respectively (this translates into the identity ip(x + y) = ip(x)tp(y)).
(b) H£(A<g>Fq,L^) = 0 (this implies the identity ^2^(x) = 0 (ip nontrivial)).
For a proof, see [19, Sommes trig. 2.7*], or use the alternate description above.
(c) Since ip is nontrivial, L^ is nontrivial; in fact, L^ is wildly ramified at
infinity with Swan conductor = 1 [19, Sommes trig. (3.5.4)]. (If 1 is the trivial
character, L\ is the constant sheaf Q^.)
If /: X —► A is an Fg-morphism, we shall use the notation
We shall sometimes write (incorrectly) L^(f(x)) instead of L^(f); for example,
if ( , ): V x V —► A is the canonical pairing of 1.1, we like to write L^((x^y))
for ( , )*L^.

154 LUC ILLUSIE
DEFINITION 1.5. Let V, V be as in 1.1. Deligne's l-adic Fourier transform
[relative to tp) is the functor
7^: Dhc{V,Qt) - D*(V\Qe)
defined by
fi^E = Rp2\(plE <8> L^((x, </))),
where p\\ V x V —► V, p2- V x V' —► V are the projections.
It is clear from 1.3 and (1.4.1) that
(1.5.1) h],^E = ?tE.
In the sequel tp is fixed, so we will drop it from the notation of the Fourier
transform.
2. Properties of the ^-adic Fourier transform. We keep the notations
of 1.5. We have (trivially)
(2.1) MQi{o}) = Qev>
(sheaf-theoretic version of (1.1.2)), and (as an easy consequence of 1.4(b))
(2.2) ?i(Qev) = Qe{0}(-d)\-2d\
(sheaf-theoretic version of (1.1.3)).
Elaborating a little on (2.2) (and using 1.4(a)), one constructs a canonical
functorial isomorphism
(2.3) ffrE ^ {a*E)(-d)[-2d},
for E G D^ (V, Q^), where a: x •—► —x is the antipodal map on V. For functions of
the type £#, (2.3) gives (1.1.4). It follows from (2.3) that Ji yields an equivalence
(2.4) ?i--D*(V,T5l)ZD*(V',Gl).
For K, L £ Dç(V, Q^), Deligne defines their convolution product
(2.5) K * L = RsifâK ® p£L)
(with s, pi, p2 as in 1.4(a)). Then one has
(2.6) 3i(/r*L) = JiKtefrL
(sheaf-theoretic version of (1.1.5)). (The simplest way to show this is to observe
that wFi commutes with external tensor-products:
(2.6.1) J,(M H N) ~ JiM H J,7V,
and that for u: V —► W a linear map of F^-vector spaces, with transpose u'',
(2.6.2) 5\{Ru\M)^u'*7{M.)
Obviously there is another definition of a Fourier transform, namely
3. (= ?.,«,): D^(V,Qe) - £>cb(V",Q£)
given by
%E = Rp^fàE ® L+{(x,y))).

DELIGNE'S i-ADIC FOURIER TRANSFORM
155
What gives these constructions their punch is the following miracle, first
observed, I think, by Verdier around 1980:
THEOREM 2.7 [10, 2.1.3]. The natural map "forget supports"
is an isomorphism.
Verdier's proof is formal: it uses the inversion formula (2.3) and the analogous
one with 9* instead of 3\ to show that £Fj —► 2* is an isomorphism on perverse
sheaves (see below). Laumon's proof (loc. cit.) is geometric and gives a little
more: for d = dim V = 1, it shows that in some strong sense the sheaf Lip((x, y))
extended by zero on PxV' (where P DV is the projective line) is locally acyclic
along x = 00 with respect to p\: P x V —► P. His argument (which works over
a more general base than Spec(Fg)) consists in analyzing the normalization of
the Artin-Schreier cover tq — t = xy near (x = 00, y ^ 0) and showing that its
projection to P is the composition of a radicial map and a smooth map. See also
Brylinski [2, 9.8] for a different argument, which works only for d = 1.
In what follows we shall identify £Fj to £F* by means of 2.7 and write it simply
J.
COROLLARY 2.8. The t-adic Fourier transform commutes with duality,
namely, there is a canonical isomorphism
DV^E ~ (yw-i)£V£)(d)[2d]
for E G D]?(V, Qe), where for a: X —► Spec(Fg) a scheme of finite type Dx
denotes the dualizing functor RJ(om(—,i2a!Q€).
This is an immediate consequence of 2.7, see [10, 2.1.5].
Now recall that a perverse Qr sheaf on an F^-scheme X of finite type is not
a sheaf, but rather a complex E G D]ï(X, Qe) which satisfies the following two
conditions (see [1, 4.0]):
(i) dim supp WE < —i for each i G Z;
(ii) dim supp JCDxE < —i for each i G Z.
In particular, JCE = 0 for i > 0. Basic examples are the following:
(a) Qex [n] for X smooth of dimension n;
(b) a punctual constructible Qrsheaf placed in degree 0 (on any X).
Let us denote by Perv(X) the full subcategory of D]?(X, Qe) consisting of
perverse Q^-sheaves. It is shown in [1] that Perv(X) is an abelian subcategory
of D^ (X, Qe) every object of which is of finite length.
COROLLARY 2.9. The functor E •—► 3E[d] induces an equivalence
Indeed, by Artin's theorem on the cohomological dimension of affine schemes
[21, X] Rp2* preserves condition (i) (cf. [1, 4.1.1]), so by duality Rp2\ preserves

156
LUC ILLUSIE
condition (ii), while p\{—) <8> L((x, y))[d] preserves both (i) and (ii) since L is
lisse and pi smooth of dimension d.
REMARK 2.10. For simplicity we have limited ourselves to a vector space
of finite dimension over Fg, but the ^-adic Fourier transform can be defined in
a more general situation; namely, one can replace Fq by an F^-scheme S which
is regular, noetherian of dimension < 1, and V by a vector bundle on S. If V
denotes the dual vector bundle, and ( , ): V x V' —► A = A^ the canonical
pairing, the Fourier transform
y,: D^(V,Qe)-*Dhc(V',Qe)
is defined as in 1.5, with L^({x, y)) denoting the pull-back of L^ by (, ) composed
with A —► Ap . All properties above extend up to minor twists, in particular
Ji = J* is still Valid (see [12, §2]).
We shall now mention or discuss very briefly some applications of the ^-adic
Fourier transform.
3. Applications.
3.1. Estimates of exponential sums (Katz-Laumon [12], Katz [11]). Let V be
as in 1.1 and let X c V be a closed, smooth, geometrically irreducible subvariety
of V of dimension n. Consider the exponential sum
Sr(y)= 2 ^«*'y»
xeX(Fqr)
for y G V'(Fq) and r an integer > 1. This is the value at y of the Fourier
transform of the function on V(Fqr) (= Vr(Fqr) if Vr = V <g) Fqr) which is 1
on X(Fqr) and zero elsewhere. In other words, this is the value at y of the
Fourier transform of the function t-~ associated to the constant sheaf Qe on
Xr = X 0 Fqr extended by 0 on Vr, which in view of (1.5.1) can be rewritten
sr(y) = *y(Q£Xr)(y),
and 3{Qtxr) 'ls Just tne pull-back of ^(Q^x) on K'- Now, by the assumptions on
X, Q^xM 'ls perverse, simple, and pure of weight n (by definition, cf. [1, §6]).
So, by 2.7, 2.9, and Deligne's fundamental theorem [6, 6.2.3], ^(Q^x)[n + C'] nas
to be perverse, simple, and pure of weight n + d. This implies that there is an
open subset U of V such that the restriction of ^(Q^x)!72 + d] to t/ is of the
form M[d\ where M is zero or an irreducible lisse Qrsheaf of rank C and weight
n. So we get a uniform estimate of Sr(y) for y G U(Fq) and all r, of the form
|Sr(»)| < Cqnr'\
This is the principle of the method used by Katz and Laumon in [12] (the
idea seems to be due to Brylinski who first noticed the possibility to derive
uniform estimates in this way in the case of Kloosterman sums, see his survey
[2, §10]). Katz and Laumon consider more general exponential sums, and also
obtain estimates which are uniform in p in some sense in situations coming from
schemes affine and smooth over an open subset of Spec(Z).

DELIGNE'S £-ADIC FOURIER TRANSFORM
157
The ^-adic Fourier transform also plays an important role in Katz's recent
study of "Kloosterman sheaves" [11].
3.2. Lefschetz theory for intersection cohomology (Brylinski [2]). Recall the
classical theory of Lefschetz for the cohomology of hyperplane sections. Let X
be a closed, smooth, connected subvariety of dimension n + 1 of some projective
space P over C. Let P' be the dual projective space (set of hyperplanes of P),
and X' the closed subvariety of P' consisting of hyperplanes tangent to X. Fix
i e Z. Then P' 3 H »-► H*(X D H, C) is, over P' - X', a local system W of
C-vector spaces. For i ^ n (n = dimX fl i/), this local system is constant, and
for i = n we have a decomposition <Kri = Fix © Van, which is orthogonal with
respect to the intersection form on Hn(X D i/), where Fix is a constant local
system, defined by
Fix(tf) = Im Hn(X) -+Hn{XnH),
and Van is either zero or an irreducible local system. If i/o is a base-point in
P' - X1, Van is the subspace of Hn(XnHo) spanned by the vanishing cycles
associated to a Lefschetz pencil passing through i/o- It is shown that this subspace
is stable under ni(P' —X', i/o), and, using the Picard-Lefschetz transformations,
the corresponding representation of -K\ on Van/(Van fl Van-1) is irreducible. But
Van fl Van-1 = 0 by the hard Lefschetz theorem. For a nice exposition of this
theory, see Deligne [5,§§4, 5].
These results have been extended by Deligne to the case of an arbitrary
algebraically closed base field, using ^-adic cohomology [5, 6].
Brylinski's contribution is the following. Let P be some projective space over
an algebraically closed field A: of characteristic p ^ ^, and let X be a closed,
irreducible, and reduced subvariety of P of dimension n + 1. Let 3Qx be the
^-adic intersection complex of X: this is the unique perverse sheaf on X which
is self-dual and equal to Qjn + 1] on the smooth locus of X {j\*Qt[n +1] in the
notation of [1, 4.3], if j is the inclusion of the smooth locus). Again let P' be
the dual projective space. Then we have:
THEOREM 3.2.2 [2, §§3, 9]. There exists a nonempty open subset U of P'
such that the restriction of H »—► Hl~l(X fl i/,3Qx\X H H) to U is a lisse sheaf
JC having the following properties:
(a) for i ^ 0, JC is constant,
(b) for i = 0, 5{° is either constant or the direct sum of a constant sheaf and
an irreducible sheaf.
In fact, for U small enough, one has 3Gx\(X D H) = JC(xnif)[l] for H G U
(see [9, §7 or 2, 3.4] in characteristic zero; in general, this follows, as Deligne
points out, from the compatibility of JC with smooth inverse images and the
generic compatibility of Rj* with base change [19, Th. finitude, 2.9]); so, over
such a £/, JC is the local system H »-► H{ (X D i/, JCxni/)- Note also that for X
smooth one has Hl-l(X D i/,3GX\X n H) = i/*+n(X n H, Q), so 3.2.2 extends
the results of 3.2.1—at least some important part of them.

158
LUC ILLUSIE
Brylinski's proof of 3.2.2 in characteristic p/0 goes as follows. Consider
in P x P' the subvariety Z defined by (x, y) = 0 (the "incidence variety" or
"universal hyperplane") and the projections p\: Z —► P, p2- Z -* P1, which are
projective bundles of rank N — 1 if P is of rank N. Define the Radon transform
Rad: Dcb(P,Q,)-,Dcb(P,,Q,)
by Rad(E') = Rp2*(p\E)[N-I]. This transformation is closely related to Fourier.
Namely, if P = (V-{0})/Gm, consider the blow-up of the origin in V, n: V —► V
and the canonical projection p: V —► P (the line bundle 0(—1)). Then one has
a canonical isomorphism
(*) ?(Rtt*p*E[N + 1])|V - {0} ~ p* Rad(£)(-l)|V - {0}.
This is just the sheaf-theoretic version of the following identity on functions:
replace A: by the finite field Fq; then a Q^-valued function / on P{Fq) can be
viewed as a function on (V — {0})(Fg), constant on the (punctured) lines, and
one can extend it to a function / on V(Fq) by putting
/(o) = E /(line)-
lines
Then an easy calculation shows that &f\V'(F) - {0} = çRad(/)|F/(F) - {0}
where Rad(/) := P2\p\f with the notations of 1.3. Now, using (*) and the fact
Fourier "exchanges" sheaves concentrated at the origin and constant sheaves, it is
not difficult to deduce from 2.9 that the Radon transform induces an equivalence
Rad: Perv(P)/(constant) -^ Perv(P/)/(constant)
(where (constant) means the thick subcategory of constant sheaves placed in
degree —N). Using furthermore that H1(P'^Qi) = 0, one finds that over some
nonempty open subset U of P', the lisse sheaf 5{° admits a dévissage of the type
] constant
M°: ] zero or irreducible
] constant
(i.e., a 3-step filtration with quotients as indicated). Finally, one can split this
filtration since 5{°[7V]|{7 is perverse and pure [1, 5.3.8].
In characteristic zero, Brylinski proceeds differently, using the theory of D-
modules and canonical transformations instead of the ^-adic Fourier transform,
but it seems that one might reduce characteristic zero to characteristic p by the
techniques of [1, §6].
3.3. The product formula for the global constant (Laumon [13, 14]). This is
probably the most important application so far.
Let X/Fq be a smooth, complete, geometrically connected curve, with
function field K. Let £ be a constructible Qrsheaf on X, lisse over some open

DELIGNE'S £-ADIC FOURIER TRANSFORM
159
subset j: U <--+ X and such that E = j*j*E. Recall that the L function of £*,
L(E,t) G QJ[£]], is defined by the infinite product
L{E,t)= 11 det(l -Fxtde^x\E)-1
x€|X|
where \X\ is the set of closed points of X and deg(x) = [k(x): Fq] (see for
example [4, §1]). By Grothendieck, this L function is a rational function of £,
which has the following cohomological description (loc. cit.)
L{E, t) = H det(l - Ft, Hl(X, E))(-Vi+\
iez
where F denotes, as in 1.2, the Frobenius correspondence and X = X ® Fq.
Furthermore, a functional equation relates the L functions associated to E and
to the dual sheaf Ë:
L(E,t) = e{E)rxL(Ë,q-lrl),
where x = x{X, E) and e(E) G Q^ is the so-called global constant, which admits,
too, a cohomological description, namely
e(E) = l[det{-F,Hi(X,E))(-iy+1
(see [4, 7]). For example, e{Qe) = q9~l where g = dimHl(X,Qx)-
Fix oj G H1(jFC), (j ^ 0, and let ip be as in 1.1. Generalizing Tate's thesis,
Deligne defined in [4] (see also [7]) a local constant1
ex{E)=ex{E,(j,iP)eQ*e
for each closed point x in X, depending on ip, cj, and the local behavior of E
near x. One has ex{E) = 1 for all but a finite number of points. For example, if
E is lisse at x, one has
^(^)=det(-F,^)^^ç-^^rk^).
If E is tame at x, ex(E) is essentially a Gauss sum.
Deligne conjectured the following formula, inspired by Langland's philosophy:
(3.3.1) e(E) = q(1-^kW H ex(E)
xe\x\
(where rk(£") is the generic rank of £*), and in [4] he proved it in a number of
cases, e.g., when the representation G&\(K/K) associated to E factors through
a finite quotient. In his Seminar at the IHES in 1980-1981, he proved it in the
case where E has tame ramification. The methods, however, did not seem to
extend to the wild case, but recently Laumon has obtained a proof of (3.3.1) in
the general case based on an entirely different idea.
1This is the constant associated to the character / i-» ^oTrfc(x)/FpResx(/u;) and the Haar
measure on Ox of total mass 1.

160
LUC ILLUSIE
Let me try to explain it in the following simple (and crucial) case: X = P is
the projective line over Fg, compactification of the affine line A = Spec Fg[x],
S C A(Fq) is a finite set of rational points, E is lisse on P — S (and such that
E = j*j*E where j: P — S <--+ P is the inclusion), and satisfies
H°(P,E)=H2(P,E) = 0
(where the bar means <8>Fg). Let A' (resp. P') denote the dual affine (resp.
projective) line, and consider the Fourier transform of E\A
7{E\A)&Dhc{A\Qt).
Denote by 0 and oo the (rational) points zero and infinity on P and similarly 0'
and oo' on P', and by 0, ôô,... a geometric point over 0, oo, — By definition
(and the assumptions made on E) we have
(3.3.2) 7{E\A)^=Hl(A,E){-l].
Reinterpreting Witten's method to prove the Morse inequalities [18], Laumon
discovered that there is for 3(E\A) some kind of a "principle of the stationary
phase," i.e., that 3(E\A)y "concentrates on S as y —► oo." More precisely, we
have the following results, where r/c (resp. r/oo') denotes the generic point of the
henselization of P' at 0' (resp. oo'):
3.3.3. (a) 7(E\A) = G[—1], where G is a constructible Qe-sheaf on A', lisse
over A' - {0'}.
(b) GfjQf admits a 3-step filtration that is equivariant under the action of
Ga^r/o'/^O')* with quotients as indicated:
} JfcK-i)
Gno,:)H\P,E)
)ESB.
where
is the fiber at s of the first cohomology group of the complex of vanishing cycles
R$(plE ® L((x, y)),P2) G D^(P x oo',Q€), L((x,y)) denoting the extension by
zero on P x P' of L((x,y)) on Ax A'.
Laumon likes to visualize this through the picture given in Figure 1. (The
circles with vertical bars represent "Milnor fibres" which might a priori give
vanishing cycles for p\E <g) L({x,y)) with respect to p2-)
The proof of (a) is very easy: by hypothesis, £"[1] is perverse, so J(£'|A)[2] is
perverse (2.9). But from 1.4(b) one deduces that there are no vanishing cycles
at 2 (i.e., at (oo,y), y ^ 0'), so J^IA^l] has to be a lisse sheaf on A' — {07},
and at 0' we have (3.3.2).
To prove (b), one looks at the vanishing cycles at 1, i.e., (oo,0'): again
by 1.4(b) one finds that R$(E ® £((z,î/)),P2)(oo,0')- = -Eôô(~~1)[~1]> ^d the
dévissage follows from the usual exact triangle for vanishing cycles.

DELIGNE'S £-ADIC FOURIER TRANSFORM
161
oo 4
0 I
Pi
0'
—f—
Vo'
[p*
Figure l
-P*
Assertion (c)—the key part—is a little more difficult. First of all, the miracle
(2.7), or rather the local acyclicity statement given by Laumon's proof, shows
that there are no vanishing cycles at 4 (i.e., (x, oo'), x G A — S) and also (by
a global argument) that there are none at 3 (i.e., (00,00')). So the vanishing
cycles are all concentrated on S x oo', and (c) follows from RTfa1 {ôô'), R$) =
ittW(W),£).
Now (b) implies that the inertia at r/o; acts unipotently on GfjQf, hence trivially
on det(GfyQ/), and therefore det(G|A' — {0'}) extends to a lisse sheaf of rank
one on A1', which we denote by K. Let Fqo' G Gal(r/0o//r/oo/)a6 correspond to
1/y G A:(r/oo/)^= Fq((l/y)) via local class field theory. Then it follows easily
from global class field theory on P' that we have
Tr(F,Kfl0,) = Tr(Foo,,KfJool),
so using (b) and (c) we find
(3.3.4)
e(E) = q-*kWdet(F,Eôë)
-2
ndet(-Foo',*i).
ses
To get (3.3.1) in this case, it remains to identify the local terms of (3.3.4) with
those of (3.3.1) for the choice of oj = —dx. Laumon does this by using a local
variant of the Fourier transform and a very nice result of Gabber and Katz [8]
to reduce to the case of tame ramification. The verification uses (3.3.1) in the
case Galois acts through a finite quotient (proved earlier by Deligne).
For applications of the product formula to the "inverse Hecke theory" (L-
functions —► automorphic forms), see [4], [7].
3.4. Simplified proof of "Weil II" (Laumon [14]). This beautiful proof does
not use the "Hadamard-de la Vallée-Poussin method" nor the calculations on the

162
LUC ILLUSIE
product of the curve by itself [6, §§2, 3]. It does rely, however, on two important
results of [6, §7], namely: (a) the theorem on the weights of the quotients of
the monodromy filtration [6, 1.8.4] and (b) the theorem on the purity of the
constituents of lisse t-real sheaves [6, 1.5.1].
The method is similar to that used in the proof of (3.3.1).
3.5. Representation theory. Deligne's Fourier transform is implicitly used by
Springer in his geometric construction of irreducible representations of Weyl
groups [17], see also [2, §11]. For a variant of this construction using the Fourier
transform for ©-modules, see [10].
Finally, let me mention that Lusztig uses Deligne's Fourier transform in his
recent study of character sheaves on reductive groups over finite fields, see [15,
Chapter 13, p. 350] and work in preparation.
ACKNOWLEDGMENTS. These notes owe much to a lecture given by P. Deligne
at the Institut des Hautes Etudes Scientifiques on June 23, 1982, where he
explained the definition and main properties of his Fourier transform as well as
Brylinski's application to the Radon transform. I am also greatly indebted to G.
Laumon for long and numerous discussions on his work. It is my pleasure here
to thank him heartily for his assistance in the preparation of this report.
References
1. A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analyse et Topologie
sur les Espaces Singuliers, Astérisque, no. 100, Soc. Math. France, Paris, 1982, pp. 1-172.
2. J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz,
transformations de Fourier et sommes trigonométriques, Preprint, Ecole Polytechnique, 1983.
3. J.-L. Brylinski, B. Malgrange, and J.-L. Verdier, Transformation de Fourier géométrique.
I, C.R.Acad. Sci. Paris Sér. I Math. 297 (1983), 55-58.
4. P. Deligne, Les constantes des équations fonctionnelles des fonctions L, Lecture Notes in
Math, vol. 349, Springer-Verlag, 1973, pp. 501-598.
5. , La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273-
307.
6. , La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137-
252.
7. , Les constantes des équations fonctionnelles des fonctions L, Séminaire Inst. Hautes
Études Sci. 1980-81, notes miméographiées.
8. O. Gabber and N. Katz, Canonical extensions of representations of fundamental groups,
Preprint, Inst. Hautes Études Sci. Publ. Math., 1984.
9. M. Goresky and R. McPherson, Intersection homology. II, Invent. Math. 72 (1983),
77-130.
10. R. Hotta and M. Kashiwara, The invariant holonomic system on a semi-simple Lie
algebra, Invent. Math. 75 (1984), 327-358.
11. N. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Preprint, Princeton
Univ., 1984.
12. N. Katz and G. Laumon, Transformation de Fourier et majoration de sommes
exponentielles, Inst. Hautes Études Sci. Publ. Math. 62 (1986), 145-202.
13. G. Laumon, Les constantes des équations fonctionnelles des fonctions L sur un corps global
de caractéristique positive, C.R. Acad. Sci. Paris Sér. I Math. 298 (1984), 181-184.
14. , Sur les constantes des équations fonctionnelles pour les fonctions L associées aux
représentations l-adiques (en égale caractéristique). I, II: suivi de une "nouvelle" démonstration des
conjectures de Weil, Preprint, Inst. Hautes Études Sci. Publ. Math., 1984; Transformation de

DELIGNE'S i-ADlC FOURIER TRANSFORM
163
Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Preprint, Orsay, 1986, Inst.
Hautes Études Sci. Publ. Math, (to appear).
15. G. Lusztig, Characters of reductive groups over a finite field, Ann. of Math. Stud., vol.
107, Princeton Univ. Press, Princeton, N.J., 1985.
16. S. Mukai, Duality between D(X) and D(X^) with its applications to Picard sheaves,
Nagoya Math. J. 81 (1981), 153-176.
17. T. A. Springer, Trigonometric sums, Green functions of finite groups and representations
of Weyl groups, Invent. Math. 36 (1976), 173-207.
18. E. Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.
19. Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4|),
P. Deligne, editor, Lecture Notes in Math, vol. 569, Springer-Verlag, 1977.
20. Cohomologie l-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie
1965-1966 (SGA 5), A. Grothendieck, editor, Lecture Notes in Math, vol. 589, Springer-Verlag,
1977.
21. Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique
du Bois-Marie 1963-64, M. Artin, A. Grothendieck, and J.-L. Verdier, editors, tome 3, Lecture
Notes in Math., vol. 305, Springer-Verlag, 1973.
Université de Paris-Sud, Mathématique-Bat. 425, Unité Associée au CNRS
NO. 752, 91405 ORSAY CEDEX, FRANCE

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Lifting Algebraic Curves, Abelian Varieties, and
Their Endomorphisms to Characteristic Zero
FRANS OORT
In the first part of this paper we give a survey of some results on the lifting
problem.
In the second part we review the theory connecting Dieudonné modules (in
characteristic p) and De Rham cohomology. This subject was started in the
thesis of T. Oda, cf. [29], and Messing, Fontaine and several others used this.
In [27] Norman used this theory in order to prove (again) lift ability of abelian
varieties. In §13 below we show that there exist abelian varieties in positive
characteristic which cannot be lifted (as abelian schemes) to characteristic zero
without introducing ramification. In order to be able to carry out the
computations we have produced some simple abelian varieties, cf. §12, and it seems an
interesting question, cf. (12.7), whether in any formal isogeny type, which is not
supersingular, there exists a (simple) abelian variety X with End(X) = Z.
In part four we discuss lift ability of endomorphisms of abelian varieties: in
general an endomorphism cannot be lifted, cf. (14.5), but if the dimension is
small, cf. (14.1), or the p-rank is large, cf. (14.2) and (14.6), endomorphisms can
be lifted. Thus it seems that the answers to all lifting questions treated here are
rather complete.
Part of this material was also treated in a series of talks entitled "Algebraic
geometry in mixed characteristics" at a symposium held September 25-28, 1984,
at Tôhoku University, Sendai, Japan. Some results of the second part were
obtained while the author visited Japan. I would like to thank the Japan Society
for the Promotion of Science for financial support, and I am grateful to Kyoto
University for hospitality and excellent working conditions.
With several colleagues I had interesting discussions on these topics; in
particular I would like to thank T. Oda, A. Ogus, B. van Geemen, P. Norman,
N. Nygaard, and K. Ueno for their interest and suggestions.
1980 Mathematics Subject Classification (1985 Revision). Primary 14K10, 14H10, 14L15;
Secondary 14D15.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
165

166
FRANS OORT
I. A survey.
1. An easy example. In [31] we find a discussion of the lifting problem,
and we find definitions (essentially due to Serre). In this section we give an
example where the lifting problem can be answered.
We recall the definition. Let A: be a field. Suppose char(A;) = p > 0 (and
throughout the paper we assume that A: is perfect). We denote by W = W^k)
the ring of Witt-vectors, i.e., the unique complete discrete valuation ring of
characteristic zero with W/pW = k. Suppose given an algebraic curve (complete
and nonsingular) Cq over A:, and an automorphism ao of Cq defined over A:. We
say that (Co,ao) can be lifted to R (where R is some domain containing VF,
admitting a homomorphism R —► A:) if there exists a (smooth, proper) curve
C —► Spec(iZ) and a G Aut^(C) such that
(C,a) ®r k = (Cb,ao);
in case this is possible with W = R, we say that (Co, ao) can be lifted in the strong
sense. Otherwise we speak of liftability in the weak sense. (These definitions
can also be given for other objects such as polarized abelian varieties, etc.; for
more details, cf. [31, §1].)
It may happen that a lifting in the strong sense (i.e., lifting to the unramified
ring W) does not exist, but that for an appropriate choice of R the lifting (in
the weak sense) does exist. In this section we provide such an example. To be
precise, suppose char(A:) = 5, and let Co be the complete nonsingular curve given
by Y2 = X5 — X. (The projective model of this, normalized, gives Co.) If A: is
algebraically closed,
#Autfc(Co) = 240
(cf. [12, p. 645]). We consider ao, /? € Aut(Co) (already defined over Fp) given
by
a° ' l Y ^ F, l Y ^ 2Y.
(Note that 22 = -1 e F5.) Clearly /T^o/? = a4; thus (a0) = N ^ Z/5 is
normal in (ao, /?) = G, and G/N = (/?) = Z/4. The following facts are true:
(l.a) Co can be lifted to W;
(l.b) (Co, ao) cannot be lifted to VF, but it can be lifted to Wfrp];
(l.c) (Co,G) cannot be lifted to any integral domain in characteristic zero.
In fact, the first statement follows either from the observation that the equation
also can be used to define a smooth complete curve over any field of characteristic
not two, or it follows from deformation theory (every smooth curve can be lifted,
cf. §3 below). The morphism
Co - Cq/N = D0
is a cyclic Galois covering, and its conductor has multiplicity 3 at
oc = P ^ Q e D0, c = SQ

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 167
(as easily follows). It can be proved that because this multiplicity 3 = n\ is
smaller than p — 1 = 4 it follows that the pair (Co,ao) cannot be lifted to W
(adjoining to W the coordinates of the branch points of a would give a ramified
extension of W of degree at most 3 on the one hand which contains the fifth root
of unity ç5 on the other hand, a contradiction, cf. [41, Appendix, Lemma A.l]
for details; it also follows from [25, §4, Theorem 3]; this shows the first claim in
(l.b). To show the second claim we produce such a lifting over VF[ç5]. Let C be
given by the equation:
Y2 = X(X-l)(X-l-ç)(X-l-ç-ç2)(X-l-ç-ç2-ç3), with? = <;5
(cf. [12, p. 645]), and let an automorphism of this curve over R = W[ç} be given
by
fX^çX + l,
It is clear that this is an automorphism, and that (C, a) ®r k is isomorphic with
(Co, ao); thus we have proved the second claim in (l.b). By the way, note that
the three fixed points of a on C are the point {1/X = 0, Y = 0) = oo, and the
points with (X = 1/(1 - ç), Y = ±(1/(1 - ç)5)1/2); we see that the conductor
under the lifting splits in three different points with multiplicity one.
Next we consider (l.c). Suppose the group G as above would appear as a
group of automorphisms of a curve C with genus g(C) = 2 over a field K with
char(ir) ^ 5. Then
C -* C/N = D-+ C/G = F.
One easily sees that C —► D ramifies at three points Qi, Q2, Q3 € D and D = P1
(use the Zeuthen-Hurwitz formula for the covering C —► D). Thus (/?) = Z/4
would operate on {Qi,Q2,<23} € P1, and this is a contradiction. This proves
(l.c).
Note that in §8 of [12] we find a list of all possibilities for Aut(C) in case
g(C) = 2, and we could have used that. Also note that it is easy to give
Co and G C Aut(Co) which do not lift to characteristic zero: any example
with #G > 84(<7 — 1) will do (and there are many of such examples in positive
characteristic, cf. [39]).
2. Some subtleties in characteristic p > 0. (2.1) We try to deform a
separable covering C —► D of algebraic curves. If there is no wild ramification
(e.g., char (if) = 0, or for every P G C the inertia group Ip has an order
not divisible by char(A:)) this deformation problem is smooth: we can deform
D and the branch locus arbitrarily and the covering follows this deformation
uniquely (this is Riemann's existence and uniqueness theorem if we work over
the complex numbers). However as soon as wild ramification appears we have
to be more careful (in characteristic p > 0, or in mixed characteristics). For
example, the Riemann theorems should be formulated with more care: otherwise
generalizations would not hold in positive residue characteristic, and in general
the related deformation problem is not smooth. In the previous section we have

168
FRANS OORT
seen the example (shown in Figure 1) that Co = SQ = conduct (Co —► A)) splits
into a conductor consisting of three different points: the related deformation
problem (in mixed characteristics) in this case is certainly not (formally) smooth.
char(fc) = 0 char(A:) = p
Figure l
Thus wild ramification gives rise to conductors with multiplicities greater than
one. Such cases have to be treated with more care than the analogous tamely
ramified ones.
(2.2) Suppose K is algebraically closed, and let N be a group scheme of rank
p over K. If char(if ) ^ p, then N = Z/p. If however, char(if ) = p, then
N = Z/p, or N = fip, or N = ap. Consider a group scheme X, and a finite
subgroup scheme N c X; we try to deform this inclusion, keeping X fixed. In
case char(jFC) = 0 the situation is rigid. In case X is an abelian variety and
N = Z/p or N = fip the situation is also rigid; hence these finite group schemes
have "the same behavior" as finite group schemes in characteristic zero (and
this "explains" the theory of canonical liftings, cf. §4). However an inclusion
olv — N C X need not be rigid, and consequently one can produce examples
y —► P1 of families of abelian varieties which are nonconstant (cf. [35, pp. 102-
103]). Thus abelian varieties which contain subgroup schemes isomorphic with
olv have to be treated with more care in deformation theory.
We now sketch some methods developed in the past, describing deformation
and lifting problems
3. Deformation theory. The ideas of Kodaira and Spencer concerning
deformation theory (cf. [15]) were worked out by Grothendieck (cf. [10, Exp. 195]),
Schlessinger (cf. [40]), and many others. This direct approach works out well in
several cases. In case of algebraic curves we have
H2(C0,eCo) = 0;
hence obstructions vanish, and the formal deformation functor is formally smooth
(cf. [10, Exp. 195, Theorem 10]). In case of abelian schemes Grothendieck showed

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 169
that the obstructions vanish (cf. [31, §2.2]), the deformation functor is formally
smooth, and the tangent space is isomorphic with
H1(X0,eXo) = TXo®Tx<).
In case dim X is at least two in general formal deformations are not algebraizable
(this also occurs in case of deformations over the complex numbers). In order to
obtain algebraic deformations we apply the Chow-Grot hendieck method: Chow
proved that an analytic manifold which can be imbedded into projective space in
fact is an algebraic variety (cf. [43,§19]); the analog in formal deformation theory
was proved by Grothendieck (cf. [11, Theorem 5.4.5]). Hence we are looking for
formal deformations which can be embedded into projective space. Consider a
polarization
Ao: ^o —► ^o-
The formal deformation problem of (Xo,Ao) is pro-represent able. Moreover if
this polarization is separable, then
dX0 : TXo —► Txt
is an isomorphism, and the "Riemann symmetry conditions" describe the
deformation space of (Xo, Ao) which in this case (a separable polarization) is formally
smooth. (This was proved by Grothendieck, cf. [31, Theorem 2.4.1]; also cf. [4].)
Thus
Method I: deformation theory
works well in the unobstructed case, e.g., for algebraic curves, for separably
polarized abelian varieties (and in particular for principally polarized abelian
varieties).
4. Canonical liftings.
DEFINITION. Let char (A:) = p > 0. We say that an abelian variety X over A:
is ordinary if
olv </+X.
An algebraic curve C over k is said to be ordinary if Jac(C) is an ordinary abelian
variety.
EXAMPLE. For every prime number p there exists hp isomorphism classes of
elliptic curves which are nonordinary, and all others are ordinary (and in fact,
[p/12] < hp < [p/12] + 2). If p = 5 the value jo = 0 belongs to a nonordinary
(=supersingular) elliptic curve, and all other elliptic curves in this characteristic
are ordinary.
DEFINITION. Let char (A:) = p > 0, and let X be an abelian variety over A:.
We see that
X[p](k) - (Z/p)', 0 < / < g = dim*,
and we call / = prank(X) the p-rank of X. It is not difficult to show that
X is ordinary <==> f = dim X.

170
FRANS OORT
Note that if X —► B is a family of abelian varieties in characteristic p > 0,
then the p-rank of the fibers is lower-semicontinuous. One could hope that any
polarized abelian variety in positive characteristic is contained as a fibre of an
irreducible family such that the generic (and hence the general) fibre is ordinary.
In fact this is true, as Mumford showed, but the proof is not easy; below we
come back to this question.
As we have argued in §2, ordinary abelian varieties in characteristic p should
behave as in characteristic zero. In fact:
THEOREM (SERRE AND TATE). Let Xo be an ordinary abelian variety over
a perfect field k of char (A:) = p > 0. Then there exists an abelian scheme
X —► Spec(VF) characterized by the fact that BT(X), the p-divisible group of X,
splits as BT(Xo) does\ moreover every endomorphism of Xq lifts to X :
Endvy(I)^Endfc(X0),
and every polarization of Xq lifts to X {and X jW is called the canonical lifting
ofXo/k, cf. [17; 21, pp. 171-174]; also see the proof by Drinfeld in [14]).
EXAMPLE (SERRE, CF. [44, p. 318-12]). The existence of the canonical
lifting is rather straightforward, but in concrete cases it is not so easy to give
it explicitly. As an example we quote some examples computed by Serre: let
A: D F5; then jo = 0 is the only supersingular ./-value, and for the remaining
values for jo in ^5 we have:
\p = 5, jo =
j (can. lift.)
1
2333113
2
_215
3
2633
4
- 21533
The canonical lifting is more arithmetical than geometric in its flavor.
Suppose Co is an ordinary curve (i.e., an algebraic curve such that its Jacobian is
an ordinary abelian variety), and let (X,X)/W be the canonical lifting of the
canonically polarized Jacobian variety Jac(Cb) = (Xo,Ao). Is this (over some
extension of W) again a Jacobian? This is a natural question if one wants to
produce curves with quite a lot of automorphisms. For g < 3 the answer is
affirmative (because any polarized abelian variety of dimension at most 3 is the
Jacobian of a "good curve" over some extension field, cf. [52, 38]). However for
larger values of g this is not the case. Recently two approaches were given to this
problem. Dwork and Ogus in their paper [8] study the infinitesimal Schottky
relations, and compare this with the properties of the canonical lifting of an
ordinary Jacobian; their result is that for p > 3 and g > 4 for the generic curve the
canonical lifting of the Jacobian is not a Jacobian. Independently Sekiguchi and
Oort showed the same result for p > 5 and g > 2(p— 1) by producing examples of
nonhyperelliptic curves which are p-cyclic coverings of P1 with low multiplicities
in the conductor; these curves with the automorphism which generates the
covering group cannot be lifted to the ring VF, and one concludes that the canonical
lifting of the Jacobian is not a Jacobian. The larger bounds in this approach
were necessary in order to be able to produce the examples, cf. [37].

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 171
Thus we have seen that
Method II: the theory of canonical liftings
works for ordinary abelian varieties.
5. Deform to ordinary abelian varieties. In 1976 Mumford developed a
program which can be used to prove liftability for polarized abelian varieties from
positive characteristic to characteristic zero. The idea is to deform (Xo, Ao)
(staying in characteristic p) to an ordinary situation, and then the theory of canonical
liftings provides the desired answer because of the following observation.
LEMMA. Let (Xq,Ao) be a polarized abelian variety, which is a fibre in a
family (JC, À) —► Spec(B), where B is an integral domain in characteristic p.
Let K be a perfect field containing B, let (Yb, /io) = (-£, A) ®b K, and suppose
that (Yo,/io) can be lifted to characteristic zero; then (Xo,Ao) can be lifted to
characteristic zero, cf. [28, pp. 430-431].
In this program the most difficult step is the deformation to an ordinary
situation. Mumford indicated in [23] how the problem can be translated into
linear algebra. The main difficulty in this aspect of the theory is the fact that
the formal deformation space of (Xo,Ao) need not be smooth. In [28] this
program was carried out, by showing that in a particular case one can give
a large (reduced) subspace of the formal moduli space with ordinary generic
fibre. The final result is:
THEOREM (MUMFORD). Any polarized abelian variety can be lifted to
characteristic zero.
Method III: first deform in characteristic p to an ordinary situation.
In analyzing the proof we see that we did not obtain information over what
kind of ring the lifting can be achieved or how much ramification is needed:
using the notation of the previous lemma, suppose that B = k[[T]] (which we
can achieve in all cases); then a lifting of (Xq, cr0) can be achieved over a domain
R contained in W^K), where K is the perfect closure of the field of fractions
of fc[[!T]]; note that this may have a large degree over Woo (A;). Thus there was
the need for a different approach, which we discuss below.
Note that the methods in [28] actually gave more information about the
moduli space Ag ® Fp of polarized abelian varieties in characteristic p. It turned
out that every component of this moduli space has dimension equal to g{g +1)/2
(as it should), and that the stratification by p-rank is very neat. Let Vgj = Vf be
the closed subset of Ag ® Fp containing points corresponding to abelian varieties
having p-rank at most /; in [28, Theorem 4.1] we find that every irreducible
component of Vf has dimension equal to {(g(g + l)/2) — g + /.
6. Dieudonné modules and De Rham cohomology. Results in the
thesis of Oda (cf. [29]) comparing the Dieudonné module of Xo[p] and H^r(Xq),

172
FRANS OORT
and methods of Messing, cf. [21], and Fontaine, cf. [9], translated the lifting
problem of polarized abelian varieties in characteristic p > 0 into a problem of
constructing certain VF-submodules of M, the Dieudonné module of BT(Xo).
Method IV: via crystalline cohomology compare the Dieudonné
module of Xq with the De Rham cohomology of X.
This enabled Norman to show (cf. [27]):
THEOREM. Let (Xo, Ao) be a polarized abelian variety over a perfect field of
characteristic p. Let R be a complete discrete valuation ring with that field as
residue class field, with ramification index e = v(p) with
2 <e <p-l.
Then (Xo, Ao) can be lifted to R.
Below we give more details on this method, plus some examples constructed
in this way.
7. Lifting automorphisms of curves. Let char(A:) = p > 0, and let
00 G A ut (Co) be an automorphism defined over this field. We would like to
decide whether this can be lifted to characteristic zero (and it seems reasonable
to expect this is possible for every automorphism of an algebraic curve). There
is the following partial answer to this question:
THEOREM (CF. [41, Corollary 5.6] ). Suppose moreover that
p2 \ order(cr0).
Then (Cb,cro) can be lifted to W[<;p].
In the proof of this the following method is used (for details we refer to the
paper itself). Any abelian covering of an algebraic curve is given, by class field
theory, by a covering with the same group of the generalized Jacobian of that
curve given by the conductor of the covering. In order to perform the lifting,
one first lifts the curve, Co/(ao) then one lifts the support of the conductor,
and then one tries to construct the whole diagram which connects the covering
of the curve and the covering of the generalized Jacobian also in characteristic
zero. This is done step by step, involving a nasty computation on deformations
of Artin-Schreier coverings (in characteristic p) to Kummer coverings (in
characteristic zero); it is at this point that the technical condition that p2 should not
divide the order of cr0 comes in; then the proof is concluded using cohomology
calculations by Breen, which prove that the desired diagram can be completed
in characteristic zero. This seems to be a very special case of:
Method V: use class field theory in mixed characteristics.
It seems that in this direction the theory should be further developed (e.g.,
arriving at some satisfactory generalization of the Riemann existence and uniqueness
theorems).

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 173
8. Survey. The examples and methods discussed in this paper can be
summarized as follows (and comparing with [31, §1], we see some of the progress
made):
lifting :
algebraic curve
separably polarized AV
ordinary AV
polarized AV
AV
Co and G C Aut(C0)
Co and <jq G Aut(Cb)
Xq and ao e End (Xq)
toW
to some R
+ 4 +
+ 4 +
+ 4 +
-
-
+
+
~~ T ~
-
+,?
~ i
cf. sections
3
3
4
5,6,10,11
6,13
1
7
14
here the "—" means: in general not liftable, and the question mark concerns
automorphisms of arbitrary order.
II. Lifting polarized abelian varieties and De Rham Cohomology.
9. Dieudonné modules. We suppose A: is a perfect field, with char(A:) =
p > 0, and we use the covariant Dieudonné module theory. Thus we take W =
Woo(fc), the ring of Witt vectors, and we work over the ring A = W[F][[V]], and
the functor M associates to a finite group scheme of p-power rank a finite length
module over this ring, and to a p-divisible group of height h a module over this
ring which is free of rank h over W.
NOTATION/APOLOGY. In [29] the contravariant Dieudonné module theory
is used; by Definition (3.12) on p. 85, and by Proposition (3.13) on p. 86 of that
paper we see that the contravariant theory "transforms F into F and V into V."
However in [27] we find the covariant theory. In most cases it does not make
much difference which of the two approaches is used. In cases we consider here
duality interchanges the two theories. For these notes I have chosen to use the
covariant notation; some diagrams look less natural than they are in fact; to
colleagues used to working with crystalline cohomology I would like to apologize
for "too many D's appearing in the diagrams." Thus we arrive at a theory where
the Dieudonné module functor "transforms F into K, and V into F." It is in
accordance with [27], and in order to compare with [19] and with [29] one has
to apply duality.
By superscript D we mean dual (in the VF-linear or in the A:-linear sense). By
X1 we denote the dual of an abelian variety (by some authors denoted by X),
and we use the notation X for the formal group attached to the abelian variety
X. We use G* for the "Serre-dual" (cf. [44, p. 318-403]) of a p-divisible group.
Let Xq be an abelian variety over A: with a polarization Ao : Xq —y Xq . Related
to this is the (covariant) Dieudonné module M = M(BT(Xq)), where BT(Xq)
is the p-divisible group of Xq. This module carries a bilinear pairing (coming

174
FRANS OORT
from Ao)
(-,-): M xM -^W
which has the properties that it is nondegenerate (i.e., the associated map M —►
MD is injective), that it is skew-symmetric, and that
(Fx, y) = (x, Vy)a for x,y G M.
By [29, Corollary (5.11), p. 131] we see that we have an exact, commutative
diagram
0 - (H°(X0^Xo/k))D - (H^R(Xo))D - (^(X0,Oxo))D ^ 0
0 — M{X[F\) — M(X[p]) — M(A-[V]) — 0
0 *- FM0 = Mo/VMo +- M0 *- KM0 = M0/FM0 *- 0,
where the symbols in the middle row are explained by the exact, commutative
diagram
0 0
X[V] = X[V]
0 ► X[p] ► X —£—► X > 0
V
0 ► X[F] ► X^/p) —^-+ X ► 0
0 0
Note that we work over a perfect field, so we have isomorphisms like M^ = M,
which are used in several places in the above diagrams.
10. Crystalline cohomology. We keep the notation of the previous
sections, and we suppose that R is a complete discrete valuation ring with absolute
ramification
v(p) =: e{R) <p-l.
Suppose (JC, A) is a polarized abelian scheme over Spec(iZ) lifting (JVq, Ao) over
A:. Then by the theory developed by Messing (cf. [21, Theorem 1.6, p. 151 and
his Definition (1.8) on p. 119]), by Berthelot (cf. [1]), by Mazur and Messing
(cf. [20, Chapter II, §9]), by Fontaine (cf. [9, Theorem 2, p. 217 and the remark
on p. 218]), and by Berthelot and Ogus (cf. [2, (7.26.3), p. 7-34]) we find:
^cris(^o) = HDR(X/R)',

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 175
further
(H^Xo))» = (M := M(X0)) ® R,
the bilinear form on M given by Ao is the same (under these identifications) as
the one on (Hdji(X / R))D given by A, and we obtain
0 ► {H\X,0X))D ► (HhR(X/R))D
V ► (M := M{BTX0)) ® R,
which we use to define V C M ® R. Note that (Lie(.T*))D = V. From the
literature given above we deduce the following theorems.
THEOREM (lO.l). Ife(R) < p— 1, if (X,\) is a polarized abelian scheme
lifting (JVq, Ao), and if
V C (Af := M{BTX0)) ® R
is defined as above, then
(i) V is free of rank g over R,
(ii) V/pV = VM{BTX0) 0 {R/pR), and
(iii) V is totally isotropic for (—, —).
Note the marvel of this theorem: from (JC, A) we deduce V C M ® R and
properties which can be stated in terms of (JVq, Ao) over A:.
The first two statements follow easily from the literature cited, and the third
follows because (—, —) given by A and (—, —) given by Aq are identified:
0 >(H\XM)D *{HhR{X))D—HH°{X^\/R))D-
-0
(Lie!
t\D
(A*)1
M®R
Lie(Z)
(LieI)D
^(^r(^*))D
->Lie(.T)
->0.
We see that V c M (
implies (iii).
) R is mapped to M * (8) R
(AoJ
M* ®R
V D by the zero map, and this
THEOREM (10.2). Let e(R) < p—l, and let (JVq, Ao) be a polarized abelian
variety over k, or e(R) < p—l and (Xo, Ao) is a polarized p-divisible group over
k which is of local-local type; suppose that y C M ® R is a W -submodule of M
satisfying the properties (i), (ii), and (iii) stated in (10.1). Then there exists a
unique lifting (JC, A) to R related to y C M ® R in the way given above (cf. the
precise references to Messing [21] and Fontaine [9] above).

176
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This (highly nontrivial) theorem enabled Norman (cf. [27]) to prove:
THEOREM (10.3). Let p > 2, let (Xq, Xq) be a polarized abelian variety over
k, and let R be as above, with 2 < e(R) < p — 1. Then there exists a polarized
abelian scheme (X, A) over R lifting (Xo, Ao).
The proof uses Theorem (10.2), which reduces the difficulties to "linear
algebra about M," and the crucial step is a choice of a good VF-base for M, cf. [27,
Main Theorem, p. 434].
COROLLARY (10.4). If a{X0) < 1, then {X0,X0) can be lifted to W. We
use the notation that a(Y) for an AV Y in characteristic p > 0 is the dimension
of the k-vector space Hom(ap, Y).
PROOF. Let M = M({BTX0)u). Because M can be generated by one
element, we can use [28, Theorem 2.2] and define V as the VF-space spanned by
Yi,..., Yn. Thus the corollary follows from (10.2).
COROLLARY (10.5). For any abelian variety X defined over a perfect field
K there exists a finite extension L D K and an abelian variety Yq defined over
L isogenous with X <g> L so that for any polarization /io on Yq the pair (Yb,/io)
can be lifted to W{L).
PROOF. By [34, Theorem 1.1] we can achieve X ® L ~ Y0 with a(Y0) < 1,
and we apply (10.4).
11. An example by Ogus (cf. [27, §4]). Let E be a supersingular curve
defined over the prime field Fp (such curves exist for every prime p, cf. [50,
Theorem 4.1.5, p. 536], take 0 = 0). Let A: be a perfect field containing Fp2, and
let H = Endk(E). Then
#®Zp = Wroo(Fp2)[F], with
in this ring. The Rosati involution given by E -^ El induces an involution on
H ® Zp, which is given by
Fv^-F, av^a? for a G ^oo(Fp2).
Let Xo := E x E. A polarization on Xq can be given as on p. 208 of [24] by
NS°{X0) ^{ae End^Xo)!^ = a} ,
and we choose
X°"(-F o)€End(x°)-
CLAIM (OGUS). The pair (X0, Ao) as constructed above cannot be lifted to
W = Woc(k) (i.e., ramification is needed).
PROOF. Note that F on E acts as V on M(BTE). We see that
M{BTE) = W e + W Ve, Ve = -Fe,
F=-V

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 177
the polarization Ao gives the pairing (—,—) on M = M(BTX0), and for M we
can choose a W-ba.se X\ = e, X2 = /, Y\ = Ve, Y2 = Vf, and the pairing is
given by
(XUX2) = 1 and (YuY2)=p;
in this way we obtain a base for M as described in [27, p. 434]. Clearly
y0 = (WY1 + WY2) ® k C M0 = MjpM
cannot be lifted to an isotropic subspace of M ; thus by Theorem (10.1) the pair
(Xo, Ao) constructed above cannot be lifted to W.
PROPOSITION (11.1). If Xq is an abelian variety, dim(X0) < 2, then there
exists a polarization Ao on Xq so that (X0, Ao) can be lifted to W.
PROOF. If prank (Xo) = 2 we conclude by the canonical lifting. If prank(X0)
= 1 then a(Xo) = 1, or in case prank(Xo) = 0 and a(Xo) = 1 we conclude by
Corollary (10.4). If prank (X0) = 0 and a{X0) = 2 then by [35, Theorem 2], we
know Xo = Ei x E2, and we can choose a principal polarization on Xq. Q.E.D.
REMARK (11.2). If prank(X0) = 0 and a(X0) = 1 it can be shown that X0
admits a principal polarization. It is easy to see that there exist abelian surfaces
with prank = 1 or prank = 2 which do not admit a principal polarization.
REMARK (11.3). If dim(X0) = g and a(X0) = 1, the local deformation
space D of (X0, Ao) contains a formally smooth subspace of dimension g{g +1)/2
over W (use [28, Theorem 2.2, p. 422]), but in general this space V is not formally
smooth.
III. Do unramified liftings exist?
12. Construction of some simple abelian varieties. In the next section
we are going to construct examples of abelian varieties in positive characteristic
which cannot be lifted to characteristic zero to an unramified domain. For this
purpose we need specific abelian varieties which have no endomorphisms not
equal to an integer. This section is devoted to the construction of such examples.
Let K be a field, char(if ) = p > 0, and let X be an abelian variety over
K. Let k be an algebraically closed field containing K. We consider the formal
group X associated with X, and we study the related formal group
G = X ®K k.
In the theory of Dieudonné and Manin certain formal groups were defined. For
integers n and m, with n > 0, m > 0, (n, m) = 1, and the pair (n = 0, m = 0)
excluded there is defined a p-divisible Gn>m. In fact this group is already defined
over the prime field Fp. Further the dimension of Gn,m is n, and the dimension
of the dual (Gn>m)* is m; in fact
for a definition of these formal groups, cf. [19, p. 35]. These groups are simple
(up to isogeny), the group Gi,o is the completion of Gm, the multiplicative linear

178
FRANS OORT
group, the group Go,i is a limit of étale groups, in fact Go,i = Zp (the pro-finite
constant group scheme), and for n > 1 and m > 1 the group Gn,m is of local-
local type. These are the only simple p-divisible groups, up to isogeny, over an
algebraically closed field in positive characteristic. Thus up to isogeny we can
write
G~ 2jGni,mi.
The set of pairs J2(niimi) 1S called the formal isogeny type of X. It has the
property that if Gn,m appears with m > 0, then Gm,n appears with the same
multiplicity ("Manin's symmetry condition," cf. [19, p. 73; 30, pp. III.19-3/4]).
Honda and Serre proved that any formal group satisfying this symmetry
condition is algebraizable, i.e., comes from an abelian variety (cf. [19, p. 76; 48,
p. 352-404]). We are going to construct first an abelian threefold with p-rank
equal to one, and with no CM.
CONSTRUCTION ( 12.1). Let E be an elliptic curve over Ki D Fp with
EndjdiE) = Z, and let E2, E$ be supersingular curves over K\. Note that we
write V\ C As for the subset of the moduli space of all polarized abelian varieties
(y,/x) with dim(y) = 3, defined over a field of characteristic p, and
prank(y) < 1.
Let W be an irreducible component of V\ containing E x E2 x Es with some
principal polarization on it. Let (F, /i) over K correspond to the generic point
of W. We would like to show that Y has no CM. In characteristic zero such a
result would follow from an easy dimension count; in positive characteristic we
have to be more careful because of the existence of continuous families of abelian
varieties such that all fibres are mutually isogenous.
CLAIM( 12.1.1). The abelian variety Y is absolutely simple.
(12.1.2) For any field k D K, Endfc(y ®K k) = Z.
PROOF. Assuming the first statement, the second follows: specializing Y to
E x E2 x Es we obtain a ring homomorphism
R = Endfcl(yfcl) ► Endk{E x E2 x E3)
Endk(E) x Endfc(£2 x Ez)
and because Hom(i£, Ei) = 0 any such endomorphism is in block form on E x
(E2 x Es)- Thus we obtain a Z-algebra homomorphism
i2 = Endfcl(yfcl)-+Endfc(£) = Z,
and y being absolutely simple, the ring R is an integral domain of finite rank
over Z, and we conclude R = Z.
Now we come to a proof of the first statement. Suppose Y is not absolutely
simple. Replacing K by some finite extension we achieve X\ C Y and X2 C Y
over K with dim(Xi) = 1 and dim(X2) = 2, and Xi x X2 -» Y. Replacing W

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 179
by a finite covering, normalizing this, we can extend Xi, X2 and Y as abelian
schemes:
Xi
y ► w
and W —► A finite onto its image. Note that W is (a finite covering of) a
component of V\ C A3; thus (cf. [28, Theorem 4.1, p. 434]) we know
dim W = 3(3 + l)/2 - (3 - 1) = 4.
From Xi —► W we obtain /1 : W —► ^, with i = 1,2. Note that either (first case)
prank(Xi) = 1 or (second case) prank(Xi) = 0. In both cases
dim(/i^) + dim(/2W0 <2
because, in the first case prank(X2) = 0, and
dim/1 (WO + dim/2 (WO < dim^i + dimV2,0 = 1 + 1 = 2,
and in the second case
dim/1 (WO + dim/2(WO < dimFi,0 + dimV2,i =0 + 2 = 2.
Take
{Z1,Z2)eImf(W)cA1 xA2
(forgetting polarizations), and let T C W be an irreducible component of all
points in W mapped onto /_1(Zi, Z2). Because of our computation dimT > 2.
Over T we have X := y \ T,
Zi xT
X -+ T
/
Z2 xT^
so that [Z\ x Z2) x T maps onto X . Let M be the kernel of this mapping, i.e.,
M :={Z1 xT)n(Z2xT)cZ;
this is a finite flat group scheme M —► T (by [22, Lemma 6.12, p. 122]). Replacing
T by a finite covering of a nonempty open subset T' of T, we can write
M = M® Mu
where each fibre of M —► T" is a local-local group scheme, and for Mi each fibre
is a direct sum of a local-étale and an étale finite group scheme.
In the first case the fact prank(Zi) = 1 implies that M = 0; thus Mi is
constant over T" (being a subgroup scheme of a constant abelian scheme, hence
being rigid). Thus in that case
X = ((ZixZ2)/Mi)xT"

180
FRANS OORT
is constant over T",
W JeOL> A3
u
finite
V < T"
and we conclude that the map T" to A3 is constant, a contradiction with dimT >
0; thus the first case does not occur.
Suppose the second case. Then
M *-> Zx x T" and M *-> Z2 x T".
Because Z\ in this case is a supersingular elliptic curve, we see that there exists
a nonnegative integer i so that
M = NixT"*->Z1x T",
where Ni is defined as the finite group scheme
Ni:=Ker(Fi:Z1-+z[pi)).
Note that Z2 = Gx H, with dim G = 1 = dimiJ, G = Zx.
LEMMA (12.2). Let E be a supersingular elliptic curve, Ni := Ker (F{ on E),
with i a positive integer, and let S be a scheme in characteristic p. If
a e Auts{Ni x S),
then locally on S, this automorphism is of the form X^=o 0tj^3 suc^ ^at for
0 < j < i — 1 we have ai G Fp2, and ai-i E T(U,0s); if moreover S is
connected, the restriction of a to Ni-i x S is constant.
The proof is an easy computation with Dieudonné modules, which we omit
here.
Suppose we are in the second case. Choose a point t ET". Then <pt: Ni —►
Zi x Z2, Ni C Zx and Ni C Z2; thus Ker (Fi on E) with Zx-E and the same
on G are identified with N{. The homomorphism <p
Ni x T" —Z-+ ZiXZ2x T"
Ni x T" c > Z2 x T"
therefore gives an automorphism a of Ni x T". We see that on the connected
part of T" containing t the restriction of a to -/V»_i x T" is constant, and o^_i
is described locally by a section in Ot" (apply 12.2). Thus there exists a closed
subset C C T" of codimension at most one which <p | (Ni x C) is constant. Then
X\C=(Z1 xZ2xC)/Im(<p\C)
is constant, and because dimT" > 2 we see dimC > 1, and we derive a
contradiction as before. This proves (12.1).

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 181
REMARK ( 12.3). It seems reasonable to expect that for any 0 < / < g, for
any irreducible component Z of Vg j C Ag (in characteristic p > 0) the abelian
variety corresponding with the generic point of Z is absolutely simple (and note
this is not so if we only suppose 0 < / and 2 < g). The method above can be
generalized immediately to the case of any g > 3 and / = g — 2, producing Y
with dim y = g, and the properties as in the claim (12.1.1/2).
LEMMA (12.4). Let n > 2 be an even integer, suppose Xq is an AV in
characteristic p > 0 with a principal polarization X, and suppose that its formal
group equals
Xo = Gn,i © Gi,n.
Then there exists a deformation (X,A) over an integral domain R D Fp such
that the formal isogeny type of the generic fibre X of X is (n, 1) + (l,n), and
such that for any field K D R we have
End{X ®K) = Z.
PROOF. We use methods developed in [28, §1]. The D-module (Dieudonné
module) of Gn,i is
M = A/A(Fn - V), with A = W[F][[V}}
and W = W(k), where k is the base field. (Remark: we use the covariant
D-module theory; hence the notation differs slightly from [19, p. 35]). Thus
with
^& = &+i, l<i<n,
F£n = Wn+1,
r?n+i = V£i.
The principal polarization À : X —► X1 is on Gn,i © Gi,n in block form, and via
the isomorphism in that matrix we identify Gn,i and (Gi,n)*. Thus X0 and À
are given by:
with
F£i = &+i)
F£n = Wn+1,
F£n+1 = 7?1,
r)i = Vrfi+1,
Vn = ^£n+l,
Vn+1 = V£l,
1 < i < n,
1 < i < n,

182
FRANS OORT
and the quasipolarization is given by
(&, *lj) = hj and (&, & =0= fa, ^).
We follow the notations as in [28, pp. 422-423]; we choose a deformation of the
quasipolarized formal group (Xo, A) (and hence a deformation (X, A) of (JVq, A))
defined by:
T = Ti2=T2i, Ttf = 0 ift+i#3;
here fc[[£]] is the ground ring, and
^& = &+i, l<i<n,
nn+i=r/i+r6,
(*) *H=Vfa2+T&),
r?i = Fr?i+i,2 <i < n,
r?n+i = V£i
and the same relations as above define the pairing (—,—) on the deformed
module. This is a deformation of the quasipolarized D-module. We show that
the deformation X —► Spec(fc[[£]]) thus obtained satisfies the requirements of the
lemma.
Let R D k[[t]] be an integral domain with field of fractions if, and let a G
End (X <8>R). We are going to show that a G Z (is an integer). We suppose that
k is algebraically closed. Because X = X ® k((t)) is algebraic we may assume
that K is finite over k((t)); thus R = k[[s]] with sm = t for some fixed positive
integer m. Let
B = W(R)[F][[V}1
and let M be the D-module of XR. This is given over B by generators and
relations as in (*). We have an exact sequence
0-+££i-+.M-+££/-+0
with £ = £n+i and £' := £modi? • £i. Note that
{Fn - V)Éi = 0 and {Vn - F)£ = 0.
Thus B • £i is the D-module of Gn,i, and B • £' is the D-module of Gi,n. Thus
the formal isogeny type of X is (n, 1) + (l,n). (This can also be proved by
computing the Hasse-Witt matrix of X from (*), concluding prank(X) = 0,
and using semicontinuity for the Newton-polygon under specialization, cf. [5,
Theorem, p. 91].) Because n > 1
Hom(Gn,i <g> R, Gi,n <g> R) = 0;
thus
(B • 6 ^ M -^ M - S • £') = 0,

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 183
and we obtain a commutative exact diagram
0 ► B ■ 6 ► M ► B ■ ?
ai
0
B-6
M
B-e
with ai := a \ (B • £1) and a' := amodB • £1.
We write (remembering £ = fn+i):
i=0
i=0
Note that
(2)
We write 9 = pn+1. Because
End(Giin) = W(Fq)[[F, V]]/(Vn - F)
(over any integral domain containing Fq) we have bo,bi,... ,bn £ W(Fg); the
same arguments give xq, Xi, ..., xn G W(Fg). Note further that
(F-Fn)-£=(TF + FT)-£i.
Thus
(3) a ((F - Vn) ■ 0 = (F - Vn) - o(0 = (TF + FT) • <*'(£')•
Using (1) we collect in the equality (3) the coefficients of F*fi, 0 < i < n. We
use:
(TF + VT) (E*^i)
n—1 n
(here a is "raising to the pth power of all coordinates of the Witt-vectors" and
n-1
^(Ea^0 =Ea?F<+1^+a»^'
0
Frjj = prjj^x, 2<j<n,
Frfn = pfn+i

184
FRANS OORT
and
VnVn = V2T)2 = Pt ~ TT^ - T'pU,
Vnr)n-X = V3T)2 = PVn ~ TT'Wn+1 - T'Vn-l (if n > 2)
Vnr,n-j = V3+2T)2 = PVn-j-1 ~ T^+V £„_,•+„ " T*'*'p*+l tn-j
if n-2>n-j > 1,
Vnfh=pt?a-TrV-1&
(we have extended R so that these entries are defined). After rearranging terms
we obtain:
(4.1) (pxn - pb°)T + [px\ - pb\n)TT = p« - p"-1^"),
(4.2) (*g - bl)T + {pxl)TT + {-jT-X-iW = aZ- p""1^",
(4.y) (*7_3)r + (p^T)TT + (-Pn+1-''Cw)rr
3 < j < n _j_ (_nn+2-jhTn mn+3-»
n+2-y
(-p-^-^Ca-y)^"
— na —r)n~1nTn
,n-l„rn
(4.(n + 1)) «_X)T + (x5 - bl )TT + (-p65 )TT = <_x - p"~X-i
(where (4.i) compares coefficients of Fl_1£i in (3)). Observe that the coefficients
on the left-hand sides of these equations are in W(Fq). Hence these equations
are of the form
u0T + mTT + • • • + UiTTt + • • • + unTrn = aCT - p^V"
(note: n > 1, thus p divides pn_1), with i^ G W(k), and a G VF(fc[[s]]); it follows
that
uq = 0 = u\ = • - - = un = a.
This we see as follows: suppose not all U{ equal to zero, let m be the largest
nonnegative integer so that pm divides all I*;, 0 < i < n. If 0 < d < m and
pd divides a, then pd+1 divides aa\ thus pd+1 divides a, and this shows that pm
divides a. Replacing the equation by considering coefficients Ui/pm and replacing
a by a/pm, we arrive at the same type of equation with not all U{ divisible by p.
Consider the lowest power in s in any of the coordinates of a. This is at least
one. Thus in aa there are no exponents smaller than p; thus all coefficients on
the left-hand side are divisible by p, a contradiction. Thus all ui = 0, 0 < i < n,
and a = 0.
Applying this to (4.1)-(4.(n + 1)) we obtain bn = 0 = 6n_i = ••• = &! and
an = an_i = • • • = ao = 0, and zg = &o> xo = ^"i tnus xo = ^o = ^o" •
Because 6q £ W(Fg), with q = pn+1, we derive 6q = ^o" = ^o" î tnus

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 185
2
60 = f>Q . Because n is even, we conclude bo = b^; hence bo € Zp = W(FP).
Note that
End(ZK) 0Zpm End(ZK)
is injective (because prank(X) = 0, its p-divisible group (= Barsotti-Tate group)
is dually isomorphic with its formal group, and we use [51, Theorem 5, p. 56]).
Thus there exists a G Z such that a = a • idx- (If a = a • id, with a G Zp, then
a <g) 1 — 1 (8) a maps onto 0 G End(Xft-), and thus a <g) 1 = 1 ® a; hence a is an
integer.) This finishes the proof of Lemma (12.4).
COROLLARY (12.5). For every prime number p, and every even integer
n > 2 there exists an abelian variety X in characteristic p so that the formal
isogeny type of X is (n, 1) + (l,n) (thus X is simple), and
End(X) = Z.
PROOF. By Honda-Serre-Tate we know (the Manin conjecture) that we can
choose Y so that
Y ~ Gn,i ® Gi,n
(cf. [48, p. 354-404]). Moreover we can choose Y so that it has a polarization
of degree p2 (use [24, Corollary 1, p. 234]). Let W with W C ^n+i,p be an
irreducible component of the stratum Vb C An+i of all polarized AV of dimension
n + 1 with prank = 0, so that (Y, /x) G W. By [28, Theorem 1, p. 434] we see
that the generic point (Z, t) eW has a(Z) = 1. Thus there is a unique ap C Z\
thus the polarization r descends to Xo = Z/ap (by [24, Theorem 4, p. 233]);
thus Xo is principally polarized, and we easily prove that
(a computation with D-modules shows that a(Z) = 1, and X0 = Z/ap implies
this). By the previous lemma we see that there exists a deformation of Xo
keeping the formal isogeny type and excluding all complex multiplications (and,
by the way, this shows that already End(Y) = Z). Q.E.D.
REMARK (12.6). In the proof of the lemma we used
Z = End(X)nZpcEnd(X),
which is correct in case prank(X) = 0. In general it is not correct; e.g., let E be
a singular elliptic curve, i.e., End(E') ^ Z and E is not supersingular (e.g., take
j(E) algebraic over Fp, and j(E) & Fp2). Then
E[p°°} = Ê © E\p°°}{k)
(if A: is algebraically closed),
End(£) = Zp = End(£[p°°](fc)),
and End(E') ®z Q is an imaginary quadratic field in which p splits; then
Z % End(£) ^ End(£) = Zp.

186
FRANS OORT
REMARK/ QUESTION (12.7): Let £Xn^ mi) be a formal isogeny type which
satisfies the symmetry condition. We say it is supersingular if (rii,mi) = (1,1)
for all i. If an abelian variety has a supersingular formal isogeny type it is
supersingular, and it has lots of endomorphisms, and if g > 2 it is not simple. In
some sense this is, or this seems to be, the only exception. In [16] it is shown that
for any symmetric formal isogeny type which is not supersingular there exists
an absolutely simple abelian variety over a finite field having this isogeny type.
Thus we are naturally led to the following question:
Given a symmetric formal isogeny type and given a prime
number, suppose that J2{niimi) ¥" 9 ' (1,1) (i-e., the type is not
supersingular). Does there exist an abelian variety X over an
algebraically closed field k of characteristic p having this formal
isogeny type such that End(X) = Z?
As we have seen, (12.5) gives an affirmative answer in a special case.
13. Ramified liftings. We study the problem as indicated in [27, p. 431],
Question C: given K D Fp and Xo an AV defined over if, can this be lifted to
characteristic zero without introducing ramification? We write W = Wqo(K).
As we saw in (11.1) the answer is affirmative in case dim(Xo) < 2, and by (10.4)
we know the same holds if a(Xo) < 1; moreover note that lifting of any (Xo, Ao)
is possible, if ramification is allowed. We show:
(13.1) For any p > 0, and any integer g > 3, there exist a field K of
characteristic p and an AV defined over K with dim(Xo) = g, such that for any
polarization A0 on X0, the pair (Xo,Ao) cannot be lifted to W = Woo(K) (in
short: in order to lift Xo to characteristic zero, ramification is needed).
For simplicity in notations we treat the case g = 3, but we shall indicate how
to generalize for arbitrary g > 3.
EXAMPLE (13.2). Using (12.1) we construct Y with properties as indicated
there. It has formal isogeny type (1,0)+ 2(1,1), and a(Y) = 1. Thus there exists
a unique X0 so that
0-+ap-+Xo^Y-+0
(and note, Y being the generic point of a component of V3,i C ^3, this does not
imply that (Xo, <p*ti) corresponds to the generic point of a component of V^i; in
fact it is not). The data thus constructed have the following properties (assume
K is algebraically closed):
(13.2.1) X0sGi,o©(GM)2;
(13.2.2) po := £>*M is a polarization of degree p2, it is not in diagonal form,
and Ker(po) contains ap;
(13.2.3) End(Xo) = Z.
Computations as done by Ogus (cf. [27, §4] and §11 above), and (13.2.3) lead
to the conclusion that for any polarization Ao on Xq the pair (Xo, Ao) cannot be
lifted to characteristic zero without introducing ramification. We leave the details
to the reader (and below we give proofs in an analogous case).

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 187
REMARK (13.3). Using (12.3) we produce X0 with dim(X0) = g > 3,
X0*{Glto)'-2®{Gltl)2,
and properties analogous to (13.2.2) and (13.2.3); in this way we settle (13.1)
for all g > 3.
EXAMPLE (13.4). Using (12.4) and (12.5) with n = 2 we choose (Y,/i),
where /i is a principal polarization, and a(Y) = 1, and by
0-+ap-+Xo^Y-+0
we have that Xq is an AV defined over if, an algebraically closed field, with
properties:
(13.4.1) X0 = G2,i©Gi,2;
(13.4.2) p0 := £>*M has degree p2;
(13.4.3) End(Xo) = Z
(and, of course, (13.4.1) implies that X0 is simple).
CLAIM. For any polarization Ao on Xo, the pair (Xo, Ao) cannot be lifted to
W = WQO(K).
PROOF. The polarization po on Xo is of the form
(> S):G3,l©G1.3-G2,l©Gi,3-
As po is "symmetric" this gives
(p0 •' -X0 —► -Xo) = (-Xo — -Xo —► -Xq);
thus <p = ip* and ip = (p* (with identifications G"2 = Gi,2 and G"i = G2,i).
We choose an isomorphism
a:Gt2l^Gl2.
thus
a<p G End(Gif2) = ^oo(F,)[[F, V]]/(F - F2)
and
aV e End(G2fi) = ^oo(F,)[[F, F]]/(F2 - V)
with g = p3. The rings End(Gi,2) and End(G2,i) are anti-isomorphic under
$: End(Gi,2) - End(G2,i), /? •- $/? := «'/^(a*)"1,
with $F = V, $V = F, and $a = a for a G VF. Note that p0 has degree p2, and
<p = tp1; thus
Ker(po) = Ker(<p) © Ker(^) = ap © av <-+ X0.
Thus we see that a^> is of the form
(crç>)(0 = (pa + 6F + cV2) • £ with a, 6, c G Wg,
where Wq := Woo(Fg) and 6^0 (modp); here
M(Gif2) = ^-£ + ^-F£ + ^- F2£, V2^ = F£,

188
FRANS OORT
and W = Woo(K). It follows that $(a<p) = a1^ G End (G2,i) is given by
(a^)(r?) = (pa + FB + Fc2) .r) = {pa + b'F + c*V2) • r?
with
M(G2|i) = W "q + W Fr) + W • F2r/, F2r/ = Fr?,
where <7 : W —> W is "raising coordinates to the pth power," and using notations
as in (12.4): n = 2, £n+i = £3 = £, and r?n+i = r?3 = rj. Elements of M(-Xq)
can be seen as WMinear functionals on M(X0) (cf.[29, p. 83]); thus fa defines a
pairing (-,-) = (—, —)Po on M = M(X0) = M(Gi|2) + M(G2|i) by (x,y) =
(poX,y). Under the identifications made we have relations like
É e M(Gif2) = M(G^) C M(X<), r? G M(G2|i) C M{X0);
then (£,r?) = 0, {£,Frj) = 0, (^,F2r/) = 1 (these relations define an element in
the dual of M(G2,i), etc.). Note that F2r? = Fr?, VÇ,V2Ç2 generate V • M0,
where M0 = M/pM, M = M(Xo); as in [27, pp. 431-434] we study the related
lifting problem. For (—,—)p mod p we compute (Fr),rj) = 0, etc., summarizing
the information we need in:
\(—,—)pmodp
Ft)
V
0
Fr)
0
€
6<r3
Fr?
0
VZ
0
V2T1
o
(and other values of (—, —)p can also be computed). Let A0 : Xq —► Xq be some
polarization on Xq. Then p^"1A0 = u G End°(X0) = Q. Let £ G Z be such that
pl exactly divides u, i.e., u = vpl:, and t; is p-adically a unit. Then (—,—)\ has
values in plW and computing (Fr/, — ) in ptW/pt+1W we have
(Fr),0x = vb(T p\ vb* eW\
and (Fr/, —)a in ptJtlW for all other base vectors. This implies by [27, Corollary,
p. 441] that (Xo, A0) cannot be lifted to W: if we could achieve this lifting, there
would exist a W-base {Xu X2, X3, Fi, F2, F3} for M so that (Xi,Xj)x = 0
and Y{ modp G F -Mo (and more properties). The residue classes of Xi, X2, X3
would generate Mo/fc • r/, so at least two of these, say X\ and X2, would give a
base, so
Xx = aFr? + /?£ 1 , , u ^ u, ,
X2= 7^ + ^/ (™d^ + ^)
with ctô — 01 ^ 0. The computation for (Fr/, —)\ then implies that (Xi, X2)\ /
0, a contradiction. Thus (X0, Ao) cannot be lifted to VF, and this concludes the
proof of the claim (13.4).
REMARK (13.5). For simplicity we have presented the construction as in
(13.4) with dim(Xo) = 3. However in an analogous way we can produce examples
for any dimension g with g > 3. In order to do this, we choose a deformation
of Gn,i 0 Gi>n as in (12.4), with g = n + 1, and the deformation is chosen such
that for the related abelian scheme we have End(Z ® K) C Zp. By the proof of

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 189
(12.4) this works for every integer n > 2. Then we take over the arguments of
(13.4) for this case, and we arrive at:
There exists an abelian variety Xo, with dim(X0) = g = n + 1,
such that for any polarization A0 on X0, the pair (Xo, Ao) cannot
be lifted to W.
REMARK (13.6). It might be that there exist better proofs for the claims
in the examples (13.2) and (13.4). Note that in these cases we have Xo = Y/ap,
where Y admits a principal polarization. Further note that ap cannot be lifted
to W (cf. [49]). This seems to indicate a proof using these facts, but we did not
succeed along these lines.
IV. Lifting endomorphisms of abelian varieties.
14. Some examples and results. The following facts are known:
(14.1) Let Eq be an elliptic curve over a field A: D Fp, and let ao G End(£*o).
The pair (£*o, ao) can be lifted to characteristic zero (proved by Deuring, cf. [6,
pp. 259-263]; also see [32]).
(14.2) Let Xq be an ordinary abelian variety over a perfect field k D Fp. There
exists an abelian scheme X (the canonical lifting) over Spec(W (fc)) so that
Ends(X)^Endk(X0)
(proved by Serre and Tate, cf. [17], also see [21, Chapter V, Theorem 3.3; 14]).
In this section we try to generalize these results.
(14.3)AN EASY EXAMPLE. We construct an abelian surface Xo in
characteristic p, and an endomorphism ao G End(Xo), and principal polarizations A0
and /io on Xo, so that the triple (X0,Ao,ao) cannot be lifted to characteristic
zero, and so that (X0, /io> #o) can be lifted. We choose A: D Fp, and n a positive
integer, E0 an elliptic curve over A:, and a,b E Endfc(i?o) with a2 = — n and
b2 = —n — 1. For example, let p = 3 (mod 4),n = p and take E0 defined by
Y2 = X3 + X (thus j(E0) = 1728); further k = Fp2, and a = F (thus F2 = -p),
and let b be defined by b = -c - F with cX = -X, cY = y/^Y, >/=ï G Fp2.
Then c2 = —1; thus (—c - F)2 = b2 = -1 - p because Fc = -cF. We consider
Xo = E0xE0 and a0 = ( ^ fe j G Endk(E0).
Then
o /—n a+b \
note a + b ^ 0. Let
F0 :=Im (al + ri) C X0.
We choose Ao as the principal polarization on Xo defined by (EoUEo) C Eq x Eq.
Claim: (Xo, Ao, ao) cannot be lifted to characteristic zero. This we see as follows.
Let A: <— R D Z with R a discrete valuation ring, and let X be over R so that

190
FRANS OORT
X ®r k = Xo, and suppose there exists a G EikIr(X) with a ®r k = ao. Then
there exists abelian schemes £ <-+ X, 7 <--+ X, so that
(£^X)®Rk = (Ex — Xo), Ex=E0x {0},
and
(7^1) ®* fc = (F0^Xo).
This is because £ := Ker(a2 + n) lifts E\ (taking kernels commutes with base
change); thus £ <--+ X is a subgroup scheme which is an abelian scheme. Thus
X/£ exists (use [10, Theorem 7.2, p. 212-17], obtaining a formal subgroup
scheme of X which is therefore algebraic). We write
X -+X/£ = 7^ X,
and clearly £ <-* X and 7 <--+ X have the required properties. For any R —► K D
Q, we have Hom(<f ® if, 7 ® jPC) = 0 because <f has CM by Q(>/-^) and J has
CM by Q{\/—n — 1). These fields are not isomorphic; hence the corresponding
curves in characteristic zero are not isogenous. Thus
thus
/? 0 fc = (F0 - X0 ^ X* -, E\ ) = 0.
This is a contradiction because Ao is in diagonal form on Xq = Eq x Fq; thus
Ker(X0 -^ -Y$ - £j) = E2 2 F0.
Conclusion: (Xo, Ao,#o) cannot be lifted to characteristic zero.
Note that E\ x F0 -^ X0 = E0 x 25o, because i?i n F0 = 0. Let /i^ be a
principal polarization on E\ = £*o, and let //q be a principal polarization on
Fo, and
fio :=diag(/i^1),/i^2)).
Note that
a0(Fi) c £7i and aQ{a2Q + n)(X0) C (ag + n)(X0) = F0;
thus a0 G End(X0) = End(£'i x FQ) is in diagonal form on Ex x F0. By (14.1)
we can lift {E\,Hq \or0 ') and (F0,/iQ , #q ); thus we can lift (X0,/io>#o) to
characteristic zero.
(14.4) It is not difficult to give an example of an abelian variety X$ and
c*o G (End(Xo)) so that for any polarization A0 on Xo the triple (Xo,A0,ao)
cannot be lifted to characteristic zero (cf. [32, Remark 7, p. 469]). In that
example it might be that for an isogenous Y0 ~ Xo we can lift ao G End(Yb)®Q-
EXAMPLE (14.5). We construct a G Endfc(X) so that for any Y isogenous
to X, for any polarization A on F, and for any positive integer N with Na =
(3 € End(y), the triple (y, A,/?) cannot be lifted to characteristic zero (in short:
a € End(X) <g) Q cannot be lifted to characteristic zero). We choose for k an

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 191
algebraic closure of Fp, we choose integers 0 < n < m with (n, m) = 1, and we
choose X, defined over k so that
■A ~ Crn>m 0 Crm,n
(this is possible by Honda-Serre-Tate, cf. §12 for references). It is known that
Q C Q(tt) CB:= Endfc(X) ® Q,
where n satisfies n2 + pn7r + pg = 0, with g := n + m; we write F = Q(7r),
and B is a central division algebra over F. We choose Q{n) = F C K so that
[if: F] = 0, with if not a CM-field, such that the two places v\ and v% above p
do not split in K (such a field K exists: take h G FP[T] monic and irreducible
of degree g, H G Z[T] monic so that i/ modp = /i, for a large integer M
the polynomial i/ + pM has nonreal zeros, and the compositum if of F and
Q[T]/(H + pM) satisfies the requirement). By [48, Theorem 1, p. 352-02], we
know that B D F is determined by the invariants n/g and m/g above v\ and
t;2, and all other invariants are zero. By [7, Satz 2, p. 118], we conclude that
K D F splits B D F, and because [K: F}2 = g2 = [B: F] we conclude by [13,
Theorem 4.8, p. 221] that
Q C F = Q(tt) C K C B.
Let a G if so that if = Q(a) and a G Endfc(X). This establishes the example,
because if = Q(a) = Q(/?) is not contained in an endomorphism algebra of
an abelian variety of dimension g in characteristic zero: use [24, pp. 201-202];
because [if : Q] = 2g we see that if C D is impossible for types I, II, and III in
characteristic zero, and type IV in this case would imply if = D, a CM-field,
which is a contradiction with the choice of if.
Besides this negative result, there is also a positive result, which generalizes
the lifting constructed by Deuring, cf. (14.1), and by Serre and Tate, cf. (14.2).
THEOREM (14.6). Let k D Fp be a perfect field, and let Xq be an abelian
variety over k with
prank(Xo) > dim(X0) — 1.
For any polarization Ao on X$, and any ao G Endfc(Xo) the triple (X0, A0,ao)
can be lifted to characteristic zero.
PROOF. If p rank(X0) = dim(X0), i.e., Xq is ordinary, the conclusion is the
result by Serre and Tate. Thus we assume p rank(Xo) = dim(Xo) — 1, and we
write
^o = Go © i/o ?
where Go is local-local, and H0 is local-étale (this is possible because k is perfect).
We write
(A | G0r= no and a0 = (/?o, 7o)-

192
FRANS OORT
LEMMA (14.7). Let Go be a formal group with
dim(G0) = l = dim(G£).
Let /?o G End(Go). The pair (Go,/?o) can be lifted to characteristic zero.
PROOF. If Ker (/3b) / 0, then Ker(l + 0O) = 0, so we assume Ker(/?0) = 0
(otherwise lift 1 + /?o and after lifting subtract again 1). Thus /?0: Go —> Go-
Let VF = Woc(k). Let M be the formal moduli functor for lifting Go to
artinian VF-algebras A —► fc. As in [18] we see that M is pro-represented by
VF[[T]]. If/?o G Zp the conclusion follows (in that case /?0 extends to the universal
deformation). Suppose /?0 £ Zp. Now we follow [32]. Let J be the formal moduli
functor of /?o : A0 —► B0 with Ao = Go = -Bo- This is pro-represented by a factor
ring of VF[[Ti,T2]]. We have the following functors:
An/
A ► MxM —^—+ M.
Note that /?o is an isomorphism; thus I -^ M. This results in a diagram of rings
and homomorphisms
W[\T}]/(0 * W[[T}}
I 4
»y[[ri,r2]]/(ri-r2) < w[[tut2}} < w[[7\]]
with /(Ti - T2) =: f G W[[T]]. The algebra WIPIIAO pro-represents the
formal moduli of deforming /?0 G End(Go), and we want to show that it has
a characteristic zero fibre. If for all positive integers n we prove pn £ (£), we
conclude W t-* VF[[T]]/(£), and we are done. Suppose not, and let n be minimal
so that pn G (0; thus n > 0, pn = £ • F, with F G W[[T]] and p f ^- This would
give p | £; thus
w[[r]]/(0 ®w fc = k[[T}}.
This would imply that /?0 extends to the universal deformation Q of Go in
characteristic p > 0. However
EndK(£®if) = Zp
for any if D k[[T]], a contradiction with the assumption /?o ^ Zp. Thus for all
integers n we have pn £ (£) and the lemma is proved.
LEMMA (14.8). Let Go be as in (14.7), and let /io be a quasipolarization on
G0 [i.e.,
(/i0: Go - GS) = (/4 • «: G0 ^ G? - G&)).
Le£ A: 6e perfect, let R -^ k be a complete local ring, and let Q be a formal group
over R extending Gq. Then there exists (a unique) quasipolarization /i on Q
extending /io.
SKETCH OF THE PROOF. As above, let M be the formal moduli problem
of Gq. Assume /io to be an isomorphism (if not, replace by /io + 1, which
r^

LIFTING ALGEBRAIC CURVES TO CHARACTERISTIC ZERO 193
is also symmetric, and if fJ>o(ap) = 0, then /io + 1 is injective on ap, hence an
isomorphism of Go). Then /io identifies the tangent spaces of Go and of Gl0. The
usual deformation theory applies; thus M equals the formal moduli problem of
(Go, no), because
dim(tangent space Go) = 1 = dim(tangent space Gq);
thus /io extends to the universal deformation. This proves (14.8).
We finish the proof of (14.6). By (14.7) and (14.8) we can choose a
complete local integral domain R -+ k with characteristic(R) = 0, and a lifting of
(Go,/io,/?o) to R. After this is chosen, we produce a lifting of BT(Xo) in a
direct-sum way, plus a lifting of BT(Xo) and of BT(ao). By the theory of Serre
and Tate this results in a lifting of (X0, A0,ao), which proves (14.6).
(14.9) EXAMPLE (Lubin AND TATE, CF. [18]). There exists a field K
(in fact a finite extension of Qp), an abelian variety E over K, and a prime
number p such that the natural map
EndK{E) <g)Z Zp *-> EndK {TPE)
is not bijective. Note that if K were of finite type over its prime field this
map would £>e bijective; note that such a map in general is not bijective for K
algebraically closed. In this example we choose a finite field A: with #fc = pa,
we choose a supersingular elliptic curve Eo over A: such that Endfc(i?o) has rank
4 over Z (this is possible if and only if a is even), and we take T = Woo (A;),
with field of fractions K = Q(T). Thus [K: Qp] = a. In the paper [18] Lubin
and Tate show in §3.5 the existence of an elliptic curve E over K with the
required property. In fact, note that the set of if-isomorphism classes of elliptic
curves E defined over K such that £ ®r k = Eo, and E = £ ®r K has CM is
countable-, proof: the set of j-values with CM is countable, and if E\ and E2 over
K have the same y-value, then they are isomorphic over an extension L D K
with [L: K] < 6. There are finitely many such extensions and #Aut(£x) < 6.
For any such curve, say Ea we have
End^(£a) =:DacD := End£(£0) = Endfc(£0) ®z Q;
note that
[Da:Z]<2 and [D:Z] = 4.
Thus the countable union
\J(Da 0 Qp) § D 0 Qp = Endfc(Go)
Ot
is not equal to the ring of endomorphisms of Go := i?o- We choose some
h e Endfc(Go) with fa £ \J{Da ® Qp).
a
By (14.7) we can lift (Go,/?o) to T, thus by Serre-Tate arriving at:
an elliptic curve E over K, with an element /3 G Endx (TPE),
and /? $ End{E)®Zp; thus EndK{E) = Z and Endx(E)<8>Zp §
EndK {TPE).
This concludes the description of such an example.

194
FRANS OORT
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Mathematisch Instituut, Utrecht, The Netherlands

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Hodge Theory

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
The Geometry of the Extension Class
of a Mixed Hodge Structure
JAMES A. CARLSON
Abstract. We review the theory of extensions of mixed Hodge structures
and its applications to curves, surfaces, threefolds, and homotopy.
1. Introduction. A mixed Hodge structure [17, 18] (see also [16, 24]) can
be viewed as an object assembled from a set of pure Hodge structures using
appropriate extension data [4, 6]. In this note we shall review first the abstract
theory of such data and then some of the geometric applications. To begin, an
extension E is a short exact sequence
(1.1) O^A^H^B^O
in the category of mixed Hodge structures MHS. Two extensions E and E' of B
by A are congruent if there is a commutative diagram in JVCKS of the following
kind:
0^A^H^B-+0
(1.2) id I 4>[ id |
0 ^ A ^ H' ^ B -+ 0
We will show that congruence classes, under reasonable restrictions, are classified
by a complex abelian Lie group (a torus). The restrictions are:
(1.3.a) For some m, WmA = A and WmB = 0 {separation)
(1.3.b) Bz is torsionfree.
To define the tori, consider, for any mixed Hodge structure H of highest
weight < 2p — 1, the group
(1.4) JpH = Hc/{FpH + Hz).
One verifies that the projection Hz to Hq/FpH is discrete, so that the quotient
is indeed a complex manifold. Two special cases bear mention. First, when H is
a pure Hodge structure of weight 2p—1, the torus is compact; if in addition H has
1980 Mathematics Subject Classification (1985 Revision). Primary 14C30; Secondary 14A20.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
199

200 JAMES A. CARLSON
level one (level=number of steps in the Hodge filtration), then the torus admits
the structure of an Abelian variety. Second, when H has pure type (p — l,p— 1),
the torus is isomorphic to (C*)n, for then JPH — He/Hz- In general JPH is
filtered by subtori
(1.5) WmJpH = JpWmH,
where
(1.6) GrZJpH = J*>GrZH.
In particular, one has canonical exact sequences
(1.7) 0 - GrJ^ PH - Wm JpH/Wm-2 - Gr£ J*H - 0.
EXAMPLE (1.8). Let X be a singular curve. Then one has the exact sequence
0 -+ Gr^ JXHX{X) -+ JXHX{X) -+ Grf JXH\X) -+ 0,
where the term on the left is a product of C*'s, and where the term on the right
is isomorphic to the Jacobian of the normalization of X.
To describe the set of congruence classes, fix an extension E and consider
sections sf and sz of 7r, the projection map, where sf preserves the Hodge
filtration and where sz preserves the integral structure:
(1.9.a) sF{FpB) C FPH,
(1.9.b) sz{Bz)cHz.
Set
(1.10) ip = sF-sz,
where we view \j) as a map of Be to Ac. Although -0 depends on the choice of
sections, the coset
(1.11) il>(x) + F° Hom(£, A) + Hom(£, A)z
does not. Here we have used the canonical mixed Hodge structure on Hom(JB, A),
where
(1.12.a) FpHom(£, A) = {ip: B -+ A \ ip{FkB) C Fk+PA for all k}
(1.12.b) VymHom(5, A) = {^: B -+ A \ ip{WkB) c ^+mA for all k}
(1.12.c) Hom(5, A)z = {tp:B^A\ ^(Bz) C Az}.
Our construction gives the required identification [4, 6]:
THEOREM (1.13). Let A and B be separated mixed Hodge structures, with
B torsionfree. LetExt(B,A) denote the set of congruence classes of extensions
of B by A. Then there is a natural isomorphism of groups,
Ext(£,A)^ J0Hom(£,A),
where the composition law on the left is Baer sum of extensions.
Note that an extension E is split over the integers if and only if there is a
section s which preserves both the Hodge filtration and the integral structure.

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 201
Figure l
Precisely in this case does the class of E vanish in Ext(B,A). Consequently,
Ext(5, A) is the group of obstructions to splitting a mixed Hodge structure over
the integers [6].
The extensions to which we shall apply the theorem are those canonically
defined by a mixed Hodge structure H:
(1.14) 0 - Wm-i ^Wm^ Wm/Wm-! - 0.
When H is the cohomology of a complex algebraic variety one can represent E
by an appropriate period matrix. To see this, observe that ip can also be written
as
(1.15) ip = rzosF,
where in general rz is an integral retraction of H onto A: rz{Hz) C Az and
rzoi = id. Thus, if w G FPB and if Q G FPH lifts w, then one has
(1.16) ^) = E(/ n)^'
where {7*} is a basis for Az and where {I\} lifts the dual basis {7^} of {Az)v
to (Hz)v.
EXAMPLE (1.17). Let X be a complete singular curve as in Figure 1, let
{pi,...,pr} be the set of nodes, and let 7; be a loop through the ith node.
One verifies that the dual basis {71,..., 7r} spans (WoH1(X))z- Let w be an
abelian differential on the normalization X. Since the integral of w over a loop
in X makes sense, one may use u to define a class fi G H1(X). One verifies that
the space of all such classes is F1H1(X), so that the coefficients of the above
formula for xj) may be viewed as abelian integrals on the normalization. To make
this completely explicit, let p+,p~ be the two preimages of a node p, and let 77

202
JAMES A. CARLSON
be a path from p to p+ which projects to the loop 7 associated with p. Then
one has
(1.18) I n = I & = abelian integral (u;,p+,p~).
Consequently the extension splits over Z if and only if these abelian integrals are
periods, that is, if and only if p+ is linearly equivalent to p~. Therefore the mixed
Hodge structure of an irreducible nodal curve with nonrational normalization
never splits.
Now let Z — (Zf,Z") be a period matrix for the abelian differentials of X
with Z1 formed from loops in the image of the projection from the normalization,
and with Z" formed from the loops associated with the nodes. Then -0, which
we have seen is given by abelian integrals, is just the right-hand period block,
Z".
Ideally, one would like to have a geometric interpretation for the extension
data whenever the underlying mixed Hodge structure comes from geometry. At
the present time there is no way of doing this, any more than there is a way of
giving a geometric interpretation to the periods of an arbitrary smooth variety.
Nevertheless, there is a part of the extension data which is accessible, and which
we now describe. For a mixed Hodge structure H set
(1.19) LpH={Gr2pH)z.
For an extension with B of pure weight 2p define a homomorphism
(1.20) u:LpB^JpA
by
(1.21) u(x) = ip{x) + FPA + Az.
When A is a mixed Hodge structure of level one, JPA is an extension of a po-
larizable abelian variety by a multiplicative torus (product of C*'s), and so is,
according to the terminology of Deligne [18, §10], a one-motif. When there are
no restrictions on A, we shall call u a generalized one-motif, that is, a
homomorphism from a finitely generated abelian group to a filtered complex abelian Lie
group. The class of such objects becomes a category if one defines a morphism
u —► uf to be a commutative diagram as below,
L -+ V
(1.22) u I K
J -+ J'
where the vertical arrows are morphisms of abelian groups and of filtered complex
abelian Lie groups, respectively. In this language one may give an invariant
interpretation to the coefficients of the homomorphism xj) which represents an

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 203
extension. For curves this results in the following:
THEOREM (1.23). Let X be a complete curve, let tt: X —► X be the
normalization, let £ be the singular locus, and let E be the preimage of E in X.
Denote by p and q the projections of E toT, and X, respectively. Define a group
Zx = {zero cycles £ on X | p*(f)?tf*(0 homologous to zero}
Let
A:Zx^Pic°(X)
be the map which associates to £ its divisor class on X. Let u be the canonical
one-motif associated to H1(X). Then there is a canonical isomorphism A = u.
For the proof of this theorem we refer to [4, 6]. In the next section we
shall state (and prove) a similar comparison theorem for one-motifs of algebraic
surfaces of a special kind.
2. The one-motif of an algebraic surface. We describe here the
geometric counterpart of the one-motif associated to the second cohomology of an
algebraic surface. To concentrate on essential ideas, we limit discussion to
surfaces of the form X = D0U Di, where the components meet transversely in an
irreducible curve D0i, and where ft1,0 (A) = 0. The case of a general surface is
treated in [7], while that of a normal crossing variety of arbitrary dimension is
treated in [5]. The present discussion is adapted from [6].
To begin, fix the notation
X[0] = Dq II Di (the normalization of X),
X[l] = An (here, the singular locus of X),
8q: X[l] —► X[0] (here, the inclusion in Di),
6i: X[l] —► X[0] (here, the inclusion in D0).
For a general normal crossing variety X[l] is the normalization of the singular
locus, <$o is the inclusion of Dij in D^ and b\ is the inclusion of Dij in D{. To
proceed, define an algebraic one-cycle Z on X which meets X[l] in a finite set
of points to be in general position. Write such a cycle as Z = Zq + Z\, where Z{
is the component of Z on A, and define the trace of Z as the difference of the
intersections with D0i:
(2.1) t(Z) = 8^Z - 6{Z = (Zi • An) - (Z0 • An).
Define the Neron-Severi group of X (written !NS(X)) to be the quotient
(2.2)
„, , _ {algebraic one-cycles Z on X[0] in general position | t(Z) = 0 in homology}
JNc>(X) =
algebraic equivalence.
Then there is an induced trace homomorphism
(2.3) r:!NS(X)^Pic0(X[l])
which sends Z to the linear equivalence class of r(Z). The motivic comparison

204
JAMES A. CARLSON
can now be stated:
THEOREM (2.4). Let X be an algebraic surface as above. Then the one-
motif u which is canonically associated to the mixed Hodge structure H2(X) is
isomorphic to minus the trace motif r.
Proof.
(a) Description of the mixed Hodge structure. We give an elementary
construction of the mixed Hodge structure on Dq U D\\ for general normal crossing
varieties see [16] or [24] and for general varieties see [18]. Consider then the
usual Mayer-Vietoris sequence of singular cohomology,
(2.5) 0 - H\X[l\) ^ H2{X) £ H2{X[0}) ^ H2{X[l\) - 0,
and define the weight filtration by
(2.6.a) WXH2(X) = 6H1{X[l\) = H1^}),
(2.6.b) W2H2{X) = H2{X),
so that
(2.6.c) Gr^ H2{X) S kernel{£* : H2{X[0]) -+ H2{X[l})}.
To define the Hodge filtration on i/2(X), consider the complex A*{X) of
differential forms uj = ujq + oji on the components which are compatible on intersections:
6*lj — ÔqUi — ôfujo = 0. Then A*(X) sits in an exact sequence of de Rham
complexes,
(2.7) 0 - A*(X) £ A*{X[0}) £> A*(X[l]) - 0
which induces a Mayer-Vietoris sequence compatible with that on singular
cohomology. Each term of Mayer-Vietoris sequence then admits a Hodge filtration
Fp, where, as usual, a form has Hodge level p if its local coordinate expression
contains at least p dz^s. The crucial technical result is the following:
LEMMA (2.8). The maps of the Mayer-Vietoris sequence are strictly
compatible with the Hodge filtration induced on cohomology.
Strict compatibility [17, 18] asserts the commutation relations below:
(2.9.a) 7T*FpH2{X) = Fptt*H2{X),
(2.9.b) tf*Fpff1(-X'[0]) = WH^Xfi]),
(2.9.c) 8FvH\X[l\) = Fp8H\X[l\).
As a consequence of these, the filtrations which Fp indices on the graded
quotients of the weight filtrations define Hodge structures. Therefore the weight
filtration (defined by topology) and the Hodge filtration (defined by de Rham
theory) satisfy the requirements for a mixed Hodge structure. The structures on
the graded quotients are identified by the following:
(2.10.a) 6: ^(Xfl]) = WXH2{X),
(2.10.b) 7T*: Gv2H2{X) s kernel{£*: H2{X[0\) — H2{X[l})}.

the extension class of a mixed hodge structure 205
Proof of the Lemma.
(i) Compatibility. We verify that 6 is compatible with the Hodge filtration,
i.e., that 6FPH1{X[1)) C FPH2{X). To begin, note that one has the
EXTENSION LEMMA (2.11). Given a complex submanifold N of a complex
manifold M and an element u G FpAk(N), there is an element u G FpAk(M)
which pulls back to u via the inclusion map.
Now let u be a closed form in j4*(X[1]) of Hodge level p and observe that
there is a form fi G FpAk(X[0]) such that 6*Q = u: take fi = (0,cD), where uj is
an extension of uj from D0i to D\ which has the same Hodge level as uj. Since d
and 6* commute, 6*dQ — d6*Q = duj = 0, so that dfi, which represents 6u, lies
in Ak+l{X). Since both fi and dQ have Hodge level p, so does 6u, as required.
We omit the verifications, similar in character to those just given, which are
necessary for the remaining maps.
(ii) Strict compatibility: We verify that which is required of 6:
(2.12) 6FpH1{X[l}) = FP6H1{X[1}) = Fp n 8Hl{X[l\).
To this end consider a closed form <\> in FpAkJtl{X) which is representable as a
coboundary. We seek a form uj in FpAk(X[l}) such that 6[uj] — \<\>\ in cohomology.
To say that <\> G Ak(X) is a coboundary is to say that there exist differential
forms uj G Ak(X[l\), Q G Ak{X[0]) such that
(2.13.a) duj = 0,
(2.13.b) £*fi = w,
(2.13.c) dfi = tt>.
We must show that a solution uj of these equations exists with the same Hodge
level as <\>. Now the last equation, which asserts that 7r*0 is exact in A*(X[0])
(but perhaps not in A*(X)), means that on each component D{ the relation
(2.13.d) dQi = (tt»;
holds. Since the exterior derivative is strictly compatible with the Hodge
filtration on a compact Kâhler manifold, there exists a solution fi to (2.13.d) whose
Hodge level is the same as that of the right-hand side provided that some
solution to (2.13.d) exists, as is the case by hypothesis. The form uj determined by
this good solution and by equation (2.13.b) is that which we seek: it satisfies
equation (2.13.a) because 6* and d commute {du — d6*Q = 6*dQ — £*7r*0 = 0),
and it has the correct Hodge level by construction.
We omit the verifications, similar in character to those just given, which are
necessary for the remaining maps.
(b) Comparison of the lattices and the tori. There are isomorphisms
(2.14) LlH2{X) = kernel^*: LlH2{X[ti\) -► LlH2{X[l})} S !NS(X),
where the first is induced by 7r*, and where the last is induced by Poincaré
duality on the D{. Note also that if A is a polarized Hodge structure of weight

206
JAMES A. CARLSON
2p — 1, then the pairing between Ac/Fp and Fp is perfect, so that there is an
isomorphism between JPA and (FpA)v/(Az)v, where Av is dual to A. Then
one has the identifications
Pic°(X[l]) = (F1i/1(^[l]))V/(^1(^[l],Z))v (Abel's theorem)
(2.15) = JlHl (X[l]) (Poincaré duality)
= JlH2{X) (Usee*).
From the above identifications one obtains the diagram below, whose commuta-
tivity remains to be shown.
LlH2{X) !> JlH2(X)
(2.16) 1 1
!NS(X) -+ Pic°(X[l])
T
(c) Calculation of the extension homomorphism ift. Choose a symplectic basis
{7a} for Hx{X[ll Z), where 7a • ^ = 0 if \a - b\ ^ g, and 7a • 7*+a = 1, for
a = 1,..., g, with g — genus(X[l]). Let {7a} be the dual basis in homology, and
let {ra} be a lifting to H2{X,Z) via the boundary map in the Mayer-Vietoris
sequence.
(2.17) H2(X[0}) =* H2(X) £ #i(X[l]) h HMO]).
Write Ta = r0a + Tia, where I\a is a 2-chain on D^, and where
(2.18) dTia = (-l)Sa.
Let Lj = (w0, wi) G #2(X[0]) be the Poincaré dual of a cycle Z G 3M8(X), and let
Q = (Q0, fii) be a lifting to a de Rham form on X such that fi; has Hodge level
one. Then, in accordance with (1.16), the homomorphism which represents the
extension class is given by the formula below:
(2.i9) </>(") = £7° / n.
.>ra
(d) Reduction of the coefficient integrals. We evaluate and interpret the
coefficient integrals of (2.19) using a special choice of the lifting fi. To construct
it, let L{ be the line bundle associated to Zi, let ai be a meromorphic section
which defines Zt-, and let |o^|2 be the squared length of ai in a suitable hermitian
metric. Then one has
(2.20) ddc\og\ai\2 = [Zi]-nu
where
ddc = (y/=ï/4ir)dd,
where fi; represents the Chern class of L;, and where [Zi] is the integration
current of Zi. Because the trace of Z is homologous to zero, the line bundle
(2.21) Loi = (ôSLJfâLo)-1
is topologically trivial. It is therefore possible to choose the metrics {hiU} on Li
in such a way that the induced metric on L0i is flat. By this we mean that there

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 207
is a system of holomorphic local coordinates on X[l] such that the local functions
hoiu = hov/hiv which give the metric are constant. Then 6qCIi = of Q0, so that
fi is a de Rham form on X.
On the complement of Zj one has
(2.22) nj=(i/47r)dd\og\aj\2.
Fix an element T G Hï{X) which lifts 7 in Hi(X[l]) relative to the boundary
map in the Mayer-Vietoris sequence, write T = To + Ti, and let U{(e) be an
^-neighborhood in rt- of the point-set (T{ • Zj). Then
(2.23) / n3- = lim(»747r) / ddlog l^f.
and Stoke's theorem gives
(2.24) / n3- = (-l)J(t747r) f d\og\a3\2 - lim{i/4*) [ dd\og\a3\2.
Jv3 J1 £^° hj-u^e)
The limit is an integer, namely, the intersection multiplicity of T3 with Zj. In
this way we obtain the formula
(2.25) f n = (»74tt) /" 9 log |a0i |2 + m(r),
where aoi = £V = G\ja^ on X[l] and where ra(r) is an integer. By Poincaré
duality on the components of X there is a cycle A on X[0] such that the
intersection number (Ao • Z0) + (Ai • Z\) is — m(T). But this intersection number is
also given by the integral of fi over A. Consequently the lifting can be modified
so that m(T) = 0. In conclusion, the coefficient integrals of (2.19) are of the
form
(2.26) f n = (i/47c) f d\og\a0i\2-
(e) Calculation of the motif. We must calculate the action of ip(w) on abelian
differentials of X[l] when uj is in LlH2(X). To this end fix an abelian differential
4> and write
(2.27) 4> = J2ia [ *>•
Use the fact that {7°} is a symplectic basis to obtain the relation below:
(2.28)
f TP(u)A<j> = (i/4n)J2 f dlogkoil2 / <t>~ f dlog|cr01|2 / <t>
JX[1\ a [J-la Jla + g J la + g J la
Let
(2.29) f(y) = f<f>
be an abelian function associated to <\>. View the cycles {7a} as giving a canonical
system of cuts for the Riemann surface X[l], so that
(2.30) ? = X[l}-\Jla

208 JAMES A. CARLSON
is a 4^-sided simply connected polygon. Then a classical argument in the theory
of Riemann surfaces yields
(2.31) / il)(u) A (j> = (i/47r) f fd log |a0i |
Jx\i] Jay
■yjywj /\(p= yif±-nj / yt/iug|uoi'
fX[l
Now /dlog|aoi|2 is of type (1,0) and is smooth on?- |r(Z)|, where \r(Z)\ is the
support of r(Z), and so /dlog |aoi|2 is closed on?- |r(Z)|. Thus, if U(e) is an
^-neighborhood of the support of r(Z), a second application of Stoke's theorem
gives
(2.32) / i/j(uj)A<t)= lim{i/4ir) [ /<91og|a0i|2.
JX[1] £_*° JdU(e)
Finally, let p be a point in the support of the trace, let zv be a holomorphic local
coordinate of X[l] centered at p, and let i/p(r(Z)) be the multiplicity of p with
respect to r(Z). Then the right-hand integral above is a sum of integrals of the
form
(2.33) (i/4ir) / /<91og/*0i|<p(T(Z))|2.
■'I*pI=«
But hoi is constant relative to a good coordinate system, so the given integral
is just
(2.34) -Vp{T(Z)){i/2x) f /31ogz = -i/p(r(Z))/(p).
J\z\=e
Summation over the support of t(Z) then gives
(2.35) J^(uj)A4>=-J2Mr(Z))f(p)-
Set t(Z) = Y2{t>i — o,) and apply the fundamental theorem of calculus to obtain
(2.36) j Tl>(u)A<t>=-Y^J'
(j) (modulo periods of (j>).
This completes the proof: if Z G 3s(S(X), if uz is the Poincaré dual of Z in
LXH2(X), then ift(uJz) acts on abelian differentials by formation of the abelian
sum with respect to t(Z). Consequently the transform of the Hodge-theoretic
one-motif u with respect to Poincaré duality is minus the trace.
3. Applications and further developments.
a. Curves. As a first application of a comparison theorem for one-motifs we
give a Torelli theorem for singular curves [4, 6]:
THEOREM (3.1). Let X be a complete irreducible curve with ordinary
singularities. Then the polarized mixed Hodge structure on Hl(X) determines X
up to isomorphism, provided that the normalization X is not hyper elliptic.
Here an ordinary d-fold singularity means a singularity of multiplicity d with
d distinct tangents. In addition, a polarized mixed Hodge structure means a pair

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 209
(i/, S*) consisting of a mixed Hodge structure H and a set of bilinear forms {S*}
which polarize the graded quotients of the weight filtration. The polarization on
the graded quotient of weight one is that induced by the isomorphism
(3.2) Gv^ Hl{X)^H\X).
Now consider the bilinear form on Zx defined by declaring the points of E, the
inverse image in X of the singular locus E, to constitute an orthonormal basis
for the lattice of all divisors supported in E. Transport this form to W$Hl using
the identification
(3.3) W0Hl{X)^ àun\ oîZx
and use this as the weight zero polarization.
To prove the theorem, use the classical Torelli theorem to construct X from
the graded quotient of weight one. Then observe that a of-fold point p contributes
an orthogonal summand Zp to Zx which is isomorphic to the root lattice Ad-i,
with root vectors given by divisors of the form qi — <72, with ^ in the inverse
image in X of p. Next, apply the comparison theorem (1.23) to the values of
the Hodge-theoretic one-motif u on the root vectors of Zp to locate the points of
7r~1(p) on X. This is possible because, with A as the Abel-Jacobi map, one has
(3.4.a) u(root vector of qi — q2)"="A(qi — q%)
and
(3.4.b) A: Symmetric square (X) —► Jac(X) via ((71,(72) ^
divisor class (<7i — q^) is 1-1 on the complement of the
diagonal, provided that X is nonhyperelliptic.
Finally, for each summand Zp, collapse the points on X which correspond to
it via u. The resulting curve is isomorphic to X.
Results of a similar kind can be obtained for irreducible curves with rational
normalization using the limiting mixed Hodge structure of the versai deformation
in place of the mixed Hodge structure of the singular curve [8]. In this situation
the limit structure [12, 37, 38] has moduli whereas that of the singular curve
does not. The Hodge-theoretic moduli are encoded as a nilpotent orbit [37] of
one-motifs, the values of which on the vectors of the root lattice of weight two
may be interpreted as cross-ratios of distinguished quadruples on X.
In the case of curves the nilpotent orbit of mixed Hodge structures coming
from the versai deformation determines (and is determined by) a corresponding
nilpotent orbit of one-motifs. These orbits are classified by spaces analogous to
the Griffiths period domains [22], and may be viewed as boundary components
for a suitable compactification of a modular quotient of Siegel space. Here
"suitable" means a Mumford toroidal compactification [1, 32]. The relation of these
compactifications to mixed Hodge theory is sketched in [8]. A much more
complete motivic description of the boundary points determined by algebraic curves
in toroidal compactifications associated to Siegel spaces is given by Hoffman [29].

210
JAMES A. CARLSON
From a geometric standpoint, the difference between the Satake-Baily-Borel [2,
36] and the Mumford toroidal compactifications is that the former records the
pure Hodge structure on the graded quotient of weight one, whereas the latter
records extension data as well.
b. Surfaces. A first application of the motivic comparison theorem (2.4)
discussed in the previous section is a Torelli theorem for certain degenerate K3
surfaces [6, 7]:
THEOREM (3.5). Let X be a degenerate quartic surface consisting of a plane
and a smooth cubic surface meeting transversely. Then the polarized mixed Hodge
structure on the primitive cohomology determines X up to isomorphism.
Here the primitive cohomology is defined to be the kernel of the restriction
map for a hyperplane section. The polarization on the weight-two quotient is
given by the sum of the cup-product forms on the components, and the resulting
lattice is isomorphic that of the Dynkin diagram E$. One applies the comparison
theorem (2.4) to the values of the motivic homomorphism on root vectors to
locate the intersections of a skew-six of lines on the cubic component with the
planar component. Since the two components meet in a plane cubic curve which
is determined by the Hodge structure W\H2(X), one is able to reconstruct the
surface from purely Hodge-theoretic data. Roughly speaking, the construction
is as follows: imbed J1H2(X) in P2 as a cubic curve £ and then blow up P2 at
the six points determined by the polarized one-motif to obtain a cubic surface
A. Let B be the hyperplane which cuts out £ on A, and set X' = A U B. Then
Xf is isomorphic to X.
There are classifying spaces for polarized mixed Hodge structures just as there
are for polarized Hodge structures (see §4). Thus, if M is the moduli space for
the singular quartics considered above and if D is the classifying space for their
mixed Hodge structures, then there is a holomorphic and horizontal period map
v: M —► T \ D. The above Torelli theorem shows that for the singular quartics
v is a degree one map.
For equisingular families of normal crossing varieties, one verifies holomor-
phicity and horizontality of v in essentially the same way as one does for smooth
families, namely, by Lie differentiation of differential forms with respect to
appropriate vector fields Z and Z which project to d/dt and d/dt respectively,
followed by restriction to the fiber. The first field is of type (0,1), so that Lie
differentiation leaves Fp invariant, establishing holomorphicity. The second field
is of type (1,0), so that dFp/dt C Fp_1, as required for horizontality. The fields
Z and Z must be compatible with the normal crossing structure: if X/T is a
normal crossing family over T, then Z and Z must be tangent to the subspaces
Di/T, where I = (z0,..., iq) and Dj = D{0 n • • • PI D{q. For then Lie
differentiation followed by restriction to the fiber acts on the Oech-de Rham complex [16]
used to define the mixed Hodge structure of the fibers Xt. Variations of mixed
Hodge structures have been studied extensively in [39, 40, 41].

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 211
Using these and other techniques, especially the work of Kulikov-Persson-
Pinkham [30, 33] and that of Mumford and others on toroidal compactifications
[1], Friedman [19] has given a new proof of the Torelli theorem of Piatetskii-
Shapiro and Safarevic [34].
Of a somewhat different character is an interpretation of the one-motif for
H2 as an obstruction for a descent problem. To describe it, consider a normal
crossing variety X for which an ordering of the components has been chosen, let
X[p] be the disjoint union of (p+ l)-fold intersections of components, and let 6{
denote the map from X[p] to X[p — 1] which maps a (p + l)-fold intersection to
the p-fold intersection obtained by deletion of the iih component. The spaces
and maps fit together to give a semisimplicial space [18], as indicated in the
diagram below:
(3.6) X[0] 1= X[l] £ X[2] g etc.
Let L[0] be a line bundle on X[0] and let cl(L[0]) be the cohornology class of L[0].
A topologically necessary condition that L[0] be induced from a line bundle L on
X is that 6*c\(L[0]) vanish in H2(X[1]). One may show that there is a further
condition which is also sufficient [5, 7]:
THEOREM (3.7). Let £ be a cohornology class in LlH2{X), where X is
a normal crossing variety. Then £ is the class of a line bundle if and only if
u(t;) = 0, where u is the canonical one-motif of H2(X).
In other words, u gives the obstruction which determines whether a Weil
divisor on X is representable as a Cartier divisor. For surfaces of the form
Do U Z?i, the arguments given in the preceding section suffice: a line bundle L[0]
on X[0] is induced from X if and only if L[0]|Z?o can be glued by an isomorphism
to L[0]|2?i, or, equivalently, if and only if (L[0]|2?i) 0 (L[0]|Do)-1 is a trivial
line bundle. Now let Z be a divisor of L[0] and observe that r(Z) is a divisor
of 6*L[Q] = (L[0]|Z?i) ® (LIOJIA))-1. Thus, L[0] comes from X if and only if
t(Z) = 0. But this last relation holds if and only if w(cl(L[0])) = 0, as required.
As an application of this theorem, we note that if L\ and L2 are skew lines on
the cubic component of the degenerate quartic surface considered above, then
the cohornology class of the Weil divisor L\ — L2 is not representable by a Cartier
divisor since
(3.8) w(class(Li - L2))"="divisor class(pi - p2) ¥" 0?
where pi is the intersection with the cubic curve of L{. Classes of the kind just
considered, given by differences of skew lines, form a sublattice of type A5 in the
full Eg primitive cohornology lattice.
c. Threefolds. In [5] we generalize theorems (1.23) and (2.4), showing that
the generalized one-motifs associated to a normal crossing variety are given by
Abel-Jacobi or trace maps, depending on the parity of the highest weight of the
cohornology considered. Here we discuss a special case of these results (see also

212
JAMES A. CARLSON
[3]) and one of its geometric applications:
THEOREM (3.9). Let X = D0 U D\ be a threefold which is the union of
two smooth 1-connected threefolds meeting transversely in a smooth 1-connected
surface Dqi. Let u be the generalized one-motif of the mixed Hodge structure
Hs(X), lefNS(X) be the group of algebraic equivalence classes of algebraic one-
cycles on Dqi which map to zero in the homology groups of Do and D\, and
let
(3.10) A:1®(X)^ Jac(X)
be the natural Abel-Jacobi map where the right-hand side is the Griffiths
intermediate Jacobian [20,21] of the normalization ofX. Then there is a canonical
and functorial isomorphism
(3.11) u = A.
PROOF. The construction (1.21) of the generalized one-motif applies to mixed
Hodge structures with even highest weight. Thus we define u to be the one-motif
associated to the dual structure on i/3(X), which has highest weight —2 and
lowest weight —3. If -0 and tjj* are the extension homomorphisms for cohomology
and homology, respectively, then the homological one-motif is defined by the
adjoint formula
(3.12) ti(7)(w) = (w,^*(7)> = (</>M,7>.
We calculate u using a description of the mixed Hodge structure of X similar to
that of the preceding section. Thus, if u is an element of F2 Gr^ H3 (X) with a
lifting of a de Rham form fi = fi0 + ^i on X of Hodge level 2 then, (1.16) gives
(3-13) ^) = E(/r n)^-
Here {7^} is the basis for i/2(X[l], Z) dual to {7*}, and I\ is a lift of 7^ along
the boundary map of the Mayer-Viet oris sequence:
(3.14) H3(X[0]) ^ H3(X) ^ H2(X[1\) h H2(X[Q]).
Evaluation of the cohomology class il){ui) on the homology class of an algebraic
one-cycle Z yields
(3.15) Ww),[Z]}=/ n,
Jd-lz
where r0 + Ti = d~lZ. Putting the pieces together, one obtains the relation
below,
(3.16) u([Z])(w) = <VM, [Z]) = I 0 = A{Z){u>),
which asserts that u(Z) acts on F2H3(X) as does the Abel-Jacobi class of
u(Z). Q.E.D.

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 213
To state the next result we recall a special case of the notion of Cartier duality
[18, §10.2]:
LEMMA (3.17). Let G be a complex Lie group isomorphic to (C*)r and let
A be a compact complex torus. Let Gv = Hom(G, C*) be the dual group, let Av
be the dual torus, and let Mot(Gv,Av) be the group of generalized one-motifs
u: Gv —> Av. Then there is a functorial isomorphism
Ext{A,G) = Mot{Gv,Av).
PROOF. Given an extension [1—> G —> J —> A —► 0] and a character \ £ Gv,
let x*{J) be the principal C*-bundle obtained from J by applying \ to the
structure group. Then define a one-motif u j : Gv —► Av by setting
(3.18) uj{X) = [class of x*( J) in Av = Pic0(A)].
The map which sends J to uj gives the required isomorphism. Q.E.D.
One may use the previous results to recast in motivic language Collino's proof
that a generic cubic three-fold is irrational [14, 15] (see [13] for the stronger but
more difficult result). Consider a family of smooth cubic three-folds {Wt}
degenerating to a three-fold Wo which has an ordinary node p as its only singularity.
Let {Xt} be the family with smooth total space obtained by blowing up p, base
extending along t »—► t2, and then normalizing the result. Then Xq = Do U Di,
where Do is a quadric three-fold, D\ is P3 blown up along a complete
intersection curve Y of type (2,3) and genus 4, and Do\ — D0 H D\ is a quadric surface.
One may identify the intermediate Jacobian of X0 with the Jacobian J2 of the
mixed Hodge structure H3(X0). The weight filtration then gives a canonical
extension
(3.19) 1 -> W2J2 -> J2 -> Gr3 J2 -> 0,
which can be identified with the generalized Jacobian
(3.20) 1 -► C* -► J2 -► Jac(F) -> 0.
Now, if the generic cubic threefold is rational, then so is X0. Consequently J2
must be a direct sum of generalized Jacobians of curves [13]. Since the
intermediate Jacobian of a smooth cubic threefold is an abelian variety of dimension
five, J2(Xq) must be the generalized Jacobian of a nodal curve Y with
normalization Y. Equivalently, the one-motif associated to the generalized Jacobian
of Y by Cartier duality must be the one-motif of the curve Y. But this motif
is also isomorphic to that of the mixed Hodge structure on i/3(X0), hence is
isomorphic to the Abel-Jacobi map
(3.21) >l::NS(Xo)-> Jac(Xo).
In the case at hand !NS is (for generic {Wt}) the group generated by R\ — R2,
where R\ and R2 are the two rulings of the quadric surface D0i. But A(R\ - R2)
in the intermediate Jacobian of X0 can be identified with A(R\ Y - R2 Y)
in the ordinary Jacobian of Y, where the (2,3) complete intersection curve Y
is viewed as imbedded in Z?qi- Following Collino, let Ri • Y = at, + bi + Ci, let

214
JAMES A. CARLSON
q be the node of the curve Y, and let q+ and q~ be the preimages of q in the
normalization Y. Comparison of the one-motifs for the nodal curve and of the
singular threefold then yields the equation below in the Jacobian of Y:
(3.22) (ax + 61 + a) - (a2 + b2 + c2) = <7+ - <T-
Vary #1 and it!2 in their linear equivalence classes so that a\ = q+ and a2 = q~
to obtain
(3.23) (6i+ci)-(62 + c2) = 0.
One observes that (&i +ci) ^ (62 + c2), so that Y carries a nontrivial linear series
of degree < 2. Consequently Y is hyperelliptic, which is contrary to fact, since
F is a smooth (2,3) intersection, hence a canonically imbedded curve. Q.E.D.
For other applications of mixed Hodge theory to three-folds, see [3] and [11].
d. Homotopy. Morgan [30] has shown that the homotopy groups of an
algebraic variety carry mixed Hodge structures. This fact places nontrivial
restrictions on the homotopy types accessible to algebraic varieties, just as does the
existence of a Hodge structure on the cohomology of a smooth variety. For 7r3
of a simply connected three-fold with torsionfree cohomology, this structure is
particularly simple, with an extension class whose generalized one-motif has a
readily identifiable geometric counterpart. To describe it, we describe first the
mixed Hodge structure of 7r3. Set G = H2(X, Z) and use the classifying map
from X to an Eilenberg-Mac Lane space K(G, 2) to construct an exact sequence
(3.24) 0 -► H3(X) -► H4{K{G, 2), X) -► H4(K(G, 2)) -► H4(X) -► 0.
From the exact homotopy sequence of the pair (K(G, 2),X) one derives an
isomorphism of HA(K(G, 2), X) with tt3(X) =(by definition) Hom(7r3(X), Z). Since
the cohomology of K(G, 2) is a symmetric algebra on G considered as a group of
elements of degree two, there is an isomorphism of H4(K(G, 2)) with the
symmetric square, S2H2(X). Applying these identifications to the exact sequence
above one obtains
(3.25) 0 -► H3{X) ^ tt3(X) -► kernel{S2tf2(X) ^ H\X)} -► 0
where h* is the dual of the Hurewicz isomorphism and where "U" denotes cup
product. Since cup product is a morphism of Hodge structures, the right-hand
term in the preceding sequence, which we abbreviate as K4(X), carries a Hodge
structure of weight four. Morgan's theorem then presents 7r3 (X) as an extension
oftf4(X)by tf3(X):
(3.26) 0 -► H3{X) -► tt3(X) -► K4{X) -+ 0.
Consequently the Hodge-theoretic one-motif has the form
(3.27)
u:L2K4{X)^J2H3{X).

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 215
To describe its geometric counterpart consider expressions
(3.28) Ç = YlmvD*®D3>
where the D{ are divisors and the m^ are integers, and let
(3.29) r(0 = £myA-.D;
denote the resulting intersection one-cycle (if it exists). Define a lattice
(3.30)
Z(X) = \ £ = ^2 mijDi <8> Dj | r(f ) exists and vanishes in homology \
where ~ denotes the relation defined by linear equivalence of divisors. For an
element £ of this lattice there exists a 3-chain T such that dT = f. Consequently
there is an Abel-Jacobi map
(3.31) A:Z{X)^ Jac(X)
defined by
(3.32) ./1(f) = [integration functional of T = d_1r(£)]-
As one guesses, there is a comparison theorem relating the two one-motifs just
defined [9]:
THEOREM (3.33). Let X be a simply connected three-fold with torsionfree
cohomology. Then there is a canonical and functorial isomorphism between the
homotopy motif
(3.34) u: L2K4{X) — J2H3{X)
and the Abel-Jacobi map
(3.35) A: Z{X)-> Jac(X).
One may use this construction to produce threefolds which are determined up
to isomorphism by the polarized mixed Hodge structure on homotopy but not
by that on cohomology [9].
Recently much progress on the mixed Hodge structure of homotopy has been
made by Richard Hain [25, 26] using Chen's theory of iterated integrals [10].
Here we cite just one result, which gives a geometric interpretation to the first
of the many extensions associated to 7Ti of a smooth curve. To describe it, fix a
smooth algebraic curve X with a base point 6, let J(X, b) be the augmentation
ideal of the integral group ring of 7Ti (X, b), and let
(3.36) N2{X,b) =Homz(J(X,6)/J3(X,6)).
Define also the group
(3.37) K2{X) = kernel {U: Hl{X) 0 Hl{X) — H2{X)}
Then there is an exact sequence
(3.38) 0 -► H\X) ^ N2{X,b) -► K2{X) -► 0
where h* is dual to the Hurewicz map. The terms on the left and right carry
natural polarized Hodge structures of weights 1 and 2, respectively, and the
middle term carries a mixed Hodge structure relative to which both arrows are

216
JAMES A. CARLSON
morphisms. Then one has a 7Ti-Torelli theorem for pointed curves [25, 26, and
35]:
THEOREM (3.39) (HAIN, PULTE). Let (X,b) and (X'',&') be pointed
curves. Suppose that there is an isomorphism of rings between Ztti(X, b)/J3 and
Z7Ti(X/, b')/J3X which preserves Hodge and weight filtrations. Then there is an
isomorphism f : X —► X' such that, with the possible exception of two points b
ofX,f(b) = b'.
Hain's result comes from his local Torelli theorem for pointed curves, while
Pulte's refinement relies on the geometry of the extension class and the
polarizations. If X is hyperelliptic, then Pulte shows that the potential exceptional
points do not occur.
A further result of Pulte relates the extension class mv for N2(X,b) to the
Abel-Jacobi class of algebraic cycles [35]:
THEOREM (3.40). Let X be an algebraic curve, and let p, q be points of X.
Let Xp denote the image of X under the Abel-Jacobi map with base point p, and
let [Xp — Xq] be the difference cycle in Ji(Jac(X))7 the intermediate Jacobian for
one-cycles on the Jacobian of X. Let mp denote the extension class of ^(X,p).
Then there is an infective homomorphism
(3.41) Ji(Jac(X)) -+ Ext{K2{X),H1(X))
which carries [Xp — Xq] to mv — mq.
One may interpret the harmonic volume of Bruno Harris [27, 28] as the
"primitive part" of mv + mq [35].
The proofs of all of the above results rely on a description of the mixed Hodge
structure on tti in terms of K-T. Chen's theory of iterated integrals (see [10] and
[25, 26]).
4. A classifying space for mixed Hodge structures. We describe
classifying spaces D for polarized mixed Hodge structures analogous to those which
Griffiths has defined for Hodge structures [21, 22, 23]. These spaces were
constructed in [6, 41]. To begin, fix a lattice Hz, an increasing filtration W. of
Hq = Hz 0 Q, a set of bilinear forms S. on the graded quotients of W., and a
sequence of integers dp. Let D be the set of all mixed Hodge structures on Hz
with weight filtration W. which have Hodge numbers dimFp = dp and which
are polarized by S.. This last condition means that the quotients Gr^ H are
polarized by the Sm. The set of mixed Hodge structures which satisfy only the
first Riemann bilinear relations relative to the S. will be denoted Dy. When the
weight filtration has length zero, these sets are the classifying spaces of Griffiths.
Given a mixed Hodge structure H in D, let GrH denote the ordered set of
Hodge structures defined on the graded quotients of W.. Since each GrJ*' H IS
classified by a period domain Dk, Gr H is classified by a point of
(4.1)
Gr£> = n£>fc,

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 217
and so the map which takes H to Gr H defines a projection
(4.2) 7r:L>->GrL>.
We can now describe the structure of D:
THEOREM 4.3. Let D be a classifying space for polarized mixed Hodge
structures. Then
(i) D is a complex homogeneous space,
(ii) 7T is holomorphic and equivariant with respect to the natural group actions
on D and GrD,
(iii) the fibers of n are homogeneous relative to a natural action of a complex
nilpotent Lie group, and
(iv) there is a natural horizontal tangent bundle which maps to that of Gr D
under 7r*.
One shows that families of mixed Hodge structures which arise geometrically
(say from equivariant deformation of normal crossing varieties) define
holomorphic maps with values in D which are horizontal in the sense of (iv), so that the
notion of variation of mixed Hodge structure is both defined and nonvacuous.
PROOF. The essence of the proof is the construction of the correct group
actions. To begin, let K be a field contained between Q and C, let GIk be
the group of linear automorphisms of Hk = Hz <8> K, and let glK be the set of
K-rational points of the Lie algebra of GIk- The choice of a reference mixed
Hodge structure on Hz determines a mixed Hodge structure on glQ [16, 17, 18,
24] which satisfies
(4.4) [Fr,Fs]cFr+s,
(4.5) [Wr,Wa]cWr+s.
Indeed, define tp to be in Frglc if ip(FpH) C Fv+rH for all p, and define <p to
be in W^glg if (p(WmH) C Wm+rH for all m. The bracket relations imply that
the subspaces WmglK are subalgebras of glK for m < 0. If m < — 1, then these
subalgebras are nilpotent.
Let p(X) be the endomorphism which sends X in Gr^ gl to lX + X, where
the transpose is taken with respect to Sm on GrJ^ H. Then p: Gr^ gl —► Gr^ gl
is a morphism of mixed Hodge structures, as is the composition p o 7r, where
7T : WoQl —► Gr^ gl is the canonical projection. Therefore g, the kernel of p o n,
carries a mixed Hodge structure with respect to the induced filtrations.
When K is R or C a subalgebra 7Tk of glK defines a unique connected subgroup
of GIk, which we denote by Afc- Elementary arguments show that the group Gc
corresponding to gc operates transitively on Dv. Construction of a good action
on D itself requires more care: the group Gr preserves the subset of mixed Hodge
structures which are split over R, and hence does not in general act transitively
(only mixed Hodge structures whose weight filtration is of length one are a priori
split over R). One verifies, however, that the group M corresponding to the Lie

218
JAMES A. CARLSON
algebra
(4.6) m = W-igc + WogK
operates transitively on D. (The algebra W_20c + Wo0r would work just as
well, since Wk is split over R modulo Wk-2.) For later use we note that if mc is
the complexification of this Lie algebra and if mK is the subalgebra of K-rational
points then htk = 0k for K = Q, R, or C, and that hiq carries a mixed Hodge
structure with respect to the filtrations induced from glK.
To give a more detailed picture of the group M, choose a splitting of the
weight filtration, i.e., a sequence of subspaces W* C Wk such that
(4.7) W*^Wk/Wk-!
is an isomorphism. Choose a basis for Hq compatible with the splitting and
represent elements of Gq as a matrices (g^) in block triangular form, where
Qij maps Wf to W* and where g^ is zero if j > i. In this representation
W-iGk is the unipotent subgroup of complex matrices defined by the condition
ga = 1, and Gr^ Gk is the semisimple subgroup of block-diagonal matrices,
isomorphic to a direct sum of orthogonal and symplectic groups. Thus M is the
semidirect product of the complex unipotent group W-\M = W-\Gc and the
real semisimple group Gr^ M = Gr^ Gr.
To give the holomorphic structure onDv fix a reference mixed Hodge
structure on Hz and consider its isotropy group in Mc, namely, F°Mc. Then
(4.8) Dw = MC/F°MC
is a quotient of complex Lie groups, hence a complex manifold in a natural way.
The space of fully polarized structures D sits as an open subset of Dv, and hence
inherits a natural complex structure.
To give the holomorphic fibrations consider the exact sequences of complex
Lie groups below:
(4.9) W^Mc -► Mc -+ Gr^ Mc
(4.10) F°W_iMc -► F°MC -► F° Gr ^ Mc
The quotient of the first sequence by the second gives a fibration
(4.11) E^DV ^GrDv
where the fiber over Gr H is the nilmanifold
(4.12) EH = W_iMc/F0W_iMc.
The restriction of this fibration to Gr D gives the required fibration for D. When
the weight filtration has length one Eh is an abelian homogeneous space, and in
fact carries a natural group structure, namely, that of Ext(GrJ^ i/, GrJ^_x H),
the group of extension classes [6] of GrJ^ H by Gr^_x H. It is therefore natural
to view Eh as a space of iterated extensions associated to the sequence of Hodge
structures Gr H.

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 219
To define a horizontal tangent bundle [22, 23] for D, note first that the
holomorphic tangent bundle is the homogeneous bundle given by
(4.13) TD = M xFoM (mc/F°mc),
where the isotropy group F°M acts on M by left translation and on mc by the
adjoint representation. The horizontal subbundle, just as in Griffiths's case, is
then defined by
(4.14) ThoTD = M xFoM (F-^mc/l^mc).
As a final remark, we observe [6] that if T is the subgroup of M which preserves
Hz, then T operates properly discontinuously on D to give a quotient which is
an analytic space, and the unipotent group W-iT operates freely on Eh to give
a smooth quotient which classifies congruence classes of iterated extensions of
Hodge structures. Thus, T\D fibers analytically over GrT\D with W-\T\Eh
as fiber.
EXAMPLE (4.15). Let Hn be a real vector space with basis {em} and let
Hz be the lattice generated by (27rz)mem, where m = 0,1,2. Define a weight
filtration by decreeing em to have weight —2m. Stipulate that Gr_2m have type
(—ra, —ra), so that Gr_2m — Z(ra), the Hodge structure of Tate. The group of
the classifying space for structures of this kind is a Heisenberg group of dimension
three:
G(K) = {
(
1 M(p,q,r) =
V
"l 0 0
p 1 0
r q 1
p,q,reK
for K a field contained between R and C, and
G(Z) = {M(p,q,r) \ p, q G 2tt»Z, r G (2ttz)2Z} .
In this case Gr D is a point, so that D is itself a nilmanifold. In fact, F°Gc is the
trivial group, so that D = Go is a unipotent group, and T\D = G(Z) \G(C) is
a principal C*-bundle over C*xC*. This latter is the natural weight fibration,
determined by the exact sequence
0 -► W-2G -► W-iG -► Gr^ G -► 0.
One should view elements of D as iterated extensions of Tate structures: form
E = Z{2)eq Z(l);
then H = £0$ Z, where $ is the matrix $ = [J]. In this "twisted direct sum"
notation the subscript is the extension homomorphism, so that E has class
[<7]GExt(Z(l),Z(2)) = C*
and H has class
[$]GExt(Z,£) = C* xC*.
Structures similar to these—with graded quotients Z, Z(l)fc, Z(2)—arise
naturally, with D homogeneous relative to a Heisenberg group of dimension 2k + 1

220
JAMES A. CARLSON
A =
B =
Figure 2
(replace the scalars p and q by vectors of length A:, etc.). For a first example,
let A U B be a union of six lines in general position in P2 (see Figure 2), three
in A and three in B. Let X = P2 — A, and let Bf = B — A. Then the mixed
Hodge structure of the pair (X, B') is an iterated extension of Tate structures
with k = 4.
One may achieve the same result with singular quasiprojective surfaces: Let
Y be the union of four planes D{ in general position in P3, let A = D0 H (Z?i U
D2 U D3), and let 5 be a union of three lines in general position relative to A (as
before). Then the mixed Hodge structure oîY' = Y — Ais isomorphic to that of
(X, B'). Finally, objects of the type just considered appear as the building blocks
of the limit mixed Hodge structures for degenerations of hypersurfaces in P3 to
a union of planes in general position. For all of these structures explicit
calculation reveals the periods to be combinations of logarithms and dilogarithms.
The analogous constructions for n-dimensional varieties yield periods which are
combinations of p-logarithms, for p < n, with classifying space homogeneous
relative to a "Heisenberg group" which is unipotent of index n.
Bibliography
1. A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactifications of locally
symmetric varieties, Math. Sci Press, Brookline, Mass., 1975.
2. W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric
domains, Ann. of Math. (2) 84 (1966), 443-528.
3. F. Bardelli, Polarized mixed Hodge structures: On irrationality of threefolds via
degeneration, Ann. Mat. Pura Appl. (4) 138 (1984), 287-369.
4. J. Carlson, Extensions of mixed Hodge structures, Journées de Géométrie
Algébrique d'Angers 1979, A. Beauville (Editor), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980,
pp. 107-128.

THE EXTENSION CLASS OF A MIXED HODGE STRUCTURE 221
5. , Generalized one-motifs of normal crossing varieties, Preprint, University of Utah,
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6. , The obstruction of splitting a mixed Hodge structure over the integers. I, Preprint,
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7. , The one-motif of an algebraic surface, Compositio Math. 56 (1985), 271-314.
8. J. Carlson, E. Cattani, and A. Kaplan, Mixed Hodge structures and compactifications of
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& Noordhoff, Alphen aan den Rijn, 1980, pp. 77-105.
9. J. Carlson, C. H. Clemens, and J. Morgan, The mixed Hodge structure associated to 7T3
of a simply connected threefold, Ann. Sci. École Norm. Sup. (4) 14 (1981), 1-16.
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11. C. H. Clemens, Double solids, Adv. in Math. 47 (1983), 107-230.
12. , Degenerations of Kahler manifolds, Duke Math. J. 44 (1977), 215-290.
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14. A. Collino, A cheap proof of the irrationality of most cubic threefolds, Boll. Un. Mat. Ital.
B (6) 16 (1970), 451-465.
15. , Footnote to a paper of Griffiths, Boll. Un. Math. Ital. B (6) 17 (1980), 748-756.
16. M. Cornalba and P. Griffiths, Some transcendental aspects of algebraic geometry, Proc.
Sympos. in Pure Math., vol. 24, Amer. Math. Soc, Providence, R. I., 1975, pp. 3-110.
17. P. Deligne, Théorie d'Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-58.
18. , Théorie d'Hodge. Ill, Inst. Hautes Études Sci. Publ. Math. 44 (1975), 5-77.
19. R. Friedman, A new proof of the global Torelli theorem for KS surfaces, Ann. of Math.
(2) 120 (1984), 237-269.
20. P. A. Griffiths, Periods of integrals on algebraic manifolds. I, Amer. J. Math. 90 (1968),
568-625.
21. , On the periods of certain rational integrals. I, II, Ann. of Math. 90 (1969), 460-541.
22. , Periods of integrals on algebraic manifolds. Ill, Inst. Hautes Études Sci. Publ.
Math. 38 (1970), 125-180.
23. P. A. Griffiths and W. Schmid, Locally homogeneous complex manifolds, Acta Math. 79
(1964), 109-326.
24. , Recent developments in Hodge theory: A discussion of techniques and results,
Discrete Subgroups of Lie Groups and Applications to Moduli (Internat. Colloq., Bombay, 1973),
Oxford Univ. Press, Bombay, 1975, pp. 31-127.
25. R. Hain, The geometry of the mixed Hodge structure of ir\, this volume.
26. , The de Rham homotopy theory of complex algebraic varieties, Preprint, University
of Utah, 1983.
27. B. Harris, Harmonic volumes, Acta Math. 150 (1983), 91-122.
28. , Homological versus algebraic equivalence in a Jacobian, Proc. Nat. Acad. Sci.
U.S.A. 80 (1974), 165-185.
29. J. W. Hoffman, The Hodge theory of stable curves, Mem. Amer. Math. Soc. No. 51
(1984).
30. V. Kulikov, Degenerations of K3 surfaces and Enriques' surfaces, Math. USSR-Izv. 11
(1977), 327-360.
31. J. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci.
Publ. Math. 48 (1979), 137-204.
32. Y. Namikawa, A new compactification of the Siegel space and degeneration of abelian
varieties. I, II, Math. Ann. 221 (1976), 97-141 and 201-241.
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of Math. (2) 113 (1981), 45-46.
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Math. USSR-Izv. 35 (1971), 530-572.
35. M. Pulte, The fundamental group of a pointed Riemann surface: Mixed Hodge structures
and algebraic cycles, Thesis, University of Utah, 1985.
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222 JAMES A. CARLSON
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80 (1985), 489-542.
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41. S. Usui, Variations of mixed Hodge structure arising from family of logarithmic
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51 (1984), 851-857.
University of Utah

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
The Local Geometry of the Abel-Jacobi Mapping
HERBERT CLEMENS
0. Introduction. The purpose of this paper is to explore the interaction
between families of curves on a complex projective manifold V and the cohomol-
ogy of V. The basic mechanism for this interaction is the so-called "infinitesimal
Abel-Jacobi mapping." Given a smooth curve C immersed in V via a mapping
/ : C —► V, we measure the first-order deformations of / into a fixed target space
V from a (possibly changing) domain curve C by the vector space H°(Nfy),
where T denotes "tangent bundle" and Nfy = J*Ty/Te- From now on, the
word curve will always mean a mapping /: C —► V, where C is a smooth curve
and / is everywhere of maximal rank. If C is actually embedded in V, we may
use the alternative notation Ncy for Nfy •
Now there is a natural contraction mapping
9: ArH°{Nfy) ® H1^1) -+ H1^)
obtained by contracting differential forms against vector fields. It is this mapping
$ that is called the infinitesimal Abel-Jacobi mapping.
If we denote by F the parameter space of a family of curves on V, then $ is
associated to a morphism of Hodge structures of type (—1,-1) called "integration
over the fibre" :
(o.i) p..g*:/r(vo-/r-2(n
where Z Ç F x V is the incidence variety of pairs (curve, point) such that the
point lies on the curve, and p and q are the natural projections. Now the tangent
space to F at a point / is naturally mapped to H°(Nfy) so that ArH°(Nfy)*
—y QrF. In this way, the mapping $ defined above gives a map
(0.2) QiH1^1)^!!0^).
It is directly verifiable that this is the same as the map induced by p* • q*.
(0.3) DEFINITION. Two projective curves, C and C;, in V are called
algebraically equivalent if there is a third curve D such that (C + D) and (C + D) are
members of the same connected algebraic family of curves. If every component
1980 Mathematics Subject Classification (1985 Revision). Primary 14C30.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
223

224
HERBERT CLEMENS
of the parameter space of the family can be made to be a rational variety, then
C and C" are called rationally equivalent
The small remark to make at this point is that C and C may be rigid, or
may have no deformations outside a subvariety on which they are not even topo-
logically equivalent, and still be algebraically equivalent on V. The prototypical
example of this latter phenomenon is obtained by letting V be the resolution of a
hypersurface of projective four-space of degree < 4 with one node, and letting C
and C be representatives of the two rulings of the exceptional locus. The curve
D which must be added to each to effect the equivalence is the proper transform
of a line through the node.
In §1 and §2, we study some elementary properties of the infinitesimal Abel-
Jacobi mapping for curves on hypersurfaces in projective space. The fundamental
relation with rational equivalence is that if the parameter variety F in (0.2) is
rational, then $* is trivial, so that we can use the rational equivalence classes in
F, rather than F itself, to parametrize cohomology of type (r + 1,1) in V.
In §3, we will make an extended study of one very special case, namely, the
case of curves on a quintic in P4. For threefolds, the infinitesimal Abel-Jacobi
mapping is simply the differential of a mapping
(H3^(V) + H2^(V)Y f
where T is a 3-chain on V such that dT = C — Co, Co being a fixed base curve in
the family F. {J{V) is called the (intermediate) Jacobian of V.) From the remark
above, it is easy to see that if C\ and C<i are rationally equivalent curves, then
0({Ci}) = ^({^2}). We will apply the results of §1 and §2 to make geometric
conclusions about the mapping (0.4) when V is a smooth quintic threefold.
1. The first commutative diagram. Let V be a smooth hypersurface of
degree m in Pn, and suppose we are given a curve /: C —► V.
(1.1) PROPOSITION (WELTERS). The following diagram is commutative:
\n-3H0(Nfv) s H0(nnn(V)) ® H°(0Pn (m)) Residue» \n-3H°(Nfy) ® H0^-1) ® H°(Nv^n)
I
\n-3H°(Nfy) ® H°(nPn(2V)) Residue» An-3H°(Nfy) ® H0^1 ® AV,Pn)
transgression
\n-3H°(Nfy)^H1(Û^-1(V)) //°(A--3^/,v)®//°(/*(n^-1®7Vv,pn))
residue transgression
An-3H°{Nfy) ® Hl(U^-2) H°{\n-3Nf,v) ® Hl{n^~l ® JV/|V)
Hl(nlc]
^ c

THE LOCAL GEOMETRY OF THE ABEL-JACOBI MAPPING 225
residue
n»n(2v)
i
nj„(2V)
^n(V)
^v>Pn+lî
NVPn ®Qpn(V)
I
NVPn <g>n£-1
i
f*(NVtI>n®n»-v
- 0
where Û means closed-forms, transgression means the connecting homomorphism
in the cohomology sequence for the corresponding short exact sequence of sheaves,
and fi and all unmarked maps are induced by the natural pairings on the sheaves
involved, fx is a perfect pairing by Serre duality since the sheaf pairing
An-3Nfy <g> (fi^T1 ® Nf,v) -► np*
is perfect
PROOF. The assertion in the lemma is a straightforward corollary of the
following series of morphisms of exact sequences of sheaves
o ► ftpn1^) ► Hp"1^) ——- npn(2y) ► o
i i
0 ► —± :— ► — ;—
on_1 nn_1
**pn **pn
S< contraction S<
o ► Tv®npn(v) ► TPn(g)npn(y)
I residue I
o ► 2v®ft£_1 ► Tpniv^n^"1
i i
0 ► NfytoQ^r1 ► Nf^pn^Q^r1
where v = 0 is a local defining equation for V. D
(1.2) COROLLARY. If the sequence
0 -► Nfy -► Nf,pn -► f*NV,Pn -► 0
is split, the infinitesimal Abel-Jacobi mapping is zero. U
As an easy geometric application of Lemma (1.1), let us consider embedded
curves C Ç Pn which we will call "M-regular"; that is,
(i) tf°(Pn;Opn(M)) -► H°{C;Oc{M)) is surjective, and
(ii)i/1(C;7Vc,pn(M)) = 0.
(1.3) LEMMA. If M = m — (n+1) andn > 4, and z/an M-regular curve C
on V has first-order deformations which generate a subsheaf of Ncy of corank
1, then the infinitesimal Abel-Jacobi mapping
9: An-3H°{NCy) -► H^O^'2)*
is nontrivial
PROOF. M-regularity gives that the map
H°{nPn(2V)) -+ Hl{^Tl <g> NCy)
occurring (diagonally) in the diagram in Proposition 1.1 is surjective. Griffith's
theory gives the surjectivity of
and so, since /z is a perfect pairing, Lemma 1.3 is proved. □

226
HERBERT CLEMENS
(1.4) COROLLARY. Suppose V Ç Pn is a smooth hyper surf ace of degree
> (n + l)/2, and suppose a divisor on V is covered by a family F of rational
curves whose generic member is protectively normal. Then the mapping (0.1)
associated to the family is nontrivial, that is, the morphism of Hodge structures
(0.1) induces a nonzero mapping
Hn-2^{V)^Hn-^°{F).
PROOF. If C is projectively normal,
Ncrn=Oc(n + 2)®(n-1\
so C is M-regular whenever M > — (n + 3). So C is always (m — (n+ l))-regular.
Also, if 2ra > (n + 1), Hn-2^(V) is nontrivial. Apply Lemma (1.3). □
The point of Proposition (1.1) is that the Abel-Jacobi mapping is measuring
the obstruction to splitting the sequence
(1.5) 0 -> Nfy -> iV/iPn -> f*NVfPn -> 0.
For example, suppose we are in a situation in which
n£_1 <g>Av,pn =ev,
for example, the case of the cubic fourfold. Then, as long as the sequence (1.5)
is not split,
ff°(n^"1®JV/fPn) = 0
so that the one-dimensional vector space
^(/•(nîT1®^»))
is (1.1) injects into
Since /i in (1.1) is a perfect pairing, this allows:
(1.6) COROLLARY. Let V be a smooth hypersurface of degree (n+ l)/2 in
Pn (n odd). Let f:C^>V be an immersed curve such that the natural map
(1.7) An-3H°{Nfy) -► H°{An-3Nfy)
is surjective. Then the infinitesimal Abel-Jacobi mapping
$: Hn-2^{V) -+ An-3H0{Nfy)*
is injective. □
So, for example, if C is a rational curve on V, and if the summands in the
line bundle decomposition of Ncy all have nonnegative degree, then C is
unobstructed in V, and so the deformation space Fy of C in V is smooth at C with
tangent space H°(Ncy). Also the mapping (1.7) is surjective, and so we have
injective
$: Hn-2^{V) -► tfn-3'0(F).

THE LOCAL GEOMETRY OF THE ABEL-JACOBI MAPPING 227
In fact, in this situation, a constant count gives that Fy deforms with V in Pn
so that, by differentiating the period mapping or invoking the irreducibility of
global monodromy, one obtains that the mapping
P* o q* : Hn-\Vf -+ Hn-3{F)
is injective, where "°" means primitive cohomology (with rational coefficients).
Finally we have:
(1.8) COROLLARY. Let F be a family of curves on a smooth hypersurface
V C Pn, such that f:C^V is maximal rank for generic f € F. Suppose
(i) the curves of F cover a divisor in V, and
(ii) H0^-1 <g> Nf9pn) = Hl{%~1 <g> Nf^n) = 0.
Then the infinitesimal Abel-Jacobi map $y is nontrivial at f.
PROOF. The transgression map
i/^nn;-1® Wv,p»)) - h1^-1 ® Nfy)
in (1.1) is bijective, An~3H°(Nfy) maps onto a nontrivial subspace of
H°(An~3Nfy) and the pairing fi is perfect. □
2. The second commutative diagram. There is a second useful
commutative diagram associated to the infinitesimal Abel-Jacobi mapping. It can be
thought of as a kind of infinitesimal version of the theory of Lefschetz pencils.
Before stating it, we need some preliminary considerations.
Suppose W Ç pn+x is a smooth hypersurface of degree m, and suppose V
is a smooth hyperplane section of W and /: C —► V is a curve on V. Let
aeH°{f*NyW) and let
1V€H1{{f*NVti>ny®Nfy)
be the obstruction to splitting the sequence
0 -► Nfy -> Nf^n -► /*AV,Pn -► 0.
Then the image T\y of ry in Hl((f*Nw^n+\)* ® Nfyy) is the obstruction to
splitting
0 -► Nftw -> W/,p~+i -► f*NWtT>n+i -► 0.
So we have commutative
ff°(/*(^®%p^)) —^ fl0(/*(nv"1«%«))
On the other hand, the contraction
H*(kn-*Nfy)®H\r(Wb®Mfy))^Hl(Uic)

228
HERBERT CLEMENS
is zero, since it factors through f*Qy = 0. Putting these two facts together, we
have commutative:
An-2H°(NftW) ® H°(r(Q^ ® NWtpn+i))
H°(An-2Nf<w) ® H^n», ® Nf,w) kn-3H°(Nfy) ® H0(f*Nv,w) ® H°(P(n^ ® Nwpn+i))
-^ x ^
H°(An-3Nfy ® f*Nv<w) ® H1 (nj», 8> JV/|ÏV)
^n-3H0{Nfv) q H0{r{Qn-l ^ Ny pn))
+ tf1^) ^ ^ H°(A»-*Nf,v)®HHnï-l®Nfy)
Using Proposition (1.1), we obtain our second useful commutative diagram.
(2.1) PROPOSITION. Let v, W, and f be as above. Then the following
diagram is commutative:
An-2H°(Nf,w) ® tf°(n£;!i (2W0) \n-*H°(Nfy) ® H°(NV,W) ® H0^1^ (2W))
H°(\n-3Nf<v ® rNVtW) ® /f°(n~tii (21V))
*vv ©Residue
\n-3H0(Nfy) q //0(nn n (2y))
*\/°Flesidue
^ i/^nj,) -<
Notice that the contraction mapping 7 in Proposition (2.1) is surjective. For
an easy application of the proposition, suppose that Fy is a family of immersed
curves on V which is smooth at some / G Fy and furthermore suppose that Fy
deforms with V as V moves in the space of hyperplane sections of W. Then
Proposition (2.1) implies:
(2.2) COROLLARY. If the infinitesimal Abel-Jacobi mapping $y for V is
nontrivial at f G Fy, then the same is true for the infinitesimal Abel-Jacobi
mapping $w for W at
f g Fw = [){Fv x {V}: V1 hyperplane section ofW).
PROOF. Suppose Qv(v,uv) ± 0 for v G An-3Tf{Fv) -> An-3H°{Nfy).
Pick ujw G J/°(njtii(2W0) and a deformation a G H°(NV,W) so that the
contraction
l{a,uw) = uy-
One needs only check that

THE LOCAL GEOMETRY OF THE ABEL-JACOBI MAPPING 229
lies in the image of a in (2.1). But this is implied by the statement that Fy
deforms with V in the direction a. U
In the other direction:
(2.3) Corollary. //, for f: C^ V ç W ç Pn+1,
$w{"l A • • • A isn-3 A(T,w)/ 0,
where a is in the image of H°{Ny,w), and v\,..., vn-z £ H°(Nfy), then
$v(i/i,...,i/n-3,rç) 7^0,
where rj is obtained by contracting uj against the normal vector field a. □
3. The universal line on the quintic threefold. We will now treat an
illustrative special case:
C = {(i,V): is a line on the smooth quintic hypersurface V Ç P4}.
By projecting onto the Grassmann variety of lines in P4, it is easy to see that £
is smooth and irreducible. The other projection
(3.1) 7r: £ —► Q = variety of smooth quintic threefolds
is proper and generically 2875-to-one [3]. Now let v\ 3 —► Q be the Jacobian
bundle, that is, the natural torus-bundle over Q whose fibre over {V} is J{V)
(see (0.4)). Define a morphism (j>: £> —y 3 over Q by the formula
./plane section of V
which is well defined since a plane section of V and 5L give the same element of
ft(7;Z)sz.
(3.2) THEOREM. The mapping (j> is everywhere of maximal rank.
PROOF. The tangent space to £ is the space of first-order deformations of
the pair (L, V), and so is the first hypercohomology IK1 of the mapping
p:Tv^NLy.
So we have an exact sequence
(3.3) 0 -► H°{NLy) -► 5C1 -► H\TV) À Hl(NLy).

230
HERBERT CLEMENS
Since Qy is trivial in this case, we have from Proposition (1.1) the commutativity
of the diagram
(3.4)
H°(NLy)®H°(0P4(b)) -
H°(NLy)®H°(np4(2V))
H0(NLy)®H1(Û:ip4(V))
H°(NL,v)®Hl{t%
-*yH0(NLy)®H°(Nv,p4)
Hl{ni)
But we also have commutative
H°{il^4(V))®H0{0P4(b))
^{QD^H^T)
H°{NLy)®H°(rNyP4)
H°(NLy)®Hl{NLy)
, Serre
^ duality
H°(Q*t(2V))
Kodaira-
Spencer
>
map
#W)
Putting these two diagrams together, we have a commutative diagram involving
the Abel-Jacobi map and the mapping A from (3.3):
(3.5)
H°(NLy) ® ff°(f#) ® Hl{Tv)
K.S.map
H^N^^H1^)
lxX
3>v
H°{NLy) ® H°(ty) ® HX(NLy)
V
c
We are now ready to compute the differential of <\> : £ —► #. If a G ÎK1 is in the
kernel of </>*, then a must go to zero in the tangent space to Q and so in H1 (TV).
So a G H°(NLy). Then 0*(a) is the linear functional $v(a, ) on jff^fi^). But
(3.4) says that the mapping 1 x A in (3.5) is surjective in this case. So, since \i
is a perfect pairing, a = 0. □
Notice that a corollary of Lemma (1.3) together with the irreducibility of the
monodromy representation is
(3.6) THEOREM. Let W be a smooth quintic hypersurface in P5 and let F
be the Fano variety of lines on W. Then the induced mapping (0.1)
H4{W)^H2{F)
is injective.
This result can also be deduced from Theorem (3.2) and Proposition (2.1)
(and, of course, irreducibility of the monodromy representation). □

THE LOCAL GEOMETRY OF THE ABEL-JACOBI MAPPING 231
Also notice that Theorem (3.2) implies Griffiths's result that the group of
algebraic cycles homologous to zero, modulo those algebraically equivalent to
zero, is nontrivial, since, for generic V, J{V) contains no abelian subvarieties,
and cycles algebraically equivalent to zero always map to an abelian subvariety
[2]. In fact, it is by constructing the corresponding mapping <\>\ 6 —► d over Q, for
rational curves of an infinity of degrees, and showing that the various mappings
6 —► Q have different branch loci, that one proves the nonfinite-generation of
homological equivalence modulo algebraic equivalence [1].
Finally, notice that the diagram (3.5) says that the Abel-Jacobi mapping
$v:H°{NLy)^Hl{tfvy
is the adjoint of the mapping
X:H\Tv)^Hl{NLy)
induced by restriction, where the pairings are given by Serre duality. (This is
true for any smooth curve on a threefold with trivial canonical bundle.)
Now it is a result of S. Katz [4] that the only possible normal bundles for lines
in quintics in Q are
% = oL(-i)eoL(-i) (*)
(3.7) =OL0OL(-2) (**)
= oL(i)eoL(-3) (***)
Since £ is smooth, Theorem (3.2), (3.3), and (3.5) give that (*) occurs exactly
at the étale points of 7r: £ —► Q. At a generic point of a component of the
branch locus of 7r, L deforms with V along a subvariety of codimension 1 so
that, by the commutativity of (3.5), (**) holds. So let R Ç £ be a component
of the ramification locus of the mapping n and let B = ir(R). Since B is a
hypersurface of Q which is invariant under the action of the general linear group
on P4, we have a Gauss map r which assigns to a general point {L,V) G R
the hyperplane in Hx(V',Tv) = Def V consisting in deformations along B. The
commutativity of (3.5) says that this map
{L,V) ^ t(L,V) tV(Hl(V,Tvy)
= p(H1(v,nly)
can also be obtained by associating to (£, V) the hyperplane
This means that there is an intimate relationship between the Nash blow-up of
B at any of its singular points {V} and the points of
T = RfMr'1{V).
(3.8) PROPOSITION. Let (L,V) be a smooth point ofT. Then we have a
local immersion
4>:T^J(V).

232
HERBERT CLEMENS
(i) If Nl,v = Ol ® Ol(—2), Men £/ie ima^e 0/ £/ie Gauss map o/0
is a component of the Nash blow-up of B at {V};
(ii) ifNLy =Ol(1)®Ol(-3), and 2/ dim(z,5V) T = 1 and
WL,fV=OL/(l)0OL/(-3)
for V near L on T, then the image of the 2nd-order Gauss map (the oscillating
map) of<f)
is a component of the Nash blow-up of B at {V}.
These phenomena seem to have very little to do with the fact that we are
studying lines on quintics but rather with the fact that we are studying curves
on a threefold with trivial canonical bundle at points where the curves don't
deform generically with the threefold (to first order). We used lines on quintics
because the list (3.7) of possible normal bundles is so small. The point is that
if Hodge's generalized conjecture is to be true for threefolds with trivial
canonical bundle, then, whenever J(V) admits a nontrivial abelian subvariety (i.e.,
subtorus perpendicular to H3>°(V)), there should also be a "versai" family of
curves with Abel-Jacobi map
e -—yd
\ /
Def(V)
such that
(i) 7T is surjective so that by (3.5)
dim$(e) = dim(DefVO;
(ii) there exists a component B(C Def V) of the branch locus of
u: 0(6) -> DefF
such that the abelian subvariety of J(V) is to be found in the mixed Hodge
structure associated to the local cohomology of B at {V}.
For V such that fi^ is ample, one might hope for diagrams (3.9) with
u: 0(6) ->DefV
generally injective, in which case one might look for the abelian subvariety of
J(y) in the mixed Hodge structure associated to the local cohomology of 7r(C)
at {V}.

the local geometry of the abel-jacobi mapping 233
References
1. H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely
generated, Inst. Hautes Etudes Sci. Publ. Math. 58 (1983), 19-38.
2. P. A. Griffiths, On the periods of certain rational integrals, II. Ann. of Math. (2) 90
(1969), 496-541.
3. J. Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), 685-724.
4. S. Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60
(1986), 151-62.
University of Utah

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Generic Torelli and Infinitesimal Variation
of Hodge Structure
DAVID A. COX
Recently several generic Torelli theorems have been proved using Infinitesimal
Variations of Hodge Structure. The inspiration comes from Donagi's success in
using these techniques to prove generic Torelli for most projective hypersurfaces
[5]. The purpose of this paper is to report on some of the current work being
done.
In §1, we recall the important idea of variational Torelli and discuss its relation
to generic Torelli. In §2 we sketch Donagi's proof in order to introduce the ideas
and techniques that are being used. §3 discusses recent developments by Saito
[15] and Donagi and Tu [7], on certain hypersurfaces in weighted projective
space, and by Green [9], on sufficiently ample divisors in varieties with very
ample canonical bundle. Finally, in §4, we present an example, due to the author
and Donagi [4], to show that variational Torelli fails for regular elliptic surfaces
with a section.
1. The basic references for Infinitesimal Variations of Hodge Structure
(abbreviated IVHS) are [1] and [14], while [2] and [10] are also useful. An IVHS
of weight n consists of a polarized Hodge structure (Hz,F',Q) of weight n, a
finite-dimensional complex vector space T, and a linear map
n
(1.1) 6: T-+0 Hom(Fp/Fp+1,Fp-7Fp).
P=i
Furthermore, setting £(£) = 0p=1 S(£)p, we require
(1.2) tf(£i)p-i o £(6)P = <H6)P-i o tf(£i)p for 6,6 e r,
(1.3)
Q(6(Ç)pfail>) + Q(<M(0n-P+iV>) = 0 for £ G T, <\> G Fp, and </> G Fn~p+1.
The idea behind this definition is as follows: let $ : S —> D be a local lifting
of a variation of Hodge structure, where S is a poly disc and D is the appropriate
1980 Mathematics Subject Classification (1985 Revision). Primary 14D20; Secondary 32G13.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
235

236
DAVID A. COX
period domain. If s G S and F' is the filtration corresponding to $(s), then the
differential of $ at s gives a map
n
$*:TS(S) - TF.(D) Ç 0 Hom(Fp,F°/Fp).
p=i
Using the infinitesimal period relations, this gives a map
n
$*: Ts(5) -0 Hom(Fp/Fp+1,Fp-7^P)>
p=i
which can be shown to satisfy (1.2) and (1.3) (see [1] and [14]).
Thus an IVHS is basically the differential of a variation of Hodge structure.
An ordinary Hodge structure has an algebraic part, the filtration F\ and a
transcendental part, the integer lattice Hz- Separately, these two parts contain little
information. In the context of Torelli problems, one must use both parts in order
to have any hope of recovering the variety one began with. However, an IVHS
(HZ,F\Q,T,6) has a substantially larger algebraic part, namely (F',Q,T,6),
and sometimes knowledge of the algebraic part is sufficient to reconstruct the
original variety. This is the basic idea of what is known as variational Torelli.
But we are now getting ahead of ourselves—we first need to review the Torelli
problem.
Suppose we are studying smooth, projective varieties of a certain type which
are classified by a coarse moduli space M. For simplicity, assume that M is an
irreducible algebraic variety. Then we have the period map
P:M->T\D,
which assigns each X G M to its polarized Hodge structure. Then we can pose
two problems:
Torelli or Global Torelli Is the period map P one-to-one?
Generic Torelli or Weak Global Torelli. Is P one-to-one on a Zariski open
subset of M?
The recent successes of the IVHS techniques have led to the formulation of a
third Torelli problem:
Variational Torelli. Can the variety be recovered from the algebraic part of
its IVHS?
Variational Torelli is interesting primarily because of its usefulness in proving
generic Torelli theorems. The general philosophy is that there should be an
implication of the form
(1.4) Variational Torelli => Generic Torelli.
Various ways have been given to make this precise, though extra hypotheses (e.g.,
the image of M must contain smooth points of T\D—see [14]) were needed. The
following proposition shows that (1.4) holds with only very mild extra
assumptions.

GENERIC TORELLI AND HODGE STRUCTURE
237
PROPOSITION 1.1. Assume that the period map P: M —> T\D satisfies:
(i) M is an irreducible;
(ii) P is holomorphic;
(iii) Generically, P is locally liftable (i.e., every point in some Zariski open
subset of M has a neighborhood on which P factors through the projection D —>
r\D).
Then variational Torelli for a generic element of M implies generic Torelli.
PROOF. We first show that generically, the differential of a local lifting of P
is injective. If this is not the case, the liftings must have positive dimensional
fibers. At a point of maximal rank, a local lifting is analytically equivalent
to a projection followed by an embedding. It follows that the IVHS is, up to
isomorphism, constant along the fiber. This contradicts our assumption that
variational Torelli holds generically.
Standard results of Griffiths (see [11] and also [1, La]) show that P has a
proper extension
P:M->T\D.
Thus P(M) is an irreducible subvariety of T\D. We will study the map M —>
P(M). _ _
Let's first show that M and P(M) have the same dimension. Let n = dime M =
dime M. Using what we just proved above, we can find a polydisc An in M that
is embedded into D by a local lifting of P. We may assume that the image of
An has compact closure in D. Since T acts properly discontinuously, we see that
P\ A" • An —> F(M) has finite fibers, and it follows that F(M) has dimension n.
Thus P: M —> P(M) is a proper surjective map between varieties of the same
dimension, and it follows that the degree of P is defined. If the degree is one, P
is generically injective, and generic Torelli follows.
If the degree is greater than one, then we can find two general points of M
mapping to the same point of T\D. In a neighborhood of these two points, P is a
two-sheeted cover onto its image. If g is the holomorphic map which interchanges
the sheets, then its differential g* gives an isomorphism between the IVHS's of
the two original points. This again contradicts our assumption that variational
Torelli holds generically. This proves the proposition.
I am grateful to Ron Donagi for suggesting that (1.4) should hold more
generally. Also, note that standard constructions often produce coarse moduli spaces
that satisfy conditions (i)-(iii) of the proposition. More precisely, suppose one
has a smooth family X —► M and a group acting on M so that M is the quotient
of M in the sense of geometric invariant theory. If M is irreducible, condition
(i) is obviously satisfied. As for conditions (ii) and (iii), consider the diagram
M £ D
-1 1
M £ T\D

238
DAVID A. COX
where M is the universal cover of M. Then P is holomorphic since P is, and
generically P is locally liftable since n is generically smooth.
2. In order for variational Torelli to be useful in proving generic Torelli
theorems, we need a way to extract information about the original variety from the
algebraic part of the IVHS. For smooth projective varieties this has been done
by Griffiths (see [2] and [12]).
Let X C Pn+X be a smooth hypersurface of degree d defined by the equation
/ = 0. The Jacobian ring R(f) is the quotient
R{f) = C[x0, xi,..., xn+i]/J(/),
where J(f) = (df/dx{) is the Jacobian ideal of /. R(f) and J(f) have natural
gradings that will be denoted by superscripts. When there is no danger of
confusion, we will omit the explicit reference to /, giving us R and J.
The importance of R and J is indicated by the following result.
THEOREM 2.1. Let f and /' be homogeneous polynomials defining smooth
hypersurfaces of degree d in Pn+1. Then f and /' are projectively equivalent if
either of the following conditions is satisfied:
(i) R(f) is isomorphic to R{f), or
(ii) j{f)d = j{f)d.
For proofs, see [13] for (i) and [1, 2, or 5] for (ii). The reason J{f)d appears
is that it is the tangent space to the orbit at / of the usual action of GL(n+2, C)
on homogeneous polynomials of degree d.
In view of this theorem, it is reasonable to try to prove variational Torelli via
the Jacobian ring, i.e., to try to recover as much of R{f) as possible from the
algebraic part of the IVHS. Furthermore, as (ii) above indicates, only certain
essential pieces of R(f) are needed.
What makes this philosophy reasonable is the work of Griffiths which shows
that the algebraic part of the IVHS is actually equivalent to partial information
about R = R(f). Let us briefly recall how this works.
Let X be defined by / = 0 in Pn+1, and consider the Hodge decomposition
on primitive cohomology
Hn(X,C)0= 0 H^.
p+q=n
Then there is an isomorphism
(2.1) Hpq ~ Ra^, a{p) = (n - p + l)d - (n + 2),
defined as follows: a homogeneous polynomial A of degree a(p) gives a mero-
morphic (n + l)-form Afi//n~p+1 on Pn+1, where
n+l
fi = J^(-l)lXi dx0 A • • • A dx{ A • • • A oten+i.
i=0

GENERIC TORELLI AND HODGE STRUCTURE
239
This form is homomorphic on Pn+1 —X, and its residue gives a class in Hn(X, C).
Since the order of pole filtration is compatible with the Hodge filtration, the
residue lies in Fp, and hence gives a well-defined element of Hpq.
In addition to the isomorphism (2.1), there is also a diagram
ffP,Q x ffQ.P _> C
(2.2) \l \l \l
Ra(p) x Ra(q) _ Rp
which commutes up to a nonzero constant, where p = a(p)+a(q) = (d—2)(n+2),
the upper map is cup product, and the lower map is multiplication in the ring
R.
Before going further, we need to recall some important facts about R. Since
/ defines a smooth hypersurface, its partials form a regular sequence, and R
is thus a O-dimensional Gorenstein ring. Furthermore, since the partials are
homogeneous, the pairing
(2.3) Ra xRb -> Ra+b
is nondegenerate when a + b < p. This classical result is known as Macaulay's
theorem [2]. Also, the isomorphism Rp ~ C in (2.2) is quite special: it comes
from the trace map for local duality in R (see [2]). Thus, when a + b = p, (2.3)
gives a pairing
Ra x Rb — C
which is perfect. This local duality theorem holds for any graded O-dimensional
Gorenstein ring. Applied to a = a(p) and b = a(q), we get, via (2.2), an algebraic
proof that cup product is a perfect pairing between Hpq and Hq>p. These facts
illustrate nicely the importance of the commutative algebra of the Jacobian ring
R.
The IVHS is also closely related to R. If M is the coarse moduli space of
smooth projective hypersurfaces of degree d in Pn+1, then one easily sees that
Tx{M)~Rd. Also, let
6:TX(M)-+ 0 Homttf™,^-1^1)
p+q=n
be the differential of the period map. Then the diagram
TX(M) x Hpq — j/p-1.9+1
(2.4) \l \l \l
Rd x Ra(p) _> Ra(p-1) = Rd+a{p)
commutes up to a nonzero constant, where the top map is induced by 6 and
the bottom map is again ring multiplication. Thus, via (2.2) and (2.4), we see
that the algebraic part of the IVHS determines various pieces of R and the
multiplication maps between them. If we apply Macaulay's theorem to (2.4), we
also see that the differential of the period map is injective. This is how Griffiths
proved infinitesimal Torelli for projective hypersurfaces in [12].

240
DAVID A. COX
Proofs of these facts may be found in [2] and [14]. The important thing is
that we now have partial information about the Jacobian ring R. In [5] Donagi
was able to use this partial information to prove variational Torelli. The precise
theorem is as follows.
THEOREM 2.2. A smooth hypersurface X in Pn+1 can be recovered (up to
projective equivalence) from the algebraic part of its IVHS provided that d\n + 2
and that we are not in the cases
(i) d = 3, n = 2,
(ii) d = 4, n = 0mod4, or
(iii) d = 6, n = 1 mod 6.
The theorem was first proved in [5] for generic hyper surf aces, and the general
proof came later in a paper of Donagi and Green [6] (see the comments following
Proposition 2.3 below). We will sketch the proof of Theorem 2.2 in order to
introduce two important ideas, symmetrizers and polynomial structures.
We first review symmetrizers, which were first defined by Donagi in [5, §6].
Suppose that we are given a bilinear map
(2.5) U x V -> W.
Then we set
B = {Te Hom([/, V) : uT{v) = vT{u) Vu, v G U).
Evaluation gives a natural bilinear map
(2.6) UxB->V,
and we say that (2.6) is the symmetrizer of (2.5).
The following proposition shows how nicely symmetrizers apply to the ring
R.
PROPOSITION 2.3. The symmetrizer of Ra x Rb -> Ra+b is canonically
isomorphic to Ra x Rb~a —> Rb provided b > a and max(a + b +1, d - 1 + b) < p.
Since R is commutative and associative, there is clearly a canonical map from
Rb~a to the vector space B of the symmetrizer. The trick is to prove that it is
an isomorphism. The proof in [5] only worked for generic polynomials, while the
proof in [6] works in general.
If we repeatedly apply Proposition 2.3 to the map Rd x Ra^ -> #d+a(p) from
(2.4), we produce a sequence of symmetrizers that terminates with Rk x Rk —>
R2k, where * = GCD(d,a(p)) = GCD(d,n + 2) (note that the sequence of
superscripts (of, a(p)) —> • • • —> (fc, k) is just a version of the Euclidean algorithm).
In order to use the map Rk x Rk —> R2k, we need to introduce our second
important idea, polynomial structures. Let S = C[xo,... ,xn+i]. Then the
natural maps <\>a : Sa —> Ra are isomorphisms for a < d — 1, so that Ra is
more than just a vector space; it has a natural polynomial structure given by
(j)a. One of the assumptions of Theorem 2.2 is that d\n + 2. Since we may also

GENERIC TORELLI AND HODGE STRUCTURE
241
assume d > 3, it follows that k = GCD(d, n + 2) (as above) satisfies k < d — 1.
Thus Rk has a polynomial structure, and the key insight is that the structure is
determined by the map Rk x Rk —> R2k.
When 2k < d — 1, this is easy to see. The kernel of Rk ® Rk —> R2k gives
a linear system of quadrics of P(Rk), and let T denote its base locus. Since
2k < d — 1, R2k is isomorphic to S2k, and it is now an easy exercise to verify
that T is the image of P(51) under the fc-fold Veronese embedding. This gives
us isomorphisms
Rk = #0(P(fl*)? 0(i)) = #0(T? o(l))
-ff°(P(S1),0(Jfe)) = Sfc,
which determines the desired polynomial structure. When 2k > k — 1, the map
Rk x Rk ^ R2k also determines the polynomial structure, but the argument is
more difficult and breaks down in the three cases excluded in the statement of
Theorem 2.2. The reader should consult [5, §5] for details.
It is now easy to finish the proof of Theorem 2.2. In light of Theorem 2.1,
we need only determine Jd, which is the kernel of the map <\>d\ Sd —> Rd. The
idea is to use the sequence of symmetrizers generated above to "bootstrap" the
polynomial structure map <\>k : Sk —► Rk into the desired map <\>d : Sd —> Rd. To
do this, note that if Ra x Rb —> RaJtb is one of our symmetrizers and we know
the maps <\>a and 06, then 0a+6 is the unique map which renders the diagram
Sa xSb -> Sa+b
Ra xRb -> Ra+b
commutative. Since we can begin with a = b = k and end with a + b = d, we
can determine 0d, and the theorem is proved.
3. We next discuss the recent work of Donagi and Tu, Saito, and Green, all of
whom draw on the ideas of Donagi's theorem. We will start with the results of
Donagi, Tu, and Saito. They seek to prove generic Torelli for certain hypersur-
faces in weighted projective space. The weights introduce several complications.
The hypersurfaces may acquire unavoidable singularities, and the Jacobian ring
is not so nice. The first difficulty is not so bad: a quasismooth hypersurface (one
whose affine cone is smooth outside the origin) is a V-manifold and thus has a
pure Hodge structure. The analog of Theorem 2.1 holds in this case, as well as
the analogs of diagrams (2.2) and (2.4). Thus the philosophy of the proof is the
same: one wants to use partial information about the Jacobian ring to determine
the entire ring. The second problem, the increased complexity of the Jacobian
ring, is more serious. In particular, Macaulay's theorem and the symmetrizer
lemma fail in general (see §4 for a specific example). Fortunately, they still hold

242
DAVID A. COX
in certain special cases, and using this, Donagi, Tu, and Saito are able to obtain
the following results:
THEOREM 3.1 (DONAGI AND Tu [7]). In weighted projective space
P(<7o,<7i,---,<7n+i), let m = LCM({q0,...,qn+i}) and s = YZ=o Qi- Assume
that <7o = <7i = 1 and that m\s. Then generic Torelli holds for quasismooth
hypersurfaces of degree d, provided m\d and d > max(3s, s + m(n + 1)).
THEOREM 3.2 (SAITO [15]). Let m and r be integers > 2. Then generic
Torelli holds for quasismooth hypersurfaces of degree d, provided m\d, m ^ d, in
the following two cases:
1. The ambient space is P(l,..., l,ra), where there are r + 1 l's, provided
(d,ra,r)^ (4,2,2) and
GCD(m + r + l,d) < m.
2. The ambient space is P(l,..., 1, m, d/2), again with r + 1 1 's, provided m
is even, d/m is odd, and
GCD(d/2 + m + r + 1, d) < m.
The two cases of Saito's theorem have nice geometric interpretations. The
hypersurfaces in Case 1 are branched covers of Pr, while those in Case 2 are
double covers of the cone over the Veronese embedding of degree m of Pr. Note
also that the first theorem covers some of the cases excluded by the second.
Specifically, suppose that m\r +1 in Case 1 of Theorem 3.2. The GCD condition
is violated, yet for d sufficiently large, Theorem 3.1 applies. So we still do not
have definitive results for these sorts of hypersurfaces. We should also point
out that Tu has studied Macaulay's theorem and infinitesimal Torelli for other
classes of weighted projective hypersurfaces (see [16]).
Green's theorem [9] is more abstract, dealing with sufficiently ample divisors
on varieties with very ample canonical bundle. More precisely, he proves:
THEOREM 3.3 (GREEN [9]). Let Y be a smooth complete variety of
dimension > 2 such that Ky is very ample. If £> is a sufficiently ample line bundle on
Y, then generic Torelli holds for smooth divisors ZG |£|.
As explained in §2, the basic goal is to prove a variational Torelli theorem. In
fact, Green shows that a smooth Z G |£| is determined up to isomorphism by
just one piece of the differential of the printed map, specifically
(3.1) T^Homtir^iT1-1'1),
where n = dim Z and T is the tangent space at Z of the coarse moduli space
of all such smooth Z's in |£|. The phrase "up to isomorphism" here means up
to an isomorphism of Y which preserves £. Since Y is of general type, these
isomorphisms form a finite group G. It follows that T ~ H°(Y,£)/(s), where
Z e |£| is defined by s G H°{Y,£).
The two key ideas in the proof are symmetrizers and Chow forms. A
preliminary lemma shows that (3.1) determines the map
t x {h°{z, nnz)/H°{Y, n?)) - h\z, n^-1)-

GENERIC TORELLI AND HODGE STRUCTURE
243
Now symmetrize this map twice. Drawing on his earlier work on Koszul coho-
mology [8], Green shows that one obtains the map
H°{Y,£ ® Kyl) x H°{Y,KY) — H°{Y,£)/{s).
Unfortunately, we only know this map as a bilinear map between abstract vector
spaces. This means that symmetrizing produces a bilinear map A x B —> T, and
there is a commutative diagram
H°{Y,£®Kyl) x H°{Y,KY) 1+ H°{Y,£)/{s)
(3.2) ?\i ?\i \i?
A x B -> T
where the question marks refer to maps we don't know. To resolve the ambiguity,
we symmetrize once more, which yields
(3.3) H°{Y,£® Ky2) x H°{Y,KY) -> H°{Y,£® Ky1),
subject to the same uncertainty as (3.2).
However, this map still contains useful information. If W Ç H°(Y, Ky) is any
linear subspace, then we can use (3.3) to determine whether the base locus of the
linear system W meets the image of Y under the embedding <\>ky determined by
Ky (this requires that £> be sufficiently ample). It follows that (3.3) determines
the Chow form of <\>ky (Y). A variant of this argument shows that we also get
the Chow form of (j>L<^K-^ (Y). Thus, in (3.2), we now know the map
(3.4) H°(Y,£®K^) x H°(Y,YK) - H0(Y,£)/(s)
up to a G-isomorphism of the source and an arbitrary isomorphism of the target.
Since we know the surjective map
H°{Y,JC®Kyl) x H°{Y,KY) -> H°{Y,q,
it follows that we know the map
fl°(y,£)^H°(y, £)/(*),
subject to the same ambiguity as (3.4). In particular, we know (5), the kernel of
this map, up to an element of G. Thus Z is determined up to G-isomorphism
and the theorem is proved.
4. In this section we discuss the joint work [4] of the author and Donagi on the
IVHS of elliptic surfaces /: X —> P1 with a section a, nonconstant ^-invariant,
and pg > 2. We will set
n = x(0x) = P9 + I-
While generic Torelli has been proved in this case by Chakiris [3], we were hoping
that a shorter proof could be obtained using IVHS techniques.
Let us first recall why such surfaces are weighted projective hypersurfaces. We
will assume that all fibers of /: X —> P1 are irreducible (this is true generically).

244
DAVID A. COX
If r denotes the involution x »-+• — x of X, then X is a double cover of the P1-
bundle X/r. The section a maps to a section E of X/r, and E2 = —2n. Thus
X/r ~ i<2n. Furthermore, if we collapse a and i£ to points, then we get a double
cover X —> F^n- Now F2n is the cone over the rational normal curve of degree
2n, so that fyn — P(l, 1,2n). The branch locus of X —> P(l, 1,2n) has degree
6n, and it follows that X is a hypersurface of degree 6n in P(l, 1,2n, 3n). After
a change of coordinate, X is defined by an equation of the form
w2 = z3 + P(x,y)z + Q(x,y)d^ F,
where x, ?/, z, w have weights 1,1,2n, 3n respectively. P(x, ?/) has degree An and
Q{x,y) has degree 6n.
As explained in §§2-3, the IVHS of X relates nicely to the Jacobian ring of the
defining equation w2 — F. Note here that the primitive cohomology H2(X, C)o
is just the orthogonal complement of the fiber and section in H2(X, C), and that
the Jacobian ring ofw2—F is just the Jacobian ring RofF since (w2 — F)w = 2w.
It is now straightforward to show that the algebraic part of the IVHS of X is
equivalent to the following data:
R7n-2 x R7n-2 _> R14n-4 _ C (cup product)?
R6n x Rn~2 -. iî7n-2 (period map).
Our hope was to recover F from this data. The following theorem shows that
this hope was doomed.
Theorem 4.1 (Cox and Donagi [4]). IfX is defined by
w2 = z3 + P{x, y)z + Q(x, y) (= F),
then for X generic, the algebraic part of the IVHS of X is determined uniquely
by PxQy — PyQx, up to a scalar.
Thus variational Torelli fails for the elliptic surfaces we're considering. In
fact, Theorem 4.1 easily implies that the algebraic part of the IVHS is constant
on three-dimensional subvarieties of the coarse moduli space.
Before discussing the proof, we should mention two interesting aspects of the
Jacobian ring R of F:
(i) Macaulay's theorem does not hold for R. For example, the map R13n-2 —►
Hom^1,^1371-1) is not injective.
(ii) The symmetrizer lemma does not hold for R. For example, R6n~3 is a
codimension two subspace in the symmetrizer of R1 x R6n~2 —► i26n_1.
Thus it is not surprising that we run into trouble when we try to follow the same
steps as in §2.
The proof starts easily enough. Since z has degree 2n, Rn~2 consists of
polynomials in x and y of degree n — 2, and this polynomial structure can be
recovered by the usual techniques. Then, using (4.1), one can describe the map

GENERIC TORELLI AND HODGE STRUCTURE
245
R1 x R6n —► P6n+1, where R1 consists of linear forms in x and y. The first
result is
LEMMA 4.2. IfuGR1,u^0, then the resulting map u: R6n -+ P6n+1 has
nonzero kernel if and only if u\PxQy — PyQx>
To prove this, we first study R in the range 6n — 2 < a < 8n — 2. Remarkably,
all of the Pa,s have the same dimension, lOn — 2, and for u fixed, the kernels of
u: Ra —► Pa+1 are all isomorphic. So we need only consider u: R6n~2 —► P6n_1.
In degree 6n — 2, only Fz = 3z2 + P enters into the Jacobian ideal J, so that
elements of R6n~2 can be written uniquely as zA(x,y) + B(x,y). In degree
6n — 1, Pa; = Pxz + <3z and Fy — Pyz + Qy enter linearly in J. Thus Ker(w) ^ 0
if and only if w|aPx + èP^, where (a, 6) ^ (0,0). This means u\aPx + feP^ and
u\aQx + bQy, which is equivalent to u\PxQy — PyQx when a and b are eliminated.
This proves the lemma.
For X generic, PxQy — PyQx has distinct factors, so that Lemma 4.1 proves
half of the theorem, i.e., that the algebraic part of the IVHS determines PxQy —
PyQx up to a scalar. It remains to prove the converse.
We can assume that the distinct factors of PxQy — PyQx are of the form
x — Ay, A G C. Note that there are lOn — 2 of these. From the above discussion
we get nonzero elements m\ G R6n and m'x G R7n~2 such that (x — \y)m\ =
(x — \y)m'y = 0. But now a bit of magic occurs: R6n and R7n~2 both have
dimension lOn — 2, and the ra^'s and ra^'s form bases of the two spaces.
Furthermore, adjusting the basis elements by scalars if necessary, one can prove
that
mAmî* = ^A" Kronecker £,
p{x, y)mx = p(A, l)m;A, p{x, y) G Rn~2.
This shows that the maps (4.1) are determined up to isomorphism by the A's,
i.e., by PxQy — PyQx modulo a scalar. This proves the theorem.
While variational Torelli fails, remember that we have been only considering
first-order variations of Hodge structure. So there is a chance that variational
Torelli may hold for 2nd order IVHS. This will require a detailed study of the
2nd fundamental form as described in [1],
Note added in proof. There have been two recent developments involving
Variational Torelli that need to be mentioned:
(i) The implication
Variational Torelli ==> Generic Torelli
is true without any extra hypotheses. This has been proved by R. Donagi, L.
Tu, and the author in "Variational Torelli implies Generic Torelli," Inventiones
Math., to appear. Thus Proposition 1.1 is now obsolete.
(ii) M.-H. Saito has recently used the methods of variational Torelli to prove a
generic Torelli theorem for hypersurfaces in compact irreducible Hermitian
symmetric spaces. (Grassmannians and quadric hypersurfaces in projective space

246
DAVID A. COX
are examples of such symmetric spaces.) These results were announced in "Ja-
cobian rings of hypersurfaces of compact irreducible Hermitian symmetric spaces
and generic Torelli theorem", Proc. Japan. Acad. 61 (1985), 321-324, and the
full details will appear in "Generic Torelli theorem for hypersurfaces in compact
irreducible Hermitian symmetric spaces," to appear.
References
1. J. Carlson, M. Green, P. Griffiths, and J. Harris, Infinitesimal variation of Hodge
structure. I, Compositio Math. 50 (1983), 109-205.
2. J. Carlson and P. Griffiths, Infinitesimal variations of Hodge structure and the global
Torelli problem, Journées de Géométrie Algébrique d'Angers, Sijthoff and Noordhoff, Groningen,
1980, pp. 51-76.
3. K. Chakiris, The Torelli problem for elliptic pencils, Topics in Transcendental Algebraic
Geometry, Princeton Univ. Press, Princeton, N. J., 1984, pp. 157-182.
4. D. Cox and R. Donagi, On the failure of variational Torelli for regular elliptic surfaces
with a section, Math. Ann. 273 (1986), 673-683.
5. R. Donagi, Generic Torelli for projective hypersurfaces, Compositio Math. 50
(1983), 325-353.
6. R. Donagi and M. Green, A new proof of the symmetrizer lemma and a stronger weak
Torelli theorem for projective hypersurfaces, J. Differential Geom. 20 (1984), 459-461.
7. R. Donagi and L. Tu, Generic Torelli for weighted hypersurfaces, (to appear).
8. M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential
Geom. 19 (1984), 125-171.
9. , The period map for hypersurface sections of high degree of an arbitrary variety,
Compositio Math. 55 (1984), 135-136.
10. P. Griffiths, Infinitesimal variation of Hodge structure, Topics in Transcendental
Algebraic Geometry, Princeton Univ. Press, Princeton, N. J., 1984, pp. 51-62.
11. , Periods of integrals on algebraic manifolds, III. Inst. Hautes Etudes Sci. Publ.
Math. 38 (1970), 125-180.
12. , On the periods of certain rational integrals, Ann. of Math. (2) 90 (1969), 460-541.
13. J. Mather and S.-T. Yau, Classification of isolated hypersurface singularities by their
moduli algebras, Invent. Math. 69 (1982), 243-251.
14. C. Peters and J. Steenbrink, Infinitesimal variations of Hodge structure and the generic
Torelli problem for projective hypersurfaces, Classification of Algebraic and Analytic Manifolds,
Birkhàuser, 1983, pp. 399-464.
15. M.-H. Saito, Weak global Torelli theorem for certain weighted projective hypersurfaces,
Duke Math. J. 53 (1986), 67-111.
16. L. Tu, Macaulay's theorem and local Torelli for weighted hypersurfaces, (to appear).
Amherst College

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
The Geometry of the Mixed Hodge Structure
on the Fundamental Group
RICHARD M. HAIN
In Hodge theory, one studies the moduli of a variety V by putting a Hodge
structure on its integral cohomology groups Hm(V; Z) and examining how the
Hodge filtration moves when the complex structure of V is deformed. In concrete
terms, one studies the periods of the differentials on V with a prescribed number
of dz's. Suppose now that one wants to study the moduli of a pointed variety1
(V,a?) using Hodge theory. This may be interesting, for example, when x is
a variety viewed as a point in a specific moduli space V. One first needs a
topological invariant of (V, x). The first that comes to mind is its fundamental
group, 7Ti(V,a:). Since this is not an abelian group, it is better to replace it by
its integral group ring Ztti (V, x) which is. Since this has infinite rank when 7Ti
is infinite, and for reasons that will soon become apparent, we replace it by a
truncation
Z7ri(V,:r)/Js+1
by some power of its augmentation ideal2 J. This is a finitely generated abelian
group. Next one needs a Hodge filtration.
In the classical case, the position of the Hodge filtration inside H'(V;C) is
given by periods of integrals over integral cycles. To do the same for homotopy,
one needs a de Rham theory for the fundamental group. K.-T. Chen has
developed such a de Rham theory using iterated integrals [5]. An iterated integral
is a linear combination of basic iterated integrals, denoted / w\ w^ • • -wr, where
each Wj is a 1-form on V. Each defines a function
{loops in V based at x} —► C.
1980 Mathematics Subject Classification (1985 Revision). Primary 14C30, 14F35, 32J25;
Secondary 30F30, 55P62.
Supported in part by grants MCS-8201642 and DMS-8401175 from the National Science
Foundation.
xThat is, a variety V and a point x of V.
2Recall that the augmentation ideal is the kernel of the ring homomorphism Ztti (V, x) —► Z
that takes each element of -k\ to 1.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
247

248 RICHARD M. HAIN
The collection of iterated integrals of length < s whose value on each loop based
at x depends only on its homotopy class will be denoted by H°(BS(V), x). Each
element of H°(BS(V), x) defines a Z-linear map
Zir1{V,x)/J*+1-+C
via integration. Chen's 7Ti de Rham theorem asserts that the integration map
H°{B9{V),x) ^Homz(Z7ri(F,x)/Js+1,C)
is an isomorphism for each s > 0. Thus the appropriate analogues of the integral
cohomology groups for a pointed variety (V, x) are the groups
Homz(Z7ri(V^)/Js+1,Z).
A Hodge filtration on H°(Bs(V),x) can be defined in the obvious way.
Namely, an iterated integral fwi---wr is in Fp if the total number of dz's
in the Wj's is > p. The weight is given by the length filtration when V is smooth
and projective:
WtiZTnWxyj9*1)* = (Z7r1(V,x)/Jl^1r=H°(Bl(V),x).
This is clearly defined over Q. If HX(V) is a pure Hodge structure of weight 2
(for example, V is a Zariski open subset of Pn), then the weight filtration on
Zin{V,x)/Ja+1 is defined by
W2t+i = W2l = (Z7n(V,x)/j'+1)* = H°(Bi(V),x).
In general, the weight filtration on H°(BS(V), x) is the convolution of the length
filtration of H°(BS(V), x) with the weight filtration on the de Rham complex of
V. Together the Hodge and weight filtrations on H°(Bs(V),x) define a mixed
Hodge structure (M.H.S.).
Our efforts would have been wasted if the M.H.S. on Z7Ti(V, x)/JsJtl did not
depend upon x. First, when 5 = 1 there is an isomorphism of M.H.S.'s
Cir1{V,x)/J2 = CeH1{V',C).
The M.H.S. on Hi (V; C) is independent of the basepoint. Thus the first possibly
interesting case is when 5 = 2. Provided that the cup product
is not injective, the M.H.S. on Ziri(V, x)/J3 will vary with the basepoint.
Sometimes this M.H.S. is very good at finding the basepoint.
(7.5) THEOREM (HAIN, PULTE). Suppose that (V,x) and (W,y) are two
pointed smooth projective curves. If there is a ring isomorphism
9:Z7r1(V,x)/J3^Z7r1(W,y)/J3
that induces an isomorphism of M.H.S. 's, then there is an isomorphism f: V —►
W such that, with the possible exception of at most two points x ofV, f(x) =
y. □

THE GEOMETRY OF THE FUNDAMENTAL GROUP
249
When V is a smooth projective curve, there is a deeper connection between the
M.H.S. on Z7Ti(V, x)/J3 and geometry. Associated with (V, x) is the Abel-Jacobi
mapping
i/*: V — Jac(V),
y
J x
The image of i/^ is an algebraic 1-cycle Vx in Jac(F). For a subset A of Jac(V),
we set A- = {—a: aGi}. The cycles Vx — Vy and Vx — V~ are homologous to
zero in Jac(V) and consequently define points in the intermediate Jacobian
T ,T x Hom(F2tf3(Jac),C)
J2(JaG) = ï/â(J^Z) *
Using techniques of B. Harris [15, 16], Pulte [25] has shown that the M.H.S.'s
on Z7Ti(V, z)/J3 (z = £, ?/) explicitly determine the points in J2(Jac) determined
byVx-Vy andVi-V^-.
The mixed Hodge structure on 7Ti also arises naturally when considering mon-
odromy representations. Suppose that V = X — D, where X is a smooth
projective variety and D is a divisor in X with normal crossings. A gl(n)-valued,
holomorphic 1-form on V with logarithmic singularities at infinity is an element
u) of Q}{X\ogD) 0 gl(n). It defines a pfaffian system on V: df = /w, where
/ : F —► Cn is a locally defined function. This system is completely integrable if
and only if duj + uj A uj = 0.
Analytically continuing the fundamental solution of such a completely
integrable system defines a monodromy representation p: 7Ti(V, z) —► GL(n). The
M.H.S. on 7Ti(V,x) controls the monodromy representation. As an example, we
state the following theorem that characterizes unipotent monodromy
representations that arise as monodromy representations of nilpotent 1-forms (i.e., 1-forms
taking values in a nilpotent subalgebra of gl(n).
(9.7) THEOREM. A unipotent representation p: 7Ti(V, x) —> GL(n) is the
monodromy representation of a completely integrable nilpotent 1-form on V with
logarithmic singularities at infinity if and only if there is an algebra homomor-
phism
p:{Cn1(V,x)/Jn}/I^g\(n)
such that the diagram
tti(V,x) - [Cin{V,x)/Jn]/I
p I [ p
GL(n) . «-. gl(n)
inclusion
commutes, where I is the ideal
F° n J + F"1 n J2 + F"2 n J3 + • • •
o/C7n(V,:r)/Jn. □
In §9 we state the main result of [30] which characterizes monodromy
representations of integrable, logarithmic 1-forms.

250
RICHARD M. HAIN
Morgan [20] was the first to put a M.H.S. on the Malcev Lie algebra g(V, x)
of 7Ti(V, x) which he did using Sullivan's minimal models. This M.H.S.
determines and is determined by the M.H.S. on the J-adic completion Q7Ti(V, x)a
of Qtti(V,x). (Q?rf is the completion of the universal enveloping algebra of 9,
while g is the set of primitive elements of Qn^.) Morgan's construction proceeds
by first putting a (not necessarily unique) M.H.S. on the Sullivan minimal model
of V. Passing to indécomposables, one gets a family of M.H.S.'s on g(V, x). This
construction is best suited to topological applications. For example, Morgan has
shown that not every finitely presented group can occur as the fundamental group
of a smooth variety. But for geometrical applications, the direct construction of
the M.H.S. on 7Ti using iterated integrals seems more appropriate.
This paper begins with a fairly complete and self-contained account of Chen's
de Rham theory for. 7ri. Iterated line integrals are defined in §1, and their
relationship to connections on trivial bundles is explored and exploited in §2. In
§3 lots of iterated integrals are constructed that are homotopy functionals and
the 7Ti de Rham theorem is proved in §4.
The study of the M.H.S. on 7ri(V, x) begins in §5 where we sketch a proof of
its existence. The extension theory for separated extensions of M.H.S. with two
nontrivial weights is reviewed in §6 and then applied to the M.H.S. on J(V, x)/J3.
General conditions under which the local Torelli theorem is true for the M.H.S.
on J(V,x)/J3 are derived in §7 and these are used to prove the almost global
Torelli theorem for pointed smooth projective curves.
Pulte's interpretation of B. Harris's harmonic volume is sketched in §8. The
relationship between the M.H.S. on J(V,x)/J3 of a smooth projective curve
and certain algebraic 1-cycles in its Jacobian is made explicit. Various results
announcing new, nontrivial restrictions on the monodromy representations of
completely integrable 1-forms with logarithmic singularities are stated in §9.
Most of the ideas in §§1-4 have been harvested from Chen's papers. However,
the elementary proof of the 7Ti de Rham theorem given in §4 is new.
We assume throughout that the reader is familiar with Deligne's mixed Hodge
theory. Two good introductions to it are the papers of Griffiths and Schmid [10]
and Durfee [8]. The accessible paper [3] by Carlson is a recommended companion
while reading §§6 and 7. An intuitive explanation of how iterated line integrals
work is given in the introduction of [12].
Finally I would like to thank everyone who has taken interest in this work,
particularly Jim Carlson, P. Cartier, P. Deligne, Alan Durfee, Herb Clemens,
and Steven Zucker.
1. Iterated integrals. Let M be a smooth manifold. Denote the set of
piecewise smooth paths 7: [0,1] —► M by FM. We will call a function F: PM —►
Rn a homotopy functional if F(i) depends only on the homotopy class of 7
relative to its endpoints. For each x G M, a homotopy functional F induces a
function 7ri(M, x) —► Rn.

THE GEOMETRY OF THE FUNDAMENTAL GROUP
251
Suppose that w is a 1-form on M. The usual line integral
w : PM -► R,
/■
/ w
J1
can be defined as follows:
[ w= [ f{t)dt,
Jq JO
where 7*w = f(t)dt is the pullback of w to [0,1]. Now fw is a homotopy
functional if and only if w is closed. Thus the usual line integral can only detect
elements of 7Ti (M, x) visible in H\ (M; R).
K.-T. Chen has discovered a generalization of the line integral that often
detects many elements of 7Ti (M) that are trivial in homology.
(1.1) DEFINITION. Suppose that wi,...,wr are smooth 1-forms on M and
that 7 G PM. Define
/ W1W2'"Wr = / '•' / fl{h)f2{t2)'"fr{tr)dt1 '"dtr,
1 0<ti<~<tr<l
where fj{t)dt = *fwj. Denote the function PM —> R, 7 —► f w\ • • • wr by
j wi - - -wr. A linear combination of such functions and the constant function is
called an iterated line integral. A functional PM —> R is an iterated line integral
of length < s if it is a linear combination of a constant function and iterated
integrals fwi--wr with r < s.
The definition given above works equally for 1-forms on M that take values
in an associative algebra A. In this case
/
wi"-wr: PM —► A.
Two cases of special interest are when A = C and A = gl(n,/), the algebra of
nx n matrices over (/=RorC).
We will denote the usual de Rham complex of C°° forms on M by E'(M).
The following naturality property of iterated line integrals is easily verified.
(1.2) PROPOSITION. Suppose that M and N are smooth manifolds, that
7 G PN, and that w\,..., wr G E1(M). If f: N —► M is a smooth map, then
/ f*Wif*W2 ' • ' f*Wr = / Wi"-Wr.
J1 J f°i
The reader should be aware that two iterated integrals that look different may
indeed be equal. The following assertion is easily verified.

252
RICHARD M. HAIN
(1.3) PROPOSITION. Suppose that wi,...,wr G E1 (M) and that 7G PM.
IffeE°{M), then
(a) / dfwi '-wr= / {fwi)w<2 • • • wr - f o 7(0) / wi'-wr,
J"y J"</ J "y
wi" -Wi-i dfwi • • • wr
(b) ^ r r
= / Wi ■ ■ ■ Wi-1(fwi)wi+i ■ ■ ■ Wr - / Wi---(fWi-i)Wi---Wr,
J "] J ~J
(c) / wi'-wrdf = /o7(l) / wi •••w;r - / w;i '-wr-i(fwr). D
J ^ J ^ J ^
Finally, we remark that some interesting functions, such as the higher
logarithms, can be written as iterated integrals. Let
_ 1 dz _ 1 dz
Wl ~ 27Ty/^ïl-z' W2 ~ 2-Kyf-t~z'
These are rational forms on P1. The fcth higher logarithm is given by the formula
\lik{z) = / W1W2 " -W2.
JO
2. Connections on trivial bundles. In this section we relate iterated
integrals to geometry via the transport of a connection on a trivial bundle. This
allows us to derive several formal properties of iterated integrals in this section,
such as their behaviour on a product of paths. This relationship allows us to
find all iterated integrals of length 2 that are homotopy functionals in the next
section and to relate the M.H.S. on 7Ti to the Riemann-Hilbert problem in §9.
Suppose that V is a connection on the trivial vector bundle Rn x M —► M.
Sections of this bundle correspond to functions x: M —> Rn. The canonical
framing of this bundle is given by the n constant functions
ti\ M —► Rn, i = l,...,n,
which take each point of M to the zth standard basis vector of Rn. Define the
connection form3 uj G El(M) ® gl(n) by
where Wij G El{M) and uj = {wij). Viewing a section of this bundle as a
function x: M —► Rn, we have
(2.1) Vz = dx — xuj.
Conversely, if uj G E1 (M) (g)gl(n), we can define a connection on Rn x M —y M
by (2.1). The connection lifts to the bundle GL(n) x M —► M by defining
VX = dX - Xuj, where X: M -► GL(n).
3This is not the usual way to define the connection form. We chose this definition so that
the transport function would satisfy T(a • /3) = T(a)T(/3).

THE GEOMETRY OF THE FUNDAMENTAL GROUP
253
Since the bundle GL(n) xM-^Mis trivial, sections of it along a smooth map
/: N —► M correspond to functions X: N —► GL(n). In particular, a section
X: [a, b] —► GL(n), along a smooth path 7: [a, b] —► M, is horizontal if and only
if
(2.2) dX(t) = X{t)^uj
for all *. If 7*u = ;4(t) dt, then (2.2) becomes
(2.3) X'(0 = X(0^(0-
Define the transport function T: PM —► GL(n) as follows. If 7: [0,1] —► M
is smooth, define T(i) to be X(l), where X: [0,1] —► GL(n) is the unique
horizontal section along 7 (i.e., solution of (2.3)) satisfying X(0) = 7. If 7 is
piecewise smooth, T(i) is defined in the obvious way.
The following properties of the transport function are immediate consequences
of the theory of linear ordinary differential equations.
(2.4) PROPOSITION, (a) // 7 G PM, then T(i) is independent of the
parametrization 0/7.
(b) If a, 13 G PM and a(l) = /?(0), then T{a/3) = T{a)T{/3).
We now give a formula for T in terms of iterated integrals of the connection
form. Recall that the definition of iterated integral applies equally well to gl(n)-
valued 1-forms.
Suppose that V is a connection on a trivial vector bundle Rn x M —► M with
connection form u and transport function T.
(2.5) LEMMA, (a) 7/7 G PM, then there exists a constant M > 0 such that
I /" -—^—.11 n(Mr\
!|A II V r\ J
so that the series
I + OJ + LJLJ +/ UJUJUJ H
J "y t/ "y J *y
T(7) = I + u>+ UJUJ+ ujujuj + • • •.
PROOF. Since T(7) is independent of the parametrization of 7, we may
assume, by reparametrizing 7 if necessary, that 7 is smooth. If 7*0; = A(t) dt,
then T(7) = X(l), where X(t) is the solution of
(*) X'(t) = X(t)A(t), X(0) = /.
Note that
/ uJl7^~u} = / • • • / A[ti)A[t2) • • • A(tr) dt.
1 0<ti<--<tr<l
converges.
(b)

254 RICHARD M. HAIN
We will solve (*) by Picard iteration (cf. [21]): X(t) satisfies (*) if and only if
X(0 = /+ / X{s)A{s)ds.
Jo
The sequence {Xr(t)} of Picard iterates is defined as follows:
Xo(0 = /, Xr{t) = I + / Xr^{s)A(s)ds.
Jo
As is well known (cf. [21]), there exists M > 0 such that
||xr(0-xr_i(0ll = o(^*r)
and Xr(t) converges almost uniformly to X(t).
First we have X0(t) = I and
X1(t)-X0(t)= f A(s)ds.
Jo
Assume, by induction, that
Xn(0-Xn-i(0= /•*•/ AM-'AMds^.-dSn
0<Si<---<3n<t
whenever n < r. Now
xr(0-xr-i(*)= / (xr_1(5)-xr_2(5))A(5)d5
Jo
= / I '" I ^(5i) ' ' ' A(sr-i)A(s) ds\ • • • dsr-i ds
0<si<---<sr-i<s
= / ••• / >l(si) • • -j4(sr)dsi • • -dsr.
0<3i<--<Sr<t
Consequently,
and
Xr(l)-Xr_!(l)= /
J*1
UJUJ • • 'UJ
T(7) = X(l) = lim Xr(l) = 1+ [ u>+ [ UU + --. D
Of special importance is the case when uj G E1 (M) ® tv, where tv denotes the
Lie algebra of nilpotent upper triangular (r + 1) x (r + 1) matrices:
fo * \
tV
[0 o)
In this case
T = I +
uj + ujuj + h UJ " -UJ.

THE GEOMETRY OF THE FUNDAMENTAL GROUP
255
For example, if w\, W2,..., wr G E1 (M) and
fO wi 0 0
0 w2 0
(2.6)
then
w
0
0
o A
o
wr
/l /iWi f W\W2 f W1W2W3
0 1 Ju>2 f W2W3
0 0 1 /w3
(2.7)
T =
f wi -wr\
fw2'"Wr
0
V
fwr
1
y
That is,
ifwi-'-Wj-x, j>i,
1, y = i,
0, y < ».
Since T(i) is independent of the parametrization of 7, (2.4), each entry of
T(7) is independent of the parametrization of 7. Applying this to (2.7) we
obtain the following generalization of a well-known property of line integrals.
(2.8) Proposition. Ifwi,...,wreEl(M)and^ePM,then$iwi-wr
is independent of the parametrization 0/7.
Applying (2.4) (b) to the transport function (2.7) we obtain the following
property of iterated integrals.
(2.9) PROPOSITION. Suppose that wi,...,wreEl(M). Ifa,(3ePM and
a(l) = 0(0), then
/ Wi ' • ' Wr = Y] Wi - - Wi Wi
Jap i=oJot J&
+ 1 '"Wr.
Here, we introduce the convention that f ip\ • • • <ps = 1 when 5 = 0. □
Recall that an iterated integral of length < s is a finite linear combination
I = \ + 22ai / wi + ^,aij / wiwj + •" X^ aj / wh " 'wjs-
\J\=s
We shall denote the value of I on the path 7 by (7,7) or 7(7). The pairing
( , ) can be extended, by linearity, to a pairing between iterated integrals and
1-chains.

256
RICHARD M. HAIN
Denote the vector space of iterated integrals on M of length < s by BS(M).
Denote the constant path at the point a; of M by ^. (That is, r)x(t) = x for all
t.) If r > 1, then
a
w1---wr,r)x) =0
for all x € M. Thus, evaluating at a constant path r)x defines a linear functional
e: BS{M) -► R,
that is independent of x. If
/ = A + ^2ai \ Wi + y£2(lii / w*w3 ^ '
then e(I) = A. Denote the kernel of e by B£(M). These are the iterated
integrals of length < s with zero constant term. Since there is a natural inclusion
i: R —► BS(M) such that soi = id, we have a natural direct sum decomposition
BS{M)^R®BS{M).
For loops a,/? G PM based at z, we can form the commutator [a,/?] =
a(3a~lP~l. Often we will denote the constant loop rjx at z by 1. Since the
paths ar/a; and r)xa differ from the loop a by a reparametrization, it follows from
(2.8) that if I is an iterated integral, then
1(a) = I{arjx) = I(rjxa).
Recall that the classical line integral satisfies
/Jw,[a,0]\=O and Ij w, (a - 1){0 - 1)\ = 0,
where a and /? are loops based at x. The following lemma is a generalization of
this fact.
(2.10) LEMMA. Suppose that wi,...,wr G El(M) and that x G M.
Suppose that ai, #2, • • • ?#s are /oops in M eased a£ £.
(a) If I G Br and r < s, then
(7,(l-a1)(l-a2)...(l-as))=0,
where 1 denotes rjx, the constant path at x.
(b) If I G Br and r < s, then
(J, [ai[a2[. ..[as_i,as]...]) =0.
PROOF. To prove (a), we need only consider the case where I = f w\ - -wr,
and 0 < r < s. The case when r = 0 (i.e., 7 = 1) can be verified directly.

THE GEOMETRY OF THE FUNDAMENTAL GROUP
257
Suppose 1 < r < s. Set
uj =
(°
Wi
0
0
0
W2
0
0 \
Wr
V
o J
Then T is given by (2.7) and
(T, (1 - ai)(l - a2) -.. (1 - aa)) = T(l - £*i)T(l - a2) • • -T(l - as).
But each T(l — ay) is a nilpotent upper triangular (r +1) x (r +1) matrix. Since
s > r, their product must vanish and (T, (1 — c*i) • • • (1 — a3)) = 0. Examining
the top right-hand entry of this matrix yields the result.
Assertion (b) follows similarly. The relevant fact is that each T(ctj) is a
unipotent upper triangular (r + 1) x (r + 1) matrix. Any bracket arrangement
of length > r of such matrices is the identity. □
Next we describe how to pointwise multiply two iterated integrals. Recall
that a permutation a of {1,2,..., r + s} is a shuffle of type (r, s) if
a"1(l)<a-1(2)< •••<a-1(r)
and
a-^r + l) <a~1(r + 2) < • • • < a_1(r + s).
The following property of iterated integrals is related to the Baker-Campbell-
Hausdorff formula.
(2.11) LEMMA (REE [27]). // wuw2,...,wr+a e EX(M) andaePM,
then
I W1W2'"Wr Wr+iWr+2'-U)r+s = y£2 W(r(l)wcr(2) •" ^(r+a) >
Jot Jot a Jot
where a runs over the shuffles of type (r, s).
PROOF. This is an exercise using the fact that
ArxAs= (J {(*,(!),..., ta(r+a)): 0 < ta(1) < < ta{T+a) < 1}.
<r€Sh(r,s)
Here we are identifying the standard n-simplex An with
{(*i,...,*n):0<*i < •••<*n<l}
andRr x Rs with Rr+S. □
For example,
/ W\ I W<2 = / W1W2 + / ^2^1-
Jot Jot Jot Jot
Two more properties of iterated integrals that will be used in the sequel follow.
Their proofs are straightforward.

258
RICHARD M. HAIN
(2.12) Proposition. lfwi,...,wreE1(M)andiePM,then
/ w\ • • • wr = (—l)r / wr • • - w\. D
J "y — 1 J ~*
(2.13) PROPOSITION, (a) Ifw1,w2eE1(M) andaj,(3j (j = l,...,g) are
loops based at x, then
'J
i=i / j=i
/a, ^1 h> wl
fa, W* hi W2
(b) // w\,..., wr G E1 (M) and a±,..., ar are loops based at x, then
(j^••^r,nK-i))=n/ wj- d
Of course the whole discussion is valid for C-valued forms and connections on
trivial complex vector bundles Cn x M —► M.
3. Homotopy functionals. In this section we show how to construct
iterated integrals that are homotopy functionals. Suppose that V is a
connection on the trivial bundle Rn x M —► M. The associated transport function
T: PM —> GL(n) is the homotopy functional if and only if V is flat.
Consequently, all of the entries of T are homotopy functionals PM —► R if and only
if V is flat. In general, these entries are infinite sums of iterated integrals as we
have seen in (2.5). However, if the connection form u of V lies in El(M) ® n,
where n is a nilpotent sub-Lie algebra of gl(n), then the entries of T are iterated
integrals.
The connection V extends naturally to the p-forms that take values in Rn by
V<p = d(p + (-l)p+ V A Lj.
Now V is flat if and only if V2 = 0 and a short computation shows that V2 = 0
if and only if uj is completely integrable. That is,
du + u; A w = 0.
For example, suppose that
[0 W\ W\2
UJ = 0 0 W2
V
in which case
(0 dw\ w\ A Vû2 + dw\2
0 0 dn)2
0 0 0
Thus u; is completely integrable if and only if
dw\ = dw2 = 0 and w\ Aw2 + dw\2 = 0.
0 0 0

THE GEOMETRY OF THE FUNDAMENTAL GROUP
259
The associated transport is
(1 f Wi JwiW2+Wi2
0 1 Jw2
0 0 1
Consequently, if dw\ = dn)2 = 0, then / w\ W2 + W12 is a homotopy functional if
and only if w\ Aw2 + dw\2 = 0.
The next result generalizes this example and is an introduction to Chen's
power series connections (cf. [5, 6]).
(3.1) PROPOSITION. Suppose that wu...,wry u e El(M) and dij G R,
1 < i, j < r. // each Wj is closed, then
^2aij I wiwj + / u
is a homotopy functional if and only if
du + Y^ ciijWi A Wj = 0.
PROOF. Denote the free associative algebra generated by Xi,X2,...,Xr, Z
by R(Xi,..., Xr, Z). Let / be the ideal generated by Xi, X2,..., Xr, Z. Set
_ -R(Xi,... ,Xr,Z)
" (X;X,-a^) + J3*
Define an A-valued 1-form uj by
uj = tfiXi + h uvXV + uZ.
Since the wy are closed,
du + u; A u = duZ + ^J ^ A WjXiXj = (du + ^J uij^i A w?)^,
and w is completely integrable if and only if du + ^ a^^ A u>; = 0.
Note that A is finite-dimensional and that the algebra homomorphism A —►
gl(A) given by right multiplication is injective. From (2.5) we have
T = l+ u+ uu
= 1 + Y2 / ™iXi + / mZ + Yl / wiw3XiXj
= 1 + ]T / w,x, + ( ]Toiy / w^ + J u\z.
Since each wy is closed and since the X7 and Z are linearly independent in A, it
follows that T is a homotopy functional if and only if
Y^aij J wi™j + J <
is. The result follows from the fact that T is a homotopy functional if and only
if uj is completely integrable. □

260
RICHARD M. HAIN
Iterated integrals of algebraic 1-forms can also be defined. Suppose, for
example, that M is a compact Riemann surface. Let w\ and w2 be abelian differentials
of the second kind on M. Define
wi U w2 = ^2 O'iPîi ai € C, Pi G M
if, locally, Wi = df and
(27r\/^ï) ResPj fw2 = aj.
If w\,..., wr are abelian differentials of the second kind on M such that
where ^ afc — 0? then we can find an abelian differential of the third kind u such
that
(271-%/—ï) ResPj. u — —aj
and u is holomorphic away from the pj. The iterated integral
y^ / aijWiWj + u
is easily seen to be a well-defined homotopy functional PM —> C. This is the
algebraic analogue of (3.1).
4. The 7Ti de Rham theorem. In this section we prove Chen's de Rham
theorem for the truncation of the real group ring R,7ri (M, x) by a power of its
augmentation ideal.
Suppose that G is a (discrete) group and that R is a ring. We shall denote
the group algebra of G over R by RG and the kernel of the augmentation RG —►
R : g —► 1 by J. The fcth power of the augmentation ideal will be denoted by Jk.
Let M be a manifold and x G M. Denote the set of elements of BS{M) that
are homotopy functionals {loops in M based at x} —► R by i/°(.B5(M),x).4
Integration induces a linear map
H°{Bs{M),x) -► Homz(Z7Ti(M,a:),R).
According to (2.10), each element of H°(BS(M), x) vanishes on Js+1 so that we
have a map
i/0(5s(M),x)^Homz(Z7ri(M,a:)/Js+1,R).
(4.1) THEOREM (Chen). For each s > 0, the integration map
H°{B3{M),x) -► Homz(Z7ri(M,a:)/Js+1,R)
25 an isomorphism.
PROOF. We will prove the theorem under the extra assumption that wi (M, x)
is finitely generated. Set G = 7Ti(M, x) and V — RG/J8*1. Since G is finitely
generated, F is finite-dimensional. Define a representation p: G —► Aut(V) by
0 —► {A —► A0}. (Here our linear maps act on the right of V.) The first step is
to show that this representation is unipotent.
4This notation is reasonable as homotopy functionals {loops in M based in x} —► R are
locally constant functions on {loops in M based at x}. If one defines the full complex of
iterated integrals (cf. [5]) then this is indeed H° of a cochain complex.

THE GEOMETRY OF THE FUNDAMENTAL GROUP
261
(4.2) PROPOSITION. The representation p satisfies (p(g) - 7)s+1 = 0 for
all g.
PROOF. First note that the powers of J induce a filtration
(*) V = RG/J9*1 > JIJ9*1 > > J9/ Js+1 > 0.
The result now follows because G acts trivially on the graded quotients J1 j' Jt+1
of V: Hue J1 and g G G, then u(g - 1) G Jt+1 so that ug = umodJ**1. □
Next form the flat bundle
V ^E
1
M
with monodromy representation p. That is, E = (V x M)/G, where (v, m) - g =
(vg, g~1m) and M —► M is a universal covering. Since G stabilizes the flag (*)
above, this bundle is filtered by flat subbundles
ED E1 D E2 D --DE9 DO,
where El -► M has fiber Jl/Ja+l.
(4.3) PROPOSITION. There is a C°° trivialization
V x M -+ £
\ /
M
of E —► M swc/i £/ia£ E* —> M corresponds to J1 /Js+1 x M —► M and swc/i £/ia£
£/ie induced connection on J1 /JtJtl x M —► M is trivial.
PROOF. This is proved by induction on s. Consider the short exact sequence
of flat bundles
0 -+ E9 -► E i £/£s -+ 0.
As a flat bundle E9 is trivial and, by induction, E/E9 has the desired type of
trivialization. Now use a splitting of p to obtain the desired trivialization of
E. D
Denote the Lie algebra of endomorphisms of V that preserve the flag (*) by
Endj(V). The connection form u of the trivialization of E —► M given by (4.3)
satisfies w G EX(M) ® Endj(F). Since every element A of Endj(F) satisfies
A3+x = 0, it follows from (2.5) that its transport T satisfies T G B8(M) <g>
End(F). Since the connection is flat, each entry of T lies in H°(B3(M),x).
Furthermore, since, by construction, the monodromy representation is the
composite
tti(M,x) -> R7ri(M,a:)/Js+1 -+ End(F),
we have
T G H°{BS{M)) <g> Rtti(M, x)/Js+1

262
RICHARD M. HAIN
and integration 7 —► (T, 7) induces the identity
R7Ti(M,a:)/Js+1 -+ R7n(M,x)/Ja+1.
It follows that the integration map
H°{B3{M)) -+ Homz(Z7ri(M,a:)/Js+1,R)
is surjective. But it is clearly injective. This completes the proof of Theorem
(4.1). D
The theorem remains true if R is replaced by C throughout and E'(M) is
replaced by the complex E'{M) 0 C of C-valued forms on M. It is desirable to
be able to replace E'(M) by any sub d.g. algebra of E'(M) that computes the
de Rham cohomology of M.
Suppose that k is R or C and that A' is a sub d.g. algebra of the complex
E'k{M) of A:-valued forms on M such that the inclusion A' —► Ek(M) is a quasi-
isomorphism. Define B9(A') to be the space of iterated integrals spanned by
fwi"-wr where each w3- G A1 and 0 < r < s. For each x G M, define
H°(BS(A'), x) to be the elements of BS(A') whose restriction to {loops based at
x} is a homotopy functional. We can now state a refined version of (4.1).
(4.4) THEOREM (CHEN). Suppose that A' is a sub d.g. algebra of Ek(M),
where k = R or C. // the inclusion A' —► Ek(M) is a quasi-isomorphism, then
for each x G M and s > 0, the integration map
H°{B9{A9),x) -► Homz(Z7Ti(M,a:)/Js+1,/c)
is an isomorphism. □
(4.5) REMARKS. One can define iterated integrals of forms of arbitrary degree
(see [5]). If u>i,..., wr G E'(M) are forms of positive degree, then f W1W2 • • - wr
is a differential form of degree XXcleg wj — 1) on the path space PM. In
particular, if each Wj is a 1-form, then / wi • • • wr is a smooth function on PM. The
space of iterated integrals / E' (M) forms a subcomplex of the de Rham complex
of PM. When restricted to {loops at x}, the formula for the differential becomes
d wi • • • wr — - ^2 / wi" ' w*+i dwiWi+i • • • wr
r-l ,
~~Y1 I Wl "'Wi-l(Wi AWt+i)Wt+2'--Wr,
i=l J
when each Wj is a 1-form. For example, if each Wj is a closed 1-form, then
d I 2^ aij / WiWj + / u ) = — 2^2 aii wi ^wj ~ du.
This gives an alternative proof of (3.1).
The full complex of iterated integrals is a canonical quotient of the bar
construction on the de Rham complex of E'(M).

THE GEOMETRY OF THE FUNDAMENTAL GROUP
263
For Hodge theory it is convenient to state a variant of (4.4). Suppose that
A' and k are as in (4.4). As we have seen in §2, B(A') = k 0 B{A'). This
decomposition restricts to give a decomposition
H°(B(Am), x) = ke H°(B(Am), x).
Denote the augmentation ideal of Z7Ti(M, x) by J(M, x). The kernel of the map
H°{Bs{A'),x) -+ Hom(J(M,x)/Js+1,/c)
is k. The following result is a restatement of (4.4).
(4.6) THEOREM. With the assumptions of (4.4), integration induces an
isomrophism
H°(BS{A-), x) -+ Homz(J(M, x)/Js+1, Jfe).
5. The mixed Hodge structure on 7Ti. In this section we put a mixed
Hodge structure (M.H.S.) on the truncated group ring Zn\ (V, x)/ Js+1 (or, equiv-
alently, its dual) of a smooth algebraic variety V with basepoint x. An equivalent
result has been proved by Morgan [20, 9.2] using different techniques. Granted
the 7Ti de Rham theorem, (4.4), our construction of the M.H.S. on
Homz(Z7ri(V^)/Js+1,C)
is direct and transparent. As we shall see in the proof of (5.1), an element <p
of this group lies in Fp if it can be represented as a sum of iterated integrals
/ w\ - - wr each with at least p dz's. The weight of an iterated integral / w\ • • • wr
is its length r plus the number of logarithmic terms in the Wj.
(5.1) THEOREM. IfV is an algebraic variety over C and x EV, then there
is a M.H.S. on
Z7n(F,a:)/Js+1
that is natural with respect to morphisms of pointed varieties. Moreover, if s >t,
then the quotient map
Z7ri(V,:r)/Js+1 -► Z7Ti(V»/Jt+1
induces a morphism of M.H.S. 's.
We will sketch the proof when V is smooth. We begin by reviewing the
construction of the M.H.S. on the cohomology of V. Suppose that V is a smooth
quasi-projective variety and that X is a smooth projective completion of V such
that X — V is a divisor D in X with normal crossings. Denote the complex of
C°° forms on V with logarithmic singularities along D by E'{X\ogD). (For a
detailed description, see [7] or [10].) The inclusion E'(X\ogD) —► E'C(V) is a
quasi-isomorphism. The Hodge and weight filtrations of the C°° log complex are
defined by
FPE'{XlogD) = {forms with > p cb's},
WtE'{X\ogD) = (forms with < / —'si ,

264
RICHARD M. HAIN
and the Hodge and weight filtrations on the complex cohomology of V are defined
by
^i/m(F;C) = \m{Hm{W^mE\X\ogD)) - Hm{V)},
FpHm{V;C) = \m{Hm(FpE\X\ogD)) -+ Hm{V)}.
One can show that the weight filtration on Hm(V; C) is defined over Q.
Define the Hodge filtration of B3(E' {X log D)) by defining FPBS(E' {X log D))
to be the subspace spanned by the iterated integrals f wi • - -wr, where Wj G
Fv* and pi + p2 + • • • + pr > p. The weight filtration of B3(E'{X log D)) is
defined by letting WiBs be the span of the $ w\ • • • wr, where wj G W\. and
h + h + \-lr+r<l. That is, the weight of an iterated integral is its length
plus the sum of the weights of the Wj's.
Note that if V is projective, then the weight filtration on E1 (V) is 0 = W-i Ç
Wo = El(y). It follows that if V is projective and smooth, then
WlB{E{y))-\Bs{E-{V)) itl>s.
The Hodge and weight filtrations of B3 (E' (X log D)) induce Hodge and weight
filtrations on H°{BaE'{XlogD),x) for each xeV.
The theorem is proved by induction on s. The rational structure will be
suppressed. We first need the following result.
(5.2) PROPOSITION. There is a natural isomorphism
CeH^ViC) ^ H°{B!{V),x),
(\,{w})^\ + Jw.
PROOF. It follows from elementary properties of line integrals that / w is a
homotopy functional if and only if w is closed. If w = df and a is a loop based
at x, then
/.
w = f o a(l) - / o a(0) = 0.
It follows that the map is well defined and an isomorphism. □
An immediate consequence of this result is that H°(Bi(V,x)) has a M.H.S.
For the inductive step we need the following result. According to (2.13)(b), we
have a well-defined map
p:H°{Ba(V),x)^®aHl{V;C)
that takes the iterated integral J to the function
^^(VjCO-C,
[ttl]® •••»[<*,]-(/, Ufa -1)
where each <Xj is a loop based at x.

THE GEOMETRY OF THE FUNDAMENTAL GROUP 265
(5.3) PROPOSITION. For all s, the sequence
0 - H\Bs^{Vlx) - H°(B9(V),x) i (S}3H\V,C)
is exact.
PROOF. We have to show that H°(B3-i,x) is the kernel of p. Suppose that
IeH°{Bs,x). Write
J2aj j w^
• --wi+r,
where o;GC and V G Bs-\. Since / is a homotopy functional, it follows from
(2.13)(b) that
p(I) = Y2aJwh ® --®wja
is closed in ®s E'(V) and that p(I) is its cohomology class. If p(I) = 0, then
where each ^ G JS1^) and /* G JS0^). From (1.3) it follows that / G
H°(BS-Ux). D
One can show, with the aid of the Eilenberg-Moore spectral sequence, that
the image of p is defined over Q and that it is a sub-M.H.S. of 0s HX{V; C).
For example, when V is projective or WiHl(y) = 0, then the image of p is the
kernel of the map
5-1
Y,Ci:®sHl{V;C)^ £ &Hl{V) ® H\V) ® ®jH\V),
i=l i+j=s—2
where
Ci(z\ <8> • • • <8> za) = z\ <g> • • • <g> Zi <g> {zi A 2t-+i) <g> ^+2 ® • • • ® 2a.
In this case it is clear that imp is defined over Q and is a sub-Hodge structure
of&H^V).
In order to complete the proof, note that the Hodge and weight filtrations of
H°(Bs(V),x) induce those of H°(Bs-i(V),x) and imp. The result now follows
from [10, 1.16]. □
(5.4) REMARK. In the proof we have shown that if Hl(y) is a pure Hodge
structure, then the graded quotients of the weight filtration of H°(B3(V), x) have
natural polarizations. This follows because they are the sub-Hodge structures
imp of & Hl(V).
As a M.H.S., the dual of the truncated group ring splits
H°(Bs(V),x) = CeH°(Bs(V),x),
so that the interesting part of the M.H.S. is H°(B3(V),x). For this reason we
record the following corollary of (4.6) and (5.1).
(5.5) COROLLARY. For each s > 0 and x G V, there is a natural M.H.S.
on
H°(Bs{V),x)^Romz{J{V,x)/Js+1,C).

266
RICHARD M. HAIN
6. Extension data. From the point of view of Hodge theory, the interesting
part of the group ring of the fundamental group of a smooth variety is its
augmentation ideal J. The filtration J > J2 > J3 > • • • of J by its powers is closely
related to the filtration of the fundamental group by its lower central series (see
[23]). For example, the function
7ri(V,x)^J/J2,
9^9-1
induces a group isomorphism Hi(V;Z) —► J/J2. Thus the first interesting
quotient of J should be J/J3. When Hi(V; Z) is torsionfree, there is an exact
sequence (over Z)
0 -► Hl(V) -+ Hom(J/J3,Z) -+ K -► 0,
where K is the kernel of the cup product Hl(V) <g> Hl{V) -► H2(V). When
H1 (V) has a pure Hodge structure, the M.H.S. on the dual of J/J3 is a separated
extension of Hodge structures. In this section we review some extension theory
of Hodge structures and give a formula for the extension data for J/J3. Full
details of the extension theory can be found in [3].
A separated extension of Hodge structures is an exact sequence
O^A^E^B^O
of M.H.S.'s, where A is a pure Hodge structure of weight m, B is a pure Hodge
structure of weight n, and n > m. Two extensions
0 -► A -► Ej: -► B -► 0, j = 1,2,
are congruent if there is an isomorphism of M.H.S.'s $: E\ —► E<i such that
0^ A -+ JSi -+ 5 ^0
id | | $ | id
0^ A ^ £2 ^ ^ ^0
commutes. Note that congruence is a finer equivalence relation than
isomorphism, where $ would be allowed to induce any automorphisms of the Hodge
structures A and B.
The set of congruence classes of extensions of B by A forms an abelian group
that we shall denote by Ext(B, A). Let F° Home{B, A) be the set of Hodge
filtration preserving C-linear maps B —► A. There is an abelian group
isomorphism
* v ' ; F°Homc(B,i4) + Homz(B,i4)
that is given as follows. If
E >B-»0

THE GEOMETRY OF THE FUNDAMENTAL GROUP 267
is an extension, choose a Hodge filtration preserving section sf'> B —► E of p
and a retraction rz : E —► A of i that is defined over Z. Composing these gives
an element \I)(E) = rz ° sp of Home{B, A). One can check that tp(E) is well
defined modulo F°Hom(B,A) + Homz(JB,A). It is not hard to construct an
inverse of -0. For details, see [3].
When B is of weight 2p, then the extension homomorphism rz°SF also induces
a group homomorphism
liiBM^Ac/FtA + Az
from the integral points of type {p,p), Bz H £p'p, of B into a complex torus.
This is called the motive of the extension (cf. [3]).
Returning to the case of interest, we first express the M.H.S. on (J/J3)* as
an extension.
(6.1) LEMMA. Suppose that (X, x) is a path connected, pointed topological
space. If Hi (X; Z) is torsionfree, then there is an exact sequence
0 - Hlz(X) -Ù Homz(J(X,x)/J3,Z) i Hlz{X) ®Hlz{X) - ff|(X).
Here i{z){g - 1) = (z,g), where g G 7Ti(X,x) and z e HX(X). If <p G (J/J3)*
and a, /? are /oops eased a£ x7 then
p(^)([a]®[/3]) = (^,({a}-l)({/3}-l)).
The reader may enjoy trying to prove this when X is a smooth curve over C.
The general result can be proved using the cobar construction [28].
Now suppose that V is a smooth variety /C and that x G V. For the rest
of this section we suppose that Hl{y) has a pure Hodge structure of weight /,
necessarily 1 or 2. Denote the kernel of the cup product Hl{y) ® HX(V) —►
H2(V) by K. Since the cup product is a morphism of Hodge structures, K is a
polarized Hodge structure of weight 21. Thus we have a separated extension of
Hodge structures
0 - H1 {V) i (J(V, x)/ J3)* ^ K - 0.
If / = 2, we have a motive
M:ffz->ffè(V)/ffà(K).
To define an integral retraction
rz:{J{V,x)/J*y^Hl{V),
choose elements a\,..., an of 7Ti (V, x) whose homology classes [<*i],..., [an] form
a basis of Hi(V; Z). Define
rzb)[^] =<p{aj-l).
Let X be a smooth completion of V such that D = X — F is a divisor with
normal crossings. One can show that the natural map
(*) H'(FpE'(X\ogD)) -> H'(E'(X\ogD))

268 RICHARD M. HAIN
is injective. Now, if
then we can choose closed 1-forms Wj G ^(XlogD) such that [wj] = Zj and
Y^*ijU>i®u>j G Fp(g)2E1(XlogD).
Since Ylaijzi A Zj = 0, it follows from (*) that there exists u G FPEX(X log D)
such that du + ^au^ A wy = 0. According to (3.1), the iterated integral
]T^Ûti / wiwj + / W
is a homotopy functional. Define
«f£ a,,*i 0 Zj) = J2 ai3 j WiWJ + / U € FPi/°(52(F)).
From (2.13)(b) it follows that pos^ is the identity on K.
Combining these we have the following fact.
(6.2) PROPOSITION. With the notation above, the extension homomorphism
ip associated with the M.H.S. on {J{V, x)/J3)* is
*l> (X^ aVZi ® Z3) lak] = / (X^ aiJwiwJ + u) ' D
We conclude this section with an example.
(6.3) EXAMPLE. Let V = P1 - {ox, o2, a3}. We will compute the motive
M0:^z-^J5rè(v)/fri(v)
associated with the basepoint t GV. Note that ni(V,t) is a free group on two
generators. Set D = {01,02,03}. The Gysin homomorphism gives a canonical
isomorphism Hl(V) = H°{D). The polarization of H°(D) given by «,ap =
6{j induces the polarization of H^ which distinguishes, up to a permutation of
{01,02,03}, the integral basis
1 (x- o7A . . ^
ofH^V).
To compute the extension data, choose loops 71,72 based at t such that
/ Wj = tfij. We may suppose that 7^ is nullhomotopic in P1 — {0^,03} when
j =^ i, 3. Since there are no holomorphic 2-forms on V, K = H1 ® i/1 and we
can define s^ by
sf f 2Z °">>3wi ® ^i) = zZ a^ / ^^i-
The ordered basis w\,W2 of ^(V) gives a canonical identification of H%(V)
with Z2. The exponential map then gives a canonical identification
Hh(V)/Hb{V) ^C'xC;
a\Wi +CL2W2 —► (exp27Tv/—îoi,exp27rv/—Î02).

THE GEOMETRY OF THE FUNDAMENTAL GROUP
269
Since, by (2.11),
rik " \Jik
we have
fi{Wj <g> Wj)[lk} = exp(7rv/ZÏ^A;) = j j '
This is not very interesting as it does not depend upon the basepoint. Again,
by (2.11),
/ W{Wj + / WjW{ = W{ Wj G Z,
J ik Jik J ik J ik
so that
ti{wi <g> Wj)[ik] = {ia{wj <g> WtJhife])"1-
Thus we need only compute fi(wi ® ^2)[72] and /i(tt>2 ® ^i)[7i]- Now
//(wi 0 W2) [72] = exp < 27TV/-Î / ^1^2 [
= exp{^î/72log[(^S)/(^)] dlog(^f)}
J\ |"(ai -a2)/{a3 -o2)l 1 r .,
the cross ratio A of the four points 01,03,02,$. Consequently, the congruence
class of the M.H.S. on J(V,t)/J3 determines the ordered 4-tuple (01,02,03,$).
Since the choice of w\ and w<i is only defined up to a permutation of
{01,02,03}, all six possible values of the cross ratio
A, A"1, 1-A, (1-A)-1, A(A-l)-1, A-^A-l)
may occur. It follows that the isomorphism class of the polarized M.H.S. on
J(V,t)/J3 determines (V,t) up to biholomorphism. □
More generally, we have the following result:
(6.4) THEOREM. IfV = Px- {oi,...,on}, then the polarized M.H.S. on
J(V,t)/J3 determines (V,$) up to biholomorphism.
This is the first hint that the M.H.S. on the fundamental group might be
interesting and our first Torelli theorem.
(6.5) REMARK. Suppose that X is a compact Riemann surface of genus g > 2.
Choose a basis of abelian differentials wi,... ,wg onX. Gunning [11] has defined
certain quadratic periods Qij{~i), where 7 is a loop in X based at x. In terms of
our notation,
Qijil) = / wiwj-
Various properties of quadratic periods such as
Qij{i) + Qjt{i) = / wi / wj
J "Y J "Y

270
RICHARD M. HAIN
follow from properties of iterated integrals (2.11), (2.12), (2.13). From our
viewpoint, the quadratic periods are giving the Plucker coordinates F2(J(X, x)/J3)*
in{J{X,x)/J3)* as
Qijil) = SF{Wi<8>Wj)[l].
(6.6) REMARK. It is shown in [14] that the M.H.S. on Zni(V,x)/Js+l
depends holomorphically on the pair (V,x). When V is fixed, the holomorphic
dependence of the Hodge filtration on the basepoint x follows by differentiating
the change of basepoint formula
/ Wi • • • Wr = ^ / wl ' ' ' Wi \ W*+l '"W3 / wi+l
• Wr
'w i>o Jl
j>0
along 7.
7. Torelli theorems. In §6 we saw that the M.H.S. on J(V,x)/J3 varies
holomorphically with (V, x) and that its isomorphism class determines (V, x) up
to isomorphism when V = P1 — {ai,...,an}. In this section we give general
conditions under which the period mapping
V^ExtiKi&iV)),
2; ^ M.H.S. on J{V,x)/J3
is injective in the case when HX(V) has a pure Hodge structure. In certain
cases this implies (almost) global Torelli theorems for (V, x) such as when V is
a smooth projective curve.
Suppose that V is a smooth variety, and that HX(V) has a pure Hodge
structure of weight /. To simplify the discussion, we also suppose that Hi(V;Z) is
torsionfree. Being the dual of Hl(V), H\(V) has a Hodge structure of weight
—/. Define the Albanese of V by
Alb(V) = ffi(VjC)
[V) FOtfxtVO+tfi^Z)'
If X is a smooth projective completion of V such that D = X — V is a divisor
with normal crossings, then integration induces an isomorphism
Hom^XlogJ^C)
of complex tori. Fix a basepoint x G V. We have a holomorphic map
9x:V^Alb{V),
J X
Define a map
$: ffi(V;Z) — Homz^tf1^))
by
(zi,A) {zj,A}
*(A) (E aiJZi ® *i) (B) = _ Zl a«i
fc.fl) fo.fl)

THE GEOMETRY OF THE FUNDAMENTAL GROUP
271
where A,B G Hi(V;Z), Z{ G Hl{V\Z), and a^ G Z. One can easily check
that $ is a morphism of Hodge structures. Consequently, $ induces a group
homomorphism
^: A\b{V) ^Ext{K,H\V)).
Denote by m: V —► Ext (if, i/1(F)) the map that takes the point ?/ of F to
the congruence class of the M.H.S. on J(V,y)/J3. Denote by vx by the map
vx:V^Vxt(K,H\V)),
y ->m(y) -m(x).
(7.1) P ROPOSITION. The diagram
Alb(V) -£ Ext^tf1^))
ex\ / vx
V
commutes.5
PROOF. Choose a path 7 in V from x to y. If <*i,..., an are elements of
7Ti (V, x) whose homology classes form a basis of Hi (V; Z), then the paths r)~laJ^
(j = 1,..., n) are elements of 7Ti (V, ?/) with the same property. The result now
follows from (6.2) and the following easily verified change of basepoint formula.
(7.2) PROPOSITION. Suppose that wi,w2 G El{V). If a is a loop in V
based at x and 7 is a path in V from x to y, then
To determine when the congruence class of the M.H.S. on J(V,y)/J3
determines y is to determine when vx is injective. Since vx factors as in (7.1), the
injectivity of 9X and $ will guarantee the injectivity of vx. We first study $.
For each u G H^(V) set
u A H\V) = {u A v G A2H^{V) : v G H^{V)}.
(7.3) LEMMA. The map $ is finite to one if and only if for all u G Hq{V)
the cup product
uAH^{V)^Hl{V)
is not injective. Moreover, if in addition Hi(V;Z) is torsionfree, then $ is
injective.6
Note that since $ is a group homomorphism, it is a submersion onto its image.
5Pulte [25] first observed this for smooth projective curves. This proposition is a very
special case of the classification of variations of M.H.S. with unipotent monodromy given in
[14].
6 See note added in proof, page 280.
/ W1W2 = / W1W2 —

272
RICHARD M. HAIN
PROOF. Since $ is a morphism of Hodge structures, and because both Hi (V)
and Hom(tf, HX(V)) satisfy F0 n F = 0, it follows that $ is a covering if and
only if
$Q: ffi(V;Q) ^HoniQ^i/1^))
is injective.
We can write iCasa direct sum of skew symmetric tensors E plus symmetric
tensors S. Now $ is injective
<* #i(V;Q) -► Hom^tf1^)) is injective
o The bilinear form
det:ffi(V)®ffi(V)->£*,
A (8) 5 —► IJ2 dijZi Azj —► ]P 0'ijZi{A)zj{B) \
is nondegenerate
<=>• For each nonzero element u of Hi(V), the map det : u A Hi{V) —► i£* is
nonzero
<* For all nonzero * in J/1^), z/\Hl(V)nE^0
o For all 2 G ^(V), the cup product zAH^(V) —► ffq(V) is not injective.
The second assertion follows from (6.1). □
The injectivity of 0X : V —► Alb(V) is sometimes well understood. For
example, if V is a smooth projective curve of genus > 2, then 9X is injective and the
conditions of (7.3) are satisfied so that $ is injective. Thus we have proved
(7.4) LEMMA. IfV is a smooth projective curve of genus > 2, then the
congruence class of the mixed Hodge structure on J(V, x)/J3 determines x. That
is, m(x) = m(y) if and only x — y. Q
A nicer result would assert that the isomorphism class of the M.H.S. on
J(V,x)/J3 determines (V,x) up to isomorphism. This may well be true.
However the following result is the best to date.
(7.5) THEOREM (HAIN, Pulte) . Suppose that (V,x) and (W,y) are two
pointed smooth projective curves. If there is a ring isomorphism
<p:Z7r1(V,x)/J3^Z7r1(W,y)/J3
that induces an isomorphism of M.H.S., then there is an isomorphism f: V —► W
such that, with the possible exception of at most two points x ofV, f(x) = y. (If
V is hyper elliptic, then no such exceptional points exist.)7
PROOF. Since <p induces an isomorphism of M.H.S.'s it will induce an
isomorphism of Hodge structures on W-1/W-2 = J/J2 = Hi. Consequently, it
induces an isomorphism of Hodge structures ip* : Hl(W) —► Hl{V). According
to (6.1), the sequence
0 -+ H\V) -+ (JIJ3Y i H\V) ® H\V) c^ Z -+ 0 is exact.
7Pulte [25] has shown that if the pointed Torelli theorem fails for X, a curve of genus g,
then (g — l)(p + q) is the canonical divisor of X, where {p, q) is the unique pair of points such
that m(p) + m{q) = 0.

THE GEOMETRY OF THE FUNDAMENTAL GROUP 273
Since p is dual to the multiplication
J/J2® J/J2 ^ J/J3,
the ring structure of Zni/J3 determines the polarization on H1. Thus the
fact that (p is a ring homomorphism implies that ip* : H1(W) —► Hl(V) is an
isomorphism of polarized Hodge structures.
By the classical Torelli theorem, there is an isomorphism / : V —► W such that
/* : H1 (W) —► H1 (V) is ±ip*. Since <p is a ring homomorphism, the induced map
on K = (J2/J3)* will be /* <g> /* = <p* <g> <p*. Note that if V is hyperelliptic,
we may assume that /* = <p*, for we may compose / with the hyperelliptic
involution if necessary.
We may now assume that V = W and that the diagram
O-ffi(V) - (J(V,y)/J3)* - K^O
±id | <p* I I id
0^HX{V) -+ (J(V,x)/J3)* -+ K^O
commutes. That is, the congruence classes m(x) and m(y) of the M.H.S.'s on
(J(V,x)/J3)* and {J{V,y)/J3Y satisfy m(x) = ±m(y). If m(x) = m(y), then
x = 2/ by (7.4). Now, if V is not hyperelliptic, then there is at most one pair of
points {x, y} of V such that m(x) = — m(y). For if m(x) + m(y) = m(p) +m(g) =
0, then it follows from (7.1) and Abel's theorem that x + y = p + q in Pic(V).
Since V is not hyperelliptic, this implies that {x, y} = {p, q}. D
Jablow [18] has proved a pointed Torelli theorem for a general pointed curve
of genus 3 with a framing of Hi.
Even in the case when
$: Alb(V) -►Ext(/r,ff1)
is not an immersion, the pointed Torelli theorem may be true. For example, let
V be the set of ordered 3-tuples of distinct points in C. That is,
V = CxCxC- {(x, y, z):{x- y){y - z)(z - x) = 0}.
When
d(x-y) d(y-z) d(z - x)
u = 1 1 ,
x — y y — z z — x
the cup product u A Hl(V) —► i/2(V) is an isomorphism. By (7.3), $ is not a
covering map. (In fact, $-1(0) = C*.) Let x = (ai,a2,a3). It is an interesting
exercise to check that the extension data of J(V, x)/J3 determines the cross ratio
[ai,a2,a3,oo]. Consequently, the M.H.S. on J(V,x)/J3 determines (V, x) up to
isomorphism. The fibers of vx are the orbits of the action of the affine group
( o" ?)

274
RICHARD M. HAIN
on V:
( n h\
: x —► ax + (6,6,6), x G K
8. Harmonie volume. Let C be a smooth projective curve over C of genus
g > 2. For each x G C we have the Abel-Jacobi map
0X:C -> Jac(C),
The image of 6X is an algebraic 1-cycle Cx in Jac(C). Set
C- = {-PeJ&c{C):peCx}.
Since the involution/? —► — p of Jac(C) acts trivially on i/2(Jac(C)), the algebraic
1-cycles Cx — Cy and Cx — C~ are homologous to zero.
Denote the group of algebraic A:-cycles in the variety V, modulo rational
equivalence, by Ak(V) and those homologous to zero by A%(V). Abbreviating Jac(C)
by Jac, we have
[Cx-Cy],[Cx-C-]eA\{lM).
The intermediate Jacobian of Jac is defined by
Hom(F2ff3(Jac),C)
J2(Jac) ff3(Jac;Z) •
There is an Abel-Jacobi map
V>:^?(Jac) -» J2(Jac),
[Z] - /r,
where dT = Z. The factorization of vx given in (7.1) has geometric meaning.
First we need the following fact from linear algebra due to Pulte [25].
(8.1) PROPOSITION. The natural map
®3hï(C) - A3^(C),
a (g) 6(g) c —► a Ab Ac
induces a surjection
KZ®H1Z(C)^/\3H^(C). D
(8.2) COROLLARY. There is a natural injective group homomorphism
J2(Jac)—Ext^tf^C)).
PROOF. First note that Poincaré Duality
PI>.:Hi{C)->H1{C)
is a morphism of Hodge structures of type (1,1). Consequently, the isomorphism
Hom(K, H1 (C)) -f Hom^ ® H1 (C), C)

THE GEOMETRY OF THE FUNDAMENTAL GROUP
275
is of type (—1, —1) and
Uomc{K®H1{C),C)
Ext{K,H\C)) =
Rom(K <g> Hl(C), Z) + F"1 Hom(tf <g> H1, C) '
Since F-1 Hom(# <g> H1, C) is the annihilator of F2(K®Hl), it follows that
v v " Homz^Off^Z)
The result now follows from (8.1). □
The following result gives a geometric interpretation of the factorization of vx
given in (7.1). It is due to Pulte [25] who exploits techniques of B. Harris [15,
16].
(8.3) THEOREM (HARRIS-PULTE). The following diagram commutes
J2(Jac)
../
Jac(C)
where A is the Abel-Jacobi map associated with the family
CxC -+ Jac(C) (y,z) -+ y-z
1 1
C z
Also V[CX - C'} = m(x) + m(y). D
So far we have not mentioned B. Harris's harmonic volume explicitly. The
technical ingredients of its construction are buried in the proof of (8.3). The
harmonic volume of C, as denned by Harris, may be recovered as follows.
Denote the primitive cohomology of Jac(C) by Pi/'(Jac). There is a primitive
intermediate Jacobian
Hom(F2PJ/3(Jac),C)
P2(JaC) = tf3(Jac;Z)
and a restriction map
r: J2(Jac) —► P2(Jac).
From (8.3) we know that 2ra(x) G J2(Jac). Thus we can restrict 2m(x) to the
primitives. One can show, either by using properties of iterated integrals or the
fact that 2ra(x) = ^f[Cx — C~], that r(2m(x)) is independent of x. Harris's
definition of harmonic volume is equivalent to the following.
(8.4) DEFINITION. The harmonic volume V(C) of C is the point r(2m(x)) of
P2(Jac).
Harris's main result, which has been incorporated into (8.3), is that
r o ty[Cx — C~] can be expressed as an iterated integral. Using this he showed

276
RICHARD M. HAIN
that when C is the Fermât quartic, C — C~ is not algebraically equivalent to
zero. Ceresa [4] had previously proved that for the generic curve C of genus
9 > 3, C — C~ is not algebraically equivalent to zero. The Fermât quartic was
the first specific curve for which this was known. This played a role in Bloch's
work [2].
It is interesting to note that Clemens has asked the following question.
(8.5) QUESTION. Is it true that a curve C of genus 3 is algebraically equivalent
to its negative C~ in Jac(C) if and only if it is hyperelliptic? More generally,
is it true that a curve C of genus g = 2k — 1 has the property that C^_1) is
algebraically equivalent to C^9~^~ in Jac(C) if and only if it has a g\.l
9. Generalizations of the Riemann-Hilbert problem In modern
language, the Riemann-Hilbert problem can be described as follows. Each mero-
morphic gl(n, C)-valued 1-form w on P1, with at worst simple poles, is of the
form
N .
uj = > dt,
where each Ak is a constant n x n matrix and tk G C. Set D = {£i,..., tjq, oo}
and V = P1 — D. (Note that oo need only be included in D if ReSoo uj ^ 0, i.e.,
Y^Ak ^ 0.) As in §2, uj defines a meromorphic connection on the trivial bundle
Cn x P1 —► P1 with regular singular points along D. Since doj + uj A w = 0, the
connection is flat and we have a monodromy representation
p:7ri(V0^GL(n,C).
The Riemann-Hilbert (or Hilbert's 21st) problem asks if every linear
representation 7Ti(V) —► GL(n) arises in this way. Lappo-Danilevsky [19], Plemelj [24],
and others have shown that the answer is yes except in some degenerate cases.
But the problem, as stated, remains unsolved (cf. [29, p. 311]).
An obvious generalization of the Riemann-Hilbert problem is the following.
Suppose that V is a smooth variety and that X is a smooth completion of V
such that D = X — V is a divisor in X with normal crossings. A completely
integrable 1-form uj G Q1 (X log D) <S> g\(n) defines a flat meromorphic connection
on the trivial bundle Cn x X —► X which has regular singularities along D. We
will refer to the associated monodromy representation p: tti(V) —► GL(n, C) as
the monodromy of (the completely integrable 1-form) uj.
The first obvious question to ask is if every linear representation 7Ti(V) —>
GL(n) is the monodromy representation of such an lj. The answer is no.
(9.1) EXAMPLE. Suppose that dimc fi^XlogD) = 1. As we have seen
previously, there is an isomorphism
(*) n1{X\ogD)^F1H1{V;C).
Choose a nonzero element w of Q1(X\ogD). Every element uj of Q1(X\ogD) ®
gl(n) is of the form uj = Aw, where A is a constant n x n matrix. Note that

THE GEOMETRY OF THE FUNDAMENTAL GROUP
277
duj — uj Auj = 0 so that all such uj are completely integrable. According to (2.5),
the monodromy representation
P:tti(V,x) ^GL(n)
is given by
P{{l}) = I + I u+ I uu+ I uu(jj H
= I + ( w]A+( / ww)A2+( www J A3 + • • •
■^(i-W3(/,-)^+à(/-),',,+-
= exp ( / wA J = exp / uj.
Here we have used the fact that if w G El(V), then
Lww-w=?.{Lw) '
f-7 '• \J1
which can be proved using (2.11) or by direct calculation.
From (*) it follows that the dual of n^XlogZ?) is HX(V; C)/F°ffi(V).
Consequently, the monodromy representation p of uj factors through the 1-parameter
subgroup of GL(n, C) generated by A:
ttiGO ± H^v^cyF0 =c
GL(n)
where 0(g) = (J w)w* and &A(tw*) = exp(L4). Conversely, if a representation
fl"i(V) —► GL(n) factors as above, then it is the monodromy representation of
Aw, where A is the infinitesimal generator of the 1-parameter subgroup a a-
We now consider two specific examples. In the first we will see how the
Hodge filtration restricts the possible monodromy representations. Suppose that
V — C/r, an elliptic curve. Consider the representation
p:7ri(V)-GL(2,C),
-(i V)-
Since the group (q *) is isomorphic to C, p is a monodromy representation if
and only if the functional
0:7n(V)-C,
/,
dz
factors through tti(V) —► Hi(V)/F°. Since cfo and dz are linearly independent
in Hl{V\ C), 0 does not factor and p is not a monodromy representation.

278
RICHARD M. HAIN
In the second example, the Hodge filtration is trivial. Suppose that V is
the complement of the cusp {(x,y): x2 = y3} in C2. Its fundamental group
is the group (a, b: a2 = b3). Viewing the symmetric group on three letters,
£3, as the set of permutation matrices in GL(3, C), we obtain a representation
p: tti(V0 -► GL(3) by defining p(a) = (12) and p{b) = (123). Since the image
of p is not abelian, p does not factor through the Hurewicz homomorphism
ni{V) —► Hi(V). It follows that p is not the monodromy representation of a
completely integrable 1-form uj G Q1(X\ogD) <g> gl(3).
By now it should be clear that the correct generalization of the Riemann-
Hilbert problem should be stated as follows.
(9.2) PROBLEM. Characterize the monodromy representations of completely
integrable 1-forms uj G n1(XlogD) <8> gl(n), where X is a smooth projective
variety and D is a divisor in X with normal crossings.
This we have done in (9.1) when dim Vtl{X log D) = 1. When dim QX(X log D)
> 2, we need a nonabelian analogue of Hi(V)/F°. It turns out to be a quotient
of the completed group ring of 7ri(V).
Suppose that G is a discrete group. As before, we shall denote the
augmentation ideal of its group ring CG by J. Denote the J-adic completion of CG by
CGC Note that if (V, x) is a pointed smooth quasi-projective variety, then by
(4.1),
Cti(V,*)"= limC7Ti(V,a;)/Js+1 = lim Eom{H°(Ba(V),x),C)
= Eom(ljmH°{Bs(V),x),C) = Eom{H°(B{V),x),C),
where H°{B{V),x) is the union of the H°(Ba(V),x):
H0(B(V),x) = {jH0(Bs(V),x)
= {iterated integrals that are homotopy functionals
{loops at x} —► C}.
In this way Ciri(V, reacquires a Hodge filtration.
According to (2.9), the coproduct
A: B{V)^B{V)®B{V),
/ Wi • • • Wr —► ^ / Wi--'Wi® / Wi+i • • • Wr
J 1=0 ^ ^
is dual to the product of Ctti(V, x). Since the coproduct A preserves the Hodge
and weight filtrations of B(V), the product
Ctti (V, x)"® Ctti (V, xf^ Ctti (V, xf
preserves the Hodge filtration. Consequently the subspace
/ = f° n J1 + F-1 n J2 + F~2 n J3 + • • •

THE GEOMETRY OF THE FUNDAMENTAL GROUP
279
of C7Ti(V, x)^is an ideal. Set
Aoo = Ctti(V, xf/I and A9 = Ctti(V, xf/I + Ja+1.
Denote the composite 7Ti(V, x) —► C7ri(V,x) —► A9 by 05(x), where 1 < 5 < 00.
Note that Ax = C 0 i/i(F)/F° and
fli(x):7ri(V,x)-.Ceffi(V)/F°,
where ft is the Hurewicz homomorphism. It is the homomorphisms 9s(x) that
generalize the homomorphism 6 of (9.1).
(9.3) THEOREM [30]. There exists a topological C-algebra
•AÇC7Ti(V,x)7J
such that
(a) im0 ÇA,
(b) p: tti(V,x) —> GL(n) w a monodromy representation of an integrable 1-
/orra on V with logarithmic singularities along D if and only if there exists a
continuous C-algebra homomorphism <p: A —> GL(n) such that
tti(V,x) 4 GL(n)
«1 1
.A - gl(n)
commutes. D
The characterization of monodromy representations given in (9.3) is not
optimal, as it does not appear to help in solving the classical Riemann-Hilbert
problem. Nonetheless, it still contributes nontrivial information when X ^ P1.
For example, let R be the kernel of the natural map
7Ti(V,x)-+C7ri(V,xp
(9.4) COROLLARY. If p is a monodromy representation, then kerp > R. □
This is a nontrivial restriction on monodromy representations as iri(V)/R =
Hi(V) when V = C2 — {(x,y): xp = y9} with p,g relatively prime. Even
when 9: wi(V) —> A is injective the theorem imposes restrictions on monodromy
representations. For example, when V is an elliptic curve 9: ni(V) —> A is
injective, but not every representation of tti(V) is a monodromy representation
(cf. (9.1)).
(9.5) REMARK. The maps 9s and 9^ were previously defined and studied by
Parsin [22] and Hwang-Ma [17] in the case when V is a smooth projective curve
of genus > 2.
When WxHl{V) = 0 (e.g., V is a Zariski open subset of Pn), F~s Js+1 = 0
for all s. Consequently 1 = 0 and A^ = C7ri(V,x)^ Under the assumption
that there exist xi,...,xj G 7ri(V,x) which are independent in Hi(V\C) and

280
RICHARD M. HAIN
generate im0, it seems reasonable to make the following conjecture:
(9.6) CONJECTURE. IfV is a variety with WiHl(V) = 0 satisfying the
conditions above, then p : 7Ti (V, x) —> GL(n) is the monodromy representation of
a completely integrable 1-form uj G Q1(X\ogD) ® gl(n) if and only if p factors
through im0:
7ri(y, x) ► im#
p\
GL(n) D
When V = P1 — D, then 0 is injective. Thus this conjecture is a
generalization of the classical Riemann-Hilbert problem. Golubeva [9] has solved the
inverse problem when V = Pn — union of hyperplanes and the generators p(xj)
of the monodromy group are sufficiently close to the identity. Her argument
can be adapted to prove (9.6) for representations for which each p{xj) — I is
sufficiently small. For nilpotent connections and unipotent representations we
can completely characterize the monodromy representations.
A gl(n)-valued 1-form uj is nilpotent if it takes values in a nilpotent subalgebra
of gl(n).
(9.7) THEOREM [30]. Suppose that V = X - D is a smooth variety. A
unipotent representation p: 7Ti(V,x) —> GL(ra) is the monodromy representation
of a completely integrable nilpotent 1-form uj G Q1(X\ogD) ® n if and only if p
factors through 9m.
a
7Ti(V,x) -3 Am
1 1
GL(m) -> gl(m) □
When WxHl(V) = 0, Am = C7n(V»/Jm+1. If p: tti(V,x) -> GL(m) is
unipotent, then p induces an algebra homomorphism
p:C7n(F,a:)/Jm+1->gl(m)
such that the diagram
tti(V,x) % Am
p [ i p
GL(ra) -> gl(ra)
commutes. Thus we have proved the next result.
(9.9) COROLLARY. IfWiHl(y) = 0, then every unipotent representation
7Ti(V,x) —> GL(ra) 25 £/ie monodromy representation of a completely integrable
1-form uj G Q1(X\ogD) ® gl(m). D
When F is a Zariski open subset of Pm, we recover a theorem of Aomoto [1].
Added in proof. In (7.3) the condition that
(a) uAH^{V)^H^{V)

THE GEOMETRY OF THE FUNDAMENTAL GROUP
281
not be injective is too strong. It should be replaced by the condition that
(b) ^aX^^^qW"
be nontrivial, where p is interior multiplication by an element tp of Hi (V) such
that <p(u) = 1. Observe that if u satisfies (a), then it satisfies (b) so that if (a)
is satisfied by all u G Hq(V), then $ is finite to one.
References
1. K. Aomoto, Functions hyperlogarithmique et groupes de monodromie unipotents, J. Fac.
Sci. Univ. Tokyo 25 (1978), 149-156.
2. S. Bloch, Algebraic cycles and values of L-functions, J. Reine. Angew. Math. 350 (1984),
94-108.
3. J. Carlson, Extensions of mixed Hodge structures, Journées de Géométrie Algébrique
d'Angers, A. Beauville, editor, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, pp. 77-105.
4. G. Ceresa, C is not algebraically equivalent to C~ in its Jacobian, Ann. of Math. (2) 117
(1983), 285-291.
5. K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879.
6. , Extension of C°° function algebra by integrals and Malcev completion of -k\, Adv.
in Math. 23 (1977), 181-210.
7. P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-58.
8. A. Durfee, A naive guide to mixed Hodge theory, Singularities, Proc. Sympos. Pure
Math., vol. 40, part 1, Amer. Math. Soc, Providence, R. I., 1983, pp. 313-320.
9. V. Golubeva, On the recovery of Pfaffian systems of Fuchsian type from the generators of
the monodromy group, Math. USSR-Izv. 17 (1981), 227-241.
10. P. Griffiths and W. Schmid, Recent developments in Hodge theory: A discussion of
techniques and results, Proc. Bombay Colloq. on Discrete Subgroups of Lie Groups (Bombay, 1973),
Oxford Univ. Press, 1975, pp. 31-127.
11. R. Gunning, Quadratic periods of hyperelliptic integrals, Problems in Analysis,
R. C. Gunning, editor, Princeton Univ. Press, Princeton, N. J., 1969, pp. 239-247.
12. R. Hain, Iterated integrals, intersection theory and link groups, Topology 24 (1985), 45-
66, and Erratum 25 (1986), 585-586.
13. , The de Rham homotopy theory of complex algebraic varieties, J. K-Theory (to
appear).
14. R. Hain and S. Zucker, Unipotent variations of mixed Hodge structure, Invent. Math, (to
appear).
15. B. Harris, Harmonic volumes, Acta Math. 150 (1983), 91-123.
16. , Homological versus algebraic equivalence in a Jacobian, Proc. Nat. Acad. Sci.
U.S.A. 80 (1983), 1157-1158.
17. S.-Y. Hwang-Ma, Periods of iterated integrals of holomorphic forms on a compact Riemann
surface, Trans. Amer. Math. Soc. 264 (1981), 295-300.
18. E. Jablow, Quadratic vector classes on Riemann surfaces, Duke J. Math. 53 (1986),
221-232.
19. I. A. Lappo-Danilevsky, Mémoires sur la théorie des systèmes des équations
différentielles linéaires, reprint, Chelsea, New York, 1953.
20. J. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci.
Publ. Math. 48 (1978), 137-204.
21. R. Narasimhan, Analysis on real and complex manifolds, North Holland, Amsterdam,
1973.
22. A. Parsin, A generalization of the Jacobian variety, Amer. Math. Soc. Transi. (2) 84
(1969), 187-196.
23. D. Passman, The algebraic theory of group rings, Wiley, New York, 1977.
24. J. Plemelj, Problems in the sense of Riemann and Klein, Wiley, New York, 1964.
25. M. Pulte, The fundamental group of a Riemann surface: mixed Hodge structures and
algebraic cycles, Ph.D. Thesis, University of Utah, Salt Lake City, Utah, 1985.

282
RICHARD M. HAIN
26. D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295.
27. R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958),
210-220.
28. J. Stallings, Quotients of the powers of the augmentation ideal in a group ring, Knots,
Groups and 3-Manifolds, Papers Dedicated to the Memory of R. H. Fox, L. Neuwirth, editor,
Princeton Univ. Press, Princeton, N. J., 1975, pp. 101-118.
29. L. Boutet de Monvel, A. Douady, and J.-L. Verdier, Mathématique et physique, Birk-
hàuser, 1983.
30. R. Hain, On a generalization of HilberVs 21st problem, Ann. Sci. Ecole Norm. Sup. 19
(1986), 609-627.
University of Washington

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Degeneration of Mixed Hodge Structures
STEVEN ZUCKER
Perhaps I should begin with the disclaimer that, as of this writing, there
really isn't much of a theory of the degeneration of mixed Hodge structures
in the abstract. Nonetheless, it would be wrong to terminate here. In order to
explain what one might want, and how it would relate to what has become known
in recent years about the geometric case, it seems like a good idea to use the
well-developed theory of degenerations of (pure) Hodge structures as a guideline;
this is, after all, a special case (where the weight filtration is trivial). It is there,
then, that we will begin. In §2, we discuss variations of mixed Hodge structure.
The third section will be an exposition of a key technique in the subject, namely,
filtered mixed Hodge complexes.
1. Degeneration of Hodge structures. We establish some notation. Let
S be a complex manifold, S a smooth partial compactification of S such that
£ = S — S is a divisor with normal crossings. Denote by j the inclusion of
S in S. Let V be a local system on S underlying a (polarizable) variation of
Hodge structure of weight k. (This presumes that V has at least an underlying
real structure.) The associated holomorphic vector bundle (locally free sheaf),
V = 0 s ®c V, with connection of, has the decreasing Hodge filtration J, subject
to horizontally:
(i) ^cnj®^-1.
For simplicity, we assume that all local monodromy transformations of V are
unipotent.
It is useful to introduce the so-called canonical extension V of V to S. In
local coordinates
t = ($i,... ,tn)
on 5, where j is the inclusion
(2) (A*)n <-+ An
1980 Mathematics Subject Classification (1985 Revision). Primary 14C30.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
283

284
STEVEN ZUCKER
(there is no loss of generality here), V is the free 0a«-submodule of j*V
generated by sections of the form
(3) v(t) = expl^.J2(logtj)NAv,
where v is a multivalued section of V, and Nj is the nilpotent logarithm of the
monodromy transformation associated to the jth factor of A* in (2). Note that
3
and therefore
(4) tfV cn|(logE)0V.
The results about degeneration of Hodge structures can be summarized as
follows:
(a) The filtration 7 of V extends to a filtration 7 of V by subbundles. This
is a reformulation of part of Schmid's result in [15, (4.12)] that 5F is asymptotic
to a nilpotent orbit, viz., a variation of Hodge structure whose Hodge filtration
is constant relative to a canonical frame (of the form (2)).
(b) Limit mixed Hodge structure. Let S = A*, and write Fo for the filtration
J(0) of the fiber V0 = V (0) of V at 0 G A. Let N denote the nilpotent
monodromy logarithm, which acts naturally on Vb- Let M = M(N) be the weight
filtration of N—shifted so as to be "centered" at k—characterized by
(i) NMtCMt-t;
(5) (ii) for / > 0, the induced mapping Nl : Grj^jVb —► GrjJljVo
is an isomorphism.
It has been remarked in [17, (2.3)] that M can be expressed as a "convolution":
M = (/f */)[-*],
where K and / are respectively the increasing filtration by kernels and images
of powers of N (the latter in negative degrees). We record the explicit formula
for the convolution:
(6) (if*/)i = E(^n/H)-
i
Then the pair of filtrations (M, F0) of Vb determines a mixed Hodge structure
on which N acts as a morphism of type (—1, —1) [15, (6.16)].
(c) Geometric realization [16, 3]. We take 5 to be a curve, and consider
families of projective (more generally, compact Kàhler) manifolds
X <-> X
(7) /1 17

DEGENERATION OF MIXED HODGE STRUCTURES
285
where / is proper and smooth, X is smooth, / is proper, and
Y =X-X = f-1{E)
is a divisor with normal crossings. Take
V = RkUC.
We have
v =* Rkf.nx/S, 3* = Rkf.F*>n-x/s,
where F is given by truncation from below.
There is a de Rham theoretic construction of the limit Hodge structure. Part
of the theory gives
(8) v = Rkun^(\og y), 7P = Rkf,Fpn^(\og y);
under the isomorphism—assume for simplicity that Y is reduced—
(9) vb = £fc(y,n^/5(iogy)®0y),
F0 is induced by F. On the other hand, in order to see M one must take an
F-filtered resolution A' of Q'— -(logY) <8>0y, large enough to admit a suitable
second filtration. (See (3.8). Also, see [20, §2] for a summary of the point of the
above assertions.)
(d) As part of the development, we include a consequence of the preceding
facts, and their proofs, concerning the degeneration of Hodge structures. Let
S be an algebraic curve, and endow it with a Poincaré metric, viz., a metric
asymptotic to the Poincaré metric of A* near a point of E. The local system V
is metrized by the Hodge metric, induced by the polarization. Then
(i) For all i,
(io) ir(s,y,v)^2)(s,v),
where the right-hand side denotes L<2 cohomology with respect to the given
metrics. The isomorphism determines a Hodge structure of weight i + k on
(ii) If V = Rkf*C as in (c), then the Hodge structure above is compatible
with that of HlJtk{X, C). The most serious case is when i = 1. We state the
simplest precise version of this assertion. The theory behind (i) also provides
a mixed Hodge structure on HX(S, V). Assume that E ^ 0 (if not, delete a
fiber!). The Leray spectral sequence for / provides an isomorphism
(11) ^(5,V) -ker{tf*+1(X,C) -> tf*+1 (fiber)}.
Then (11) is an isomorphism of mixed Hodge structures.
We sketch the highlights of the proof. To prove (i), one needs a "Poincaré
lemma," local about the points on E, for the L<i cohomology. It goes more or
less by direct calculation, with use of the norm asymptotics.
\\v\\2 - (- log |*|)'-fc if v e Mt - M_x [15, (6.6)]; see also [19, (3.7)],
||d*A||2~iog2|*|

286
STEVEN ZUCKER
(see [19, §6]). For (ii), one considers the complex (cf. (4))
(13) V ^n|(logE)0V,
quasi-isomorphic to Rj*V. The filtration F, where Fp is defined as
yp->n£(iogE)®yp-1 (cf. (1))
induces the Hodge filtration of the mixed Hodge structure on H1(S, V). The
weight filtration M on (13) begins with
V->(imd),
which is quasi-isomorphic to y*V, and then the higher GrfM,s are supported on
E. Using (c), one sees how the preceding is derived from fi^-(log Y) and its usual
nitrations (W,F) (see [19, §15]).
When S is of arbitrary dimension, part (i) has been generalized in the form
(14) IH'(S,V)czH-w(S,V),
with IH denoting (middle perversity) intersection homology [10, §6], in [2] and
[14], by rather elaborate calculation, using the several variable norm asymptotics
obtained in [1] and [13]. In the direction of the analogue of (ii), one can combine
the results of [23] and [24].
2. Variation of mixed Hodge structure.
(2.1) We take as initial position the stance that a variation of mixed Hodge
structure is a more complicated variant of the pure case. As such, we can and
will adapt most of the notation from §1. Also, the abstract theory should be
modeled on what one knows from the variations arising from geometry.
For general reference, we refer the reader to [17] and its bibliography.
(2.2) The basic object of study is a graded-polarizable variation of mixed Hodge
structure on S. This consists of:
(i) a local system VonS, with an increasing weight filtration W, defined at
least over R;
(ii) a decreasing Hodge filtration 7 of V that obeys (1);
(iii) for each k G Z, the filtration induced by 7 determines on Grj^ V a
polarizable variation of Hodge structure of weight k.
(2.3) To see what one might expect from a general theory, we turn to the
geometric case over a curve, as in (7) of §1, but where / need not be either
smooth or proper. This has been studied by several mathematicians [6, 9, 11,
17]. One obtains, by methods based on those mentioned in §l(c):
(i) a limit Hodge filtration i*o;
(ii) a limit mixed Hodge structure (M, Fq) on Vb, filtered by W, inducing
those of §l(b,c) on each Grj^Vb.1
1 * In §3 we will discuss filtered mixed Hodge complexes, the very useful technical device for
proving such statements.

DEGENERATION OF MIXED HODGE STRUCTURES
287
(2.4) From (2.2,i), it follows that the monodromy logarithm TV respects W.
The weight filtration M of (2.3,ii) is shown to be M (TV; W), the so-called weight
filtration of TV relative toW. By definition, this is the filtration of Vb for which
(i) NMk C M*_2;
(ii) M induces on Gr^Vb the weight filtration M(Gr}yTV) (centered at /).
(Because TV need not respect W strictly, M (TV; W) is usually not induced by the
"absolute" weight filtration M (TV) [centered at 0].)
(2.5) We note that in the setting of abstract nilpotent transformations of
filtered vector spaces, the relative weight filtration fails to exist generically, though
it is uniquely determined when it does exist. As such, geometric monodromy
logarithms are, by (2.4), very special. We give some illustrations of the kind of
"equations" that they satisfy [17, §2]:
(i) if Grj^ V is nonzero for only two consecutive values of k (occurring
geometrically, e.g., for complements of smooth subvarieties of smooth proper varieties),
then there is an TV-invariant splitting of W [17, (2.16)];
(ii) if Grf TV = 0 (i.e., NWk C Wk-{) for all Jfe, then in fact NWk C Wk-2 (so
M = W) [17, (2.14)].
(2.6) We continue to assume that S is curve. In a sense, the features described
in (2.3) and (2.4) at the points of E, namely,
(i) limit Hodge filtrations,
(ii) relative weight filtrations,
(iii) limit mixed Hodge structures, filtered by W (in the sense of (2.3, ii)),
are a good set of conditions at infinity to impose on a variation of mixed Hodge
structure. (In fact, as was remarked by Deligne, if one assumes that the limit
Hodge filtrations (i) induce those of §l(b) on each Gr^ Vb, then (iii) above is
a consequence of (i) and (ii); see [17, (A.9)].) As evidence for the preceding
allegation, we cite the following generalization of §l(d): there is a functorial
construction of mixed Hodge structures for H'(S, V) and //"(S,y*V), filtered by
W, reducing to those of §l(d) in the pure case. They are compatible with the
Leray spectral sequence of j [17, §4]; moreover, in the geometric case (cf. (7)),
the mixed Hodge structures are compatible with those of H'(X, C) [21].
For the extension to higher-dimensional 5, via a "curve test," see [22].
(2.7) It seems obvious that one would want (2.6, i-iii) to be the consequence
of more fundamental hypotheses on a variation of mixed Hodge structure. In [5,
(1.8.15)], Deligne posed the problem of setting up conditions that would allow
the nilpotent orbit theorem of Schmid (for S = A*) to generalize to the mixed
case. (Note that the notion of a nilpotent orbit (see §l(a)) still makes sense.)
That some hypotheses are necessary is quickly illustrated by simple examples:
(i) there are nilpotent orbits that have well-behaved limit Hodge filtrations,
though (2.6, ii) is missing [17, (3.15)];
(ii) there exist variations of mixed Hodge structure with no limit Hodge
filtrations, even though the relative weight filtration exists [17, (3.16)].

288
STEVEN ZUCKER
(2.8) It is generally agreed that the missing ingredient in the theory of
variations of mixed Hodge structure is a satisfactory notion of a polarized mixed
Hodge structure, one which provides coherency to the polarizations of the
individual pure subquotients. The notion of graded-polarizability is just too weak.
Perhaps this is best illustrated by using the description of a mixed Hodge
structure as an iterated extension of pure Hodge structures of different weights (cf.
classifying spaces for mixed Hodge structures; see [18]). A graded-polarization
simply provides no control over the extension data; in particular, the latter could
have essential singularities on E.
3. Filtered mixed Hodge complexes. The material described in this
section, which first appeared in [7], is inherently technical, but it provides the
tool for producing filtered mixed Hodge structures. There are several occurrences
and applications in algebraic geometry.
(3.1) To help put the definition of a filtered mixed Hodge complex in
perspective, we first recall Deligne's formalism of (ordinary) mixed Hodge complexes [4,
(8.1)], which axiomatizes the way that mixed Hodge structures on cohomology
groups arise in practice.
Let K' be a complex of sheaves of C vector spaces on the space Z. Abusing
language just a little, we denote by KT(K') the complex of global sections of a
resolution of K', to which one always tacitly passes, for the cohomology groups
of RT(K') give the hypercohomology
(i) E{z,k-).
We suppose further that K' has an increasing filtration W and a decreasing
filtration F. It is also required that K' be W-filtered quasi-isomorphic to the
extension of scalars of some complex of R (or Q) vector spaces, thus imparting
a real (or rational) structure to (1), as well as the filtration induced thereon by
W. Since most of the "action" gets expressed in terms of K' itself, one tends to
suppress the real complex (cf. [20, §1]).
The condition on the bifiltered complex K' that defines a cohomological mixed
Hodge complex is that the filtration induced by F on
(2) L-k = G$K-
determines a so-called cohomological Hodge complex of weight k, viz.:
(i) on IP{Z, L'k), F induces a Hodge structure of weight i + k\
(ii) the spectral sequence
fEpa = H^(Z,G^FL-k) => H>>+«(Z,Lk)
degenerates at E\ (cf. classical Hodge theory).
(3.2) EXAMPLE ([4, §3]; see also [20, §1]). If D is a divisor with normal
crossings in the compact Kàhler manifold Z, let
(3)
K- = n-z(iogD).

DEGENERATION OF MIXED HODGE STRUCTURES
289
For RT(K'), we may take the global C°° log complex,
r(z,£'z(iog/?)),
or the Cech resolution of (3). We have that F is the Hodge filtration (cf. §l(c)),
W is the weight filtration by number of singularities. Via A:-fold residues, one
has
(4) GrJT/r^n^, [-*](-*),
where D^ denotes the disjoint union of the A:-fold intersections of the
components of D (Z)(°) = Z), and the bracketed numbers indicate a shift in degree and
filtration. One sees then that K' is a cohomological mixed Hodge complex, with
(5) Hm{Z,Km) c* Hm{Z - D,C).
(3.3) When one has a cohomological mixed Hodge complex, it follows that the
filtrations induced by W and F on (1) define a mixed Hodge structure.
Specifically, this requires that on
(6) GtVW(Z,K') = ^^{mZ,WkKl^mZ^
the filtration induced by F (of IP(Z,K')) defines a Hodge structure of weight
i + k.
The proof goes roughly as follows. Consider the spectral sequence of W. Its
first differential is
(7) du W{Z,GtflC) c* wE^k'i+k - wEîk+u+k czIP^iZ.GT^K').
The second superscript in the term of the spectral sequence conveniently gives
us the weight of the Hodge structure (3.1,i). One observes that d\ is compatible
with F, and also with the underlying real structure, hence is a morphism of
Hodge structures. Thus, one gets Hodge structures on the E<2 term by taking
cohomology with respect to d\. Next,
is a "morphism" of Hodge structures in the direction of decreasing weight, and
this forces o^ to be the zero mapping. Likewise, all higher differentials vanish.
This gives an isomorphism of (6), i.e., wE^, and w^, providing the desired
Hodge structure.
(3.4) The last sentences of (3.3), beginning at (8), were a little too quick.
The filtrations induced by F on \yEr that are for natural reasons compatible
with the dr's, and also the one we want for (6), are not (in the total generality of
bifiltered complexes) the same as the ones that were obtained recursively through
the spectral sequence. However, the fact that we also have (3.1, ii) is sufficient
to imply equality; this uses the so-called lemma on two filtrations [4, (1.3.17)].
We repeat for emphasis that one needs some extra hypothesis in order to be able
to manipulate freely with one filtration in the spectral sequence of another.

290
STEVEN ZUCKER
To get some idea what the problem is, suppose that A/B is a subquotient of
KT(K') that occurs in the spectral sequence of W. The filtration induced by F
would be
(9) FP{A/B) = image{FM -► {A/B)}.
If in the spectral sequence, an isomorphism like
(10) {A + C)/{B + C) ~ A/B {AnCcB)
occurred, then the left-hand expression in (10) would give a larger expression for
Fp than (9) whenever
FpA + B + C % FP{A + C) + B + C.
(3.5) Filtered mixed Hodge complexes defined. Suppose that the cohomological
mixed Hodge complex K' has a third filtration (increasing), denoted Q, also
defined over R. For simplicity, assume that (W,F) imparts to each QiK' the
structure of a cohomological mixed Hodge complex, for this is the case in practice.
Clearly, the filtration induced by Q on H?{Z, K') is a filtration in the category
for mixed Hodge structures. One seeks reasonable conditions under which the
spectral sequence of Q:
(11) qE™ = Hp+q{Z,Gr®plC) =>Hp+q{Z,lC)
is one of mixed Hodge structures, such that the filtrat ions defined recursively by
W and F on qEqq coincide with the ones induced directly on (6), as in (3.3).
Certainly, in order to get started, one wants to assume that each Gr^ K'
becomes a cohomological mixed Hodge complex. One then examines separately
the filtrations W and F in the spectral sequence of Q, taking a cue from (3.4).
Since the spectral sequence for F in a mixed Hodge complex degenerates at E\
[4, (8.1.9, v)], we may apply the lemma on two filtrations directly. On the other
hand, that for W degenerates only at i£2- Here, one can make use of a trick used
by Deligne in [4, (1.3.3)], replacing W by the filtration Dec W, defined by
(12) (Dec WOj/P = {xG Wt-iK* : dx G W^i^1Ki+1}.
Its formation is left adjoint to convolution with the filtration by degree; it
induces, up to shifts of indices, the same filtration on cohomology, but its spectral
sequence degenerates one term earlier than that of W, so in the case at hand, at
Ei.
In order to apply the lemma on two filtrations to Dec W (on the level of
RT(K')) in (10), one must know that
(13) the operation of forming Dec W commutes with passage to Gr^,
so this is the condition that one imposes in a filtered mixed Hodge complex [7].
REMARK. The discussion is somewhat simpler if one knows that the spectral
sequence (11) of Q degenerates at F2, e.g., when Q is the limit of the weight
filtration in a degeneration of mixed Hodge structure (as in (3.8)).

DEGENERATION OF MIXED HODGE STRUCTURES
291
(3.6) A criterion. It does not take long to realize that one would prefer not
to compute explicitly with Dec W. Fortunately, there is a reasonable criterion
[17, (6.8)], which always seems to hold in practice, for when the condition (13)
is satisfied: if there is an isomorphism of complexes
(14) GtfQiK- c* 0Gr^ Gr9 K\
3<l
According to [21, (A.l)], (14) holds automatically whenever W is of the form
P * Q (recall §1(6)) for some filtration P of K\
(3.7) The following is the most basic illustration of a filtered mixed Hodge
complex. Take
/r = n^(iogz?),
as in (3.2), and suppose that we specify a decomposition of the divisor with
normal crossings D &s EUY. Then we have filtrations W{E) and W{Y) of K',
given by weights with respect to E and Y respectively. Clearly,
W = W{E)*W{Y),
hence K' becomes a filtered cohomological mixed Hodge complex if we put Q =
W{E) orQ = W{Y).
(3.8) We next sketch the proof of the results stated in (2.3) and (2.4) in the
case where the general fiber is nonsingular (see [17, §5] for details). In the
geometric set-up of §1(7), we take:
S = A*, S = A, X = Z-{DUY), X = Z-D,
where Z is smooth and proper over 5; D is a relative divisor with normal
crossings over S such that D U Y is also a divisor with normal crossings. For fGA*,
we write
(15) Xt = Zt-Dt = f-1(t).
The cohomological mixed Hodge complex, denoted A'(Z,D), that defines the
limit mixed Hodge structure for H'(Xt,C) is given by a double complex with
terms
(16) A"'"(Z, D) = fiP+*+1(log(D U Y))/W{Y)q (p, q > 0),
with (l,0)-differential given by exterior derivative, and (0, l)-differential given
by exterior multiplication by f*(dt/t). (When D = 0, (16) gives the complex
called A' in §l(c).) The Hodge filtration F is determined by truncation with
respect to p; for reasons not to be explained here, the weight filtration is given
by
(17) WkAp'"(Z,D) = image of Wk+2q+1npz+9+1 (log(D U Y)).
We presume as understood the case where D = 0, from [16].
One introduces a third nitration W(D) of A'(Z, D), defined in the most naive
manner:
(18) W{D)lAp'q{Z,D) = image of W(D)tnpz+q+1{\og(DU Y)).

292
STEVEN ZUCKER
It is not hard to see that W(D) is a convolutant of W on A'(Z,D), and hence
the mixed Hodge complex is filtered by W(D) (3.6). We note that
(19) Gr?iD)A'{Z,D)*A'{DW,0)[-t\{-l),
in the sense of complexes filtered by W and F, so is covered by results in [16].
The spectral sequence of W(D), being the limit of that of W on Q'Zt(\ogDt),
degenerates at E^. On
(20) w(D)E:Ui+l = IP{GxY{D)A-{Z,D)) ~ tf'"'^^^)),
F determines the limit Hodge filtration for D^l\ appropriately shifted. In
passing to the Ei term of the spectral sequence, one uses the fact that all exact
sequences of polarizable variations of Hodge structure split; hence the
recursively induced filtration F on w{D)E2 = w(D)Eoo is the limit Hodge filtration
foTGT?Hm{XuC). Since we are dealing with a filtered mixed Hodge complex,
the natural filtration F coincides with the former.
The discussion of W is a little more interesting. We first note that the most
natural formulation of the weight filtration in a mixed Hodge structure is in
terms of Dec W (on RT(K')), and not W itself, since it is really this filtration
that one uses (note the weight of the Hodge structure on (6)), and which is
strictly compatible with morphisms. With this said, (Dec W)k induces on (20)
(taken as the rightmost expression) Wk-i-i, which determines absolute weight
filtration level (center at zero now) k — (i + I) for the monodromy logarithm
for D^l\ In particular, the monodromy weight level determined by (DecW)k
on w(D)EÏ'q is independent of p. The differential di, being a morphism of
mixed Hodge structures and preserving g, is thus strictly compatible with the
monodromy weights. (One sees this also by using splittings again.) Passing to
i?2, one gets that on
w(D)Eïl'i+l * GvY(D)H\Z,A\Z,D)),
(DecW)k necessarily induces monodromy weight level k — (i + /); equivalently,
Wk induces weight level k — /, giving (2.4).
(3.9) We mention briefly some other places where filtered mixed Hodge
complexes appear:
(i) the mixed Hodge theory for H'(S,V), when V underlies a graded-polariz-
able variation of mixed Hodge structure on the curve 5, mentioned in (2.6);
(ii) mixed Hodge theory for the Malcev completion of the fundamental group
of a variety, limit theory, etc. (higher homotopy in the simply connected case);
see [12];
(iii) asymptotic behavior of Abel-Jacobi integrals (treated first, in a different
manner, by Clemens) [17, (5.28)];
(iv) the mixed Hodge theory for the cohomology of the Milnor fiber associated
to a singularity, in which the third filtration Q is determined by a polar curve
[25];
(v) the mixed Hodge structure of the cohomology of a reducible variety, or of
its complement in an ambient space.

DEGENERATION OF MIXED HODGE STRUCTURES
293
References
1. E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge structures, Ann. of Math.
(2) 123 (1986), 457-535.
2. , L2 and intersection cohomologies for a polarizable variation of Hodge structure,
Invent. Math. 87 (1987), 217-252.
3. C. H. Clemens, Degeneration of Kahler manifolds, Duke Math. J. 44 (1977), 215-290.
4. P. Deligne, Théorie de Hodge. II, III, Inst. Hautes Études Sci. Publ. Math. 40 (1971),
5-57; 44 (1974), 5-77.
5. , La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137-
252.
6. P. duBois, Structure de Hodge mixte sur la cohomologie évanescente, Ann. Inst. Fourier
35 (1985), 191-213.
7. F. El Zein, Complexe de Hodge mixte filtré, C. R. Acad. Sci. Paris Ser. I. Math. 295
(1982), 669-672.
8. , Dégénérescence diagonale. I, II, C. R. Acad. Sci. Paris Ser. I. Math. 296 (1983),
51-54, 199-202.
9. , Théorie de Hodge des cycles évanescents, Ann. Sci. École Norm. Sup. (4) 19
(1986), 107-184.
10. M. Goresky and R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983),
77-129.
11. F. Guillen, V. Navarro Aznar, and F. Puerta, Théorie de Hodge via schémas cubiques,
1982.
12. R. Hain and S. Zucker, Unipotent variations of mixed Hodge structure, Invent. Math, (to
appear).
13. M. Kashiwara, The asymptotic behavior of a variation of polarized Hodge structure, Publ.
Res. Inst. Math. Sci. 21 (1985), 853-875.
14. M. Kashiwara and T. Kawai, The Poincaré lemma for a variation of polarized Hodge
structure, Proc. Japan Acad. Ser. A. Math. Sci. 61 (1985), 164-167.
15. W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent.
Math. 22 (1973), 211-319.
16. J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1976), 229-257.
17. J. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math. 80
(1985), 489-542.
18. S. Usui, Variation of mixed Hodge structure arising from family of logarithmic
deformations. II: Classifying space, Duke Math. J. 51 (1984), 851-875; Supplement by Saito, Shimizu
and Usui, 52 (1985), 529-534.
19. S. Zucker, Hodge theory with degenerating coefficients: L<i cohomology in the Poincaré
metric, Ann. of Math. (2) 109 (1979), 415-476.
20. , Degeneration of Hodge bundles {after Steenbrink), Topics in Transcendental
Algebraic Geometry, Ann. of Math. Studies , vol. 106, Princeton Univ. Press, Princeton, N. J. and
University of Tokyo Press, Tokyo, 1984, pp. 121-141.
21. , Variation of mixed Hodge structure. II, Invent. Math. 80 (1985), 543-565.
22. M. Kashiwara, Variation of mixed Hodge structure, Preprint RIMS-539, Kyoto Univ.,
1986.
23. M. Kashiwara and T. Kawai, Hodge structure and holonomic systems, Proc. Japan Acad.
Ser. A. Math. Sci. 62 (1986), 1-4.
24. M. Saito, Modules de Hodge polarisables, Preprint RIMS-553, Kyoto Univ., 1986.
25. J. Steenbrink and S. Zucker, Polar curves, resolution of singularities and the filtered mixed
Hodge structure on the vanishing cohomology, Preprint no. 18, Univ. of Leiden, 1986.
Johns Hopkins University

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Enumerative Geometry

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Hilbert Scheme of Points:
Overview of Last Ten Years
A. IARROBINO
/ dedicate this to the memory of Gerry McCollum, 1944-1984, who was a
participant in the Bowdoin Algebraic Geometry Conference of 1967. His liveliness
and spirit through adversity is an inspiration.
Abstract. The article summarizes progress by many persons; it is
accessible to nonexperts. My goal is to point to what is available, and give
some feeling for the subject.
When X is a nonsingular surface, Hn = Hilbn X parametrizes the
length-ra zero-dimensional subschemes of X, and it is a natural desingu-
larization of the symmetric product Sn = Symn X. I will begin the
survey in an elementary way by introducing H2, H3, and the neighborhood
(H3, zq) parametrizing schemes near To = Spec (Op/raj*), where p is a point
of X = P2. I will describe how this neighborhood is a local flattener of
a map germ. Similarly, if C is a plane curve through p with order of
vanishing there at least two, To is a subscheme of C, and the
neighborhood (HilbnC, zq) is a local flattener; we specify the local equations of
(Hilbn C, z0) in (Hilbn P2, z0). This leads to the use of Hilbn C to com-
pactify the Jacobian of C.
I mention other progress here concerning the fibres of the map from
Hilbn X to the Albanese of X, the homology groups of Hilbn P2, and the
use of Hn in enumerative problems.
I will touch on the progress in understanding the local punctual Hilbert
scheme Hilbn 0P, 0P a local ring on Pr, that arose from a chaining method:
the irreducibility of key "t>th order" subschemes when r = 2, the study of
the Hilbert scheme near complete intersection subschemes of Pr. Likewise
I will touch on local methods pertaining to the geometry or geography
of Hilbn Pr when r > 2 (see summaries in print, [14, 16]). I also mention
progress in foundations, the equations defining the Hilbert scheme, the map
Hn to Sn when char k = p.
Artin algebras and modules—hitherto camouflaged—make their
appearance. I describe some recent work here, and a problem arising from vector
bundles on P3.
1980 Mathematics Subject Classification (1985 Revision). Primary 14C05; Secondary 14C25,
14B05, 14M10, 13C05, 13H10, 57D35.
Key words and phrases. Hilbert schemes, zero cycles, Artin ring, finite map germs,
deformations, Hilbert function, curves, surfaces, compactified Jacobian, irreducibility, generic
singularity.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
297

298
A. IARROBINO
0. Introduction.
0.0. Collision of points on a surface. When X is a nonsingular surface over an
algebraically closed field k, the symmetric product Sn = Symn X, the quotient
of Xn by the symmetric group, is singular at the large diagonal D parametrizing
sets of n points not all distinct. The punctual Hilbert scheme Hn = HilbnX
parametrizes length- n zero-dimensional subschemes of X, and it is a natural
desingularization of Symn X. The projection
7rn:HilbnX->SymnX
takes a point zt of Hn parametrizing the subscheme T of X, to the point tt(zt)
of SnX parametrizing the support of T. The map 7rn is an isomorphism on
the open set Un = Hn — D — 7r~1(Sn — D) parametrizing nonsingular length-n
subschemes of X—those whose support are sets of n distinct points of X.
How does Hn desingularize Sn? Let Op be the local ring of a point p in
the projective plane P2, and consider the fiber Hilb2 Op of 7r2 above the point
"2p" of S2. The local Hilbert scheme Hilb2 Op parametrizes the length-2 algebra
quotients of Op. If x,y are local parameters on X at p, the geometric points of
the fibre are
{Hilb2 Op} = {z{L) = zT{L) | T = Spec(Op//L), where IL = (L, m2),
and L = ax — (3y mod k* |a, (3 G A:},
RiltfOp-P1 =P(mp/m2p).
Below on S2 the tangent space r2p has length 5. When X is the affine plane
A2 C P2, the tangent space r2p arises from the following motions of two points
on A2 toward p = 0:
a. The centroid of two points moves toward 0: \imt-+o{Pi (/?*? oit), pi (2/3t, 2at)}.
These generate a two-dimensional subspace of r2p.
b. Spiral approach so the centroid is 0: spiral part of motion in
One-dimensional subspace of r2p.
lim{Pl(M2),p2H,-*2)}.
•—*-•—*~ \ y i *—*" ~*—*
(a) (b) y (c)
FIGURE 1. Independent motions of two points on A2 toward p.
c. Symmetric approach along the line L = 0, so the centroid is 0:
\imo{Pl{/3t,at),p2{-/3t,-at)}.
Two-dimensional subspace of r2p. (See Figure 1.)
Hilb2 P2 is just the blowup of Sym2 P2 along the diagonal (except in
characteristic 2; see [CH]); the tangent space to the point z(L) on the fibre Hilb2 Op

HILBERT SCHEME OF POINTS
299
of H2 above "2p" is t(z(L)) = Hom(/L,Op/iL), and it has dimension four. Two
points of A2 approaching 0 symmetrically along the line L = 0, so the centroid
is 0, determine a flat family of subschemes of A2 approaching T(L): the tangent
space r(z(L)) on H2 contains only a one-dimensional subspace arising from
motions c above. Thus, the added information in z(L) G H2 over that in "2p" in
S2 filters the ways two moving points of P2 can approach p.
The fibre Hilb3 Op of tt3 : Hilb3 P2 -> Sym3 P2 over "3p" contains two kinds
of points. There is a special point zq that parametrizes the subscheme T0 =
Spec(Op/ra2); and an open dense subset of the fibre consists of points z(a,b),
a, 6 G A:, parametrizing the subschemes
T{a, b) = Spec(Op/J(a, 6)), /(a, b) = {x + ay + 6*/2, m3).
If a is a fixed element of A:, then on Hilb3Op \imt)-+00z(a,b) = 2o- Three
distinct points in P2 moving generally enough toward p determine a family of
subschemes converging to the special subscheme T0. However, if the three points
move quite specially toward p, moving up to third order at p along the parabola
x + ay + by2 = 0, the family of subschemes they determine converges to T(a, 6),
the more generic kind of length-3 subscheme with support "3p". This
seeming paradox—more generally moving points converging to p determine a more
special kind of limit scheme—is quite central to the punctual Hilbert scheme.
Yet how a moving system of distinct points on Pr converging to p determines
the limit scheme at p is poorly understood: what invariants of the moving system
allow us to simply determine the limit scheme? One might expect that many
questions about Hilbn Pr would be answered through studying such convergence.
What was known about the punctual Hilbert scheme in 1975? Quite generally,
A. Grothendieck had defined the Hilbert scheme Hilb^ Y parametrizing sub-
schemes of the quasiprojective variety Y having Hilbert polynomial Q [GROl].
For X a nonsingular surface, Hilbn X was known to be nonsingular (Fogarty,
1968); and the Picard group
Pic(Hilbn X) = Z © Pic(Symn X)
was known for char A: ^ 2 (Fogarty [F2]). For p G X, the local punctual Hilbert
scheme 7T"1 ("np")= HilbnOp had been shown irreducible by J. Briançon in
a tour de force [BRI, BR2]. He and the author had separately studied the
stratification of HilbnOp by the Hilbert function h{T) = £>i(T)**, where
hi(T) = /(mj,0T/mp+10T), the length of the z'th graded piece of 0T = Op//(T),
and I(T) is the ideal in Op defining the subscheme T of X concentrated at p.
They showed that for each suitable numerical polynomial h in z, the strata
Zh = {zT G Hilbn Op \h{T) = h}
is irreducible, and locally an affine space of known dimension [BR2, 12]. Let
order / denote the integer v such that rri^D I but m^1 ^> I; a corollary is that
the t;th-order subscheme Z(v) C Hilbn Op, parametrizing subschemes of X at p
defined by ideals of order v, satisfies dimZ(v) = n — v.

300
A. IARROBINO
For higher dimensions, if F is a connected r-fold, Hilb^ Y was known to be
connected (R. Hartshorne [HI], 1966); a short proof that Hilbn Y is connected
is in [F2]. In general, for r > 3 and n large enough, Hilbn Y was known to have
several irreducible components [13].
We divide the progress over the last ten years into the following four main
areas.
0.1. Punctual Hilbert scheme of a surface X, and of a curve C on X.
J. Briançon, M. Granger, and J. P. Spéder studied the Hilbert scheme of a curve
on a surface [BGS]; a series of articles by D'Souza, C. Rego, A. Altman, and S.
Kleiman studied Hilbn C in Hilbn X, and applied that study to compactifying
the Picard scheme of X, as suggested by D. Mumford and A. Mayer. M. Huibreg-
ste studied the fibres of the natural map from Hilbn X to the Albanese variety
of X showing them irreducible for high n. Several authors studied the homology
of HilbnP2; recently, G. Ellingsrud and S. A. Str0mme used Bialynicki-Birula's
theory of C*-actions on projective varieties to determine the homology groups
of HilbnP2. P. Le Barz applied the punctual Hilbert scheme to study classical
enumeration formulas for n-secants to curves C in Pr; his article joint with G.
Elencwajg [E-L3] is an introduction. S. Kleiman, S. Colley, and Z. Ran have
used related approaches to study multiple point formulas for maps.
I will outline the work on curves in a surface, after introducing the concept
of Hilbert scheme as flattener (as in [BGS, Gl, G2]).
0.2. The local punctual Hilbert scheme Hilbn Qp and mapping germs. There
have been striking advances here, particularly in studying a neighborhood,
denoted (Hilbn Op, zt), of a point zt, where T is a complete intersection subscheme
of Pr concentrated at p. The proofs depend on a chaining method of T. Gaffney
and R. Lazarsfeld, used also by M. Granger; they also depend on interpreting
the neighborhood (HilbnOp,^T) as the flattener of a suitable mapping germ
(see [BGS] and §1.1 below). I will describe these advances briefly, having
summarized them in [14] with a view to suggesting directions for further study.
M. Granger's memoir [GR] is a definitive source for what is known now about
the local punctual Hilbert scheme; T. Gaffney's article [GA2] is the reference for
the chaining method, which he extends to singular Y. J. Damon and A. Galligo
have used Hilbert scheme methods to study mapping germs.
0.3. Geometry o/Hilbn Y, and foundations. The geographer of Hilbn Y studies
the irreducible components, their dimensions, and intersections—a first
approximation to their geometry. J. Briançon and the author bounded the
dimension of components, while J. Emsalem and the author introduced a notion of
compressed algebra to give examples of components (see [16] for a summary).
J. Emsalem, G. Mazzola, D. Happel, A. Dimca and C. Gibson, and C. T. C.
Wall have studied either the Hilbert scheme or, equivalently, finite map germs
in low colengths.
The foundations of the subject have been improved by G. Gotzmann's study
of global equations for Hilb^ Pr, making more explicit one of A. Grothendieck's

HILBERT SCHEME OF POINTS
301
constructions of this Hilbert scheme. A. Neeman has studied the normality of
the Chow scheme of n-points on Pr, and given a very explicit construction of
the Chow morphism.
0.4. Extensions to modules, applications, vector bundles. H. Kleppe and
A. Miri studied the scheme Quotn(Op) parametrizing length-n quotients of a
free Op-module. J. Emsalem viewed a scheme as a set of punctual distributions,
using this to write a singular curve as a generalized quotient of its normalization.
P. Kergin and A. Hirschowitz's studies of interpolation are related to the Hilbert
scheme; R. Buchweitz used components of Hilbn Pr to construct super-rigid
singularities in higher dimensions. D. Ferrand began a study of multiple structures
on curves related to vector bundles on Pr, developed by C. Banica, O. Forster,
and N. Manolache; this motivated A. Hirschowitz and J. Brun's stratification of
Hilbn P2 by postulation. Le Potier and A. P. Rao have noted the appearance of
certain components of the Hilbert scheme of points as moduli spaces for vector
bundles.
I should like to thank the authors of the surveyed work for teaching me about
it, and S. Kleiman, H. Hirschowitz, J. Briançon, and A. P. Rao for their
comments.
1. Hilbert scheme of a surface X, and of a curve C on X.
1.1. The neighborhood (HilbnX, zT) as local flattener of a map germ. We
assume X is a nonsingular surface; most of the flattener statements extend to
Hilbn(yr). We use as example the neighborhood (Hilb3 P2, z0) of Hilb3 P2 near
the point zo parametrizing the scheme T0 = Spec(Op/ra2). We will first identify
the neighborhood with the local flattener of a suitable map germ; the local
flattener had been introduced by H. Hironaka, M. Lejeune, and B. Teissier [HLT],
and the Nice group of J. Briançon, M. Granger, and H. P. Speder noted the
relationship to Hn. Following the Nice group, we use the flattener to study the
Hilbert scheme of plane curves [BGS].
We let k = C, and assume the subscheme T of X is concentrated at p G X,
hence is defined by an ideal / of the local ring Op whose maximal ideal mp =
(x, y). Order the monomials of Op in x, y so that (jl < v implies /i// < vja' for any
monomial //. The particular order is not important for the flattener, but may
be important when r > 2 for speed in calculations, or in regularity questions (see
[Gl, B-Sl, B-S2]). Here we order the monomials first by total degree, then by
x-degree, so • • • y2 < xy < x2 < y3 • • -, and we form a standard basis of generators
/o, • • • ? fv for the ideal /, where v = order /. The standard basis has the property
that the initial monomials (in /0,..., in fv) = (yv, yv~1xkv~1,..., xk°) generate
the initial ideal in J = {in/ | / G /}. Let A = Qp/I be an Artin algebra, and
let A = Ai,..., An be a monomial cobasis for in(7) in R = Op; the classes
Ai,..., An in A are a basis for A, and #{A n Ri} = hi(T), the length of the
zth graded piece A{ of A* = Grm(A). Consider the mapping
/:(C2,0)-(C"+1,0), f(x,y) = (f0,...,fv),

302
A. IARROBINO
and define a deformed mapping
F = (F0,...,FV): (C2 xCvn+n,0)^(Cv+1,0),
Fi = fi + ^2 Cis^si s = 1,..., n; Cis coordinates on Cvn+n.
Let (p be the projection <p: Z = F~x(0) —► Cvn+n = 5 of germs at the origin;
the local flattener <pf of (p is a flat morphism £>f- Z x$ P -^ P with P an
analytic germ of space over 5, satisfying a universal property for flat projections
ip'\ Y —► P1 from F C (C2,0) x P' whose special fibre is defined by /. Let
Sf C S be the image of P in 5, and Z = ^^(P): m effect nere> the local
flattener (Pf is the restriction of (p over the largest open set Sf C S over which
it is flat.
THEOREM [BGS]. The image Sf of the local flattener <pp is the germ of
HilbnX at zt; the map (Pf- Z —> Sf is the projection from the universal sub-
scheme Z C (HilbnX) x (C2,0) to (HilbnX,2T).
Briançon, Granger, and Speder give the following method for obtaining
explicit equations for (HilbnX, zt). Form a basis {Ej = (Ejq,. .. ,Ejv), j =
1,..., v} of relations among the /», so Yl Ejifi = 0- Now
(1) Form M = {Mj \ Mj = Er F = ££^FJ.
(2) Reduce Mj in the standard way mod the Ff.
M3=G30{x,y',c)F^Y,Gji{x',c)Fl+H3,
i>0
with Hj = YlHjsAs, and Hjs G k[c] = k[{cis}].
THEOREM: LOCAL EQUATIONS FOR THE HlLBERT SCHEME [BGS]. The
equations {Hjs = 0, s = l,...,n; j = l,...,v} define (HilbnX,zt). Note
that the standard way of reducing Mj uses k[x, y; c] first to eliminate any terms
involving multiples of inP0 = V° from Mj, then uses fc[x;c]Fi,..., k[x;c]Fv to
further reduce to H3 in fc[c](A). The fibres of Z C P_1(0) have constant length
n over points of Sf-
EXAMPLE. We let / = m2, where v — 2, the standard basis (/c/1,/2) =
(y2,xy,x2), the cobasis A = (l,y,x), and the relations are E\ = (0, — x, y),
E2 = (x,-*/,0). Then
Mi = -xFi + xF2
= -x(xy + en + C122/ + ci3x) + y{x2 + c2i + c22y + C23X).
One reduces Mi by adding —C22P0 + (<?i2 — £23)Pi + C13F2. The remainders H\,
H2 satisfy
Hx = Hn + H12y + i/i3x, tf2 = #21 + H22y + #232
where
#12 = -Cll + (C12 - C23)C13 + C13C23 - C22C03,
#13 = C21 + (C12 - ^23)^12 + C13C22 - C22C02,
#23 = Coi + (Ci3 ~ C02)C13 + Ci2C03 ~ ^03^23-

HILBERT SCHEME OF POINTS
303
The terms Hn and H21 are in the ideal of k[c] generated by the other terms,
and H22 = H13. Thus, the local equations of the germ (Hilb3 P2, zq) are i/12 =
His = #23 = 0. Note that the germ is nonsingular, of codimension 3, and hence
dimension 2n = 6, as expected.
For a different approach to local coordinates in the special case of Hilbn P2
only, any zt, see §4D of [12], which gives a synthetic construction from the
standard generators and relations of the ideal / defining T.
1.2. Hilbert scheme of a plane curve. Consider a curve C in P2 passing through
p with local equation /. Suppose / is a colength n ideal of Op containing /, and
consider the ideal / in Op/(/) defining the subscheme T = SpecOp/7 of C.
The local punctual Hilbert scheme (Hilbn C, zt) is the locus in (Hilbn P2, zt) of
points zt, lying on C—whose defining ideals I(V) contain /. Write / = Yl 9ifi-
Define a deformed relation Mf = Yl 9iFi, and reduce it as before in the standard
way by i*o,..., Fv to a remainder in k[c](A),
Hf = YlHf*A*> Hfsek[c}.
THEOREM: LOCAL EQUATIONS FOR Hilbn C ([BGS]; parts b, c also in
[AIK]). a. The local equations of the Hilbert scheme H(C) = (Hilbn C, zt) on
(Hilbn P2) are {Hfs = 0, s = 1,..., n}.
b. Each component of H(C) has dimension n, and Hilbn C is a local complete
intersection in Hn = Hilbn P2.
c. Each zero-dimensional subscheme T of C has a nonsingular deformation
on C—to a scheme parametrizing n distinct points of C.
d. H(C) is singular at zt iff the ideal I in Op/(/) defining T is not principal.
e. /// has order v' in Op, then the fibre Hilbn(Op/(/)) of H{C) above "np"
on Symn C satisfies dim(Hilbn Op,c) > inf {v' - 1, n - 1).
PROOF. Part a is a direct consequence of the method of §1.1. Since Hilbn P2
is smooth of dimension 2n, each component of H(C) has dimension at least n.
Since the fibre of HilbnP2 over "np" has dimension (n — 1), and the
singularities of C are isolated, it follows that no component of H(C) has dimension
more than n, and any component of dimension n must have an open dense
subset parametrizing smooth length-n subschemes of C. Part d has an
algebraic argument. Part e involves a detailed study of the strata of Hilbn Op,c =
Hilbn(Op/(/)) by the Hilbert function.
EXAMPLE. Consider the plane curve / = x2 — y3 on A2, and the point zo of
H(C) = Hilb3 C defining the subscheme T0 = Spec Op/ra2 at p = 0 on C. We
write f = f2 — yfo; in reducing Mf = F2 — yF0 one substracts C12F0 + C13F1
and obtains the local equations
Hf,i = -C21 - C12C01 - C13C11,
Hft2 = C11 - C22 ~ C12C02 ~ C13C12,
Hf,3 = c23 ~ C12C03 - c23.

304
A. IARROBINO
Note that the first equation is singular, as c2i is a quadratic expression in the
local coordinates C02, C03, C12, C13, c22, C23 of (HilbnP2,2o). The ideal / has
generators xy, ?/2mod(/), so is not principal.
In the same article [BGS], the Nice group describes the irreducible
components of Hilbn C when C has several components; they give an example
f(x,y) = y4 — x5, I = {x,y2), where the germ (H(C),zt) is not analytically
irreducible, although the germ (C, 0) is.
There is nothing in the proof of the above theorem special to the plane P2;
the natural context is curves on a nonsingular surface X.
1.3. Compactified Jacobian of a curve C; Albanese of a surface X. D. Mum-
ford and A. Mayer had proposed constructing a compactified Jacobian P of an
integral curve as the moduli space of torsion free sheaves of rank one and Euler
characteristic (1 — pa), where pa is the arithmetic genus. C. D'Souza, A. Altman
and S. Kleiman, C. Rego, then H. Kleppe and S. Kleiman developed this idea,
using the Hilbert scheme.
D'Souza constructed the scheme Pm, the moduli space of torsion free rank one
sheaves of Euler characteristic m, and he showed that the natural morphism, the
"Abel map"
an: m\bnC - Pn, n' = (1 -pa - n)
dnizr) = ideal sheaf 3(T) of the scheme T on C
is smooth when n » 0 and C is a Gorenstein curve—for example, when C is
planar.
Altman and Kleiman suggested developing the compactified Jacobian of a
family C/S, S a base scheme, in the spirit of Grothendieck's development of the
Picard scheme. They used the dualizing sheaf u and took Quot(w/C/5) as source
of the Abel map (this is different from Hilbn C when C is not Gorenstein). In
[AIK] the authors show
THEOREM: COMPACTIFIED JACOBIAN. Let C/'S be a flat projective
family of geometrically integral curves of arithmetic genus pa lying on a smooth
quasiprojective family X/S of surfaces parametrized by S. Let P denote the
compactified Jacobian of X/S. Then P/S is flat; its geometric fibres are
integral, Cohen-Macaulay, local complete intersections with dimension pa; and P
contains the Jacobian Pic£>/5 as an open Zariski-dense subscheme.
The proof uses the smoothness of the Abel map, and the properties of the
Hilbert scheme mentioned above in §0. The authors show a globalized, abstract
version of the Local Equations Theorem of §1.2 and [BGS]. Suppose C is the
zero locus of the section s of an invertible sheaf C on X, let Z denote the universal
subscheme in (Hilbn X) x X, and let <p denote the projection <p: Z —> Hilbn X,
and q the projection q: Z —► X.
THEOREM: HILBERT SCHEME OF A CURVE [AIK]. Hilbn C is the locus of
zeros of a section a of the locally free rank-n sheaf ip+(q*£). Furthermore, the
section a is transversally regular over S.

HILBERT SCHEME OF POINTS
305
They also produce an example of a curve C on P3 for which Hilbn C is not
irreducible. In fact all the "bad" zero-dimensional schemes T in Pr, r > 3, can
be embedded in suitable curves on Pr : such schemes T include those determining
high-dimensional components of HilbnPr (see §3.1).
C. J. Rego [RE] in 1977 showed that for C a reduced, irreducible curve over
k, the compactified Jacobian P defined by D'Souza satisfies
THEOREM A. If the singularities of C have embedding dimension two, then
P is irreducible. If C has a singularity of embedding dimension at least 3, then
P is reducible.
THEOREM B. When C has planar singularities, the boundary (P — Pic°C)
of P is the union of m = ^€C (multiplicity Oc,P — 1) irreducible divisors.
To show A he first gives a new self-contained proof that when p has embedding
dimension two, the dimHilbnOp = n — 1; and he uses extensively the duality
properties of Gorenstein rings, as well as a sort of linking. The proof of B depends
on Briançon's theorem that Hilbn Op is irreducible when p is planar [BR2].
Hans Kleppe and S. Kleiman also showed that if X does not lie on a smooth
surface, then the compactified Jacobian is reducible; their work uses the Quot
scheme as the source of the Abel map (see [K-KL]; for the Abel map see A.
Altman and S. Kleiman [AK]).
Suppose X is an irreducible nonsingular surface; let AlbX denote the Al-
banese variety of X, and let fn be the sum map fn: SymnX —► AlbX.
M. Huibregste has studied the composite mapping $n = fn • 7rn : Hilbn X —►
AlbX in [HU1, HU2]. He first shows that for n large, the fibers of $n are
irreducible of dimension 2n — g, where q = dim AlbX. His method is to show
that for n large the fibers of $n are birationally equivalent to those of fn, known
to be irreducible for large n. When X = AlbX is an Abelian surface, he shows
that the fibration $n is a twisted product of base and fiber: the twist is by
an isogeny of X. In the sequel he shows that $n is smooth for n » 0 only if
/ : X —► Alb X is smooth. The converse, / smooth implies $n smooth, is true
over the complexes, or for Albelian surfaces X = AlbX when chark\n\ but he
shows $n is singular for any irregular Abelian surface X when char k = p, p \ m,
and n = m2. His work is based on the remark that $n is singular at zt iff some
nonzero holomorphic 1-form wa on Alb X determines a symmetrized differential
W[n] = $* (wa) zero at the subscheme T of X. He also studies Weierstrass points
of the cotangent bundle ^X, points p G X where the projection jp: r(Q^) to
the ra-jets at p is not of maximal rank. If /i = dimc({w G r(fi^) | W[n] is zero
everywhere on ^>~1(np)}), then
THEOREM [HU2]. The point pe X is a Weierstrass point of order n - 1 iff
/i > max(<7 — n(n + 1), 0).
This generalizes to surfaces a result for curves C, that the zeros of symmetrized
differentials on Symn C represent the special divisors of degree n on C (see [MT-
MY]).

306
A. IARROBINO
1.4. Homology of the Hilbert scheme of a surface X. When X is a nonsingular
surface, from the fact dim(Hilbn Op) = n — 1, one can show that the Picard group
of Hn = Hilbn X satisfies Pic(Hilbn X) <g> Q = Q 0 Pic(Symn X) <g> Q: the first
factor arises from the branch locus B = 7r~1(D) of HilbnX over SymnX (see
[II]). J. Fogarty calculated the Picard group of Hn, finding
Pic(Hilbn X) = Pic(X/Jfc) 0 Ca(X) ®Z (n > 1),
where CS(X) are the symmetric divisorial correspondence classes on X (see [F3];
the characteristic 2 case is corrected in [F4]).
A. Hirschowitz showed that the homology group (equivalently, Chow group)
A*(Hilb3 P2) is free of rank 22, with homogeneous components of ranks (1,2,5,6,
5,2,1) in dimensions zero through six; his method is a theorem on equivariant
Chow groups [HR1]. G. Elencwajg and P. Le Barz gave geometric generators
and intersection relations among them for the Chow ring A*(Hilb3 P2), the first
nontrivial case (see [E-Ll], and [E-L3]).
Recently, G. Ellingsrud and S. A. Str0mme, in a remarkable and elegant
example of finding the right tool for the problem, determined the homology
groups (or Chow groups) of Hilbn P2 when k — C. Their tool was the Bialynicki-
Birula theory of C* actions on nonsingular projective varieties; in particular
THEOREM [BI]. Let X be a smooth projective variety with an action of Gm.
Suppose that the fixpoint set (xi,..., xn) is finite, and let
X{ = {x G XI lim tx = x^}.
Then
(i) X has a cellular decomposition with cells X{.
(ii) The tangent space TxiiXi = (jx^)* > the positive part of the tangent space
to Xi in X.
Let S denote the graded Z-algebra freely generated by ci,..., cn, c^,..., c^,
and c^,...,^.!, where the degree of c^, c[ and c" is i. Let Sk denote the
graded part of S in degree k. Let b^k be the rank of the dimension 2k homology.
Ellingsrud and Str0mme showed
THEOREM [E-S]. The homology groups of Hilbn P2 over k = C are free
Abelian groups. If 2k < n, then 62fc(Hilbn P2) = rk^ Sk, and b^n-^k — &2fc-
They also determined the homology of Hilbn A2 and Hilbn Op, for p e P2.
In their proof, the fixed points of the natural C* x C* x C* action on Hilbn P2
are 2t(j) f°r certain monomial ideals / in the homogeneous ring k[w,x,y] of
P2. They used equivariant resolutions of the ideals to obtain the dimension over
C of Txx., and hence the homology degree in which to count the cell X{. For
example, when the cell corresponds to a fixed point x = zt(i), where the scheme
T(I) is supported at a point p of P2, and is defined by the ideal / C Op, they
showed that the dimension of r^ x is n — e — 1, where e is the socle degree of the
Artin algebra Op/I (satisfying rrip<£I but m^1 C /).

HILBERT SCHEME OF POINTS
307
They conjectured that the Chow ring of Hilbn P2 is generated by the Chern
classes of Eq, Ei, E<i, the bundles E{ = <p*q*{Of>2(i)), where tp is the
projection from the universal subscheme Z to Hn, and q the projection Z to P2. In
support, they remark that since ci(^) = 2c\(Ei) — ci(Eb), the algebra S maps
surjectively to the subalgebra of A*(Hilbn P2) generated by the Chern classes of
the Ei's. See also the article [E-L3] for a discussion.
1.5. Hilbert scheme in enumerative problems. P. Le Barz initiated the use of
the curvilinear part Hilb" Pr of the Hilbert scheme—parametrizing subschemes
T of Pr lying locally on smooth curves—to prove or generalize classical
enumeration formulas for n-secants to curves or surfaces in Pr. A curvilinear subscheme
T of Pr is one such that at each point p of the support of T, there are local
parameters xp = x(p) and yp = y{p) such that the local algebra QP/I(T)QP of
T at p is isomorpic to an Artin algebra k[xp, yP]/{yP,xp p)) ~ k[xp]/(xp') of
length n(p), and YlPn(p) ~ n-> tne length of T. This is a local criterion. The
variety Aln Pr parametrizes aligned n-tuples T of points in Pr : subschemes T
lying on some line L of Pr. Thus, Aln Pr is fibred over the variety Grass(l, r)
parametrizing lines L in Pr by Hilbn L = Symn L. If x, y are coordinates of
A2 C P2, the scheme T(a, b) = fc[x, y]/(x + a?/ + by2,y3) is curvilinear but not
aligned, when 6 G C is nonzero.
Le Barz defines the cycle of n-secant lines to a curve C in Pr as z* (Hilb™ C) in
the Chow ring A*(A\n Pr), where i defines the inclusion of Aln in Hilb" Pr. In
a foundational article [LB2] he shows that if C deforms in certain flat families
to C;, then the corresponding family deforming Hilb^C to Hilb^C' is flat, a
statement no longer true for Hilbn C. (The counterexample is n = 3, and the
family C C C^ x^ x Cx with ideal J = {xy,x{w-\),y{w + \),z2-\2): Hilb3 C0
has the "parasite" component Hilb3 Q ~ P2, where Q = Spec(00,c3/^o) C ^o)
The flatness property of Hilb^ C allows calculating intersections with i* (Hilb^ C)
through deformation or degeneration. A typical classical formula proven in this
way is
THEOREM [LB1]. If C is a smooth curve in P3 with h "apparent" double
points, then the number of trisecants meeting a fixed line is t(C) = h(n — 2) — (3).
G. Elencwajg and P. Le Barz give enumerative applications of their calculation
of the Chow ring A*(Hilb3 P2) in [E-L2] and [E-L3].
Related approaches have been used by S. Kleiman, S. Colley, and Z. Ran in
studying multiple point formulas for maps. Let Un C Hilbn Pr be the irreducible
component parametrizing smoothable length-n subschemes of Pr. The iterative
approach introduced by S. Kleiman uses a cover Un of Un by a variety that
distinguishes the order of distinct points in the support of a scheme T (see
[Kl]). The resulting formulas sometimes require genericity assumptions needed
for the iteration, and involve an extra (n — 1)! factor coming from the cover. He
proposed a second method using the Hilbert scheme to give refined formulas.
Z. Ran has continued this approach, introducing a scheme Wn C Un containing
the restriction Un \ Hilb^ Pr as open subscheme; he obtains general formulas for

308
A. IARROBINO
the analogous cycle of n-secant lines [RAI]. Le Barz studied the normal bundle
of the Hilbert scheme Hilbn C C Hilbn X of a curve C inside that of a surface X
in order to count the multiplicity of lines on the surface as multisecants [LB3].
Z. Ran extended this work to more general settings in [RA2]. Both S. Colley and
Z. Ran have studied multiple point formulas with the refinement of specifying
multiplicities in the fibres of a map (see [C]). W. Fulton surveys the special case
of double point formulas in [FU].
In general, for r > 3, the Hilbert scheme Hilbn Yr contains points zt
parametrizing subschemes T of Y that cannot be the limit of a family of smooth length-
n subschemes of Y. This occurs for Y = Pr, and {r,n) = (3,388) or (4,8).
Even the closure Un parametrizing smoothable subschemes can be complicated:
for example its intersection Un H Hilbn Op with the local Hilbert scheme may
have several components [16]. Thus, a key problem in generalizing this work is
the choice of convenient subschemes of Un—or a related scheme—with which to
work.
2. The local punctual Hilbert scheme Hilbn Op and mapping germs.
The tie between the local punctual Hilbert scheme Hilbn Op, p a point of Yr, and
germs of differentiate maps has developed in the last seven years (see [GA2,
GR, D-G2, 14]). The link is the concept of Hilbert scheme as local flattener,
developed by the Nice school and discussed above in §1.1. There has been a
rich interaction. For example, a chaining method developed by T. Gaffney
and R. Lazarsfeld [GA-L] was combined with a connectedness theorem of A.
Grothendieck to study the local Hilbert scheme around a complete intersection
(T. Gaffney [GA] and M. Granger [GR]). M. Granger combined this with parts
of the proof of Briançon's local irreducibility theorem (Hilbn Op is irreducible if
p is planar) to greatly generalize it. Recall that / has order v in Op if ra£ D /
but rap+1 ^ /; fixing n, we let Z(v) parametrize the length-n quotient algebras
of Op defined by ideals of order v.
THEOREM [GR]. // the point p is planar (p lies on a smooth surface X),
and v>l, the Zariski closure Z(v) contains Z(v + 1).
The irreducibility of Hilbn Op is equivalent to Z(l) D Z(v + 1). The articles
[G2], [GA1], and [14] in the AMS Areata Singularities volume survey this work;
the last article proposes still open problems. One problem concerns peelable
subschemes T of X at p, those subschemes having a deformation in steps to n
distinct points: in the first step one peels T to T\ of support (n — l)pUp', where
p' ^ p; in each succeeding step one lowers the multiplicity of p by one, and
adds another distinct point to the support of T{. Recall that Un parametrizes
smoothable schemes. T. Gaffney shows
THEOREM [GA2]. Any nonembedded component o/C/nnHilbn Op in Hilbn Pr
has dimension at least (n— l)(r — 1). Any irreducible component having the
minimal dimension consists of points zt parametrizing peelable schemes.

HILBERT SCHEME OF POINTS
309
The peelable conjecture asks whether there is a unique such component,
namely, the Zariski closure Hilbc Op of the family of curvilinear schemes
concentrated at p^1)
The Memoir of M. Granger [GR] is a good source for what is now known
about the local punctual Hilbert scheme, and includes some discussion of singular
Yr. T. Gaffney's article [GA2] is the most complete discussion of the chaining
method.
Let R denote the reals and suppose / = (/i,..., ft) : (Rr, 0) —► (R*, 0) is the
germ of a differentiate map; the mapping algebra of / is
Q(/) = R{xi,...,xr}/(/i,...,/t),
quotient of the local ring at 0 of Rr. Is Q(f) a topological invariant of /? J.
Mather showed that right-left (domain-source) equivalency of stable mapping
germs / corresponds to isomorphism of the mapping algebras [M]. In 1975, J.
Damon and A. Galligo showed that if Q{f) is planar—isomorphic to fc[x, y]/I—
and of discrete algebra type (only finitely many isomorphism classes of algebras
nearby) then the Hilbert function h(Q(F)) is a C° invariant for C°°-stable germs
/, with r < t. Furthermore, if (r, i) are also in the nice dimensions, then Q(f)
is a topological invariant [D2]. Finally, when Q(f) is planar, J. Damon and A.
Galligo show that the Hilbert function of the mapping algebra Qp{f), p G Rn,
/ C°°-stable, partitions Rn into smooth submanifolds. If / is planar, if Q(f) ^
k[x,y]/I with v(I) = v, and p — n > v — 1, then the Hilbert function of Q(f)
is a topological invariant for C°-stable germs / (see [D-G2]). Their proof uses
the known smoothness of the strata in the partition of Hilbn Op, p G A|, k = R
or C, by Hilbert function [BR2, 12]: they use the Hilbert scheme to construct
canonical blow-ups of singular submanifolds of the jet space.
There has been some work in classifying mapping germs in special cases, such
as when the length of Q{f) is small, that carries over to the local punctual
Hilbert scheme, and vice versa (see §3.1). The key difference here is that for
the algebraic geometer a nearby algebra Qf has the same length as Q{f)—the
deformations are flat; for the differential geometer nearby germs /' may have
algebras Q{f) of smaller length. See [I-E], [14], or [GA2] for discussion and
H. Kurke, B. Martin [KU-M] for k arbitrary.
3. Geometry of Hilbn F, and foundations.
3.3. Geography, geometry o/HilbnPr. An elementary component of HilbnPr
parametrizes subschemes T of Pr each with a single point p(T) of support. An
arbitrary component arises from a partition n = ^ n;, and elementary components
of each HilbniPr. Near the beginning of this period J. Emsalem and the author
used a "small tangent space" method in [I-E] to determine elementary
components of Hilb8 P4 and Hilb14 P6 (the latter required computer verification—
Example 6 of [16]). Thus Hilb8P4 has two irreducible components, the closure
(*)In some earlier articles, like [14], what we here call curvilinear was termed "aligned." In
mapping germ articles, the curvilinear germs are "Morin singularities."

310
A. IARROBINO
U$(P4) parametrizing smoothable schemes T and the family Z described
below. The authors also showed the existence of other components determined
by algebras in the deformation space of certain compressed Gorenstein Artin
algebras—having maximal length given the embedding dimension r and socle
degree j (so A = Op/I with I ^> mp but / D m£+1). The compressed
Gorenstein Artin algebras have a symmetric Hilbert function hi = min(/(i^), l(Rj-i)),
with R = Op (shown also by E. L. Greene in [GRE]), and lend themselves to
parametrization by a variety
C(r,y)cHilbnOp, n = Y^hi-
The dimension of C(r,j) is greater than rn for r > 4 and j large enough: then
C(r,j) is not in the closure of C/n(Pr)—the general compressed (Gorenstein)
algebra is not smoothable. When j is even, the order v of I satisfies j = 2v — 2;
such Gorenstein algebras are called extremal, and had been previously studied
by P. Schenzel (see [SC]). Later, the author, R. Frôberg and D. Laksov, then
the author and J. Elias extended the compressed concept to Artin algebras, to
graded Cohen-Macaulay (CM) algebras, then arbitrary CM algebras [15, F-L,
EL-1]. M. Miller and B. Ulrich have studied the extremality of graded compressed
algebras in their linkage class [M-U]; J. Emsalem and the author parametrize
relatively compressed Gorenstein Artin algebras (Theorem 1.4 of [1-7]). All
known nonsmoothable components of the punctual Hilbert scheme arise from
compressed or relatively compressed Artin algebras (see [1-6]).
The lowest length example of a component of a Hilbn Pr not in the closure of
Un(Pr) is the family Z C Hilb8 P4 parametrizing schemes TP(F) concentrated
at a point p of P4, defined by algebras
Qp(F) = R/(F) = R/(fu...,f7),
where F = (/i,...,/V) is the vector space spanned by seven general enough
quadratic forms in R2. (Here R = Op, and QP{F) is the mapping algebra
<3(/), where / = /i,..., f7 is a choice of basis of F.) The Hilbert function
hiQpiF)) = (1,4,3) is of length 8. The fibre Zp of Z above p G P4 is an open
dense subset of the Grassmannian Grass(7, #2) of dimension 21 = (n — l)(r — 1).
T. Gaffney's theorem quoted above shows 21 would be the minimum possible
dimension of a nonembedded component of Us H Hilb8 Op; however, a tangent
space argument here shows that QP{F) in general has no deformation outside
of the family Z, so Zp <jL Us (see [I-E]; it is open whether the lower bound
(n — l)(r — 1) on dimension extends to all nonembedded components of Hilbn Op).
The algebra Qp(F) is a compressed algebra of socle type 3z2: it is a maximal
length quotient of R having three-dimensional socle (0 : m), all in degree 2. Do
the compressed algebras—as here—have a special role in directly determining
elementary components of HilbnPr? Although they usually—unlike QP(F)—
are parametrized by large-dimensional families so are in general—like QP(F)—
nonsmoothable, most algebras are nonsmoothable: we just don't know much

HILBERT SCHEME OF POINTS
311
about most Artin algebras! To be locally compressed—at each p—is neither an
open nor a closed condition on Artin algebras parametrized by Hilbn Pr.
J. Briancon and the author studied the dimension of HilbnPr, and showed
that the highest-dimensional component satisfies
an2~2/r < dim Hilbn Pr < bn2~2/r a, b constants.
The proof uses a Borel fixed point theorem concerning the action of the upper
triangular group in Sl(r), then a combinatorial argument to bound the dimension
of the tangent space t(zt) — Hom(/,Op/7) to zt at the fixed points, which
are determined by certain monomial ideals J in Op. An open problem when
r > 3 is to prove a conjectured sharp bound on the dimension 1(t(zt)): when
n = l{Op/rrip) the conjectured bound is (let T(n) = Spec(Op/rap))
l(T(zT)) < l(r(zT(n))) = l(Rs-i) • l(Rs)
(see [B-I, BE]).
There have been a number of studies of Hilbn 0P for small lengths n < 8. A
history of early work is given by D. Happel [HA]. Tables of isomorphism class and
specialization for length n < 5 appear in Briançon's article [BRI]. G. Mazzola
uses a medley of approaches in listing the finite number of isomorphism classes
and their deformations for n < 6 [MZ]. He also shows that for n < 7, the family
Hilb™ 0P of curvilinear algebra quotients of 0P is dense in Hilbn 0P, and that this
is not so for n > 8 (see the explicit example QP{F) above). The first length where
there are moduli for isomorphism classes of length-n Artin algebras is n = 7,
Hilbert function h = (1,3,3,0): the algebras are quotients of R = k[xi,X2,xs]
defined by ideals generated by three quadratic forms. The geometry of this case
is related to elliptic curves, and was studied by J. Emsalem and the author in
[E-I]. C. T. C. Wall classified these succinctly for both k = R, k = C in [WAl],
and went on to study nets of conies in P4 (algebras with h = (1,4,3,0), see
[WA2]). There is a virtually complete picture of isomorphism classes of Artin
algebras and their deformations for n < 8, but that picture has never been
written down in one place. Potentially useful is the computer calculation of
deformation spaces, using the method of §1.1 above, and a modification of, say,
the Bayer-Stillman "Macaulay" program for finding algebra resolutions [B-S3].
From the mapping germ approach, A. Dimca and C. Gibson have classified
up to contact equivalence all planar unimodular mapping germs; this means
they have classified up to algebra isomorphism those algebras R[x,y]/I such
that nearby algebras have only one parameter moduli (of isomorphism classes).
C. T. C. Wall surveys part of the classification problem of finite map germs
(Q(f) Artin) in [WA3].
Explicit deformations of large classes of algebras C[x,y]/I to simpler ones
occur in the articles of J. Briançon and M. Granger. In the real case, J. Briançon
and J. Damon remarked that D. Eisenbud and H. Levine's work [EI-L] on the
degree of map germs shows Hilbn R[x, y] is not irreducible (see §5B of [12]).

312
A. IARROBINO
A. Hénaut studies the exceptional locus to the blow-up of Cr,0 relative
to an ideal / = (/i,..., /r+5); he defines its cycle Bj in pr+s~1 as Bj =
Proj(r(/*(Cr),0)). Let e(I) denote the multiplicity of
/ = (r!) • lim (l(00/n/vr);
v—>oo
he shows that / —► Bj(I) defines a germ of analytic map /3j: (Hilb£Oo,2r) —►
Chow^_1(Pr+s~1, Bj) from the equimultiple strata of Hilbn to the cycles of
degree e and dimension r — 1 (see [He]).
3.2. Foundations. A. Grothendieck constructed the Hilbert scheme Hilb^ Yr
for a polynomial Q, and projective F as a projective limit of certain Quot
functors—parametrizing coherent quotients of free Oy sheaves. Hilb^ Pr
parametrizes graded ideals I in R = k[xo,. •., xr] such that the algebra A = R/I has
Hilbert polynomial Q (meaning, h{(A) = q(i) for i large enough), mod the
equivalence J ~ /' iff I{ = l[ for i sufficiently large. In fact, given Q, there is an integer
j = jQ such that Ij = Vj implies I ~V (see [GROl]). If jQ be known, Hilbg Pr
can be given quite natural global equations, inside a Grassmannian (see below).
D. Mumford studied m-regularity, giving an explicit bound for the degree j
needed. More recently, G. Gotzmann gave the minimum j'q needed; for constant
Q = n, that degree jn is n. To describe the equations, we need to recall a result
of F. H. S. Macaulay (see [MC] and later accounts by F. Whipple [WH] and
M. DeMazure [DE]). Let M(d,j,r + 1) denote the vector space spanned by the
first d monomials x£, xJ0~1Xi,..., of degree j in xo,..., xr in lexicographic order.
Macaulay showed that if V is a d-dimensional subspace of Rj, then the vector
space RiV = ({xv \ x G R\,v G V}) satisfies
l(R\V) > l(RiM(d,j,r + l))=d' (depending on d,j,r).
def
G. Gotzmann showed, in 1978,
THEOREM [GO]. For vector subspaces V of Rj, the equality 1{R\V) = d'
(the minimum possible value) implies equality at the next step: l(R2V) = (d')'.
For such a vector space V, the size of the ideal / = (V) in each degree
matches that predicted by the suitable Hilbert polynomial Qd,j,r> Consequently,
the Hilbert scheme Hilb^ Pr parametrizes such vector subspaces of Rj, having
minimal growth l(R\V) = d!. Recall that the Grassmann variety Grass(d, Rj)
parametrizes d-dimensional vector subspaces of Rj = (xq, xj0~ x\,..., x3r).
THEOREM (A. Grothendieck, G. Gotzmann [GO]). Let dn = l(Rn) - n.
Then d!n = l{Rn+i) ~ n. The Hilbert scheme Hilbn Pr = Hn is the sub-
scheme Hn C Grass(dn,Rn) x Grass(d^,it!n+i) parametrizing pairs of spaces
{(V,W) \V cRn, W cRn+u l{V)=dn, l(W) = d'n, and W = RiV}. Thus,
Hn c Grass(dn, Rn) is defined by the vanishing of the (d'n + \)-minors of R\V.
Further work was done by D. Bayer [BA]; a discussion by M. DeMazure
generalizes the work to sheaves [DE].

HILBERT SCHEME OF POINTS
313
A. Grothendieck noted the existence of a natural morphism 7rn : Hilbn Yr —►
Symn Fr, taking the point zt parametrizing the subscheme T of Yr to its support
[GROl]. J. Fogarty constructed a map -k'u : Hilbn Pr —► Chow(n, r) to the Chow
scheme in [Fl]. M. Nagata had shown that when char A: = 0 the Chow scheme
is normal; then the morphism 7r^ lifts to define 7rn. Otherwise, B. Iversen's
theory of linear determinants [IV] can be applied to the symmetric product
Sym^Z over H = HilbnPr of the universal subscheme Z C HilbnPr x Pr.
The section sn : Hilbn Pr —► Sym# Z that Iversen constructs composes with the
projection Sym# Z —► Symn Pr to give the morphism 7rn, carrying out the
construction Grothendieck noted. (I thank S. Kleiman for pointing this out to me.)
A. Neemann studied the normality of Chow(n, r) in characteristic p, showing that
if r, n > p +1, Chow(n, r) is not normal; he also gives a very explicit construction
of the morphism 7rn in [N]. See also [HU1].
There has been progress in deformation theory of local algebras. After work
by M. Andre, D. K. Harrison, S. Lichtenbaum and M. Schlessinger, and
others in the 60's and 70's, M. Schlessinger and J. Stasheff, S. Halperin and L.
Avramov, A. Laudal, and others have developed connections between
deformation theory of local algebras and rational homotopy theory [AV, A-H]. M.
Schlessinger and J. Stasheff describe a differential graded algebra T(A) they call
the tangent Lie algebra T(A) to the local algebra A; higher-order obstructions
to deforming A appear as Massey-Lie brackets on T(A). A. Laudal reconstructs
A up to isomorphism from some extra structure involving the Massey products
on Andre cohomology, when A is complete [LA]. Any Artin algebra of length
n + 1 is a deformation of the trivial Artin algebra A = fc[xi,..., xn]/m2, where
m = (xi,..., xn). Its tangent algebra T(A) is DerL/adL, where L is the free
Lie algebra on n variables (see [S-S]). The base ring B of the miniversal
deformation space Spec B of A is B = /;({# G Der1 L/ad L})/([d, #] = 0), a quotient
of the polynomial ring on (Der1 L/ad L) variables divided by the indicated
homogeneous quadratic equations [S-S]. See also the survey [PAL]. The Hilbert
scheme is closely related to Spec B; one hopes for development. An unsolved
puzzle is whether there are any rigid Artin algebras over A:, besides k © • • • © fc.
4. Extensions to modules, applications, vector bundles.
4.1. Quotients of free modules. C. Rego suggested the study of the family of
length-n quotients of 0p2 as an aid to that of the compactified Jacobian [RE].
H. Kleppe studied both local and global versions Quotn Op and Quotn Of>r for r =
2 in [KL1]. Recently, A. Miri studied the subfamily of Quotn Op parametrizing
compressed length-n quotient Op-modules—those having maximal length, given
n, r and degrees of socle elements. He extended results of [F-L, 15], and for
r = 3 found a length-526 quotient module of Op with one-dimensional socle
(0: m), having no deformation to any cyclic module Op/If. To show this he
uses H. Kleppe's result [KL2] that when r = 3 every Gorenstein quotient of
Op is smoothable; the family Miri finds has dimension too large to consist of
smoothable modules, but any cyclic deformation Mf (more general) of M must

314
A. IARROBINO
also satisfy 1(0: m) — 1, so would be Gorenstein (so smoothable). His study
of the case p planar gives results consistent with a conjecture that Quotn Op be
irreducible when r = 2; Briançon's irreducibility theorem is the case 5 = 1.
4.2. Applications of "thick points." Thick points or furry points (points épais,
points fourrés—schemes concentrated at a point) have appeared in enumerative
problems (see §1.5), in desingularization of curves, in interpolation, and in the
study of multiple point structures on lines that are jump lines of vector bundles
(see §4.3).
J. Emsalem defines a vector space X of finitely supported punctual
distributions D for any scheme X of finite type over k. X has a basis consisting of pairs
(p, A), where p G X and A is in the topological dual to 0P in the rap-adic
topology (so A is a linear operator, 0P —► A:, continuous in the rap-adic topology); X
is in fact the topological dual to the product Yl Op | P G X of completed local
rings. A morphism /: X —► Y of schemes entails a homomorphism f:X—>Y.
The morphism / is strongly surjective if / is surjective, equivalently, if for every
closed point j/GF, the canonical homomorphism tjjy : 0^ —► ([[ 0X \ x G f~1{y))
is an injection. The morphism / is a generalized quotient if / arises from an
equivalence relation on X.
THEOREM [E2]. If /: X —► Y is a finite k-morphism that is dominating
as a map of schemes, then f is a strongly surjective passage to a quotient (a
generalized quotient).
In particular, a singular curve F is a generalized quotient / : X —► Y of its
normalization X. Here, Emsalem finds a finite set of linear algebraic identities
that a morphism g : X —► Z must satisfy in order to factor through /. Note that
each pair (p, A) corresponds to a Gorenstein thick point concentrated at p (see
[El, E2]).
P. Kergin studied interpolation of Ck functions on Rr in a way suggesting a
connection with the Hilbert scheme [KE]. A. Hirschowitz studied interpolation
in two variables: at how many points can one specify the values of a given
degree polynomial, together with derivatives to order one or two? He relates the
problem to a maximal rank problem for punctual schemes of P2, studied using
the semicontinuity of l(H1(Oz,C)) for a suitable sheaf on the Hilbert scheme.
Kergin studies families D of distributions whose general element corresponds to
a point of Un in Hilbn(Rr), and whose special element D0 is concentrated at
the origin. Understanding the convergence of nonsingular punctual schemes to
a singular one is a key open problem (see §0.0).
There has been recent study of ideals determined by sets of distinct points
in Pr, by P. Maroscia, A. Geramita, E. Davis, and others, who are interested
in the Hilbert function of the defining ideal—related to maximal rank problems.
This work so far uses Un = SymnPr — D (where D is the large diagonal), or
SymnPr (talk of B. Harbourne) and not the special properties of the Hilbert
scheme; we consider this work outside the scope of this report, but expect it will
develop connections to study of the Hilbert scheme.

HILBERT SCHEME OF POINTS
315
When the holomorphic function / : Cr —► C has an isolated singularity V at
the origin, then the moduli algebra
A(V) = Go/ J(f) = R/(f, df/dxu..., df/dxr)
is Artinian; such algebras are quite special, and are sparse among all length-n
quotients of R, even when r = 2. J. Mather and S. S.-T. Yau showed [M-Y]
there is an isomorphism of sets
isolated hypersurface 1 J Artin algebras R/J 1
singularities in (Cn, 0) J 1 that are moduli algebras J
When r = 3, Yau studied the Lie algebra of derivations L(A(V)), and showed it
is solvable; he goes on to attach a generalized Cartan matrix or Kac-Moody Lie
algebra to V, which is an analytic invariant [Y]. In general, the Lie algebra of
derivations of an Artin algebra is little understood (compare §3.2).
R. Buchweitz notes that if zt is a generic point of a component of Hilbn Pr,
then the deformation space of the Artin scheme T has a super-rigid singularity.
He also studies the relation between the deformation spaces of two singularities
that are linked [BU].
4.3. Vector bundles. The study of vector bundles recently has shown some
ties with that of the punctual Hilbert scheme. D. Ferrand initially, C. Banica,
C. Banica and O. Forster, N. Manolache, and others have been led to consider
multiple structures—thickenings—on analytic subspaces of complex varieties (see
[FE, B, H2] and [HR3], an overview). This study has also motivated J. Brun
and A. Hirschowitz's study of the stratification of Hilbn P2 by the sets Wld —
{zT | (/(#°(P2,3{d)) - [{d + l)(d + 2)/2 - n]+) = »}; they study these mod
codimension three in HilbnP2 (see [BR-H]).
The Hilbert scheme has appeared in special cases as moduli spaces of vector
bundles (talk of Le Potier).
An amusing example is pointed out by A. P. Rao. Consider the family of
instanton bundles E on P3—stable rank two bundles for which c\ = 0 and
c2 = 3. The module M(E) = 0 J/^P3,^)) has for generic E the resolution
(here R = k[xo, •.. ,£3])
0 -► R{-b) -► R7{-3) -► R3{-2) 0 #3(-l) -► Rs -+ R3{1) -► M -+ 0.
Let Mv denote HomH(M,/c) and M'{E) = Mv ® i2(—5). Then M'{E) has
resolution beginning R7(—2) —► R —► Mf(E) —► 0, and is an Artin quotient
of R defined by an ideal / = (/1,..., /V) generated by seven quadratic forms.
The family of Artin algebras M'{E) that occurs is in fact an open dense subset
of Zp C Hilb8Op, p = 0 G P3, the family of algebras QP{F) having Hilbert
function (1,4,3) described in §3.1—the first component of HilbnOp not in the
closure Un C HilbnP3. The moduli space M(0,3) for the instanton bundles is
known to be irreducible and smooth; while Zp is an open dense set in the Grass-
mannian Grass(7, R2) = Grass(7,10). Questions that arise in viewing M(0,3)
this way are, does the generic resolution M(E) have ghost summands in addition

316
A. IARROBINO
to those listed (no, by an example), and do the bundles E having nongeneric
resolutions (so not giving an Artin algebra R/M'(E)) form a component in M(0,3)
(no, by what we already know).
In studying similar modules M, the following problem occurs. Parametrize
the family of surjective homomorphisms <p: Rs(a) —► R* such that a specified
term Rk(b) (or sequence of terms) occurs in specified places in the minimal
resolution of ker (p. This seems difficult, and is closely related to further study
ofHilbnOp.
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NORTHEASTERN UNIVERSITY

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Intersection Theory and Enumerative Geometry:
A Decade in Review
STEVEN L. KLEIMAN
WITH THE COLLABORATION OF ANDERS THORUP ON §3
Preface. For the siblings—Intersection Theory and Enumerative
Geometry— the past decade or so has been exceptional; it has been a period of
mushrooming interest, profound insights, and extensive development, unprecedented
in this century. The major events are reviewed here. More information will be
found in other articles in these proceedings and in the general literature. My
wish here is simply this: may not these few pages serve to whet thy appetite!
1. Introduction. Intersection Theory has been a central part of algebraic
geometry ever since it was founded in 1720 by Maclaurin (cf. [55, pp. 552-554,
607-608]). On the basis of examples, Maclaurin asserted that two plane curves,
defined by equations of degrees r and s, intersect in rs points. Unsatisfactory
proofs were given by Euler in 1748 and Cramer in 1750. Then in 1764 Euler and
Bezout independently developed a more refined method of eliminating one of the
two variables from the two equations, producing a polynomial in one variable
of minimal degree, mn. Bezout went on to treat the case of m equations in m
unknowns, and Maclaurin's assertion and its higher-dimensional generalizations
have become known, of course, as Bezout's Theorem.
Bezout's Theorem was used creatively. Maclaurin in 1720 used it to show
that an irreducible plane curve of degree r has at most (r — l)(r — 2)/2 double
points, where he counted an ordinary A:-fold point as k(k — l)/2 double points;
then he introduced the notion of "deficiency" (which Clebsch refined in 1865 and
called the "genus" ) as the maximum number of double points minus the actual
number. Braikenridge in 1733 (see [94, p. 266]) used Bezout's Theorem to show
that if the sides of a triangle ABC rotate about 3 fixed points while A and B
trace fixed curves of degrees r and s, then C will describe a curve X of degree
2rs; he enumerated the points of intersection of X with an arbitrary line.
1980 Mathematics Subject Classification (1985 Revision). Primary 14N05, 14N10.
Supported in part by the National Science Foundation.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
321

322
STEVEN L. KLEIMAN
Bezout's Theorem was used (see [27, pp. 2-4]) by Plucker in 1839 to obtain
his celebrated equations, (4.18) below, for a plane curve. Salmon in 1847,
generalizing Plucker's work to a surface with a singular curve, ran into and solved a
new problem, "excess intersection" ; he had to find the number of isolated points
in the intersection of three surfaces containing the curve.
Bezout's Theorem was used (see [47]) by Steiner in 1848 and 1854 implicitly
and by Bischoff in 1859 explicitly to find the number of conies tangent to 5
others. They correctly found that the 6 coefficients of the equation of a curve
of degree r tangent to one of degree s themselves satisfy an equation of degree
s(s + 2r - 3), but they incorrectly concluded that the number is 65 = 7776.
Cremona in 1864 explained the error: every double line appears tangent to any
curve, and there are infinitely many double lines. The problem is again one of
excess intersection, and 65 has no enumerative significance.
A revolutionary change in Intersection Theory was begun by Schubert [84].
He explicitly introduced the first intersection rings and implicitly the operations
of pullback and pushforth. He computed the rings of the projective plane and
3-space, of the Grassmannian of lines in 3-space, and of the universal line
bundle over the Grassmannian, in each case giving "Poincaré" dual additive bases.
His method was ingenious and prescient: in effect he found a natural Kunneth
decomposition of each diagonal. He also worked brilliantly in the intersection
rings of sophisticated compactifications of the varieties of plane triangles, conies
in 3-space, quadrics, and twisted cubics. He proceeded by finding expressions
for elements in terms of others through explicit degeneration and the
Correspondence Principle. Thus Schubert was able to systematize and simplify much of the
earlier work in enumerative geometry and to solve problems that had previously
defied attack.
Schubert's ideas were slowly interpreted and made more precise, more formal,
and more rigorous. Just some of the many, many mathematicians involved were
Severi, van der Waerden, Ehresmann, B. Segre, Todd, Weil, Zariski, Chevalley,
Samuel, Hodge, Pedoe, Chow, Serre, and Grothendieck. By 1960 an acceptably
rigorous theory was available. It centered on intersection rings. They were
constructed for smooth quasi-projective varieties as rings of cycles modulo rational
equivalence. The product of two classes was defined by moving a cycle
representing one class until it met a representative of the other "properly" (dimensionally
transversally). Then the intersection cycle was defined using an algebraic theory
of local intersection multiplicity. Given a map /: X —> Y and a class y on Y,
the pullback f*y was defined by representing y by a cycle, moving the cycle so
that its cartesian product with X meets the graph of / properly, and forming the
intersection cycle on X x Y but viewing it on the graph. The technical aspects
of the theory were fussy, and minor errors were common.
A second revolutionary change in Intersection Theory began about 10 years
ago. The new theory is due primarily to Fulton and MacPherson. It has a new
point of view and new techniques. It is significantly simpler and cleaner, more

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 323
refined and more general, more flexible and more suggestive. In short, it is more
powerful!
The new point of view is, in a nutshell, this: Intersection Theory is not a
theory of intersection rings of cycle classes on smooth X but a theory of operators
on groups of cycle classes on arbitrary X. The revolution, however, brings us
back practically to Schubert's original point of view. Indeed, Schubert [84,
p. 3] explicitly defines the sum and product of symbols representing numerical
equivalence classes of conditions on complete families of figures.
A fundamental new operator is the pullback operator along a regular
embedding (one defined locally by a regular sequence). To construct it, degenerate
the ambient scheme into the normal bundle of the subscheme. Correspondingly,
degenerate a representing cycle into a cycle on the bundle. It is easy now to
move the degenerating cycle until it meets the 0-section of the normal
bundle properly—or even better, until it becomes vertical—and then to define the
pullback. This device of reduction to the normal bundle was developed within
traditional intersection theory by Verdier, who thus made a key step forward.
One important example of a regular embedding is the diagonal map d: X —>
X x X of a smooth X. The intersection product of two classes on X is now
defined as the pullback along d of their cartesian (or external) product; in fact,
the intersection product is a well-defined class on the intersection of the supports
of any two representing cycles. There is no need for a general moving lemma.
Hence, X need not be quasi-projective. And the base need not be a field; a
discrete valuation ring serves as well. In fact, for some purposes, a more general
base will do, and sometimes, no base is needed at all.
There is also no need for an a priori algebraic theory of local intersection
multiplicity. Indeed, such an algebraic theory falls out of the geometric theory.
Thus, P. Roberts [77] (and independently Gillet and Soulé [31], who did a similar
thing using algebraic K-iheory) proved Serre's long-standing conjecture about
the vanishing of the local intersection multiplicity by establishing the commuta-
tivity of the local Chern character of a complex of vector bundles on a scheme
of finite type over a regular scheme.
§§2 and 3 below are devoted to the new intersection theory. The fundamental
reference for both sections is Fulton [28]. This brilliant book also treats
topics, like the Schubert-Giambelli-Thom-Porteous Formula and the Riemann-Roch
Theorem, left out of the two sections.
§2 summarizes the basic notions and results of the new intersection theory.
It discusses the rational equivalence groups, pullback along a flat map of pure
dimension, pullback along a regular map, pushforth along a proper map, the
Chern and Segre operators, and the Residual Intersection Formula.
§3 presents a variation on Bivariant Intersection Theory, which was written
in collaboration with Thorup. This theory offers, first of all, a convenient way
to systematically treat the more refined properties of the operation of pullback
along a regular map. However, the theory acquires a life of its own, because it

324
STEVEN L. KLEIMAN
provides a remarkable analogue of relative (or local) singular cohomology. In
particular, each X, whether smooth or not, has associated a natural
commutative intersection ring, which carries Chern classes. The section treats
Alexander duality, orientation classes, orthocyclicity (a notion implicit in [28, 17.3.1,
p. 324], and external products.
Enumerative Geometry is a fascinating and beautiful subject. Its goal,
according to Schubert [84, p. 1], is to answer all questions of the following form:
how many [algebro-] geometric figures of a fixed type satisfy certain given
conditions. Such questions are timeless. Apollonius about 200 B.C. wrote a two
volume work, On Contacts, on the several circles through i points, touching j
lines, and touching 3 — i — j circles. The ancient Greeks were interested in
construction. However, the finesse of Enumerative Geometry comes in finding the
number of figures without finding the figures themselves. The numbers,
moreover, are often very large. Indeed, Steiner in 1848 gave 7776 as the number of
conies touching 5 others just to indicate how much more complex Apollonius's
problem becomes with arbitrary conies in place of circles. Steiner was mistaken
about the value. The correct value is 3264, and it was first published by Chasles
in 1864. Certainly, 3264 is still large enough. However, its determination requires
more sophisticated methods.
Enumerative Geometry is interesting and important, though, not because of
the values of numbers like 3264, but because of the way they are found. Solving
enumerative problems has led to a deeper understanding of basic geometry, to the
sharpening of old tools, and to the development of many new ones. Enumerative
Geometry grew in substance and sophistication during the last century, until, at
the turn of the century, the ingenuity and complexity of its methods had so far
outstripped the capacity of the available technical mathematics to clarify and
justify their use that no longer could everyone harness the power of its methods
nor even become convinced of the accuracy of the results of others. Interest
waned, though it never vanished. Then, about 10 years ago, the current great
revival began.
The work of the last decade is reviewed in §§4-9. §4 centers on the "ranks" of
a projective variety, its basic extrinsic numerical invariants. They are also called
the "classes" and are also indexed differently. (Caporali, posthumously 1888,
called them the ranks; Severi [85, fn. 1, end of §1, p. 46] changed the name.)
§4 also reviews the progress in the theory of the duality of projective varieties;
nearly half the material reviewed concerns the ranks. The section closes with a
review of the work in the theory of Plucker formulas; basically, these formulas
are relations among the ranks and numerical invariants of the punctual and
tangential singular loci. More information on about 2/3 the material of §4 is
found in the author's surveys, [46, 52],
§§5 and 6 discuss multiple-point theory for maps, allied theory, and
applications. Given /: X —► F, an r-fold-point formula enumerates the points P of X
for which there exist r — 1 other points Q such that f(Q) = f(P). An early

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 325
example is the following formula of Clebsch (1865), which refines Maclaurin's
formula above: given a plane curve of degree d and geometric genus g with d
nodes and k cusps,
(1.1) 2d + 2k = r2 - 3r + 2 - 2g [or g = (l/2)(r - l)(r - 2) - d - Jfe].
The left side is the degree of the double-point cycle ra2 of the map / from the
normalization X of the curve into Y = P2. Indeed, 2d is the number of P in
X for which there exists a Q "distant from P" such that f(Q) = /(P), and
2A: is the number of P for which there exists a Q "infinitely near" P such that
f(Q) = /(P). A more formal discussion of (1.1) is given in §4.
Multiple-point theory has had a rich history. The current interest in it among
algebraic geometers arose around 1974. It was stimulated by preprints of Peters-
Simonis and Holme. Peters-Simonis consider a smooth n-fold X in p2n+x and
give a formula for the number of bisecants through a general point A; basically,
it is a numerical double-point formula for the projection /: X —► P2n, from A.
Independently, Holme found a numerical obstruction for a projection /: X —► PN
to be an embedding. It turned out that the obstruction is just the degree of the
double-point cycle m2(/), and so, effectively Holme's work recovers Peters and
Simonis's. In turn, Holme's work is related to some work of Severi [85, §3],
whose approach was made rigorous by Catanese [8].
A general multiple-point theory, inspired in particular by Todd (1940) and
Ronga (1973), was developed (1976-80) by Fulton, Hansen, Holme, Johnson,
Laksov, J. Roberts, and the author. It is sketched in §5. More information
about the theory and its history may be found in the author's surveys, [46, 50].
Some recent refinements, due to Colley, Roberts, and Ran, are mentioned at the
end of §5.
§6 reviews the applications of multiple-point theory and of some allied theory.
The applications are to the enumerative theory of singularities of projective
varieties, the enumeration of multisecant lines, the enumeration of higher-order
contacts of plane curves, and the enumeration of exceptional secant planes of
a curve in PN and of a surface in P3. Some typical examples are included.
The allied theory is related to multiple-point theory in that its goal too is to
enumerate families of subschemes of finite length, or families of more refined
objects, like triangles and tetrahedra. However, the objects are not viewed as
lying in the fibers of a map but simply as satisfying the conditions at hand. An
introduction to triangle theory and to its use in the enumeration of contacts is
given in [87].
§7 reviews the work on two aspects of the enumeration of contacts: the
Contact Theorem and the Galois group. The group is defined for virtually any
enumerative problem, but it has been considered only in cases involving
conditions of contact, including incidence and multiple contact. The theory was
begun last century and then seems to have gone dormant until Sept. 1977 when
Serre (private communication) inquired about the Galois group of the 5 conic
problem.

326
STEVEN L. KLEIMAN
The Contact Theorem gives a formula for the number of varieties in an ra-
parameter family that touch m fixed varieties in general position, and it addresses
the nature of the contacts it enumerates. Various special cases of the formula
were given in the last century, but a definitive general form seems not to have
appeared until 1982. And, needless to say, little attention was paid before to the
enumerative significance of the formula. The theorem reduces the enumeration
of the varieties in the family to the determination of the family's characteristic
numbers, namely, the numbers of those that touch ji z-planes for 0 < i < N and
for all j{ > 0 such that jo + \- Jn-i = ™- Introductions to the theorem are
given in [51] and [87].
§8, which deals with quadrics and correlations, is more of an introduction
than a review. The section begins with a discussion of Zeuthen's (1865) method
for finding the characteristic numbers of families of conies. The modern
interpretation and justification is motivated by a discussion of the early nineteenth
century geometric theory of polarization. In that theory, a conic corresponds
to a plane correlation, namely, a linear transformation from the plane to the
dual plane, or put more algebraically, a bilinear form in 3 variables; the conies
correspond to the symmetric bilinear forms.
The theory of polarization also motivates the enumerative theory of plane
correlations, initiated by Hirst (1874-90). It is similar to the enumerative theory of
conies, and contains it. Both theories were generalized right away, and over the
years, they have probably attracted more attention than any others in
Enumerative Geometry. In particular, about 1980, they began to receive a great new
burst of attention. The old results have been reworked and new ones discovered.
The work is reviewed briefly at the end of the section. More information will be
found in the historical surveys, [57] and [59].
§9 deals with the work on the enumeration of curves of higher degree,
particularly plane and twisted cubics. First, it briefly reviews the work in general.
Then it concentrates on a promising new method. The method is somewhat
untraditional; it does not proceed via the construction of a suitable parameter
space and the determination of its intersection ring. Rather, its spirit is similar
to that of the (pre-1879) approach of de Jonquiéres, Chasles, Cremona, Zeuthen,
Cayley, Halphen, etc. It centers on the 1-parameter families defined by various
combinations of the contact conditions and the determination of additive
relations among the contact conditions and the degeneracy conditions. Using the
method and the Contact Theorem, Kleiman, Str0mme and Xambô [54] obtained
the first rigorous verification of Schubert's value 5819539783680 for the number
of twisted cubics tangent to 12 given quadric surfaces, and established its
enumerative significance.
Hilbert's 15th problem concerns Intersection Theory and Enumerative
Geometry (see [45]). Hilbert wrote:
"15. Rigorous Foundation of Schubert's
Enumerative CALCULUS. The problem consists in this: To establish

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 327
rigorously and with an exact determination of the limits of their
validity those geometrical numbers which Schubert [84]
especially has determined on the basis of the so-called principle of
special position, or conservation of number, by means of the
enumerative calculus developed by him.
"Although the algebra of to-day guarantees, in principle, the
possibility of carrying out the processes of elimination, yet for
the proof of the theorems of enumerative geometry decidedly
more is requisite, namely, the actual carrying out of the process
of elimination in the case of equations of special form in such a
way that the degree of the final equation and the multiplicity of
their solutions may be foreseen."
It is tribute to Hilbert's vision that he could foresee that the rigorous
development of Intersection Theory and Enumerative Geometry would be one of the
great mathematical endeavors of the twentieth century.
Securing Intersection Theory was certainly the first job, and a most difficult
one at that. It took so long, well over half a century, that work in
Enumerative Geometry and work on Hilbert's 15th problem came to mean to many
simply work in Intersection Theory. However, Enumerative Geometry is a
separate subject and Hilbert make clear his interest in establishing "those geometric
numbers" ; he says that "decidedly more is requisite" than an "in principle"
determination. Tremendous progress has been made over the last decade, and now
for the first time in over a century, a full understanding and rigorous treatment
of all the material in Schubert's book [84] appears close!
2. Basic Intersection Theory. Remarkably few hypotheses must be
imposed on the ambient schemes X. Here, the X will be assumed, at a minimum,
to be separated, noetherian, and universally catenary. And the maps /: X —► Y
will be assumed to be of finite type. Thus, for example, it is acceptable to work
in the category of schemes of finite type over a Cohen-Macaulay base. This
minimum is probably a good one, although with care it is possible to weaken
the hypotheses here and there. For example (Thorup, private communication,
April 18, 1986), it is virtually always sufficient to assume that the /: X —► Y
are essentially of finite type, and this weaker hypothesis is often technically
convenient. (However, it is doubtful that the first part of Example 20.1.3, p. 396,
[28], works exactly as stated.) Although the case of a base field extension of
infinite degree is technically not covered, it will be clear that such an extension
is compatible with the operations of intersection theory.
A prime cycle on an X is a symbol [V], associated to a closed integral sub-
scheme V. An arbitrary cycle is a finite Z-linear combination of prime
cycles. A closed subscheme Z (resp. a coherent sheaf F) has a fundamental cycle
[Z] := J3nv[^] (resp. [F] := J3rcv[V]) m which the V are the irreducible

328
STEVEN L. KLEIMAN
components of Z (resp. of the support of F) and ny is the length of the local
ring Ozy of Z at the generic point of V (resp. of the stalk Fy).
A nonzero rational function / on a closed integral subscheme Z of X has an
associated cycle, [/] := J3ordv(/)[V], where V ranges over the 1-codimensional
closed integral subschemes of Z and ordy is the order function; namely, if / = a/b
with a, b in Oz,v, then by definition
ordv(/) := length(Oz,v/a) - \ength{0 Zy/b).
Similarly, a (Cartier) divisor D on Z has a cycle, [D] := J3ordv(/)[^], where /
is a function on Z defining D at the generic point of V.
Two cycles on X are called rationally equivalent if their difference is a linear
combination of cycles of functions on various closed integral subschemes. The
group of all rational equivalence classes is denoted AX. The class of a prime
cycle [V] will also be denoted by [V].
Suppose the X are of finite type over a (separated, universally catenary
noetherian) base S whose local rings are all equidimensional. For example, they
are equidimensional if S is Cohen-Macaulay, or if every connected component of
S is irreducible. Then the AX may be graded by relative dimension. Indeed, by
definition, if /: V —► S and V is integral, then
dim(vyS) := tr.deg(Jfe(V)/Jfe(/V)) - dim(05JV).
It is easy to prove additivity: if /: V —► S factors through an integral W, then
dim(vyS) = dim{V/W) + dim{W/S).
It follows that if the cycles are graded by relative dimension, then the cycle of
a rational function is homogeneous; whence, the AX inherit a grading. It also
follows from additivity that the pullback, pushforth, and Chern class operators
defined below are homogeneous.
If there is no such 5, then doubtless the AX have no useful grading, contrary
to the assertion made in [49, 1.6, p. 25]. That assertion was based on the
theory developed there in §1.2, which extends to the relative case the theory of
codimension in EGAIV2 and rests on it. However, a basic result, Proposition
(5.1.9), p. 88, is false (for example, over a field in P3, take F to be a line, x a
point on F, and X a plane through x not containing Y). The error comes in
the application of (0,16.1.4), which is false but corrected in EGAIV4 (Erriv, 5),
p. 347; however, (5.1.9) appears not to have been corrected. On the other hand,
for many purposes, a grading is unnecessary.
Let /: X —> Y be proper. It induces a pushforth operator /*: AX —> AY,
which is defined on a prime cycle [V] by
(2.1) f.[V}:=deg(f\V)[fV},
where the degree deg(/|V) is, by convention, 0 when the restriction f\V is not
generically finite. If Y is viewed as a base scheme, then set
x := f*x for x in AX.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 329
This alternate notation is used especially when AY = Z • [Y], the free group
on [Y], For example, Y might be the spectrum of a field, or Z, or a discrete
valuation ring. Then f x is viewed as an integer and called the degree of x.
Let g: Yf —► Y be flat and of pure dimension (that is, every fiber is of pure
dimension). Then g induces a pullback operator g*: AY —> AY', which is defined
on a prime cycle [W] by
(2.2) g*[W] := [g^W].
For example, if g is the structure map of a vector bundle, then g is flat of pure
dimension, and it is a theorem that g* is a group isomorphism.
Pushforth and pullback commute in this sense: Given a cartesian square
x C x'
1/ 1/'
Y S- Y',
if / is proper and g is flat of pure dimension, then so are /' and </, and
(2.3) g'f. = fW.
Formula (2.3) is proved on the level of cycles. On that level, the theory may
be generalized as follows (Thorup, private communication). For arbitrary / and
g, define /* by (2.1) but with fV replaced by its closure, and define g* by (2.2)
but with [<7-1W] replaced by the sum J3 length(Og-ijy,v) restricted over those
components V of g~lW that dominate W. Then (2.3) holds.
When / is proper, then /* preserves rational equivalence. When g is flat of
pure dimension, then g* preserves rational equivalence. The hypothesis "of pure
dimension" may be eliminated here if the definition of g* is modified further
so that the sum ^2length{Og-\Wy) is taken over only those V that dominate
W and are of dimension dim(W) + max.dim(g~xy). However, if this is done,
functoriality will be lost; for example, if Yf is the union of two irreducible
components, if the two restrictions of g are flat of different dimensions, and if h is
the inclusion of the one of smaller dimension, then h*g* =0 but {gh)* ^ 0. If
Y' is the disjoint union of the two components, then there is an obvious better
definition of g*. Similarly, if g is smooth but not necessarily of pure dimension,
then there is a better definition of g*. It would be good to have a definition that
would work more often.
Let E be a locally free sheaf of finite rank on X. Associated to E are certain
Chern and Segre operators on AX, which generalize the more traditional classes.
The operators play a more fundamental role than the classes used to, and the
theory is simpler and easier to develop. They are defined as follows.
First, suppose that E is invertible. Then its first Chern operator c\E on AX
is defined on the class [V] of a prime cycle by
[ciE)[V] := [D] if Ov{D) = E\V.

330
STEVEN L. KLEIMAN
Here, any divisor D onV that satisfies the condition on the right will do. The
operator is not well defined on the level of cycles.
Next, consider the structure map of the associated projective bundle, p: P(E)
—► X. It is flat and of pure dimension r — 1, where r is the rank of E (which may,
harmlessly, be allowed to vary from one connected component of X to another).
For each i > 0, define the ith Segre operator as the operator composition,
(2.4) S^:=p,(ClO(l)r+r-y.
Then S{E = 0 for i > n = dim(X). Define the total Segre operator, sE := Yl siE.
Finally, define the total Chern operator as the formal reciprocal
(2.5) cE := l/sE\ where E* := Hom(£, Ox).
Define the ith Chern operator as the component of cE of degree i.
There is some confusion, unfortunately, about conventions. The scheme P(E)
carries two different objects conventionally denoted by the same symbol 0(1):
one is a sheaf; the other, a scheme; and 0(1) = V(0(1)*). Both sheaf and
scheme are assigned the same first Chern operator. In fact, more generally,
(2.6) cV{E*) = cE.
However, the Segre operators of V(J5*) are defined using the projective bundle,
P(V(£*)) = P(£*),
of line subbundles of V(E*), or of invertible quotient sheaves of E*. Hence,
(2.7) cV{E*) = l/sV(£*) and sV(£*) = sE* (not sE).
Changing notation so that V(E) and P(E) would stand for Spec(Sym(i£*)) and
Proj(Sym(i£*)) would destroy the usefulness of these symbols when E is an
arbitrary quasi-coherent sheaf. If the lack of symmetry in (2.5)-(2.7) is intolerable,
then it would be better to alter the right side of (2.4) by a factor of (—l)1 while
keeping the analogous definition in the case of a vector bundle as is. Also, the
word "bundle" should be reserved for a scheme and never used as a synonym for
"locally free sheaf."
Let /: X —► Y be a closed embedding, / the ideal. Consider the normal cone,
7V/:=Spec(0/n/7n+1).
Degenerating X into Nf induces a "specialization" map, AY —► ANf, which is
defined on the cycle level by assigning to a prime cycle [V] the fundamental cycle
of the normal cone of the restriction, f~xV —► V.
Suppose further that f:X —► Y is regular; that is, X is locally defined in Y
by a regular sequence of elements. Then Nf is a vector bundle over X, so AX
is naturally isomorphic to ANf. Hence, there is a natural pullback map,
/*: AY -+AX.
Call an arbitrary map f: X —>Y regular, or a factorable l.c.i. (local complete
intersection), if there exists a factorization / = gh such that g is smooth and h

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 331
is a regular embedding. If / is regular, define an operator and a virtual bundle
by
/*:=hV and Nf := Nh - h*Tg (where Tg := V(flJ)).
Given another factorization f = gfhf such that </ is smooth and ft' is an
embedding, necessarily: (1) ft' is regular; (2) /* = ft'V*; and (3) Nf = A^, - 0'*7V.
For example, /: X —► Y is regular if X and F are smooth over some S.
Indeed, the projection g: X Xg Y -^ Y and the graph ft: X-^Xx^ Y work.
In practice, most every / possesses a factorization / = gh such that g is the
structure map of a projective bundle and ft is an embedding. Call such an /
strongly quasi-projective. The question becomes: is ft regular?
Suppose /: X —► Y is regular, and consider a cartesian square,
X <— X'
1/ 1/'
y <— y.
Define a "refined" pullback /*: AXf —► AF' as follows. First assume that / is
a (regular) closed embedding. Then the normal cone Nf> sits in the pullback
Nf Xx Xf. The latter is a bundle over X'; so, if p denotes its structure map,
then p* is bijective. Define /* as the composition of the specialization map, the
pushforth along the inclusion, and p*_1:
/*: AY'-+ANr -► ANf xx X' -+ AX'.
In general, there exists a factorization j — gh such that g is smooth and ft is a
regular embedding. Define /* := h*(g x Y')*. Then /* is independent of the
choice of factorization.
The Residual Intersection Formula (an old formula revived in 1976 by Laksov
and refined by Fulton, Laksov, MacPherson, and the author) deals with the case
that X1 splits into a closed subscheme W and a residual scheme R:
R := P(I(w))
X' ^W.
There is another description of R in a key case. Suppose there is a closed
embedding ft: X' —► H such that the embedding hw: W —► H is regular. Let
b: B —► H denote the blowing-up along W. Then R is naturally embedded in
5, and its ideal is determined by the following equation:
I{R/B) • I{b-lW/B) = Iib^X'/B).
Note that b~1W is the exceptional divisor, so I(b~1W/B) is invertible.
The Residual Intersection Formula holds under the following assumptions:
(2.8)(i) /,/i := /V, and w\ := f'w are regular; (ii) cod(/) =cod(/;); and
(iii) / is strongly quasi-projective and Yf embeds in a regular scheme.

332
STEVEN L. KLEIMAN
Assumption (iii) is made for technical use in the proof. It may be weakened
or replaced with another. However, it usually obtains in practice.
Fix Y"/Y' and y in AY". Set
n := cod(/) - cod(wi) and cn(f/wi) := cn(w*g'*Nf - NWl).
The Residual Intersection Formula is this:
(2.9) f*y = ufty + w*cn{f/w{)wly.
A companion formula, which is a corollary of the proof, is this:
(2.10) r*(ClOH(l)<7i*</) = -WtCn+kif/wJfty for k > 1.
When W — X', the Residual Intersection Formula reduces to the Excess
Intersection Formula:
f*y = cnif/nfy.
It relates /* and /'* when /' is regular too, but of excess dimension.
If also g and gf are regular, then clearly cn (///') = cn(g/g'). Thus, if g is
a regular closed embedding, if the ideal is the quotient of a locally free sheaf
of finite rank, and if / is the blowing-up, then the Excess Intersection Formula
reduces to a modern form of Grothendieck's Key Formula to the structure of
AX.
3. Bivariant intersection theory. (The following variation of the theory
in [28, Chapter 17] was written in collaboration with Thorup.) Let /: X —► Y
be a separated map of finite type between arbitrary schemes. For each separated,
universally catenary noetherian F-scheme Yf, set Xf := X Xy Yf. Denote by
Hf or H(X/Y) the group of systems of operators,
ay: AY' -+ AX' for all such Y'/Y,
that are compatible with proper pushforth; that is, for each cartesian diagram
X" -► Y"
W ÏQ
X' -► Y'
in which g is proper (so g' is too), the following two compositions are equal:
(3.1) ayg* = g*aY"-
Alternatively, denote ay by aY>/Y or simply by a. There are two basic examples:
(1) if / is regular, define /* as the system of refined pullbacks; (2) if / is flat of
pure dimension, note that fy: X' —► Y' is too and define /* as the system of
pullbacks fy,.
Given any g: Y\ —► F, the restriction, or pullback, of a in Hf, denoted
a\Y\ or ga or g*a, is the system in HfYx defined by (a|Yi)(y//y) := aY>/y.
Given a factorization f = gh such that h is proper and g is of finite type,

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 333
the pushforth, denoted h\a or h+a, is the system in Hg defined by (h\a)Yf :=
{hY')*CLY'> Given g: Y —► Z of finite type and a system b in Hg, the composition,
or intersection product, denoted ab or a. b, is the system in H(gf) defined by
(ab)zf := ayxZ'&Z'- Given g: W —► Z of finite type, given 6 in A<7, and given a
base scheme S (for example, Spec(Z)), the external (cartesian or cross products),
a x b = a Xs b and b x a = b Xs a, are the systems in #(/ x g) defined by
axb:={a\Y xZ).{b\XxZ) X xW a{YxW ) y x W
[b\XxZ [b\YxZ
bxa:= {b\Y x Z). (a\Y xW) Xx Z a]YxZ >Y x Z
If b x a = a x b, say a and b commute and call them compatible.
A cochain is a linear function that assigns to each cycle on each Y1 a cycle
on X' and that is compatible with proper pushforth of cycles, at least modulo
rational equivalence; that is, the analogue of (3.1) holds modulo rational
equivalence. For example, if / is flat of pure dimension, then /* was constructed
essentially as a cochain. If X is a principal divisor in Y and / is the inclusion
map (this is the key case of a regular embedding), then /* is induced by this
cochain: given a prime cycle [V] on Y', set f*[V] := [V] if V' :=V xX' ^ V,
and /*[V] := 0 if V = V. A cocycle is a cochain that preserves rational
equivalence. A coboundary is a cochain whose values are all rationally equivalent to
0. Obviously, every system a in Hf may be represented by a cocycle, and two
representatives differ by a coboundary. Moreover, by linearity and
compatibility, a suitable representative may be specified by simply assigning values to the
fundamental cycles [Y'] of the integral Yf. Thus Hf may be viewed reasonably
as a group of "cohomology" classes.
The operations of restriction, proper pushforth, composition, and external
product obviously make sense for cochains, and they preserve cocycles and
coboundaries. In this connection, the following result is useful: if a cochain
a in Hf commutes up to a coboundary with g* whenever g\W —► Z is flat of
pure dimension or is the zero-section of the affine line over the base, possibly Z,
(that is, a x (g*) and (g*) x a differ by a coboundary), then a is a cocycle and
it commutes up to a coboundary with g* whenever g: W —► Z is regular.
There are two important subgroups of Hf. The first, called the group of
bivariant classes and denoted by Af or A(X/Y), consists of all the systems
a that commute with g* whenever g: W —► Z is flat of pure dimension or is
regular. The second, denoted Bf or B(X/Y), is the subgroup generated out
of these same g* via repeated restriction, proper pushforth, composition, and
external product. It is a fundamental theorem that Bf is a subgroup of Af.
Obviously, A(X/X) is a ring under composition, and B(X/X) is a
commutative subring. Moreover, B(X/X) contains the Chern and Segre classes of bundles
E on X, which are defined by {ciE)[X'\ := (ciExf)[X'], etc. Indeed, it obviously
suffices to prove that if L is a line bundle on X, then c\L is in B(X/X). Now,
if i: X —► L denotes the zero section, then i is regular and c\L = z*z*.

334
STEVEN L. KLEIMAN
Call /: X —► F oriented when it is equipped with a class /* in Bf such that
f*[Y] = [X]; call /* the orientation class. If / is flat of pure dimension (resp.
is regular), call the natural orientation class the fundamental class (resp. the
Thorn-Verdier class). If / is oriented, say that Alexander duality obtains if, for
any g:W —► X of finite type, composition with /* is bijective:
-.r-.Ag^ A(fg), or -. /*: A{W/X) ^ A{W/Y).
(The names stem from an analogy of the Af with relative, or local, cohomology.)
Note that if /: X —► Y and g:Y —► Z are oriented (resp. and satisfy Alexander
duality), then the composition gf is (resp. and does) too.
The most important case in which Alexander duality obtains is this:
(3.2) PROPOSITION. ///: X —► Y is smooth, then Alexander duality
obtains, and the isomorphism, -./*, from Ag to A(fg) restricts to one from Bg to
B(fg).
Indeed, the inverse isomorphism (as the analogy with topology suggests) is
given by a kind of slant product with the Thom-Verdier class of the diagonal:
W -► XxW -f W
I I Ï9
X -i XxX S X
Ïp If
X U Y
given b in A(fg), the slant product d*/b is denned by
d*/b:={d*\XxW).(b\X).
Since / is smooth, so are p and q, and d is a regular embedding; hence, the
theorem of independence of the factorization yields: d*q* = 1 and d*p* — 1.
Also, q* = f*\X because / is smooth. Therefore, commutativity yields:
(<r/6). (/• ) = (<r\x x w). (f*\w). b = ((<r. oi«0 • » = ft,
d*/(c. /*) = (<f |X x W){c\X x X)p* = {{c\X x X)|X)dV = c.
Finally, note that if b is in B(fg), then d*/b is in 5/. Thus (3.2) is proved.
Call Y orienting if every /: X —► Y may be oriented in one and only one way.
Call Y orthocyclic if Bf = Af and if evaluation at [Y] is an isomorphism,
EY: A{X/Y)=Af ^ AX.
These two different-sounding conditions are equivalent! Indeed, if Y is ortho-
cyclic, then the unique system in Bf corresponding to [X] is, obviously, an
orientation and the only one. The converse is covered by this criterion:
(3.3) PROPOSITION. // every projective map g: V —► Y with V integral is
oriented, then Y is orthocyclic and orienting.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 335
Indeed, consider an arbitrary map g: V —► Y with V integral. By Chow's
lemma, there exists a commutative diagram
v à- y
x C v"
in which ft and g" are projective, ft is birational, i is an open embedding, and V"
is integral; set g* := h\(i*g"*). Now, given f: X —>Y and a cycle v = X^nv[V]
on X, form Fv := ^2 nv3\Q* m ^/? where j: V —► X is the inclusion and g := /j.
If v' is a cycle rationally equivalent to v, then Fv' = Fv; indeed, given any map
ft: W -► y with W integral,
(Fv)[»y] = (Fv)h*[y] = h*(Ft/)[y] = h*v = ftV = • • • = {Fv')[W}.
Thus, F induces a map F: AX —► £/, and EyFv = v. Finally, given any a in
Af and any g: V ^ Y with V integral, set v := Eya — a[Y]. Then
(FEYa)[V) = (Fv)g*[y] = g*(Ft;)[y] = g*{EYFv)
= g*v = g*a[Y)=ag*[Y)=a[V).
Thus F Eya = a, and a is in 5/. Thus y is orthocyclic.
For example, if y is the spectrum of a field or is a 1-dimensional regular
scheme, then Y is orthocyclic and orienting. Indeed, given g: V —► Y with V
integral, either (1) g is flat of pure dimension or (2) g = jft, where j is the
inclusion of a closed point. If (1), then g is naturally oriented. If (2), then j is
regular and h is flat of pure dimension; so, set g* := ft*,;*. Finally, apply (3.3).
Similarly, if y is a 2-dimensional regular excellent scheme, then Y is ortho-
cyclic and orienting. Indeed, let g: V —► Y be any projective map with V
integral. Then there exists a commutative diagram
y <X- V
[q [q'
Y ( h y
in which Y' is regular, ft is projective, gf is flat, V is integral, and ft; is proper
and birational. For example, let U be a nonempty open subscheme of gV such
that g~1U —> U is flat. Take the induced map U —► Hilby/y, and close its
graph subscheme in Y x Hilby/y. Since the graph is excellent of dimension < 2,
it has a desingularization Y'. Let V be the pullback of the universal scheme.
(Alternatively, use the flattening theorem of Raynaud-Gruson [76, 5.5.2] and
resolution.) Now, say ft = pi where p: P —► Y is a projective bundle map and
i is a closed embedding. Since Y' and P are regular, i is a regular embedding.
Set g* := ftj(^*2*p*) and apply (3.3). Of course, what really was proved is this:
if y is an n-dimensional regular excellent scheme and if every excellent scheme
of dimension < n has a desingularization; then Y is orthocyclic and orienting.

336
STEVEN L. KLEIMAN
If /: X —► Y is smooth and Y is the spectrum of a field or is a 1-dimensional
regular scheme or is a 2-dimensional regular excellent scheme, then X is ortho-
cyclic and orienting by (3.2), by the above, and by (3.4) below. So, again by
(3.4), any map between two smooth F-schemes satisfies Alexander duality.
(3.4) PROPOSITION. Let Y be orthocyclic. Given f: X —► Y, Alexander
duality obtains iff X is orthocyclic.
To prove (3.4) take any g: W —► X and consider this commutative triangle:
Ag ~-^ A(fg)
By hypothesis, Ey is an isomorphism; so -. /* is one iff Ex is.
Suppose Y is orthocyclic. If /: X —► Y has a factorization, / = /i • • -/n,
such that each fi is oriented (for example, fi might be flat of pure dimension or
be regular), then
Indeed, /£ • • • /* is obviously an orientation of /, but / can be oriented in only
one way because Y is orienting. Similarly, other identities may be checked by
evaluating both sides on [Y]. For example, if /: X —► Y is proper and birational,
then /,/* = 1 because f*f*[Y] = f*[X] = [Y].
The orthocyclic Y are characterized by another attribute, the existence of
external products: given two F-schemes X and Z, the external product is a map,
(3.5) AX ® AZ -► A(X xY Z)
such that (1) as Z varies, the system of operators [X] ® -: AZ —► A(X x Z)
is an orientation of X/Y and (2) for every proper map g: W —► X, pushforth
induces a commutative square. If external products exist, then Y is orthocyclic
and orienting by (3.3). Hence, by (2), the products are unique. Conversely, note
that the external product of systems always yields a natural map,
B{X/Y) <g> B{Z/Y) -► B{X xy Z/Y).
If Y is orthocyclic, identifying the terms yields the product (3.5). Thus:
(3.6) PROPOSITION. External products exist over Y iff Y is orthocyclic,
iff Y is orienting. If the products exist, then they commute with all systems in
all Ag, and they commute with interchange of the two factors (that is, the map
AZ (g> AX —► A(Z x X) obtained formally from (3.5) is equal to the external
product).
Suppose X is orthocyclic. Then, in particular, Poincaré duality obtains:
EX' A{X/X)^AX.
Since A(X/X) = B(X/X) and so is a commutative ring, AX is one too.
Furthermore, given any g: Z —► X, the natural action of A(X/X) makes AZ an

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 337
AX-module; the product of x in AX and z in AZ is denoted by {g*x) .z or
x. gz. If g is proper, the projection formula obtains: g*{{g*x). z) — x. g+z.
Indeed, generalize by varying x in AX' and X' over X and using the external
product. Then both sides are in A(X/X). Evaluating each at x = [X] yields
<7*z; whence, the assertion.
Suppose f:X —► Y is smooth. Fix g: Z —► X. Then Alexander duality, (3.2),
and the external product of systems are compatible; that is, if of: X —► XyX
is the diagonal map, then the following diagram is commutative, where e :=
-.(d*|ZxX):
£(X/X) <g> fl(Z/X) —+ #(^P0
(3.7) I I
B{X/Y) <g> B(Z/y) -► £(X xy Z/y) -^ B(^/y)
Indeed, take a in B(X/X) and 5 in B(Z/X). Then
Z -► XxZ -► XxZ -► Z
i i I lb
X -i XxX -► XxX -^ X
|p [v 1/
x A x X y
(d*|X x Z). (6. /* xy a. /*) = (d*|X x Z). (6|X x X). p* . a. /*
= b.d* .p* .a.f* = b.a.f*.
Thus (3.7) is commutative.
Identifying the terms in (3.7) yields this:
(3.8) PROPOSITION. ///: X —► Y is smooth and Y is orthocyclic (whence
X is too by (3.1) and (3.3)), then for any Z/Y, the AX-module multiplication
on AZ is equal to the composition,
AX&AZ-+ A(X xy Z) -► AZ,
of the external product and the Thorn- Verdier class of the diagonal map.
Observe that it does not suffice for compatibility that X and Y be orthocyclic,
despite (3.4). For example, let y be a regular, 1-dimensional scheme, and X
smooth over a closed point of Y. Then X and Y are orthocyclic, but the external
product, AX ® AX —► A(X Xy X), is equal to 0.
All the above theory works, of course, virtually without change (but then
it is less refined) if the groups AX, Af, and Bf are replaced by their tensor
products with the rationals, AqX, Aq/, £q/, or again if the Bf, resp. £q/,
are replaced by the larger groups Cf, resp. Cq/, of all systems in Hf that
commute with all systems in all Ag, resp. Aç^g. P. Roberts [77] proved that
localized Chern characters lie in the Cq/; probably a variation of the proof will
prove that they lie in the £q/.

338
STEVEN L. KLEIMAN
Suppose Y is regular. Then any projective map g: V —► F with V integral can
be CQ-oriented as follows. Say g — pz, where p: P —► Y is a projective bundle
map and z is a closed embedding. Then P is regular, so CV has a resolution i£.
by locally free sheaves on P. Set n := cod(F/P) and 0* := chn(i£.)p*. Then
(3.3) and (3.6) yield:
(3.9) PROPOSITION. Any regular scheme is Cç^-orthocyclic and Cq-
orienting and supports arbitrary external products.
Thus the coarser Q-form of the "main missing ingredient," requested by
Fulton [28, p. 393], has been supplied. It is possible that chn(E.) is definable in Bi\
it is probable, if Y is excellent. When it has been defined in Bi, then the more
refined ingredient will be available.
4. The ranks—duality—Plucker formulas. Let X be an n-dimensional
closed subvariety (= integral subscheme) of the TV-dimensional projective space
P^ over an algebraically closed field of arbitrary characteristic p.
The basic extrinsic numerical invariants of X are its N ranks, r* = r»(X)
for 0 < i < N. (They are also indexed by j := n — i.) They may be defined
geometrically as follows: r^ is the number of simple points P of X such that (1)
P lies on a general z-codimensional linear space section of X and (2) if 0 < i < n,
then the embedded tangent space TpX and a general (z + 2)-codimensional linear
space A meet in an (n—z — 1)-dimensional space; that is, TpX+A is a hyperplane.
Ifn <i < N, then r* = 0.
The polar locus X(A) is defined as the closure of the scheme of all simple
points P satisfying (2) alone; in other words, it is the scheme pullback under the
Gauss map of the appropriate Schubert variety. Therefore,
(4.1) r; = degX(A); in particular ,rn = degX.
If the characteristic p > 0, then X(A) is on occasion not reduced, and then the P
satisfying (1), (2) must be weighted by a power q of p; however, q is independent
of P. Trivially, if i = n, then q = 1 and X(A) = X; thus, rn = degX.
The following useful comparison theorem of Severi [85, §1, p. 45] was given
its first modern treatment by Piene [67], (4.1) and (4.2):
(4.2) Piene-Severi Comparison Theorem. Let Y be the section ofX
by a general hyperplane M, let Y\ be the projection of X to M from a general
center; then
ri(Y) = ri+l(X) and r^Yx) = n{X) forO<i<N-2.
Here Y and Y\ may be viewed as subschemes of PN or of M, a P^-1; their
ranks are the same either way.
Call ro the class of X. Then, by (4.2), r^ is the class of a general z-codimen-
sional linear space section of X.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 339
Several proofs of (4.1) are available. One [52, p. 189] is based on this
observation: if B is a general (i + l)-codimensional linear space and if the
intersection B.M is equal to A, then Y(B) = X(A).M\ dually, if A is general
containing the center and A! denotes the projection of A, then Y [A') is the
projection of X(A). Both observations are clear set-theoretically and may be
checked scheme-theoretically by considering the corresponding Gauss maps and
Schubert varieties. (A remarkable case occurs in characteristic p = 2: if X is a
smooth quadric surface in P3 and if i = 1, then X(A) is reduced, but Y (B) is a
nonreduced point. By contrast, if p ^ 2, if X(A) is reduced, and if 0 < i < n,
then Y (B) is reduced.) Another proof (Holme, private communication, Oct. 11,
1985) may be based on a comparison of the conormal varieties of X in PN with
those of Y and Y\ in M and on (4.3) below.
The conormal variety CX is defined as the closure in PN x PN* of the set of
(P, H) such that P is a simple point of X and H is a hyperplane containing TPX.
Clearly, dimCX = N - 1. Let pi,P2 denote the projections from PN x PN*.
Let A' denote the linear space in PN* of hyperplanes containing A. It is
intuitively clear and not hard to prove that p\ carries p^"1(X/. Af) birationally
onto X{A). Hence, [X(A)\ = p^p^tX' .A% Therefore, for 0 < i < N,
(4.3) rt = J ClO(iy . [X(A)\ = J ciO(l)*. \pup^ (Xf. A%
Now, the embedding of p^1 (X'. Af) in CX is generically regular of codimension
N — 1 — i, the codimension of A!\ hence, [p^1 A'} = p^A']. Hence, for 0 < i < N,
(4.4) n = y pîciO(ir .pSdOU)"-1-*. [CX].
This formula is sometimes used to define the r^.
The dual variety of X is the image X1 \— p<iCX in PN*. Set
n'^dimX' and c := n + n' - (N - 1).
Then, as first observed by Holme [42]: r^ = 0 for i < n — c. Indeed, A! is general
of codimension N — 1 — i, so doesn't meet X;, and (4.3) yields the result.
Let H be a hyperplane. The H-contact locus Xh is the scheme fiber p~^xH
of CX, viewed via p\ as embedded in X. It is the subscheme of X of all points
at which H touches X. Suppose H is a general hyperplane touching X; that is,
H is represented by a general point of X1. Then obviously c = cod(X# /X), and
an elementary argument shows that c = 0 iff X is a linear space. Moreover, as
noticed in characteristic p = 0 where degX# = 1 by Holme [42, Theorem 2.1],
(4.5) rn_c = degX/fdegX/.
Indeed, if i = n — c, then A' meets X' in degX; distinct general points H. So,
by (4.3), rn_c is the sum of the degX#. Finally, degX# is independent of H
by the "Principle of Conservation of Number" (as Schubert termed it in 1874);
namely, p2*{p\ciO{l)l[CX}) = m[Xf] for some m, and m = degX# because

340
STEVEN L. KLEIMAN
the embedding g: 77 —► PN* is regular and g* commutes with P2* and with
Pîci0(l)\
Thus Ti = 0 for i < n — c and i > n, and T{ ^ 0 at the extremes. In fact,
(4.6) n^O iff n-c<i<n.
The following is a modified version of a proof found by Holme around 1978
(private communication, Nov. 11, 1985). If r^ = 0, but r^i ^ 0, then by
repeated application of (4.2), we may assume that ro(X) ^ 0, ro(Y) = 0, and
n > 0. Then, by (4.5), X' is a hypersurface, but Y\ is not. On the other hand,
obviously, in any case, Y\ contains the cone over p<ip~{xY with vertex M. In the
present case, P2\P\Xy is a finite map, because X' is a hypersurface and F is a
general hyperplane section. Hence, Y\ is also a hypersurface, a contradiction.
Let Lj be a ^'-dimensional linear space for 0 < j < N. Clearly, ri(Lj) = 0 if
i ¥" 3 by (4.6), and r^L») = degL* = 1 by (4.1). Now, let 7 denote (the graph
of) the point-hyperplane incidence correspondence. Then the classes
PÎdOur.pSdOU)"-1-*.!/]
form a basis of the (N — l)-codimensional classes in AI because 7 is a projective
subbundle over PN of the trivial bundle PN x PN*. Hence, by (4.4), the [CLj]
form the dual basis of (N — l)-dimensional classes, and
(4.7) [CX] = r0[CL0] + --- + rN_1[CLN_i] in AI.
There are two other proofs of (4.7). One involves degenerating X into a
hyperplane, and the other, the Schubert-Giambelli-Thom-Porteous formula; see [29,
pp. 175-177; 52, III-7, pp. 213-219].
Call X reflexive if CX = CXf. In practice, X is usually reflexive. It always
is in characteristic p = 0. Indeed, apply the following criterion.
(4.8) M ONGE-S EGRE-Wallace Criterion. X is reflexive iff CX -► X'
is separable.
To prove (4.8), use rudimentary Lagrangian geometry. This is essentially what
Monge (1805) did for curves and surfaces over C, C. Segre (1910) did in any
dimension over C, and Wallace (1956) did in any dimension and characteristic.
Indeed, identify 7 with the projectivized cotangent bundle of PN, resp. of
PN*. Let w, resp. w*, denote the contact form. (In dual systems p,q of
homogeneous coordinates, w = pdq_ and w* = gdp.) An elementary computation
shows that CX is a Lagrangian for w (that is, an (N — l)-dimensional solution
of w = 0). So CX is one for w*, because w + w* =0 (differentiate, 7: p. q = 0).
Now, an elementary computation shows that a subvariety D of 7 is contained in
C (P2D) if (1) w*\D = 0 and (2) D —► p^D is separable (= generically smooth).
Therefore, CX = CX1 if CX —► X1 is separable. The converse is trivial.
Let 77 be a general tangent hyperplane of X. If X is reflexive, then the
contact locus X# is a linear space; in fact, it is immediate from the definitions
that Xh is the linear space dual to the tangent space ThX' . Conversely, if Xh
is a generically reduced linear space, then by (4.8) X is reflexive.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 341
The following lovely result is an immediate consequence of (4.4).
(4.9) PlENE-URABE THEOREM. IfX is reflexive, ri(X) = rN-i-.i(X') for
all i.
The converse holds too. In fact, if the equality holds just for i = n — c, then
X is reflexive. Indeed, by (4.5) degX/j = 1; hence, Xh is a generically reduced
linear space, and by (4.8), X is reflexive.
A remarkable restriction on the number c is given in this theorem:
LANDMAN-ElN PARITY THEOREM. IfX is smooth and reflexive, but X is
not a linear space and Xf is not a hyper surf ace, then c is even; in fact,
c = 2ciN(Xh/H) where N(Xh/H) is the normal bundle.
Landman (see [52, II-(19), p. 195]) proved the evenness of c using Picard-
Lefschetz theory. Ein [18, II, 2.1] established an isomorphism
N(XH/H) ^Rom(N(XH/H),0(l))
from which the formula for c follows, since Xh is a linear space. The isomorphism
is given by the Hessian matrix of X.H in X at the various points of Xh\ the
matrix is nonsingular for a general tangent H iff X is reflexive.
There has been some recent work on classifying the nonreflexive plane curves
X in characteristic p > 0. Years ago, Wallace (1956) found essentially that X
satisfies a homogeneous equation of the form,
(4.10) G := a(x, y, z)px + 6(x, y, z)py + c(x, y, z)v z = 0.
Note that the partial derivatives Gx, Gy, Gz are ap, IP, cp. Hence, conversely,
if a plane curve satisfies G = 0, then by (4.8) it is not reflexive. Moreover, the
equation G = 0 defines a smooth curve iff a, 6, c have no common zero on the
curve.
The defining equation F = 0 of a nonreflexive plane curve need not be of
the form (4.10), although F must divide such a G. For example (Hefez, private
communication), in characteristic p = 2, consider
G := (x + z)2x + (y + zfy = (x2 + xy + y2 + ^2)(x + y);
the first factor F on the right defines a smooth (nonreflexive) conic.
Rathmann (private communication, Nov. 22, 1985) observed this. Let /i,...,
/jv-i define a smooth curve in PN. Let xo,..., xn be a system of homogeneous
coordinates. Let fa denote the jth partial derivative. Set & := xoff0 -+••• +
xxfiN- Then the gi define a smooth curve C. So a general projection A^ in
P2 has only ordinary nodes. Moreover, the Gauss map of C is obviously purely
inseparable. So the Gauss map of X is too; that is, X is not reflexive. However,
the defining equation F = 0 of X cannot be of the form (4.10) if p > 3, because
in the expansion of G about a point, if there is no constant or linear term, then
the leading term has degree at least p.

342
STEVEN L. KLEIMAN
On the other hand, suppose X is a smooth and nonreflexive plane curve of
degree d. Pardini [64] proved this: if p > 3, then (1) the polynomial defining X
is of the form (4.10) and (2) if d = p + 1, then X is projectively equivalent to
the curve
(4.11) Y : xd + yd~lz + zd~ly = 0.
For d = pe + 1 with e > 2, the curve Y of (4.11) is rather interesting (it was
considered by Wallace, 1956). While smooth but not reflexive, nevertheless Y
satisfies reciprocity: (Y')f = Y. In fact, Y1 has the same equation as Y.
Homma [43] proved this: suppose both X and X' are smooth; if d > 4, then
d = pe + 1 for some e > 1 and X is projectively equivalent to (4.11); if d = 3,
then p = 2 but the normal form of the equation depends on the ^'-invariant.
Homma observed (private communication, Dec. 11, 1985) that the curve with
equation
xpmx + zpmy + (xptzpm~pt + ypm)z = 0 with 1 < i < m
is smooth, but reciprocity fails (whence reflexivity fails); in fact, the curve's
double dual is a line (t = xp% is a separating transcendental for the dual, convenient
for computing the double dual).
It would be good to have an example of a smooth (nonreflexive) curve X such
that every tangent makes 2 or more contacts (in other words, CX —► X', while
inseparable, is not purely inseparable) or to prove that such X do not exist.
Suppose now that X is smooth of arbitrary dimension n. Then, clearly,
(4.12) CX = P{N*{X/PN){1)),
where N*(X/PN) is the conormal bundle. Hence, by (4.4) and (2.4),
(4.13) n = J ClO(iy . sn-i(N*(X/PN)(l)). [X].
For example, if X is a hypersurface, then N*(X/PN) = Ox{—rn)', so
(4.14) ri=rn(rn-l)n-\
A version of (4.14) for plane curves was given by Goudin and du Séjour in 1756.
The tangent bundle-normal bundle sequence of X in PN and the "Euler"
sequence relating 0(1) and the tangent bundle Tpn yield:
(4.15) u = f>l)J r^lj JciOiiycn-jiTxm
Similarly, the intrinsic invariant C{(Tx) can be expressed in terms of ciO(l)
and the [X(A)]; originally, Todd (1937) introduced the canonical classes of X in
essentially this way.
Singularities necessitate correction terms. The terms required by nodes and
cusps on a plane curve were found by Poncelet (1822) and Plucker (1836).
Teissier (1975) found the following lovely generalization.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 343
(4.16) TEISSIER'S PLUCKER FORMULA. If X is a hypersurface with
isolated singular points P and if ep denotes the multiplicity at P of the Jacobian
ideal (that is, the Fitting ideal Fn(fi^ P)), then r0 = rn(rn — l)n — YleP-
For example, if X is a plane curve with d simple nodes and k simple cusps,
then a simple computation yields
(4.17) r0 = ri (ri — 1) — 2d — 3k in characteristic p ^ 2.
If p = 2, the coefficient of A: is 4 instead of 3.
Piene [67, Corollary 3.7, p. 266] and Dolgachev (1985, private communication)
gave formulas like (4.16) for hypersurfaces with nonisolated singularities.
Correction terms for (4.15) with i = n — 1 were found by Pohl (1964). For
curves (n = 1), Pohl's result is this.
(4.18) POHL'S PLUCKER FORMULA. IfX is a curve, Z the normalization,
g the genus of Z, and kç the multiplicity at Q of the Fitting ideal F°(n]Lx),
then
r0 = 2g-2 + 2rl-YJkQ.
For example, if X is a plane curve with k simple cusps, then
(4.19) r0 = 2g - 2 + 2rx - k if p ^ 2.
If p = 2, then the coefficient of A: is 2.
If X is a reflexive plane curve, then its normalization Z is also the
normalization of X', and by (4.9), t-0(X) = ri(X') and ri(X) = r0(X'). Hence, applying
(4.17) and (4.19) to X' yields two new formulas. In particular, if X has only
simple nodes, cusps, inflectional tangents, and bitangents, say d, k, d', and k' of
them, then eliminating g yields Plucker's 4 original formulas:
(4 20) r° = fl(ri " 1} ~ 2d " 3K ri = r°(r° ~ 1} " 2d' " 3A;/' if v * 2
[ ' } k' = 3n(fi - 2) - 6d - 8Jfe, k = 3r0(r0 - 2) - 6d' - 8k' VT '
The left and right formulas are dual. Plucker, one of the leading pioneers of the
Principle of Duality, made a point of treating X and X' equally.
For surfaces X in P3 with various types of singularities, Salmon (1847-49),
Cayley, Noether (1871-73) and Zeuthen (1876) developed a series of formulas
like Plucker's (4.20). Piene [66] gave an extensive modern treatment.
For curves X in PN, Cayley (1845) and Veronese (1882) developed a series
of formulas generalized Plucker's (4.20). In fact, the formulas were obtained by
applying Plucker's to the N — l plane curves derived from X via various
combinations of these 3 commuting associative operations: (1) general central projection
into PM; (2) section by a general m-codimensional plane of the developable
variety ruled by the osculating m-planes; and (3) formation of the "strict" dual
curve, the curve in PN* of osculating hyperplanes.
The formulas relate 4 types of numerical characters: (i) the numbers of
singularities of these curves, which may be interpreted as the total numbers of actual
and apparent double osculating spaces; (ii) the osculating ranks rm, the number

344
STEVEN L. KLEIMAN
of osculating m-planes cutting a general (m + l)-codimensional plane; and (iv)
the stationary indices ki, the numbers of hyperosculating i-planes, those making
excessive contact with X. Piene [65] gave a modern treatment of all this theory,
assuming that the characteristic p is 0 or is so large that only "classical"
osculation occurs. Under essentially the same assumption, Eisenbud and Harris [19]
reprove one of the formulas and use it in an important study of general linear
series on general curves. Classical osculation will also occur, according to the
remarkable Theorem 2.7 of Neeman [63], for embeddings of degree b + kpm for
suitable 6, m > 0 and all k > 0.
The nonclassical case too is remarkable. The notion of osculating i-plane as
the (unique!) i-plane with maximal contact must be replaced by the more
refined notion of associated i-plane to preserve continuity. Moreover, the maximal
contact of an i-plane at a general point may be greater than i + 1. For example,
a plane curve is nonreflexive in characteristic p > 2 iff the contact of the
tangent line at a general point is greater than 2 (in which case, it is equal to the
inseparable degree of the map CX —► X').
There is an intelligent replacement for Pohl's formula (4.18) in the case of
the ith associated map, which carries a point P to the osculating i-plane at P,
viewed as a point in the Grassmannian embedded via the Plucker embedding.
The associated map is given by a refined Wronskian, so the stationary points are
also called Wronskian points. When X is embedded by the canonical series, then
the theory of refined Weierstrass points, initiated by F. K. Schmidt in 1939, is
recovered; so the refined stationary points of the osculating hyperplanes of any
embedding are also called Weierstrass points.
The modern theory was developed notably by Laksov [56, 58]. Related work
was done by others. Homma [43] thus got the results quoted after (4.11). Stohr
and Voloch [89] got an elementary proof, like Stepanov's, of Weil's Riemann
hypothesis and obtained some improvements on it, like those of Stark, Drinfeld-
Vladut and Serre, when the genus is large enough with respect to the number of
elements in the field of definition. They enumerate the points P whose images
under the Frobenius map lie on the osculating hyperplane at P.
5. General multiple-point theory. Let /: X —► Y be a proper map
between separated, universally catenary noetherian schemes. Form this diagram:
X2 = {X/Y)2 := P(/(d)) where I{d) is the ideal of d
ir
X J^- X Xy X «— X where d is the diagonal embedding
Y J- X.
Set /i := p2r and m2 := fi*[X2]. Call f\: X2 —► X the derived map of / and
m2 = rn2(f) the double-point cycle.
Consider the map r. It is an isomorphism over the complement of the diagonal
subscheme dX, because the subscheme's ideal 1(d) is equal to the structure sheaf

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 345
there. Its behavior over dX depends on the ramification of /; in fact,
r-1dX = P(fi}).
It is not hard to see that X2 is canonically isomorphic to the total space of the
universal flat family of length-2 subschemes in the fibers of /.
Set D := P2TX2. Its geometric points P are obviously of two types: (1) the
strict double-points of /, those for which there is a second geometric point Q such
that Q ^ P and f(Q) = f{P)\ and (2) the cuspidal or ramification or stationary
points of /, those such that fi}(P) ^ 0. The complement X — D consists of the
points P such that f~l fP = Spec(A:(P)). So X — D is the maximal open set on
which / is a closed embedding, because / is proper. Thus D merits its name,
the double-point set
Often in practice, / is well behaved at the generic point P of each component
of D\ namely, f~lfP is finite and dim(Q^(P)) < 1. Then, obviously, D is the
support of the double-point cycle ra2 = wi2(/) := /i*[X2].
Suppose / is regular of codimension n. Denote the nth Chern operator of the
virtual normal bundle Nf by cn(f). Then the Double-Point Formula is this:
(5.2) m2 = rU[X)-cnU)[X\.
For example, (5.2) yields Clebsch's formula (1.1). Indeed, suppose that Y =
P2 over a field fc, that X is a smooth curve, and that /: X —► Y is finite
and birational onto its image, fX. Then, obviously, //*/*[X] = r\ where
n = deg/X. Also, Cl(/) = c1(rTY) - Cl(Tz); so, f cx{f)[X] = 3n - (2 - 2g).
Thus, the right side of (5.2) yields that of (1.1).
Now, ra2 is equal to the cycle of the subscheme whose ideal is the conductor.
This fact is given a computational proof in [26, Theorem 3]. It would be good
to have conceptual proof. Perhaps one may be based on the fact that X2 is the
total space of the universal flat family of length-2 subschemes in the fibers of /.
Possibly some proof will work in higher dimension.
At any rate, if fX has only simple (rational) nodes and cusps, then ra2 can
easily be computed directly, because X2 is the residual scheme in X Xy X of
the diagonal, which is a divisor in X x^ X. For instance, if P is a ramification
point, then the local ring of X2 at (P, P) is equal to fc[[s,£]]/(F,G), where F :=
(s2 - t2)/{s - t) = s + t and G := (s3 - t3)/{s - t) = s2 + st + t1. Notice
that the multiplicity of ra2 at P is 2, although dim(fij.(P)) = 1; in other words,
while r: X2 —► X Xy X is an embedding at (P,P), the image sticks out of the
diagonal. Thus, (5.2) yields (1.1).
The general Double-Point Formula, (5.2), is valid when:
,- „x (i) / and /1 are regular; (ii) cod(/i) = n [:= cod(/)]; and
(iii) / is strongly projective and X embeds in a regular scheme.
If X is Cohen-Macaulay, then /1 is regular if / is regular and cod / = cod /1. So
if X and Y are smooth and projective over a field, the only new requirement is
that cod/ = cod/i. The hypotheses (5.3) are those of the Residual Intersection

346
STEVEN L. KLEIMAN
Theorem, (2.8), and it yields the Double-Point Formula as follows:
m2 =pi*r*[X2] = Pi*(/*[X] -<W/)[X]) = rMX}-Cl(f)[X}.
The first equality holds because, obviously, the switch involution of X Xy X
leaves r*[X2] invariant.
A fruitful way to develop higher-order multiple-point theory is to iterate.
The procedure is motivated by the following observation of Salomonsen (private
communication, July 3, 1976): A geometric point (P, Q) oiXxy X with P ^ Q
is the image of a double-point of /i : X2 —► X iff there exists a second such pair
(it!, Q) with it! t^ P, iff (P, Q, P) is a strict triple-point of /; that is, it is a triple
of distinct geometric points of X with the same image in Y. Now, set
m3 :=/i*m2(/i).
Then, the Double-Point Formula for ra2(/i) yields this Triple-Point Formula:
(5.4) m3 = /7.m2 - 2c„(/)m2 + I ]T 2n->c,-(/)c2_i(/) J [X].
To obtain the right side requires (2.10) and an expression for the virtual normal
bundle Nfx in terms of Nf. Finally, (5.4) is valid if (5.3) obtains and if the
derived map /2: X3 —► X2 of /i is regular of codimension n.
Multiple-point formulas of all orders, s, may be obtained similarly by turning
the crank. However, at present, each formula has to be worked out separately
for each s > 4 and each n > 0. There is one exception:
(5.5) The Herbert-Ronga Formula. ms = /* f*m9 - scn(f)ms iff
is unramified.
The various multiple-point formulas are valid if (1) (5.3) obtains and if (2)
for 1 < i < 5, the derived map fo of fa-i is regular of codimension n.
Condition (2) is a problem. It may be weakened trivially by requiring only
that there be a subset S of Y such that cod(/_15/X) > sn and the restriction
f\(X — f~xS) satisfy the condition. However, even this weaker condition
cannot obtain when s > 4 if there are 52-singularities (points P of X such that
dim(fi}.(P)) > 2) unless 5 = 4 and n = 0,1,2,3 or s = 5 and n = 0, 1 or s = 6
and n = l. It may well be that outside this range the presence of S2-singularities
necessitates new terms in the formula for ras.
On the other hand, instead of seeking new terms, it might be better to try
to enumerate only the curvilinear subschemes and their limits in the fibers of
/. A cu7vilinea,7 scheme over a field is one of embedding dimension < 1 at each
point P; in other words, it is, locally at P, analytically embeddable in P1, or
equivalently, dim(Qi(P)) < 1. The notion was introduced by Le Barz, and has
been used with success by him and by others; see below and §6.
Condition (2) should really be weakened to the condition that the composition
fs-i • • • /i be regular of codimension n. One way to do this, at least in principle,
for smooth varieties over a field k was proposed in [46, pp. 389-390]. Namely, it

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 347
is clear that (X/Y)3 is contained in [X/k)9 and is equal to the residual scheme
with respect to a certain divisor of the preimage of the r-fold diagonal of (Y/k)X3.
So the Residual Intersection Formula will yield the class \{X/Y)9], and pushing
the formula down to X will yield an s-point formula. (Ran [73, end of §3] kindly
points out an error on p. 390 concerning a related matter: the scheme Zr there,
here (X/k)s, does not always, if r > 4, parametrize a natural flat family of length
r-subschemes in the fibers of /.)
Ran [73, 74] refined the above approach as follows. He uses the following
cartesian diagram, where X is embedded inXx^ Y via the graph map:
(X/Y)9 - (X/k)s
1 1
(X/k)s xkY=(X xk Y/Y)8 - (X xk Y/k)a
He applies the Residual Intersection Formula recursively to get a formula for
[(X/k)3] in A((X xk Y)/k)8. The formula is valid because the schemes involved
are smooth of appropriate dimensions; the dimension of (X/Y)i for 2 < i < s is
irrelevant. Then Ran restricts the formula to (X/k)s. This approach appears to
be better for computations, and Ran has made some. However, nothing explicit
like (5.4) has come out yet.
Another approach to higher-order multiple-point theory was discussed at the
end of [50]. The idea is to apply the Residual Intersection Formula, (2.9), to the
case that Yf = Hilby."1 and W is the universal family. Then R parametrizes the
universal flat family of length-1 extensions, in the fibers of /, of the family W/H.
For 5 = 2, this approach is just the one used to get the Double-Point Formula,
(5.2). For s > 3, the formulas obtained this way are more refined than those
obtained via iteration, because the (s — l)-tuple of other points Q associated to
an s-fold point P is taken unordered.
In the preceding way, a more refined form of (5.5), modeled on Formula (1) in
[39, §2], and an analogous more refined form of (5.4) may be proved. Moreover,
in the latter case, the actual computation involved is substantially shorter. The
formulas derived this way are valid if (1) (5.3) obtains, (2) p^r is regular of
codimension n, and (3) P2W is regular, necessarily of codimension 0. Condition
(2) is analogous to the condition that the composition /s_i • • • /i be regular of
codimension (s — l)n. Condition (3) is automatic if (a) / is unramified, or if
(b) there are no ^-singularities, or if (c) s < 3. If s > 4 and if there are 52-
singularities, then p^w will not be regular. In that case, it may pay to replace
Hilby--1 with the subscheme parametrizing the curvilinear subschemes in the
fibers of /.
Stationary multiple-point theory concerns the enumeration of the s-tuples
(Pi,..., Ps) of points of X such that (1) the P{ all have the same image under
/ and (2) for a given partition (a) = (ai,..., am) of s (where a\ < • • • < am),
the first a\ P's lie infinitely near each other, the next a^ P's lie infinitely near
each other, etc. The theory has been advanced recently. Roberts [78] began
the recent work, Colley [9] has completely generalized the method of iteration,

348
STEVEN L. KLEIMAN
and Ran [73, 74] has been especially concerned with weakening the hypotheses
required for validity in the curvilinear case.
6. Applications of multiple-point-theory—allied theory. Let Z be
a closed n-dimensional subvariety (= integral subscheme) of PN over an
algebraically closed field of arbitrary characteristic p.
A number of applications of multiple-point theory have been made to the
study of numerical characters of Z. The theory is applied to the natural map
/: X —► Y where X is the normalization of Z and Y = PN.
Multiple-point theory was used in this way in Piene's treatment [66] of the
Salmon-Cayley-Noether-Zeuthen formulas for surfaces Z in P3, in Piene's
modern version [68] of Noether's own proof of his famous formula,
l2E{Ox) = c\+c2,
for a smooth surface X, and in the treatment by Piene and Ronga [71] of the
analogous formula for a smooth 3-fold. In the latter two cases, X is given and
Z is taken as the image of X in Y = PN under a general projection /.
Multiple-point theory has been applied similarly to enumerating the
singularities of various types on suitable varieties ruled by a 1-parameter family of linear
spaces by Roberts [78], Kleiman [50], and Colley [10]. For example, Colley
(Example 5.6) and Roberts (p. 93) show in this way that a plane form in P4 of
degree r3 and sectional genus g has, just as Roth (1931) found,
2r\ - Ugr3 - 24g + 24
triple-points at which 3 branches meet but two generating planes coincide.
Another application of multiple-point theory has been to enumerate the mul-
tisecant lines of the variety Z in PN. In this application, X is the set of pairs
(P, L) where P is a point of the variety and L is a line through the point P,
and /: X —► Y is the projection to the Grassmannian. For example, Colley [11,
Theorem 6.8, p. 67] takes (a) = (2,2) and proves that the number of bitangents
to a smooth generic surface Z in P4 is
(6.1) \rl(r2 - l)2 + r\ - r2r1{r2 - 1) - 24r| + 150r2 - 15n - 18c?,
where c\ is the first Chern class of (the tangent bundle of) Z. However, the
formula may be put in a more nineteenth-century form by replacing c\ by its
value in terms of the "Oth type" so, which is the number of tangent planes
through a general point. (In general, the iih type of Z is the degree of the
"polar locus" of points P whose tangent space TP meets a general (2n — i)-
codimensional plane. If Z is smooth, then this locus represents the (n — i)th
Chern class of the twisted conormal bundle, N*(Z/PN)(1).)
It is tempting to try to prove (6.1) by applying the Double-Point Formula
(5.2) to the restriction of / to the subset of pairs (P, L) such that L is a tangent
line at P, as was done in [6, pp. 151-166]. However, as Colley [11, p. 68] says,
the wrong formula comes out, because the derived map of the restriction is not

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 349
regular of codimension 1; the Double-Point Set D includes the 1-dimensional
family of (P, L) such that L is a flex tangent.
A similar situation occurs for plane curves. In that case, the point-tangent
line pairs form the conormal variety, and the restriction of / is the projection to
the dual curve. The Double-Point Formula counts both the double tangents and
the flex tangents; see (1.1) and the paragraphs following (5.2).
Similarly (see [11, §4, pp. 55-59]), the Stationary (3)-Point Formula applied
to the map / itself yields the Plucker formula (4.20) for the number k' of flexes.
However, contrary to what Colley says on p. 59, the (3)-Point Formula is the same
whether or not the curve is singular. Indeed, the normal bundle of / depends
only on the normal bundle of the curve in P2, and it reflects only the degree of
the curve. Furthermore, local computations made by the author show that, in
characteristic p ^ 2, each simple flex, node, and cusp count with multiplicities
1, 6, and 8, just as they should.
For an arbitrary Z, given a partition (a) = (ai,... , am) of s > 2, it is not
hard to see that the corresponding subscheme of X3 parametrizes the universal
flat family of length-s subschemes of X (i) that are alined (that is, they are also
subschemes of a line L) and (ii) that are the union of k ordered components, the
first of length ai, the second of length a2, etc. The case in which Z is a curve in
P2 or a curve or surface in P3 was studied in the last century and is discussed
in [84, §§33-34, 41-43].
In a recent series of impressive articles, Le Barz has extensively developed the
theory of alined and curvilinear subschemes of finite length in PN via Hilbert
scheme techniques, and he has applied it to enumerate multisecants. For
example, in [62], he not only (correctly) established (6.1), but he generalized it to the
case of a surface with ordinary double points. Moreover, he proved that if the
surface contains an isolated line whose self-intersection number is t, say, then
the line counts as 4(3J*) bitangents.
The full Hilbert scheme HilbmP2 has recently proved tractable and useful.
It is smooth (Fogarty, 1968), and its intersection ring A(HilbmP2) has been
determined additively, and for m = 3, also multiplicatively by Fogarty [25],
Hirschowitz [41], Le Barz and Elencwajg [20, 21, 22], and Ellingsrud and
Str0mme [23, 24]. Hirschowitz uses a theorem on rational equivalence. Le
Barz and Elencwajg use direct geometric means. Ellingsrud and Str0mme use
Bialynicki-Birula's theorem that a smooth projective variety having a Gm-action
with finitely many fixed points has a cellular decomposition; for m = 3, they
find some vector bundles whose Chern classes generate multiplicatively, and they
find the relations.
Here is an interesting application of the above work made by Le Barz and
Elencwajg. A curve Z in P2 determines two curves in Hilb3P2: one is the
closure of the locus of a point representing the (curvilinear) length-3 subscheme
supported by a simple point of Z and contained in Z, and the other is the locus of
a point representing the "amorphous" length-3 subscheme supported at a point

350
STEVEN L. KLEIMAN
of Z (namely, the one defined by the square of the maximal ideal of the point
in P2). Similarly a 2-parameter family of curves in P2 determines a divisor:
the union of the Hilbert schemes of the various curves. Say that Z is smooth of
degree r\, that a general member of the family is of degree of, and that / members
pass through two general points. Then Z is osculated by
(6.2) 3ri/(d + n-3)
members of the family, and there are
(6.3) 3n/(d-l)
singular points of members located on Z. These formulas are established by
expressing the corresponding cycles on Hilb3P2 in terms of a canonical basis and
then computing their intersection product. The proof of (6.3) makes effective
use of the amorphous subschemes.
In the same vein as the work on Hilb3P2 is the work in the enumerative theory
of triangles, initiated by Schubert (1880), revived by Semple (1954) and Tyrrell
(1961), and recently advanced by Roberts and Speiser [79, 80], and by Collino
and Fulton in work in progress. It respects duality!
The most basic variety of triangles lies in P2 x P2 x P2 x P2* x P2* x P2*.
It is the subset W of 6-tuples (Pi,P2,Pz,Li,L2,Lz) such that L\ contains P<i
and P3, etc. To provide second-order information, assign to each nondegenerate
triangle the point of the Grassmannian parametrizing the 2-dimensional linear
system of conies through its vertices, and define W* as the closure of the graph
of this correspondence. Then W* is smooth (Semple), and its intersection ring
AW* is just what Schubert found.
Collino and Fulton determine the ring using Bialynicki-Birula's theorem, and
they use its structure to prove this: given two curves of degrees r and s and
classes r' and s', the number of triangles inscribed in the first and circumscribed
about the second is
\r(r - l)s'{s' - l)(2rr' - 3r - 3s' + 4).
Cayley (1871) found the formula by direct means. Schubert rediscovered it using
his triangle calculus, but (according to Collino and Fulton) he made an error and
arrived at the wrong formula.
The intersection theory of W* is approached by Roberts and Speiser via some
auxiliary triangle varieties of Semple's and a key new one. They [80, II, §4] use
it, for example, to prove a more general version of (6.2) and to prove this: given
two general 1-parameter families of plane curves, the number of points at which
a curve in one family osculates one in the other is
(6.4) Z0*o + hk* + k0l* + fci/J + 3/0/5 + 3/i/Î,
where /o is the number of curves in the first family through a point, l\ the
number tangent to a line, k\ the degree of the cusp locus, k0 the class of the
locus enveloped by the flex tangents, and Iq, /*, A;*, k$ are the corresponding
characters of the second family. The amorphous subschemes appear as points of

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 351
curvature oo, or cusps, and are dual flexes. Zeuthen (1879) found (6.4) by direct
means, and Schubert (1880) recovered it using his triangle calculus.
Another application of multiple-point theory has been to enumerate the
exceptional multisecant linear spaces of arbitrary dimension of a curve Z in PN;
the linear spaces may also be required to satisfy some Schubert conditions. Here
X is taken to be the set of pairs (P,£), where P is a point of the
normalization Z* of Z and L is a linear space of given dimension in PN containing the
image of P, Y is the Grassmannian, and /: X —► Y is the projection. Since
Z* is a smooth curve, there are no S2-singularities. Hence, X3 embeds in the
product (X/y)xs, and given a partition a of s, X(&) may be analyzed in XXs or
in (X x Y)xs. In fact, with care, Z may be varied in a family. If, for example,
Z is embedded by the canonical series, then the exceptional secants correspond
(under pullback) to special divisors. In this way, Ran [73] recovered results of
Beauville's and of Mumford's about special divisors.
Instead of Xs, the 5-fold symmetric product of Z* may be used. It was
by MacDonald (1958, 1962), Mattuck (1965), and Arbarello-Cornalba-Griffiths-
Harris [2, Chapter VIII]. Different techniques are involved. The intersection ring
of the symmetric product is described, and the class of the image of (Z*/fc)(a)
found. The image of (X/F)(a) is identified as the intersection of the image
of (Z*/fc)(o) and the degeneracy locus of a map from a free bundle to a
certain "secant" bundle, obtained by pulling back O(l) to the universal family
of length-5 subschemes and then pushing it forth to the symmetric product.
(The notion of secant bundle was introduced by Schwarzenberger, 1964.) By
the Schubert-Giambelli-Thom-Porteous formula, the fundamental class of the
degeneracy locus is a certain polynomial in the Chern classes of the secant
bundle, and they may be found using the Grothendieck-Hirzebruch-Riemann-Roch
theorem. Finally, some involved computations are required.
Ran [73] shows that if the secant bundle is pulled back to Xs, then it has
a convenient recursive description and the Grothendieck-Hirzebruch-Riemann-
Roch theorem is no longer needed. In fact, Ran gives the description for a variety
of arbitrary dimension. Moreover, he ties in some work of his on a normal bundle
and he ties in some related work of Le Barz.
Another way to enumerate the exceptional secant planes is to study their
locus in the Grassmannian. This method was used by Levi, Tanturri, and others
around 1900 and again recently by von zur Gathen [93] and Gruson and Peskine
[34]. So far this method has not been as fruitful as the other ways.
Closely related to the enumeration of exceptional secant planes is de Jon-
quieres's formula. De Jonquieres himself considered a plane curve with simple
nodes, say of degree ri and geometric genus g. By degeneration and application
of the Correspondence Principle for rational curves, he found a formula for the
number of curves of degree d that meet it with multiplicity a{ at some point P{
for 1 < i < m and that pass through an appropriate number of auxiliary points,
some lying on the curve and some infinitely near others. Set v := dr\ — Ylai-

352
STEVEN L. KLEIMAN
Let Sj(xi,...,xm) denote the yth elementary symmetric function. Then the
number is
m+v / \
(6.5) ((^ai)/vl)£{m + v-j)ljllgj\S,ia1-l,...,am-l).
If Z is reembedded via the dth Veronese map, then the curve sections become
hyperplane sections, and (6.5) enumerates the exceptional secant hyperplanes of
type (a). Nowadays, de Jonquieres's formula is usually stated and proved in the
corresponding form for an abstract linear system.
For example, take m = 1 and a\ = a. Then (6.5) reduces to
(6.6) a[dn + (a-l)(0-l)]
because So(x) = 1, S\(x) = x, and Sh{x) = 0 for h ^ 0,1. For instance, if a = 2
and d = 1, then (6.6) yields 2{r\ + # — 1) as the number ro of lines through
a point and tangent to the curve Z, in agreement with (4.19) because k = 0.
Plugging this result in (6.6) yields
(6.7) Ua)(a ~ 1)fo + a(d ~ a + l)^i,
a formula obtained by Cayley (1868) using his "functional" method. If a = 3
and d = 1, then (6.7) yields
(6.8) 3(7-0-7-!)
as the number kf of flexes, in agreement with (4.19) and its dual (assuming the
curve is reflexive). If a = 3, then (6.7) agrees with (6.2) because / = 1 since the
system of curves of degree d is linear. It is an open problem to find a common
generalization of (6.7) and (6.2) with a and / arbitrary.
If a = 6 and d = 2, then (6.7) yields
(6.9) 3(5r0-67-i)
as the number of conies that satisfy the irreducible 5-fold condition of making
a 6-fold contact with the curve. The points of contact are called the sextactic
points of the curve.
Formula (6.9) counts each flex as a sextactic point. So Halphen (1876)
subtracted (6.8) from (6.9) to get 3(4t*o — 5ri) as the number of proper sextactic
points. Halphen obtained the formula using a Wronskian, like the one mentioned
at the end of §4 in connection with the enumeration of exceptional osculating
spaces. This method has several advantages. (1) It enumerates points, rather
than curves, and there may be infinitely many curves. For example (Hefez,
private communication, Spring 1983), there are infinitely many conies that intersect
the curve with equation y = x + x7 + ys with multiplicity 6 at the origin. (2)
It is not hard to compute multiplicities of appearance. For example, it is not
hard to see that each flex counts as a single sextactic point, or that if the curve
has cusps, then each cusp counts as a double flex in (6.6) with a = 3 and d = 1.
(3) In positive characteristic, when nonclassical behavior appears, the refined
Wronskian may be used.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 353
A variant of the preceding approaches to the proof of de Jonquieres's formula
was developed by Vainsencher [91]. In the direct product Zxm x P^*, he
identifies the locus of points (Pi,..., Pm, L) such that the divisor a\ P\ H hamPm is
contained in the divisor cut out by L; it is equal to the locus of zeros of a section
of a bundle of the appropriate rank. The bundle has a natural filtration, whose
successive quotients are readily identified. So its Chern classes may be
computed without the Grothendieck-Hirzebruch-Riemann-Roch theorem. In fact,
the natural setting for the method is a family of divisors on a family of smooth
projective varieties. Earlier, Lascoux [60] had already proceeded similarly in the
case that m = 1 and Z is a curve or a surface.
No explicit general formula like de Jonquieres's has been found in higher
dimensions. Nevertheless, Vainsencher proceeding as above recovered the following
old formulas among others:
1. (Salmon, about 1848) The number of tritangent planes to a
surface of degree d in P3 is
^d9 - 6ds + 15d7 - 5M6 + 204d5 - 339d4
+ 770d3 - 2056d2 + 1920d).
(This formula was also recovered by Piene [66] and by Ronga
[81] using two other methods.)
2. (Schubert [84, 19, p. 236]) The number of doubly
inflectional lines of a surface of degree d in P3 is
d{d - 4)(d - 5)(d3 + 3d2 + 29d - 60).
3. (Salmon, about 1848) The number of planes in P3 cutting
a general surface of degree > 4 in a curve with a cusp and a
node is
4d{d - 2){d - 3){d3 + 3d - 16).
4. (S. Roberts, 1867) The surfaces of degree d in P3 that
have at least 2 (possibly infinitely near) double points form a
variety of degree
2{d - l){d - 2)(4d3 - Sd2 + Sd - 25).
Vainsencher establishes the enumerative significance of these numbers too!
7. The contact formula—the Galois group. Work in PN over an
algebraically closed field of arbitrary characteristic p.
Let X be a closed subvariety of PN. The linear form
r(X) := r0(X)l0 + • • • + rN-X(X)lN_u
where r^(X) is the ith rank of X (see §4) and l{ is an indeterminate, is called
(following Chasles, 1864) the module of X.
Let V vary in an irreducible m-parameter family of closed subvarieties of PN,
without repetitions. Let Xi,... ,Xm be fixed closed subvarieties of PN. And

354
STEVEN L. KLEIMAN
let (#1,..., gm) be a general ra-tuple of linear transformations of PN. Then [29,
52]:
(7.1) THE CONTACT THEOREM. I. (The formula.) The weighted number
#ofV tangent to g\Xi,..., gmXm is finite and given by the formula
# = r(Xi).r(X2)...r(Xm).
The product is evaluated by expanding it formally, interpreting each monomial
jjoiji i3n-\
lo li ' 1n-i
as the corresponding characteristic number of the family, namely, as the number
ofV tangent to ji general i-planes for 0 < i < N — 1. If for some i and most V
dim V + dimXi <N-2 or dim V" + dimX^ < N - 2,
where V and X[ are the dual varieties in P^*, then # = 0. In particular,
W ' • ' lJN-l = ° ifJi >0 andi<N-l- dim V.
II. (The significance) A. Consider a tangent V. Then V is a general member
of the family. Fix i and say V is tangent to g{X{ at P. Then P is a general
point on V, and P is a general point on g{X{; in particular, it is a simple point
on both varieties. Moreover, the sum of the tangent spaces
H-.^TpV + TpgiXi
is a hyperplane; in fact, it corresponds to a general point of giX[ and to a general
point ofV'. Also, P is an isolated point in the intersection of the contact loci,
Xh '(giXi)H- If y and g{X{ are reflexive, then this intersection is dual to the
sum ThV + TngiX'i, and it is the hyperplane in PN* dual to P. Finally, if
dimV + dimXi = N — 1, then P is isolated in the intersection V . g%Xi; in fact,
this intersection is finite.
B. Suppose the characteristic p = 0. (1) If dimV + dimX^ > N, then the
intersection V.giXi has an (isolated) nondegenerate double point at P; that is,
the tangent cone at P is defined in the Zariski tangent space, which is equal
to the intersection TV .TpgiXi, by a quadratic form whose Hessian matrix is
nonsingular. (2) V and g{X{ are not tangent at a second point, distinct from P.
(3) V counts once in #.
C. Suppose p > 0. Then each contact counts with the same multiplicity q, and
q = pe for some e > 0. If p\#, then q = 1. If q = 1, then B. (1) obtains. If
p t^ 2 (and probably also if p = 2) and if all the Xi and almost all the members
of the family are reflexive, then B. (2) obtains.
The idea of the proof of (7.1) is this. Let S be any completion of any nonempty
open subset So of the parameter space of the family. Let I denote the (graph
of the) point-hyperplane incidence correspondence of PN. Let UCVxm denote
the closure in 7Xm x S of the set of (m + l)-tuples
((Pi,ffi),...,(Pm,ffm),V)

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 355
such that (i) V is in So, (ii) Pi is a simple point of V, and (iii) Hi contains the
tangent space TPiV (that is, (Pi, Hi) lies in the conormal variety CV). Let pi
denote the iih projection of 7Xm x S. Then
# = JpWCqM - "VUCQmXm] • [UCVxm].
Hence the Contact Formula follows from (4.5). The remaining assertions follow
after a little work from general transversality theory [44,90] and from Goldstein's
[32] theory of a generalized second fundamental form of two smooth varieties
meeting in a third.
If initially there are repetitions in the family, replace the parameter space by
its image in the Hilbert scheme. Otherwise, the weight of each tangent V would
be divisible by the number of times V is repeated, namely, the degree of the map
into the Hilbert scheme. In [52, III-(5), p. 201, and III-(9), p. 206], this factor
was inadvertently left out. Also, regret ably, it is stated there that the Contact
Formula is valid whenever # is finite; of course, this statement may be false if
the parameter space is not complete.
For instance, suppose that the V are curves in P3. If some Xi is a point, then
# = 0 by the vanishing assertion; basically the reason why is this: it is a double
condition that V be tangent to a point, that is, pass through it. If some Xi is a
curve and V is tangent to g{X{ at P, that is, if it cuts it there, then the assertion
that H := TpV + TpQiXi is a (hyper)plane just says that the two tangent lines
are distinct.
In an interesting and famous example, first considered by Steiner in 1848, V
varies in the 5-parameter family of all plane conies and the Xi are nondegenerate
conies. By (4.14), r0(X*) = 2 and n(Xi) = 2. So, by (7.1),
# = (2/0 + 2/i)5 = 25(/£ + 5/g/i + 10/g/f + •••),
where /$, IqIi, ... are interpreted as the characteristic numbers of the family. If
p ^ 2, then these numbers are 1,2,4,4,2,1 (see §8); hence # = 3264, just as
Chasles (1864) claimed, not 7776 as Steiner suggested.
Each of the 3264 conies V is nondegenerate, and none is bitangent to any
of the 5 fixed conies. The V are distinct, and none osculates any of the fixed
conies, by the general theory, if p ^ 3, 17. If p = 3, 17, then the statement is
still true [30, §7]; in fact, if no two Xi are tangent, if there is no pair of points
such that each X» contains one or is tangent to their join, and if there is no pair
of lines such that each X» is tangent to one or contains their intersection, then
there are 3264 conies meeting the Xi, and each is nondegenerate and counts with
multiplicity
5
(7.2) H(4-aud(XVi)).
i=i
Suppose p = 2. Then the number of conies V tangent to 5 nondegenerate
conies in general position is 51. The V are nondegenerate and distinct, and none

356
STEVEN L. KLEIMAN
is bitangent nor osculates. This was proved by Vainsencher [90], who worked
with the blowup of the P5 of conies along the hyperplane(!) of double lines.
From the present point of view (see [52, III-5, pp. 209-211]), the characteristic
numbers are 1,2,4,0,0,0 and # = 25 • 51; each of the 51 V counts in # with
multiplicity 25 = 32. Moreover (Higman, private communication, Spring 1978),
under the reduction from p = 0 to p = 2, half of the 3264 conies tangent to 5
nondegenerate conies in general position degenerate, and the other half coalesce
in 51 groups of 32 each to the 51 conies tangent to the reductions of the 5.
There is a group associated to any enumerative problem that, like the contact
problem, has variable algebro-geometric conditions. The group measures the
independence of the solutions. By definition, the group is the Galois group of
the field of definition of the solutions over the field of definition of the generic
conditions. However, it may be viewed as a monodromy group, at least over C
(see [35]): varying the parameters of the conditions along various closed paths
presents the group as a subgroup of the group of permutations of the initial
solutions.
The solutions are fully independent when the group is the full symmetric
group. The group is the full symmetric group when it is doubly transitive and
contains a transposition. Simple transitivity holds when, over a dense open
subset of the parameter space, the space of solutions is irreducible. Double
transitivity holds when the corresponding fibered self-product minus the diagonal
is irreducible too. The group contains a transposition when there is a simple
point of the parameter space at which exactly two of the solutions coalesce and
the solution space is analytically irreducible at the double solution. This theory
was developed over C by Harris [35], who used some topological methods, and
algebro-geometrically in arbitrary characteristic by Ballico and Hefez [4].
The group of the 5 conic problem is the full symmetric group. The double
transitivity was proved by Higman (private communication, Jan. 1978), the
existence of a transposition follows from the formula of Fulton and MacPherson,
(7.2), and the ingredients were processed by Harris [35] in the machine he
developed. The work was refined and extended to the case of plane curves of degree
> 2 touching the appropriate number of general curves X{ of various degrees
> 2 by Hefez and Sacchiero [38] and to the case in which the X{ are smooth
curves in general position (that is, replaced by g%Xi for general &) by Ballico [3].
These results were claimed only in characteristic 0, but they are valid in every
characteristic aside from the expected exceptions.
The group was also determined over C by Harris [35] in these cases: the flexes
and bitangent s of a plane curve, and the lines on a hypersurface of degree 2N — 3
in PN. The group is the full symmetric group except in the initial cases: the
flexes of a cubic, the bitangent s of a quart ic, and the lines on a cubic. In these
cases, the groups were found last century. The theory is pretty.
In addition, Harris [36] proved over C that the group of the points in a
hyperplane section of a curve in PN is the full symmetric group. This result is the

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 357
key to his bounding of the geometric genus. The result was proved for a smooth
reflexive curve in characteristic p ^ 2 by Ballico and Hefez [4]. The reflexivity is
equivalent to the existence of a hyperplane that cuts with exactly two coincident
points; so reflexivity guarantees the existence of a transposition. Rathmann [75,
Chapter 3] observing that a weaker result suffices to bound the genus, studied
the curves with small Galois groups; thus he established Castelnuovo's bound
on the genus of a smooth irreducible nondegenerate curve in PN in arbitrary
characteristic.
8. Quadrics and correlations. Work in PN over an algebraically closed
field of arbitrary characteristic p.
Zeuthen (1865), inspired by Cremona (1864), developed a powerful method
for finding the characteristic numbers (see §7) of nearly arbitrary families of
conies in P2 (over C but p ^ 2 suffices). The method is based on the following
result of Chasles (1864):
(8.1) CHASLES'S DEGENERACY RELATION. I. Given a (complete,
irreducible, flat) 1-parameter family of conies and p ^ 2, the (weighted) number of
double lines e\, if finite, is related to the family's characteristic numbers l0 and
h by
h = 2/0 -ci.
Zeuthen's first example was the determination of the characteristic numbers
of the family of all conies. It went essentially like this. First, the condition
on (the coefficients of) a conic that it pass through a point is obviously linear;
hence Zq = 1. Now, consider the 1-parameter family of conies through 4 points
in general position. It obviously contains no double lines. Use (8.1):
llh = 2/g - Zfci =2-0 = 2.
Next, consider the 1-parameter family of conies through 3 points and tangent to
a line in general position. It obviously contains no double line. Use (8.1):
ZgZ? = 2l$h - llhei =2-2-0 = 4.
Finally, since p ^ 2, a conic is reflexive, and its dual is a conic by (4.14), (4.5)
and (4.8); whence, the remaining characteristic numbers are 4,2,1.
Relation (8.1), the preceding determination, and allied theory were made
rigorous through the efforts of many mathematicians over the last 120 years. (It
is a fascinating story; see [47].) The upshot is this. To prove (8.1), denote by
T the parameter space of the 1-parameter family, by t: T —► P5 the natural
map into the 5-space of conies, and by V the Veronese surface in the P5 of
double lines. Define e\ as the multiplicity of the ideal of t~1V'. Replace T by its
normalization. Then t~1V is a divisor, and e\ is its degree.
Consider the blowup B of P5 along V, and let E\ denote the exceptional
divisor. Since T is normal, t lifts to a map t: T —► B, and obviously t~lE\ =
t~xV. Then (8.1) results from this corresponding relation among divisors on B:

358
STEVEN L. KLEIMAN
(8.2) Chasles's Degeneracy Relation. II. LetL0 be a point, ZL0 the
proper transform of the divisor of all conies through Lo; let L\ be a line, ZL\
the proper transform of the divisor of all conies tangent to L\\ then
[ZLi] = 2[ZL0) - [Ex] in AB.
Moreover, the computation ofl$l\ andï$l\ may be interpreted rigorously as taking
place in the intersection ring AB.
Chasles found (8.1) using the Correspondence Principle as follows: fix a line
in general position, and say two points on it correspond if they lie on a conic in
the family; the number of coincidences is h + ei, and (8.1) follows.
In the same spirit, (8.2) could be rigorously established as follows: apply the
Correspondence Principle globally to the family of all conies not containing the
line to show that some relation holds among the 3 divisors and then determine the
coefficients by a direct local computation, possible with a specific 1-parameter
family, or possibly by applying the relation in several cases and solving for the
coefficients. Grayson [33] in more or less this way justified two of Schubert's
proofs of the Contact Theorem for 1-parameter families of plane curves.
Today the standard proof of (8.2) involves more knowledge of the geometry
of B. For this, recall the theory of polarization (developed early last century).
Fix a nondegenerate conic C. A point P determines a line L, called its polar: if
P is off C, drop the two tangents from P to C and join their points of contact
to get L; if P is on C, the tangent at P is L. Dually, a line M determines a
point <3, called its pole: if M is not tangent to C, take the two points of M on
C and intersect the corresponding tangents to get Q; if M is tangent to C, the
point of contact is Q. (The names reflect this remarkable fact: if P varies on M,
then L rotates about Q.) In dual systems of coordinates for P2 and P2*, the
first correspondence is represented by a symmetric 3 x 3-matrix, u say, and the
second by the matrix of signed 2 x 2-minors, f\ u. Two points P, Q are called
conjugate if each lies on the polar of the other—in coordinates, PuQtT = 0. The
locus of self-conjugate points is C.
Thus, the P5 of conies may (if p ^ 2) be viewed as the P5 of symmetric
3 x 3-matrices u, and the duality correspondence (the rational map sending a
point of P5 representing a conic to the point of P5* representing its dual conic)
is given in coordinates by u »-► f\ u. Now, the Veronese surface V of double
lines is defined by the equation f\ u — 0. Hence, the blowup B of P5 along V is
equal to the closure in P5 x P5* of the graph of the duality correspondence. So,
a point of B represents a conic completed by a "de jure" dual. For this reason,
B is usually called the variety of complete conies.
The proper transform ZL\ on B is equal to the inverse image of a hyperplane
in P5*—in fact, it is the hyperplane of conies in P2* passing through the point
representing L\—and (8.2) is an instance of an easily derived relation that holds
on the blowup of the common zeros of a set of sections of a bundle (see [53, §2]).
To obtain (8.1) from (8.2), it must be proved that U — f t*[ZLi}. It is easy to
reduce the problem to proving that, in the notation of §6, if m = 5, if V varies

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 359
in the family of all nondegenerate conies, and if S = B, then
P2*ttPl[Cli}).[UCV*1}) = [ZLi}.
Now, the two cycles obviously have the same support, and [ZLi] is reduced.
Moreover, the cycle on the right is reduced too because the universal family
parametrized by S is a linear system.
From the point of view of polarization, it is natural to look (as Chasles did)
at the characteristic numbers lo, h of a 1-parameter family of conies as follows.
Fix two points P, Q and two lines L, M. Define Hp& as the closure in P5 of the
locus of nondegenerate conies for which P and Q are conjugate; define Hl,m
dually. In coordinates, Hp,q: PuQtT = 0, a linear equation; hence Hp& is a
hyperplane. Similarly, Hl,m is a hyperplane. In particular, Hp,p is obviously
the hyperplane of conies through P, and Hl,l is obviously the hyperplane of
duals of conies tangent to L.
Consider a complete, irreducible, flat, 1-parameter family of conies, whose
general member is nondegenerate. Let T denote the parameter space, t: T —► P5
the natural map, and t'\ T —► P5 the dual map. Clearly, if the points P, Q are
in general position, then /0 = /**[#p,q]- And if the lines L,M are in general
position, then l\ = ft*[HL,Nf}- Now, as a nondegenerate conic varies in the
family, the polars of P envelop a curve (possibly not reduced nor irreducible),
whose class is the number of polars through Q; hence, fo is equal to this class.
Similarly, the poles of L trace a curve, whose degree is the number of poles on
M; hence l\ is equal to this degree. For example, the center of a conic is the pole
of the line at infinity. Hence the centers of the conies inscribed in a quadrilateral
trace a line (called Newton's line by Lemoyne, 1923), because the number of
conies tangent to 5 lines is 1.
The geometry of B is beautiful. The projection B —► P5* blows up the
Veronese surface of double lines in P2*, and the exceptional divisor, E^ say, is
equal to the proper transform of the divisor in P5 of line-pairs in P2. The natural
action on B of the linear group of P2 has 4 orbits: an open orbit, B — (E\ +#2);
a closed orbit, I := E\. E^\ and two orbits of codimension 1, E\ — I and E% — I.
Both Ei and E2 are smooth, and they intersect transversally. Their intersection,
7, is isomorphic to the point-line incidence correspondence.
Sometimes I is called the locus of Halphen conies, because Halphen (1876,
1878, 1879) found some subtle conditions—each defines a divisor on B—
(Ei H- E2) whose closure contains I—and he extended Chasles's enumerative
theory of conies to deal with them—effectively, he considered a series of blowing-
ups, the first centered at I. Halphen's work generated a lot of interest and some
basically philosophical controversy: a 1-parameter family can contain
degenerate conies that satisfy these subtle conditions, and Chasles's formula includes
them in the enumeration—is this situation acceptable, or should it be
rectified, as Halphen did? Recently, Casas and Xambô [5, 6, 7] rigorously justified
Halphen's theory and carried it a good deal further. In particular, they deal with
indecomposable multiple conditions, generalizing in Halphen's spirit Cremona's

360
STEVEN L. KLEIMAN
(1864) extension of Chasles's work. De Concini and Procesi [16] have done some
related, but more abstract, work.
An enumerative theory of linear maps from one plane to another, strikingly
analogous to Chasles's theory of conies, was founded by Hirst (1874). In
particular, he generalized (8.1) to a 1-parameter family of correlations; in a correlation,
the target is viewed as a dual plane, and the map, as carrying points to lines and
lines to points. Now /o is the number of correlations carrying a fixed point P to
a line that contains another fixed point Q; and l\ is the number of correlations
carrying a fixed line L to a point that lies on another fixed line M; and e\ is the
number undefined on some line, or equivalently, whose image is a point (in other
words, they are of rank 1). If the correlations are the "polarities" of the conies
in a 1-parameter family, then (8.1) is recovered. Hirst went on to determine the
"characteristic" numbers /§> ^i> The computations may be interpreted as
taking place in the intersection ring of a certain smooth variety of "complete
correlation," namely, the blowup of the (8-dimensional) projective space of all
3 x 3-matrices along the smooth subvariety (isomorphic to P2 x P2) of all those
of rank 1.
The enumerative theory of conies and the greater theory of plane correlations
were generalized to higher dimensions right from the start. They have continued
to attract the attention of one generation of mathematicians after the other,
more so than any other theories in enumerative geometry. Around 1980 a great
new burst of activity began. Some of those involved to a greater or lesser extent
are: Abeasis, Casas, De Concini, Demazure, Drechsler, Finat, Gianni, Goresky,
Ihle, Kleiman, Kzawa, Laksov, Lascoux, MacPherson, Procesi, Springer, Sterz,
Strickland, Thorup, Traverso, van der Waerden, Vainsencher, and Xambô. The
details of who did what, when and where may be found in [59]. The upshot is
that now there is a much sharper picture of the variety of complete linear maps
and of its subvariety of complete quadrics, two similar varieties of rare beauty.
The picture is briefly described next.
The two varieties may be described in some 5 different ways:
1. Each is the closure in the product of the appropriate projective spaces of the
set of M-tuples (u, /\ u,..., /\ u) where u varies over the set of linear maps,
resp. symmetric linear maps, of maximal rank M + 1 (that is, M = min(JV, Nf)
ifu: P^^P^').
2. Each is defined in the product of projective spaces by certain explicit
bilinear equations, reminiscent of the quadratic equations of a Grassmannian.
3. Each is the final blowup in a sequence of M — 1. The first is the blowup
of the projective space of all u along the smooth subvariety of u of rank 1. The
ith is the blowup of the (i — l)st along the proper transform of the subvariety
of u of rank < i\ this center is smooth and its normal bundle may be described
explicitly.
4. Each is the minimal marvelous compactification (in the sense of Luna and
Vurst) of the homogeneous space of u of maximal rank. "Marvelous" means

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 361
that the closures of the orbits are smooth and intersect transversally. "Minimal"
means that any other marvelous compactification dominates this one. In the
present case, the 2M subsets I of the interval [1,M] index the orbit closures
B(I); moreover, codB(I) = card/, and B(I) .B(J) = B(I + J). Furthermore,
B(I) is fibered over the product of a flag manifold of type I and one of type CI,
the complement of /, resp. over one of type 7; each fiber parametrizes strings
of complete maps from the first successive quotients to the second, resp. of
symmetric maps on the successive quotients. The orbit closures of codimension
1 are the proper transforms from the parameter space of all maps u, of the
subvarieties of u of rank < i for 1 < i < M.
5. Each parametrizes complete maps, resp. quadrics, defined a priori and not
as limiting cases. The definition is recursive: a complete map is an ordinary
nonzero map u plus, if u is not of maximal rank, a complete map from its kernel
to its image; resp. a complete quadric is an ordinary quadric V plus, if V is a
cone, a complete quadric in its vertex.
Each of the two varieties has two natural bases of its Picard group (divisors
modulo rational equivalence): a Z-basis, the pullbacks of the hyperplane classes
from the M projective spaces; and a Q-basis, the orbit closures of codimension
1. There are M basis-change relations generalizing (8.2) and its dual (Schubert,
1894). By using the basis-change relations, the structure of the orbit closures,
and induction, it is possible mechanically to work out all the characteristic
numbers. A good algorithm is given in De Concini-Procesi [15]; it was implemented
on two computers for quadrics in P4 and P5 by De Concini, Giani, and Traverso
[13].
The intersection rings of the two varieties may be found at least in principle in
three different ways. One is to proceed iteratively, using the blowup description,
3 above. Another is to use one of the natural cellular decompositions that are
associated to the fibrations of the orbit closures described in 4 above. A third
and new way is presented in De Concini, Goresky, MacPherson, and Procesi
[14]. It is described on p. 10 as "conceptual non-iterative in the sense that
Borel's calculation of the cohomology of the flag variety or Danilov's calculation
of the cohomology of a toric variety are." The method is developed over C, and
it yields the intersection ring tensored with Q.
The preceding theory supports a rigorous determination of the number #
of quadric surfaces tangent to 9 general others in P3 in characteristic p ^ 2.
Indeed, by (4.14), the iih rank rt of a smooth quadric is equal to 2 for i = 0,1,2.
Hence, the Contact Theorem, (7.1), (and a little fuss if p > 2) yield:
# = (2/o + 2/i + 2/2)9 = 29 (/g + ... + /J + ... + 1%),
where the monomials l^l^l3^ represent the characteristic numbers. Thus, # =
666,841,088 just as Schubert [84, p. 106] claimed!
The preceding theory works in any characteristic—in fact, over any base—
except, of course, that the theory of quadrics (or symmetric maps) fails in
characteristic 2. Also, some care must be taken here and there. For example, the

362
STEVEN L. KLEIMAN
theory of marvelous compactifications is based on representation theory, and it
has not been worked out in detail in positive characteristic, never mind over Z.
In addition, some care has to be taken in formulating the definition of families
of complete maps (whether symmetric or not). In this connection, in the case of
quadrics, it would be interesting to know the answer to the following question.
Let S be the variety of complete quadrics, So the open subset of nondegenerate
quadrics V, and F the scheme of complete flags in PN. Let /0: So —► Hilb(F)
be the map that assigns to each nondegenerate V the locus of its tangent flags.
Then, does /o extend to a map /: S —► Hilb(F)? If so, then / is necessarily an
embedding because of Description 1 above. And then, the intuitive conception
of S as a universal space will be justified.
9. Cubics. Chasles in 1864 established the Contact Formula for 1-parameter
families of curves of arbitrary degree, and via examples involving families of
conies, he gave a procedure for using the theorem with m-parameter families
(see [52, III-6, pp. 207-208]). Thus the main problem in the enumerative theory
of curves of degree > 3 became the determination of the characteristic numbers
of families of them.
Seven years passed before any progress was made. Then Maillard (1871)
and Zeuthen (1872) independently but in the same step-by-step way found the
characteristic numbers of the cuspidal plane cubics, then those of the nodal
cubics, and finally those of the smooth cubics. Then Zeuthen (1873) found the
characteristic numbers of the nonsingular plane quartics via the determination
of those of one family of singular quartics after the other. Finally, Schubert
(1874-1875), encouraged by an 1873 prize offering of the Royal Danish Academy,
refined the work and then found the characteristic numbers of the twisted cubic
space curves; the only published account appears in [84], and it ends with the
determination of the number 5,819, 539,783,680 of cubics tangent to 12 general
quadric surfaces. Essentially no other case is known.
The matter attracted relatively little attention over the next 100 years or so.
About 6 years ago the situation changed. The field has become rather active,
and significant progress has been made. Virtually all of it concerns cubics.
However, Coray and Vainsencher [92] have proved that there are 105 rational
quintics through 10 general points by parametrizing the quintics via pencils of
cubic surfaces doubled along a common line.
First, consider the plane cubics. They form a P9, and starting with it, Sterz
[88] has constructed and studied a sequence of 5 blowups. He works over C
and describes each blowup in coordinates. He gives 3 bases of the homology
group #i6, the basis change relations, and some of the intersection numbers.
On a different tack, Sacchiero [82, 83] verified the characteristic numbers of the
cuspidal cubics and of the nodal cubics in characteristic p ^ 2,3. In both cases,
he works with cycles on the conormal variety of P2 embedded in P9 by the third
Veronese map.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 363
Now, consider the twisted cubics in characteristic p ^ 2,3. Let S be the
(homogeneous) space of all nondegenerate cubics, H S its closure in the Hilbert
scheme. Piene and Schlessinger [72] proved that HS is smooth. Ellingsrud (see
[70, pp. 332-333]) proved that Pic(S) = Z/2 and Pic(HS) = Z x Z. Ellingsrud
and Piene (see [70, p. 333]) proved that there is a smaller smooth compactifica-
tion of S, obtained by contracting the divisor of nodal cubics to the point-plane
incidence correspondence by sending a nodal cubic to the pair (P, H) where P
is its node and H the plane of its reduction. Harris [36] enumerated the orbits
of the linear group on HS.
Piene [69], interpreting "degeneration" as flat specialization of subschemes,
studied Schubert's [84] 11 first-order degenerations of the triples consisting of
a twisted cubic, its locus of tangent lines, and its locus of osculating planes.
Alguneid (1956, 1959) had already treated the degenerations as cycles. Piene
[70, p. 335] observed that the two corresponding compactifications differ, because
the former has a larger Pic.
Abeasis [1], viewing the linear group of P3 as the parameter space of
parametrized twisted cubics, studied a sequence of blowups of an equivariant compact-
ification of the group. Vainsencher [92] has studied a compactification of a
blowup of S, obtained as the parameter space of the pencils of quadrics through
a variable line containing a fixed point. Thus he recovered the elementary result
that there is 1 cubic through 6 general points, and he verified that there are 20,
10, and 5 through 5 general points, cutting a lines, and touching 2 — a planes
respectively for a = 0,1,2, just as Schubert found.
An effective "new" way of obtaining characteristic numbers was introduced in
Kleiman-Speiser [53] and used to verify the numbers in the case of the cuspidal
plane cubics. The case of the nodal plane cubics is currently being written up.
Meanwhile, Kleiman, Str0mme, and Xambô [54] have used the method to verify
the characteristic numbers of the twisted cubics; whence, by virtue of the Contact
Theorem, 5,819,536,783,680 too. In fact, the method can be used to verify all
of Schubert's values for the numbers of cubics through i general points, cutting
j general lines, and touching 12 — 2i — j general planes. However, it is still an
open problem to verify others of Schubert's numbers, such as the number 120 of
twisted cubics touching 3 general lines and cutting 3 others.
The new method has 3 main steps: (1) Find a smooth partial
compactification U of the parameter space S such that (i) the curves of intersection of the
contact divisors are complete and (ii) the boundary U — S consists of irreducible
"degeneracy" divisors. (2) Determine an appropriate number of relations among
the contact divisors and the degeneracy divisors, so that the desired
characteristic numbers may be expressed, via the curves, in terms of the characteristic
numbers of the several families of degenerate figures. (3) Determine as many of
the latter numbers as needed, one way or the other.
For example, consider the cuspidal plane cubics in characteristic ^ 2,3. The
nondegenerate ones are parametrized naturally by a 7-dimensional subvariety of

364
STEVEN L. KLEIMAN
P9, which is an orbit under the linear group of P2. The boundary of S has
a unique 6-dimensional orbit, A\ say, which parametrizes the reducible cubics
made up of a smooth conic plus a tangent line. The normalization of the union
of S and A\ is a good choice of partial compactification U. Indeed, a direct
elementary analysis shows that U is smooth and that A := U — S is irreducible.
Moreover, the formation of U is symmetric with respect to duality (note that the
dual of a cuspidal plane cubic is also one); in fact, if S is embedded in P9 x P9*
via the duality correspondence, then U is equal to the normalization of the union
of S and a unique irreducible 6-dimensional orbit in its boundary.
Fix a point P and a line L. Let ZP, resp. ZL, denote the closure in U of the
set of nondegenerate cuspidal cubics passing through P, resp. touching L; so
ZP is the pullback of a hyperplane in P9, and ZL of one in P9*. Fix a general
7-tuple (#1,..., g7) of linear transformations of P2. Then the following scheme
intersection is a complete curve:
Ti := (9lZL)... (fc-iZL). (gi+iZP)... (g7ZP).
Indeed, its image in P9 x P9* is complete by transversality considerations.
The key relation is this:
(9.1) [ZP] + [ZL]=2[A] in AU.
It is due essentially to Maillard and Zeuthen; they argued directly using the
Correspondence Principle. Schubert suggested a lovely alternative derivation,
based on a study of the triangle formed by the cusp, the flex, and the meet of
the cusp and flex tangents of a cuspidal cubic. Schubert's derivation was made
rigorous in Kleiman-Speiser [53].
Another way to prove (9.1) was suggested by Str0mme (private
communication, Jan. 31, 1983, inspired by a lecture of Speiser's on Schubert's derivation).
The idea is this. There is a natural smooth compactification of 5, namely, the
bundle over the point-line incidence correspondence I of P2 whose fiber over
(<3, M) is the P4 of cuspidal cubics with cusp at Q and cusp tangent M. In it,
consider the union of S and the unique irreducible 6-dimensional orbit in the
boundary of S; this union is isomorphic to U. Str0mme obtained a number of
divisorial relations in AU by considering the family of blowups of the ambient
P2 at the variable point Q.
The zth characteristic number, 1 < i < 8, may be defined as the sum of local
intersection numbers,
Ni := JlgiZL]... [gi-xZLUgiZP]... [g7ZP].
Then obviously, for 1 < i < 7,
Ni = jin.lgiZP) and Ni+1 = J[Ti\.[9iZL].
Although U is not complete, N{ is a priori independent of the choice of </t-. Indeed,
varying Qi replaces QiZP and QiZL by rationally equivalent divisors; whence, N%

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 365
is invariant because T{ is complete. Similarly, N{ is independent of the choice of
Qj for all ij.
Now, (9.1) yields, for 1 < i < 7,
^ + ^+1=2|[T,].[A].
The numbers /[Tt-].[i4] are 42,87,141,168,141,87,42. They may be found by
using the following compactification A' of A: let B0 be the natural parameter
space of the nondegenerate conies, B the variety of complete conies (§8), and I
the point-line incidence correspondence; embed A in Bo x / as the union of the
conormal varieties of the nondegenerate conies V, and define A' as the closure
of A in B x I (in the notation of §6, A1 := UCVxl). It is easy to identify the
divisors defined by P and L, and compute the required intersection numbers
using the knowledge of the intersection rings of B and I.
Finally, the Principle of Duality yields N{ = Ng-i for 1 < i < 8. Hence,
2N4 = N4+N5 = 2x 168 or N4 = N5 = 168,
N3 = N6 = 2 x 141 - 168 = 114,
N2 = N7 = 2x87-114 = 60,
Nx = Ns =2x42-60 = 24.
Thus Maillard's and Zeuthen's values are confirmed.
Consider, finally, the twisted cubics in characteristic ^ 2,3. A good partial
compactification U of the 12-dimensional space of nondegenerate cubics is the
open subset of the closure of U in the Hilbert scheme, obtained by adjoining the
two 11-dimensional orbits under the action of the linear group of P3, A and B
say, which parametrize respectively the nodal plane cubics plus nilpotent at the
node and the cubics composed of a nondegenerate conic plus a unisecant line.
Then U is smooth by the Piene-Schlessinger theorem.
Fix a line L and a hyperplane H. Let ZL, resp. ZH, denote the closure
in U of the set of nondegenerate twisted cubics cutting L, resp. touching H.
Fix a general 7-tuple (</i,...,</7) of linear transformations. Form the scheme
intersection
Ti := (giZH)... (fc-iZff). (gi+iZL)... (g7ZL).
Then Ti is a complete curve.
Indeed, in the Chow variety of cycles in the point-plane incidence
correspondence I of P3, form the set representing the cycles of the conormal varieties CV
of the nondegenerate cubics, and let T be the normalization of its closure. Then
U embeds in T, and the boundary T — U is of dimension 10; in fact, T consists of
6 orbits, which are easy to enumerate using the following specialization theorem
[29, Proposition (a), p. 179; 40, Corollary 421, p. 241]: given a specialization
V -► V* and its lifting CV -► C\
[C*]= E ™w[CW]
w<v*

366
STEVEN L. KLEIMAN
where (a) the W include every component of W* and (b) if W is not a component,
then W lies in the singular locus Sing(V*). (While this theorem is not valid
for every specialization V —► V* in positive characteristic, the proof in [52,
(2.5), p. 16] does go through in the case at hand.) Therefore, by transversality
considerations, T{ is a complete curve.
The key relations in AU, due essentially to Schubert, are
(1) 2[ZL] = 3[A] + [B], (2) 3[ZH] = 2[ZL] + 2[B].
They may be established by studying the normalization of the total space of the
universal family of curves parametrized by U and computing some Chern classes.
Now, just as Schubert found,
(9.2) yVi]. [A] -12,960,
(9.3) /[Ti]. [B] = 121,440, 180,240, 236,160,....
The number 12,960 may be found via a straightforward calculation on this
natural compactification of A: the variety of triples (Q, M, V) where Q is a point, M
a plane through Q, and V a cubic in M with a singular point at Q. (This variety
is just the union of the conormal varieties of the planes in P3, reembedded by
the 3rd Veronese map.) The verification is similar to that of Schubert's value
92 for the number of conies in P3 cutting 8 lines, given in [36, p. 26] and in
[27, Example 14.712, p. 275 and Example 3.2.22, p. 63]. It is also similar to the
verification of the number, 12, of nodal plane cubics through 8 points in [83].
The numbers (9.3) may be verified by using the compactification of B obtained
by closing the natural embedding of B in the product of the space of complete
conies and the point-line incidence correspondence. The verification is like the
one discussed above for cuspidal plane cubics.
The characteristic numbers of the twisted cubics may be defined by
Ni := f[giZH]... [gi-iZH]. [9iZL]... [g7ZL] for 1 < i < 13.
Then, obviously, for 1 < i < 12,
Nt = J[Tt}.[9lZL} and Nl+1 = j [T%]. [9iZH\.
So the relations (1) and (2) above yield
2^ = 3^^]. [A] + J[Ti\. [£], 3JV<+i = 2Nt + 2^]. [B].
Therefore,
Nx =(3xl2,960 + 121,440)/2 = 80,160,
N2 = 2(80,160 + 180,240)/3 = 134,400,
JV3 = 2(134,400 + 236,160)/3 = 2,366,160,
etc.

INTERSECTION THEORY AND ENUMERATIVE GEOMETRY 367
Finally, the rank r^ of a smooth quadric is equal to 2 for i = 0,1,2 by (4.14).
Therefore, the Contact Theorem, (7.1), yields (after a little fuss in characteristic
p > 3) that the number of twisted cubics tangent to 12 quadric surfaces in
general position is
(2/0 + 2/i + 2/2)12 = 212(/1+/2)12
= 212(iVx + (112)^2 + (g2)^ + ■■■) = 5,819,539,783,680,
just as Schubert found!
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Massachusetts Institute of Technology

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Completed Quadrics and Linear Maps
DAN LAKSOV
1. A short historical background. On the following few pages we shall
describe two lines of thought that led to the conception of complete conies and
that evolved into more general theories of complete geometric objects. We do
not intend to give a complete historical account; all we hope to do is to give some
motivation for the presentation of the general theory in the following sections.
For a historical presentation of the developments before the turn of the century
we refer to [24] and in particular to the article of Zeuthen. A more specialized
survey of complete conies is Kleiman's article [38], and more scattered historical
remarks can be found in [39].
During the first part of the previous century several advances were made in the
study of duality properties of curves. A "principle of duality" was established,
and it became part of a geometric way of thinking, always to consider figures
together with their duals. Since antiquity, one of the main sources of inspiration
for the study of duality was the conies. A simple example will convince the
reader how the complete conies appear naturally in this study. The dual of the
conic defined by the equation x2 + ay2 — z2 with a ^ 0 is the conic x2 +y2 /a -
z2. When a tends towards zero, the first conic degenerates into the two lines
x — z and x + z intersecting in (0,1,0) and the dual curve into the double line
y2. However on the double line there are two distinguished points (1,0,1) and
(1,0,-1) corresponding to the above lines. If we let a tend to oo, we obtain
dually that the first curve degenerates into the double line y2, but with two
distinguished points on it. Hence in order to retain a principle of duality there
must in every family of conies be degenerate ones belonging to the two classes:
(i) the double lines with two distinguished points,
(ii) two lines intersecting in a point.
It is the nondegenerate conies together with the two degenerate forms that
have become known as complete conies.
The importance of the complete conies was further emphasized by their
appearance in the study of contact, or tangency, problems for families of conies.
1980 Mathematics Subject Classification (1985 Revision). Primary 14M99; 14N10; Secondary
14M17, 14C17.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
371

372
DAN LAKSOV
Such problems were intensively studied and debated around the middle of the
century. We shall again illustrate how the completed conies appeared by
resorting to a simple example.
Consider the problem of determining the number Nm of conies that pass
through m given points and that are tangent to 5 — m given lines. It has been
known since antiquity that N$ = 1, and it is easily seen that N4 = 2. A natural
procedure to determine the remaining numbers is then to substitute, one at a
time, the condition of "passing through a point" by the condition "being tangent
to a line." It is, following this method, tempting to conclude that Nm is 25_m.
However, it is easy to see, and follows by the duality principle, that No = 1. The
failure of the above reasoning seems to have been found puzzling (see Zeuthen's
article [67, p. 303]) until Cremona [12] pointed out that one must be careful to
consider a double line with two points to be tangent to a line only when the line
passes through one of the points. Then there are no degenerate conies tangent
to five lines. If one does not take this into account, as in the above reasoning,
all double lines will be tangent to 5 given lines, and the numbers obtained have
no geometric significance.
Chasles [8, 9] put everything together. He introduced, in order to retain
a "principle of duality," for each 1-parameter family of complete conies two
numbers, called characteristic numbers,
fi = the number of members that pass through a given point,
v — the number of members tangent to a given line.
He observed that, for a large number of simple contact conditions imposed
upon a given 1-parameter family of conies, the number of solutions could be
expressed in the form
afi + /?*/,
where a and /? depend only upon the given conditions. Also, he observed that,
if we let
A = the number of degenerate members of type (i) above, and
7T = the number of degenerate members of type (ii) above, then
(1) A = 2/i — i/, 7T = 2v — fi.
These observations were, as we shall see, tremendously important for the
development of the theory of completed objects. They were immediately carried
over to space quadrics by de Fauque de Jonquières [16-18], Chasles [10-11],
Zeuthen [66], and Schubert [47]. In the following years they were extended to
higher-dimensional spaces, mainly by Shubert [47-50].
About ten years after Chasles introduced the two characteristic numbers for
systems of conies and gave the formulas (1) for the degenerate conies, T. A. Hirst
[31, 33-35] observed that a similar theory could be built up around correlative
plane figures. The theory was extended to dimension three and partly four by
Hirst [32, 36], and by Visalli [63, 65]. Again, mainly through works of Schubert
[49, 51], G. de Prête [20], and G. Z. Giambelli [27], the theory was extended to
higher dimensions.

COMPLETED QUADRICS AND LINEAR MAPS
373
We shall not at this point enter into the definitions and properties of
correlative figures, partly because we have given elsewhere [41] a geometric description
of correlations and collineations similar to that given for conies above, but more
importantly because we shall give a more up-to-date version below. From the
following presentation of quadrics on the one hand and the correlations and
collineations on the other, it will become apparent that the two theories are
completely analogous. We want however to point out that, even if this might
have been obvious from the outset, traditionally the geometers were interested
in the finer geometric aspects of the different theories and tended rather to stress
the particularities of each subject than their similarities.
At the turn of the century the picture of completed objects was rather clear.
The geometry of families of quadrics, correlations, and collineations was well
understood, and a powerful and flexible symbolic calculus had been developed
to handle the enumerative problems. Also a large number of special enumerative
problems about the number of completed objects satisfying a variety of special
conditions had been solved. Moreover some general enumerative formulas for
computing characteristic numbers were known.
During the present century an intersection theory has been developed that
gives a satisfactory understanding of the symbolic calculus. In order to apply
this intersection theory to obtain the enumerative results of the previous century,
a considerable amount of work has been spent to construct parameter spaces for
completed geometric objects. The purpose of the following sections is to expose
some of this work.
2. The parameter spaces for complete quadrics and bilinear forms.
We shall in this section give a short description of the complete quadrics,
correlations, and collineations in spaces of any dimension and indicate the attempts
of this century to find a parameter space for these.
First we turn to quadrics in an r-dimensional projective space Pr in which
we have chosen a set of coordinates. A nondegenerate quadric hypersurface Q
in Pr is the zero locus of a quadratic form.
r
where q = (qij) is a symmetric (r + 1) x (r + l)-matrix of rank r + 1. In the
case of conies we saw that we had to consider a conic together with its dual. To
generalize this we must consider for m = 0,1,..., r — 1 the locus Qm of m-planes
in Pr that are tangent to Q = Qo- We can consider Qm as a subspace of the
Grassmann manifold Grassm(r) of m-planes in Pr or, via the Plucker embedding,
as a subspace of the projective space Pr(m) of dimension
r(m)=(T + \)-l.

374
DAN LAKSOV
As a subspace of Pr(m) the space Qm is the intersection of Grassm(r) with the
quadric hypersurface defined by the quadratic form
/ ^ ^to,tl,...,»m,io,il,-..,imajtO,tl,---,*majio,il,---,im?
where
m+1
f\ Q ~ teo,il,-,imjojl,-jm)
is the symmetric (r(ra)+l) x (r(ra)+l)-matrix whose entries are the determinants
of the (ra + 1) x (ra — l)-submatrices of q (and written in the basis of Pr(m)
obtained from the lexicographic ordering of the products of the basis of Pr). The
matrix Am+1 Q ls called the rath adjugate of q. To see that Qm is the above locus
we observe that an ra-plane L is tangent to Q if and only if the form restricted
to L is degenerate. Let b: L —► Pr be an (ra + 1) x (r + l)-matrix giving L as
a subspace of Pr, with respect to some basis of L. Then the condition that Q
is degenerate on L can be expressed by the condition that the matrix b • q • b* is
singular or equivalently that the adjugate
m+1 m+1 m+1 m+1
A(*-« ->tr)= /\b- /\q- A>tr
is zero. However Am+1 ^ *s the point of Pr(m) corresponding to L and the
condition expresses that this point is on the quadric defined by Am+1 Q-
We see that, if we let Mn denote the projective (r(ra)2+3r(ra))/2-dimensional
space of symmetric (r(ra)H-l) x (r(ra) + l)-matrices, we can represent the quadric
Q together with all the tangential loci Qo, Qi, • • •, Qr-i by the point
of the space
M = Mi x M2 x • • x Mr.
A naive way of degenerating a family of nondegenerate conies, and in fact a
way that is close to the traditional approach, is to consider the images in M of
the members of the family by v and then to take the closure of the image in M.
Following this naive approach we define the space of complete conies Qq as the
closure of the image of the map v : U —> M, where U is the open subset of Mo
of nonsingular matrices.
The construction of Qq is easily done in a coordinate-free way as follows:
Let V be a vector space of dimension (r + 1) over a field k and let P(V) be
the r-dimensional projective space of 1-quotient spaces of V. Denote, for any
vector space W, by S2W and /\m W the 2nd symmetric power, respectively the
rath exterior power, of W. Then Mm = P(S2 /\mV) represents the space of
symmetric maps /\m V —► /\m V*. There is a natural map
v: U — MQ =P{S2V) xP [S2 f\V) x - - - xP [S2 f\V]

COMPLETED QUADRICS AND LINEAR MAPS
375
which sends a point of P(S2V) corresponding to a nonsingular symmetric map
a: V —► V* to the point (c*i, c*2, • • • , &r) of M, where
m
am:S2/\V^k
corresponds to the rath exterior power
mm m
of a. We now turn to correlations and collineations. Two figures, one in Pr and
one in the dual space Pr* (respectively Pr), are said to be correlative
(respectively collinear) if the points of the two figures "satisfy" a bilinear form
r
with respect to fixed coordinate systems of Pr and Pr*, and b = {b{j) is a
nonsingular (r + 1) x (r + l)-matrix. As in the case of quadrics one may define
the tangential structure of such figures. We shall not do this here, but enter
directly into a coordinate-free construction of a parameter space for bilinear
forms, which generalizes both the case of correlations and collineations.
Let W be an (s + l)-dimensional vector space over A:, where r < s. The
projective space P(V <8> W*) represents bilinear maps from V <8> W* into A:, or
equivalently linear maps from VtoW. There is a natural map from the open
subset U of P(V <8> W*), corresponding to maps of rank r + 1, into the product
mb = p{v®w*)xp (A^A^*)x* xP (A^A^*
which maps the bilinear form b corresponding to a map (3:V-+W into the point
i/(6) = (6i,...,6,),
where the bilinear form
m m
bm- f\V®f\W*^k
corresponds to the rath exterior power /\m /?: f\m V —► /\m W of ft.
We define the parameter space Qb of complete bilinear maps to be the closure
of the image of U in Mb-
When W = V* or W = V we obtain parameter spaces Qcp and Qcl for
complete correlations, respectively complete collineations.
The above definitions of Qq and fis are clearly useless unless we are able to
obtain precise information, not only on the set v(U), but also on the points of its
closure. Such information was provided by J. G. Semple [53] who gave an explicit
covering of Qcl by affine spaces. Semple's observation has been of tremendous
importance for the study of the parameter spaces, and we shall present it for
nB.

376
DAN LAKSOV
Let Ax = A<r+1>r/2 and A3 = As(r+1)-r(r+1)/2 be the affine spaces whose
coordinate functions are the coordinates of the (r + 1) x (r + l)-matrices
*(*) =
/ i
Xlfl 1
^2,0
V xr,0 ■ ■ ■
\
0
Xr,r-1 1/
respectively the (s + 1) x (s + l)-matrices
(\ Z0,i •••
1
Z(z) =
0
1 ^r,r+l
1
V
■Zo,s\
0
0
and Ai = Ar the space whose coordinates are the functions u\, u^,..., ur of the
(r + 1) x (s + l)-matrix
/l 0
2/1
Y(y)
0\
yiV2
0
V
2/12/2 •••2/r 0
oy
If we denote by [f\ m Y(y)] the matrix obtained from the (m — l)st adjugate of
Y(y) by removing the common factor
„,m—l„,m —2
Vl V2 '"Vm,
we obtain a map A = A\ x A2 x A3 —► Mb by sending a point ((a* j), (6»), (c^-))
to the point
2
X(a)[y(6)]Z(c),/\X(a).
Arw
f\Z(c),...,
r
f\X(a)-
Ayw
Azw >
Then Semple [53] claimed and J. A. Tyrrell [59] proved (for quadrics and
collineation) that the above map sends A isomorphic ally onto an open subset of

COMPLETED QUADRICS AND LINEAR MAPS
377
Qb and that fi# can be covered by such open affines. In the case of quadrics
the same methods give that Qq can be covered by affines corresponding to the
matrixX(z)-y(2/)-X(a;)tr.
The existence of open coverings of Qq or fi# of the above type immediately
gives the following properties of these spaces (we shall refer to these spaces simply
as fi when we describe common properties).
(i) fi is a smooth space.
(ii) For each member / = (ii, i2,..., ii) of the set N of all of the subsets of
{1,2,..., r} there is a closed nonsingular subspace fi(7) of codimension /. In the
above open subsets of fi these spaces are defined by {^ = yi2 = • • • = yit = 0}.
(iii) We have for I and J in N that
n(/)nn(j) = n(/u J).
Hence in particular fi(J) and fi(J) meet transversally. The spaces fi(7) are
called degeneration spaces. Particularly important are the first-order
degeneration spaces fi(ra) for m = 1,2,..., r.
The work of Semple and Tyrrell has been of great inspiration to later workers
in the area because it suggests several different approaches to the subject.
First of all the above embeddings of U in the spaces M, by taking the products
of the subdeterminants of all sizes of a matrix and then closing up the image,
strongly suggest [53] that fi can be obtained by successively blowing up deter-
minantal loci in the spaces P(S2V) or P(V <8> W*). This approach was taken by
I. Vainsencher [61, 62]. He shows that fi can be defined inductively as follows:
Start with either of the two projective spaces P given above and the sequence
of determinantal varieties
A2 c A3 c • • • c Ar c P,
where Am = Am(i) is the locus where the points of P have rank strictly less
than m. Then if we have defined a sequence of spaces
Am+i(m) C Am+2(ra) c • • • C Ar(m) C P(m),
we define a sequence
Am+2(m + 1) C Am+3(m + 1) C • • • C Ar(m + 1) C P{m + 1)
by blowing up P(m) in Am+i(ra) and letting A;(ra + 1) be the strict transform
of At(ra) for i = m + 2, m + 3,..., r. Then P(r) = fi, and the first-order
degeneration spaces fi(ra) are the inverse images of the Am+i(ra) for m = 1, 2,..., r.
The difficulty with this approach is to prove that the successive centers Am+i (m)
are smooth. Vainsencher not only overcame these difficulties, but even described
the normal bundles of these spaces.
A second approach, which is suggested by the embedding of U in M and by
the local description of fi, is to consider the action of SL(r + 1) on fig or of
SL(r + 1) x SL(s + 1) on fi# and to consider the problem of finding a
parameter space fi as a general problem of finding certain minimal compactifications

378
DAN LAKSOV
associated to algebraic groups. This approach was followed by Demazure [21]
and Drechsler and Ihle [22] and was further generalized and developed by De
Concini and Procesi [14]. They prove the following result:
Let G be a semisimple adjoint, simply connected algebraic group over the
complex numbers with an automorphism a: G —► G of order two. Moreover
let H = Ga. Then there exists a canonical variety X with an action of G
such that
(i) X has an open orbit isomorphic to G/H.
(ii) X is smooth with finitely many G orbits.
(iii) The orbit closures are all smooth.
(iv) There is a 1-1 correspondence between the set of orbit closures and the
subsets of N = {1,2,..., rank(G/J/)}.
(v) Denote by X(I) the orbit closure corresponding to I G N. Then
codimX(J) = card(J) and X{I) n X(J) = X(I U J).
We see that these statements generalize the above results about fi. The results
of De Concini and Procesi are closely related to results of D. Luna and T. Vust
[44], and the space X represents in some sense a minimal compactification in
the theory of the latter authors. A simpler and more general presentation of the
results of De Concini and Procesi was given by Uzawa [60].
In the above constructions it is not clear from the outset that the points of the
parameter spaces really represent completed geometric objects in the traditional
sense. Indeed, they are by definition simply nice compactifications of the
nondegenerate objects, and it is a nontrivial task to give a geometric interpretation of
the points outside of the nondegenerate locus. Of the three approaches sketched
above it is easiest to obtain such an interpretation from the open covering of fi
given by Semple and Tyrrell. For example, when r = 2, their result shows that
after a suitable base change every point in Qq can be represented by matrices
in one of the three forms
(1
0
lo
fl'
0
v°
(1
0
lo
0
0
0
0
2/i
0
0
0
0
(A
0
°y
fl
x 0
\o
(A (i
0x0
o) [o
0>
0
°>
\ I1
x 0
! [o
0
2/2
0
0
0
0
0
0
0
°\
°
oj
°\
0
°J
o\
0
°y
corresponding to yi = 0, respectively 2/2 = 0, respectively yi = y2 = 0. However,
it is clear that the three presentations (i), (ii), and (iii) can be interpreted as

COMPLETED QUADRICS AND LINEAR MAPS
379
saying that after a suitable coordinate change:
(i) Every element in fi(l)\fi(2) can be written as a double line {x2 = 0} with
two points {y2 + y^z2 — 0} on the line.
(ii) Every element in fi(2)\fi(l) can be written as two intersecting lines
{x2+yiy2=0}.
(iii) Every element in fi(l) n fi(2) can be written as a double line {x2 = 0}
with a double point {y2 =0}.
An approach that starts with the geometrical interpretation was indicated by
Laksov [41] and performed in [42]. He starts by defining complete linear maps
over an arbitrary space, or equivalently families of such maps, as follows;
Let E and F be vector bundles of ranks r + 1, respectively s + 1, over a space
S and let r = (ri, r2,..., rj) be an /-tuple of integers satisfying the inequalities
r = n > r2 > • • • > ri > 0.
A complete S-linear map between E and F consists of S-linear maps
olj : Ej —► Fj (g) Lj from j = 1, 2,..., /,
where Lj are line bundles on 5, such that, if we denote by I(m,aj) the ideal
in Os generated by the determinants of order m +1 of a3;, the following three
conditions hold:
(i) For y = 1,2,...,/ the ideals in the sequence
I(rj - rj+1 - 1, a3) Ç • • • Ç 7(1, tti) Ç 7(0, a3) = 05
are invertible.
(ii) For y = 1,2,...,/- 1 we have that
l{rj-rj+1,aj) = 0.
(iii) We have that E\—E and F\—F and that ify+i = ker ol3 and
Fy+i 0 L3 = F7 (8) Lj/(I(rj - r7-+1 - 1, a,) : imc*,) for j = 1,2,...,/- 1.
The above definition is inspired by the work of Semple and Tyrrell. Indeed
the definition is designed in such a way that it makes it possible to obtain open
coverings of the type given by Semple and Tyrrell. A crucial step in this approach
is to define morphisms between completed families such that they become a
category. One then shows that fi is a final object in this category. This theory
is designed to hold over any base scheme and Qb becomes a smooth scheme
over the base which is covered by relative affine spaces over the base. A similar
"functorial" approach can be given for quadrics.
Another approach which is even more refined and general has been made by
Kleiman and Thorup [40]. Their point of departure is the equations for QB in
Mb in the following generality:
Let E and F be quasi-coherent sheaves on a scheme S. A sequence (u1, u2,...,
ur) of invertible quotients
i i
u{ : f\ E <g> f\ F -► Li for i: = 1,2,..., r

380 DAN LAKSOV
satisfying
D(u\ui+1) = 0 fromz = l,2,...,r- 1
is called a complete form.
Here D(ul,ulJtl) = 0 are equations that define a closed subscheme of the
product
P(£<g>F)xP (/\E ® /\f\ x • • x P (/\E ® /\f\
which then is the scheme Qb of complete bilinear forms. The equations are
defined by
D{u\ u{+j) = D'(u\ v?+j) - D"(u\ u{+j),
where
D'(u\ u{+j) = {u* ® ui+j) o (£>, 1)
and
D'V, ti*+') = (ti* 0 tf+J) o (1, D)
and where finally D is defined by considering the exterior product and coproduct
maps
i j i+j i+j i j
f\: f\E®f\E^ [\E respectively \J : f\E^ f\E®f\E
and taking the composition
i j+k i j k
Dijk f\E® f\E ^ f\E®f\E®f\E
l\\
j i k j+l
f\E®f\E®f\E XM f\E®f\E
one lets D = Dl>h% and
i i i+j i+j i i i+j i+j
(D,l): /\E®/\F® /\E® f\F f\E ®/\F ® /\E ® /\F
ai ai
i j+i i i+j i i+j i i+j
f\E® f\E®f\F® f\F £®î f\E® f\E®f\F® /\ F.
Similarly one defines (1,-D).
Equations for Qb in Mb had been used earlier by van der Waerden [66]
in dim 2, by Semple [52] (see also [55]), and by Alguneid [2] in dimension 4,
however without any indication of how to generalize them to higher dimensions.
The work of Kleiman and Thorup is still in progress and will certainly reveal
more of the secrets of the parameter spaces fi.
At last we would like to mention that a vast amount of special knowledge
about the geometry of Qq and fis in low dimension has been amassed in this
century. For such results we refer to [1, 2, 3, 23, 52, 55, 56, 57, 66].

COMPLETED QUADRICS AND LINEAR MAPS
381
In the general case Finat [25] takes up the point of view of Semple, Tyrrell,
and Vainsencher, and Tjurin [58] points out connections to the theory of moduli,
theta characteristics, and Enriques surfaces.
3. Intersection theory on the parameter spaces fi. In §1 we mentioned
that the important numbers of a one-parameter family were the characteristics
fi and v and the numbers A and tt of degenerate members of the family. These
numbers are easily generalized to characteristic classes and degeneration classes
in an intersection ring of fi. The characteristic classes fxm for m = 1,2,..., r
are the classes induced on fi by the hyperplane sections of P(/\m V <S> f\m W*)
or P(S2 /\mV) and the degeneracy classes 6m are the classes of Q(i) for i =
1,2,...,r.
In all the presentations mentioned in the previous section it is easy to show
that one has the following equivalent set of relations, which generalize the relation
(l)of§l:
(2) fjLm+i = -6m - 2£m_i m£i + (m + l)A«i for ra = 0, l,...,r
or
-6m = AWi ~ 2/Km + A*m-i for m = 1,2,..., r
with /zr+i = /i0 = 0. It is also easy to see that the Picard group, that is,
the classes of codimension 1 families, is free on the generators /ii,/i2, • • • ,A*r
or equivalently £i,<$2,..., £r-i?A*i- Hence any simple condition on completed
objects of fi can be expressed in the form aifii + ol^i + ' ' ' + arVr, where
the ai depend exclusively on the given condition. Hence the completed objects
satisfying a certain number c of simple conditions can be represented by the class
c
I"J(ai^l + a2i^2 + ' " + arifAr)
when the solution has a geometric significance. We see that to solve this kind of
enumerative problems one has to find the numbers ctji and the classes
(3) ^'^-^-
We shall not discuss the determination of the numbers ctji. They have been
determined for thousands of different conditions in the previous century, and the
reader can get a taste of these kinds of computations from the articles of the
reference list. We will however mention that for contact problems they can be
expressed in terms of the ranks of the spaces involved and refer the reader to
[39] or [26].
For the determination of the classes (3) the relations (2) are most important
because they reduce the problem to computations of the degenerate loci that are
points, lines, planes, This was observed immediately after the appearance of
the relations (1) for conies by Cremona [13] and Zeuthen [67]. Later algorithms
for determining the classes (3) by this method were given for space quadrics
by Zeuthen [68, 69] and Schubert [45, 51] and in four-dimensional space by

382
DAN LAKSOV
Schubert [46, 51]. Finally Schubert [49, 50] used inductive methods to find
algorithms in higher dimensions. At the end of the century it seems to have
been part of the folklore that these methods gave an algorithm for determining
the classes (3), either by reduction to the loci fi(ra) and inductive procedures
or by the reduction to the most degenerate spaces fi(l) n fi(2) PI • • • PI fi(r).
The latter space, called the Halphen locus, is a flag manifold over the base and
consequently all numbers can be calculated in this space. However, an algorithm
based on the latter method was first stated explicitly by De Concini and Procesi
[15].
Schubert [49-51] also gave explicit formulas for some of the classes (3) when
mi + rri2 H \-mr = dim fi.
These numbers are evidently the most important for enumerative problems and
are called the characteristics of the elementary systems. The most spectacular
of the formulas given by Schubert is the following:
Let
L0 C Li C • • • C Lr C Pr and M0 C Mi C • • C Mr Ç Pr*
be sequences of linear subspaces of the projective r-dimensional spaces Pr,
respectively Pr*, where dim Li = a; and dimM; = bi for i = 0,1,..., r. Denote
by
fi(a) = fi(ao,ai,... ,ar) and fi(6) = fi(60,&i, • • • , M
the Schubert cycles in the intersection ring of the Grassmannians G\ = Grassr(n)
and G2 = Grassr(n) of r-planes P(V) in Pr and P(W*) in Pr* such that
(4) dim(P(Vr) PI Li) > i, respectively dim(P(W*) PI M{) > i
for i = 0,1,... ,r.
Moreover, let D(a, 6), be the determinant of the matrix
M{a,b) =
Then if m = ]C[=o(a* + M + r> we have, in the intersection ring of E =
Gi x G2 x fi, that
D(a,b) = /i7l-n(a)-n(6).
Here the right-hand side represents the number of r-planes P(V) and P(W*)
with a complete correlation a: V —► W such that P(V) and P(W*) satisfy the

COMPLETED QUADRICS AND LINEAR MAPS
383
Schubert conditions (4) and such that the intersection of P(V) and P(W*) with
m-fixed hyperplanes of Pr, respectively Pr*, are pairwise correlative under the
map P(W) —-► P(V), induced by a.
The assertions of Schubert were generalized and proved by Giambelli [25].
His main result is the following generalization of Schubert's formula.
Let p be an integer such that 0 < p < r, and let mi,m2,...,mP+i be non-
negative integers satisfying the conditions
r
m\ + ra2 H h rap+i = ]P(a* + bi) + r
i=0
and
q-l
mi+m2 + -' + mq> ^(ar_g+i+i + br-q+i+i) +q+ 1 for g = 1,2,... ,p- 1,
i=0
where the a^ and 6* are the numbers of the preceding result.
Moreover, let
•WO-(>)+!.■ (>)+■•• + •■({)•
Then the number
considered in E is equal to
lroi • 2m2 • • -pm" • (p+ 1) • D(o,6) - E^p(mp+i,mp+1
r—p
- E(ft> + ki) -{r-p+l))- D{H,K) • D{N\H,N\K),
i=o
where the sum is over all increasing sequences H = (h0^i, • • • ,hr-v) and
/C = (fco,fci,...,fcr-p) taken from A^ = {1, 2,... ,r + 1}, and D(H,K) and
D(N\H,N\K) are the determinants of the submatrices of the matrix M(a,b)
above, taken from rows H and columns K, respectively rows N\H and columns
N\K.
A. Lascoux observed that the above formula of Schubert would follow easily
from the theory of Schur functions and Gysin maps that he had pioneered in the
theory of Schubert varieties. This observation was communicated to Vainsencher,
who included a proof of Schubert's result along these lines in [62].
A proof of the more general formula of Giambelli, along the lines sketched by
Lascoux, is presented in [43].
4. Problems raised by Halphen's examples. We mentioned in §1 that if
we use P = P(S2V) as a parameter space for conies, we would have difficulties
solving even the simple problem of deforming the number of conies tangent to
five lines. Indeed, the first characteristic fx of ffo comes from P, and there it
represents the conies passing through a fixed point. The condition that the conies
of P are tangent to a line is then expressed by 2/z and the condition that they are

384
DAN LAKSOV
tangent to 5 lines by 25/z5, which gives the number 16 of such conies. We also
remarked that the reason for the incorrect answer is that the locus of double lines
in P represents degenerate conies that are all tangent to the five lines and that
the number 16 therefore has no geometric significance. Passing to Qq we saw
that there were only a finite number of complete conies tangent to 5 lines, and
this number is given by i/5 = 1. Cremona was the first to point out the reason
why arguments of the above type may lead to erroneous results. He advocated
strongly that great care had to be taken in solving enumerative problems, so
that the solutions always have geometric significance. His warnings were taken
most seriously by his contemporaries. However it was not until recently that a
more systematic approach to such questions was taken. Particularly Kleiman
has pioneered techniques in this direction, and his transversality result [37] has
been used over and over again to check that solutions to enumerative problems
have geometric significance.
Cremona also suggested that there might even be problems for families of
conies that cannot be solved on Qq, in the above sense. Such examples were
first found by Halphen. In fact he found a whole sequence of examples of intricate
conditions that, when imposed upon complete conies, had all the double lines
with one double point of the Halphen locus as solutions. Moreover, his examples
proved that it was necessary to perform an infinite sequence of "blow ups" of
Qq to solve all these examples. He also suggested a procedure for solving such
problems [26-28]. Similar problems for surfaces were discussed by Halphen
[29], del Pezzo [19], and Burali-Forti [7, 8]. Halphen's examples for curves were
taken up by Semple [54], who discusses the parameter space obtained from Qq
by blowing up the Halphen quadrics.
Recently De Concini and Procesi [15] have taken up the same problems in
the more general setting of their group-theoretical version of completed objects
referred to above. They define a space with an intersection theory by a
limiting process and thereby obtain a theory generalizing that of Halphen. A more
down-to-earth treatment for conies is given by Alvero and Xambo [4, 5]. They
work on Qq itself and define a Halphen ring of complete conies and thereby
an intersection theory for which they are able to find a basis and a
multiplication. Hence they have a tool for solving enumerative problems corresponding
to Halphen's suggestion. They also generalize the theory to conies in higher-
dimensional spaces [6].
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Royal Institute of Technology, Sweden

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Local Chern Characters and Intersection Multiplicities
PAUL C. ROBERTS
Abstract. Using the theory of local Chern characters, it is possible to
prove the vanishing of intersection multiplicities in several cases, including
the case of modules over a regular local ring conjectured by Serre.
1. Introduction. One of the recurring problems of intersection theory has
been to define intersection multiplicities in the greatest possible generality so
that the definition still has all the properties which intersection multiplicities
should have. Serre [10] gave a purely algebraic definition for subschemes of a
regular scheme in terms of the module structures of the local rings at the point
of intersection; more generally, if M and N are finitely generated modules over a
regular local ring A, and if the supports of M and N meet only at the maximal
ideal of A, he defined the intersection multiplicity
X(M,N) = ^(-lriengthCror^M, JV)).
i>0
For the definition to make sense, one needs to know that the lengths of the
Tor's in the sum are finite and that almost all of them are zero. The finiteness
of the lengths follows from the hypothesis that the supports of M and N meet
only at the maximal ideal, and they are almost all zero because every module
over a regular local ring has finite projective dimension. Serre proved that the
condition on the support implies the following inequality on Krull dimensions:
(1) dim M + dim N < dim A.
In addition, if A contains a field, he showed that the above definition of
\{M,N) gives the same answer as one gets by the more classical method of
imbedding into the product and intersecting with the diagonal, and that this
implies the following connections between the Krull dimensions and the
intersection multiplicities:
(2) If dim M + dim N < dim A, then x(M, N) = 0.
(3) If dim M + dim N = dim A, then x(M, N) > 0.
1980 Mathematics Subject Classification (1985 Revision). Primary 13H15, 14C17; Secondary
13D15, 13D25, 14F15.
Supported in part by a grant from the National Science Foundation.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
389

390
PAUL C. ROBERTS
Serre proved these statements, in fact, when A is unramified, and he
conjectured that they hold over arbitrary regular local rings. Statement (2)
("vanishing") is used in showing that a good intersection product can be defined for
arbitrary regular schemes, and statement (3) ( "positivity" ) implies that this
product has the properties one intuitively expects. We show in this paper that
the vanishing conjecture holds not only for regular local rings, but also when M
and TV are modules of finite projective dimension over a complete intersection
or a local ring whose singular locus has dimension at most one. Most of these
results were announced in Roberts [9]. The main tool in the proof is the theory
of local Chern characters of Baum, Fulton, and MacPherson [1]. These results
have been proven independently in the case of complete intersections by H. Gillet
and C. Soulé using Adams operations on Grothendieck groups of complexes with
given support; we refer to their paper [6] for a discussion of these methods.
It had been hoped that the theory of local Chern characters would prove
even stronger versions of the vanishing and positivity conjectures. If A is any
local ring, M is a module of finite projective dimension, and TV is any finitely
generated module such that M ®a N has finite length, then x(A^ N) is defined,
and it was asked whether statements (2) and (3) hold in this generality. Perhaps
the best way to illustrate this idea is to describe a very simple case where it
works. Suppose that dimTV = 0 (so that TV has finite length), and let
0->F*:-> ► F0 -► M -► 0
be a finite free resolution of M. Let
ch0(M) = ^(-l)^rank(FO.
Then Tor^(M, TV) is the homology of the complex F* ® TV, and using the fact
that the Euler characteristic of a complex is the same as that of its homology
when both are defined, we have
X(M,TV) = ]T(-l)Mength(i^(F, ® TV))
= X^"1)' length(^ ® N) = cho(M) length(TV).
It is easy to show that ch0(M) ^ 0 <-► ch0(M) >()<-► dim M = dim A, so
that statements (2) and (3) follow in this case.
The idea was to extend ch0(M) to a sequence of invariants ch0(M),chi(M),
..., and to extend the invariant "length (TV)" to a sequence of invariants To (TV),
ti(TV), ..., Td(TV) where d = dim(TV), and to use these to prove statements (2)
and (3) in general. This has been done in the case of graded modules over
a graded ring by Peskine and Szpiro [7]; in this case ch^(M) are the Chern
characters of a graded resolution of M considered as a complex of locally free
sheaves over projective space. It has also been done by Foxby [4] when dim TV =
1, using the MacRae invariant for chi(M). (This result has also been proven by
Dutta [2] using different methods.) However, Dutta, Hochster, and McLaughlin
[3] have recently constructed a local ring A of dimension 3, a module M of finite

LOCAL CHERN CHARACTERS
391
projective dimension and of Krull dimension 0, and a finitely generated A-module
N of dimension 2 with x(Af,7V) = -1. The part of the above argument that
does not work is that, to prove the theorem, one needs to know that
ch^(M) = 0 for i < codim(M) = dim A - dimM.
This holds if i — 0 or 1 or if M is graded over a graded ring, but in the above
example, we have
codim(M) = 3, but ch2(M) ^ 0.
We show below that in spite of this kind of example, it is possible to prove
the vanishing conjecture using this method in many cases where both modules
have finite projective dimension. In the next section we outline the properties
of local Chern characters which we need, and which we use in §3 to prove the
vanishing theorems. In §4, we prove those properties which have not been proven
elsewhere.
2. Local Chern characters. The basic reference for this section is Fulton,
Intersection Theory [5], in particular, Chapter 18 for the construction of local
Chern characters. We shall not attempt to describe this construction in detail
here, but we outline their main properties.
Let F be a scheme of finite type over a regular scheme. In our applications,
Y = SpecA, where A is a local ring; the assumption that Y is of finite type
over a regular scheme means that A is a homomorphic image of a regular local
ring. This condition is necessary for the intersection theory to work properly
(see Fulton, Chapter 20 for details of how this works when Y is not of finite type
over a field).
We next describe the rational Chow group of Y. Let ZkY, the group of cycles
of dimension k, be the free Q-module on the set of integral subschemes of Y of
dimension k. If V is an integral subscheme of dimension k, we denote the
corresponding basis element by [V]. Let AkY be ZkY modulo rational equivalence;
if W is an integral subscheme of dimension k + 1, and if r is a rational function
on W, the cycle
E ordv(r).[K]
VÇW
dimV=k
is defined to be rationally equivalent to zero; rational equivalence is the
equivalence relation thus generated. Let A*Y, the rational Chow group of F, denote
0fc AkY. The image of [V] in AkY will still be denoted [V].
The Chow group has certain functorial properties:
1. A proper map f:Y' —► Y defines, for each k, a map f*:AkY' —► AkY
called "proper push-forward."
2. A flat map /: Y' —► Y of relative dimension n defines, for each A:, a map
/*: AkY -► Ak+nY' called "flat pull-back."
3. A Cartier divisor D defines, for each A:, a map AkY —► Ak_iD called
"intersection with D."

392
PAUL C. ROBERTS
Now let E* be a bounded complex of locally free sheaves on Y\ such a complex
will be called perfect. In the context of the introduction, E+ will be a free
resolution of a module of finite projective dimension. Let X be the support of
E+\ this is the set of points where E+ is not exact, or, equivalently, the union of
the supports of the homology of E+. If E+ resolves a module M, X is just the
support of M. The local Chern character of E*, denoted
ch(£*) = ch0(£*) + chi(£*) + • • •,
is a sort of "intersection operator" on A*Y. The properties of such an operator
are formalized in the notion of a "bivariant class" (cf. Fulton, Chapter 17). Let
X be a closed subscheme of Y. A bivariant class c of codimension i with support
X is, for every map /: Yf —► Y and for every integer A:, a map
compatible with the above operations on A+Y. This means:
1. If / is proper, then f*(c(a)) = c(f*(a)) in Ak-i{X) for all a G -4*1".
2. If / is flat, then /*(c(a)) = c(/*(a)) in ^^(/"HX)) for all a G AkY.
3. If D is a Cartier divisor, then D n c(a) = c(D n a) in Afc_i_i(Z) n X) for
aeAkY.
It follows from these conditions that a bivariant class commutes with ordinary
Chern classes of locally free sheaves (Fulton, Proposition 17.3.2). The Abelian
group of bivariant classes of codimension i with support X will be denoted
AlxY. The functorial properties of the local Chern character can now be stated
by saying that cht-(J5*) is an element of AlxY, where X is the support of E+.
Note that there is a product mapping
given by composition of operators.
The actual definition of ch(i£*) is given by the graph construction, which
we summarize very briefly. To define ch(J5*), we must define its action on a
generator [V] of AkYf for a map: Yf —► F. We first pull J5* back to V; then,
changing notation, we assume that V = Y, that Y is integral, and that we are
computing ch(J5*)([y]). Let n = dimF, and let ej = rank(^) for each j. The
graph of dj\ Ej —> Ej-i is a locally free subsheaf of Ej 0 ^-i of rank ej with
locally free quotient, and thus defines a section s of the Grassmannian
Gras8ej(Ej © J5y_i)
Y
Putting these together for all j, this gives a section s of
3

LOCAL CHERN CHARACTERS 393
The result of applying the graph construction to [Y] is (roughly) the following
sequence of operations: the image s(Y) in Ge is a cycle of dimension n; deform
this to a cycle lying over the support X of E+, act on this by
£(-l)»ch(&),
where ^ is the tautological bundle of Grasse>(^ 0 F7-1), and push the result
down to X. In general, this is a complicated process, but in certain cases it can
be computed fairly easily.
EXAMPLE 1. If E* of the form
0^Oy{-D)^Oy ->0,
where D is a Cartier divisor, then
ch(F*) = D - D2/2\ + D3/3\ ,
where D denotes intersection with D (or with the pull-back of D to Y'). In
particular, if Y = Spec A for a local ring A, then D is defined by a principal
ideal and ch(F*) is simply intersection with D.
EXAMPLE 2. If Y = SpecA, where A is a Cohen-Macaulay local ring, and if
F* is a Koszul complex defined by a regular sequence Xi,...,X* in A, then
ch(£,)M = [^l inAn_*(H0,
where W = Spec(A/(Xi,... ,X*)), and [W] = X^n»[W»l? where W^ is a
component of W corresponding to a minimal prime ideal Qi and
ni = \ength{A/{X1,...,Xk))Qi.
The local Chern characters have the following properties:
1. (Additivity.) If
0 -+ El -► E* -+ E'l -+ 0
is short exact sequence of perfect complexes, then
ch(£*)=ch(£;)+ch(^)
in A^Y, where W is the union of the supports of E*, E'+, and E".
2. (Multiplicativity.) If F* and F* are perfect complexes with supports X and
W respectively, then
ch(F* <g> F*) = ch(F*) ch(F*) in A*XnwY.
This means that for each integer A:, we have
chk{E* <g>F*) = ]T ch^F^ch^F*).
3. (Commutativity.) If c is a bivariant class in AlwY, if F* is a perfect
complex, and if j is any integer, then
c{chjiE.)) = ch(F*)c in A^y.

394
PAUL C. ROBERTS
4. (The local Riemann-Roch formula.) Let M be an A-module, and assume
that A is a homomorphic image of the regular local ring R. Let G* be a resolution
of M over i2, and define
t(M) = ch(G*)([Spec#]) in A>(SuppM).
Let E* be a perfect complex over A such that E* <g> M has homology of finite
length. Then
X(£*®M)=ch(£*)(r(M)).
In this formula, ch(F*)(r(M)) is an element of A*(p), which we identify with
Q (= A0(p)), where p is the closed point of Spec A. In particular, if A is itself
regular, we have r(A) = [Y], where Y = Spec A; we denote this also by r(Y). It
follows from Example 2 that this holds also if A is a complete intersection. In
general, if n = dim A, then r(A) and [Y] have the same component in dimension
n, but they may differ in lower dimension.
Properties 1 and 2, as well as a more general version of 4, can be found in
Fulton [5, Chapter 18]. We prove property 3 in §4 of this paper, and we also
give a more direct proof of property 2.
3. The vanishing of intersection multiplicities. We now prove the
vanishing theorems promised in the introduction. Let E* and F* be perfect
complexes with supports X and W respectively. Let Y = Spec A, where A is a local
ring, and denote the closed point of Y by p. Let n = dimY. The theorems
below concern the vanishing of the Euler characteristic
X(E* ® F«) = ]T(-l)i length(#i(£* 0 F*)).
The theorems for modules M and N of finite projective dimension follow by
applying this to resolutions E* and F* of M and N, since then Hi(E* ®F*) =
Tor*(M, TV), so
X(E*®F*) = ^(-l)Mength(Tor,(M,7V)) = x(M,7V).
THEOREM 1. Suppose that
i. xnw = p.
2. dimX + dimW <dimY = n.
Then if a is any element of AnY, we have
ch(£*<g>F*)(a) =0.
PROOF. By multiplicativity, we have
chn(£*®F*)(a)= ]T chi{E*)chjiF*){a).
We claim that every term of the sum on the right-hand side is zero. Suppose
first that j <n — dim W, so that dim W < n — j. Then chJ(F*)(a) is an element
of An-j(W), which is zero. Similarly, if i < n — dimX, using commutativity, we
have
chiiEjchjiF^ia) = chi(F+)chi(F+)(a) = 0.

LOCAL CHERN CHARACTERS
395
Now the hypothesis that dim X + dim W < n implies that for all i and j with
i + j = n, we must have either j < n — dim W or i < n — dim X. Hence all terms
are zero, so
chn(E* ® F*){a) = 0.
Using Theorem 1 and the local Riemann-Roch formula, we can easily prove
the vanishing theorem when r(Y) has nonzero components only in dimension n.
THEOREM 2. Suppose that r(Y) has nonzero components only in dimension
n. Suppose that E* and F* are perfect complexes, and that their supports X and
W satisfy the hypotheses of Theorem 1. Then
x(£*(g>F*) = 0.
In particular, this holds if A is regular or, more generally, a complete intersection.
PROOF. By the local Riemann-Roch theorem, we have
X(£*®F*) = ch(£*®F*)(r(Y)).
Since r(Y) is an element of AnY by hypothesis, Theorem 1 says that
ch(£*®F*)(r(Y))=0.
Thus x(£* ® F*) = 0.
We next prove the vanishing theorem when the singular locus of Y has
dimension at most one. The main point here is that if Z is a regular scheme and X
is a closed subscheme of Z, then A*XZ = 0 for i less than the codimension of X
in Z (this could be used to give another proof of Theorem 2 when A is regular).
This implies that, with hypotheses as in Theorem 1, any nonvanishing terms in
the sum
Ç chiiEJchjiF.)
i+j=k
must be supported on the singular locus, and if the singular locus is small, it
can be shown that they vanish also.
THEOREM 3. Let Y be a regular scheme of dimension n, and let X be a
closed subscheme ofY. Then AlxY = 0 for i < dim y — dimX.
PROOF. Let c be an element of AlxY', with i < codimX. Let k be an integer,
and let V be an integral subscheme of Y of dimension fc. Let G* be a resolution
of Oy by locally free sheaves on Y. Then
\V] = chn-k{G.)[Y].
Hence c([V\) = c(chn_fc(G*)[Y]) = chn_fc(G*)(c[Y]) by commutativity.
But c{[Y]) is in An-i{X), and n - i > dimX, so c[Y] = 0. Thus
c([V))=chn-k(G*)(0)=0.
In the next theorem, Y may have components of different dimensions, so
the statement on the dimensions of the supports of E* and F* is a little more
complicated than in Theorem 2.

396
PAUL C. ROBERTS
THEOREM 4. Let E+, F*, X, and W be as above. Assume that the singular
locus of Y has dimension at most one. Assume also that
1. XnW =p.
2. For every component Z of Y, dim(X D Z) + dim(W n Z) < dim Z.
Thenx(E*®F*) = 0.
PROOF. As above, the local Riemann-Roch theorem states that
X(£*®F*)=ch(£*®F*)(r(y)).
We shall show that for any integer k and any a € A* y, we have
chfc(£*®F*)(a) = 0.
Since any integral subscheme lies in some component, we can assume that Y
itself is a component, and, by hypothesis 2
dimX + dimW < dimy
As above, we have
chfc(£*(g>F*)(a)= ^2 chiiEJchjiFJia),
i+j=k
and we may assume that j < dimy — dim VF. Let S denote the singular locus
of y. Let Y = Y - S, and let W = W - {W D S). Then Y is regular, so, by
Theorem 3, chJ(F*)(a) restricted to W = 0. There is an exact sequence
Ak.3(S DW)^ Ak-jiW) - Ak-,-{W) ^ 0.
Thus chj(F*)(a) is the image of an element ji of Ak-j(S 0 W). We have a
commutative diagram
A0{p) = A0{p)
ch,-(£7.)î îch,(^)
Ak-,iSnW) » Ak-j{W)
P ^ ch3iF.)(a).
Hence chi(E*) chj(F+)(a) = chi(E*)(/3). Now the dimension of S D W is at most
one, so if this element is not zero, we must have i = k — j = 0 or 1. However, by
Roberts [8, Theorem 3], we have
ch0(£*) = 0 if dimy - dimX > 0,
chi(£0 = 0 if dimy - dimX > 1.
Thus, using hypothesis 2 of the theorem, the only way this could fail to be zero
is if
dimX = dimy-l and dimW = 0.
But then dim(S fl W) = 0, so that this term is zero in that case also, so we have
shown that every term is zero, and this completes the proof.
We remark that crucial use is made in Theorem 4 of the fact that ch; (E* ) = 0
when codimX > i for i = 0 and 1, and that this is false if i > 2. Thus this proof

LOCAL CHERN CHARACTERS
397
cannot be extended. However, the question of whether the vanishing theorem
holds for two modules of finite projective dimension over an arbitrary local ring
is, at present, still open.
4. The commutativity and multiplicativity of local Chern
characters. We show in this section that local Chern characters satisfy properties 2
and 3 of §2. Property 2 is proven in Fulton [5, Example 18.1.5], but the proof we
give here is more elementary in that it does not rely on the Riemann-Roch
theorem. The main construction, which is also due to Fulton [5, Example 18.3.12],
is to blow up the original scheme so that the pull-back of the original perfect
complex will have a filtration with quotients of the form
0 -+ Li -+ Li-x -+ 0,
where Li and Li-\ are invertible sheaves. Multiplying by L~\, this complex
becomes
Li_i ® (0(-Z?) - 0)
for some Cartier divisor D. This reduces questions about local Chern
characters to questions about divisors and ordinary Chern characters, both of which
are more elementary and which we assume known. This process is somewhat
analogous to the "splitting principle" for ordinary Chern classes.
In what follows, the phrase "E is a locally free subsheaf of F" will include the
condition that the quotient F/E is also locally free.
We now describe the constructions. Let y be a scheme, and let E* be a perfect
complex in Y with support X. We wish to prove formulas involving ch(i£*); as
usual, we assume that Y is integral and that we are computing the action of
ch(i£*) on [Y], We assume also that X ■£ Y\ if X = Y, the formulas follow from
properties of ordinary Chern characters.
We shall construct a proper birational map f:Y —► Y such that f*{E*) has a
filtration as above, and such that the supports of the homology of the quotients
map into X.
We first assume that Ek has an invertible subsheaf, where fc is the highest
integer with Ek i1 0. We can do this by replacing y by a closed integral subscheme
of the projective bundle P(Ek) (= Proj(Sym(i5fc))) which maps birationally onto
y.
Let L «-► Ek be an invertible subsheaf (with locally free quotient, as we are
always assuming), and consider P(Ek-i) with its tautological invertible subsheaf
0(—1). Let p:P(Ek-i) —► y be the projection, and consider the composition
p*L - p*(Ek) ^ ?•(£*_!) - p'iEk-J/Oi-l).
Let P be the closed subscheme of P(Ek-i) where this composition vanishes.
On P, p*L maps into 0(—1), and we wish to find a closed subscheme of P
which maps birationally onto Y. To achieve this, we note that on Y — X, E+ is
locally split, so dfc(L) is an invertible subsheaf of Ek-i, and it defines a section
s: Y - X -+ P{Ek-y) with s*(0(-l)) = dh(L). Thus on s(Y - X), dk maps p*L

398
PAUL C. ROBERTS
into 0(-l), so s{Y - X) Ç P. Let Y be the closure of S(Y - X) in P; it is
clear that Y maps birationally onto Y, and that the pull-back of E* to Y has
the subcomplex p*L —► O(-l), which is of the desired type. Also, the support
of this complex maps into X.
Repeating this process, one arrives at a birational map f:Y —► Y such that
f*E* has a filtration with quotients
EilEi~l ï* Li® (Q(-Di) ^ 0)
for invertible sheaves Li and Cartier divisors Di. Thus, by Example 2.1, multi-
plicativity for ordinary Chern characters (see Fulton [5, Proposition 18.1.(c)]),
and additivity, we have the formula
ch(£*) = ^ch(L;)ch(-A),
i
where we denote ch(-A) = A - D?/2l + D%/3\ .
It follows immediately that any bivariant class c commutes with ch(F*) on
y, since c commutes with ordinary Chern characters and with intersection with
divisors. The commutativity on Y now follows from compatibility of bivariant
classes with proper push-forward: let c E A\yY, and let j be an integer. We
have
cichjiE.W})) = dchjiE.KMY}))
= c(/.(cM£,)[Y])) = /.(c(ch,(£.)[Y]))
= f*(chj(E*)c[Y]) by commutativity on Y
= chj(E.)(c(MY})) = chj(Et)(c{Y)).
Thus we have proven commutativity; we now prove multiplicativity. Let E*
and F* be perfect complexes with supports X and W respectively. Arguing
as above, using the compatibility of local Chern characters with proper push-
forward, we can assume that both E* and F* have filtrations with quotients of
the form
L*®(0(-A)->0).
The tensor product of these filtrations induces a (double) filtration on E*<8>F*;
thus, using additivity and the multiplicativity with ordinary Chern characters,
we are reduced to proving the results when
E*=0{-D)^0 and F* = 0{-D') -* 0
for Cartier divisors D and D'. The tensor product E* <g> F* is then
0(-D - D') -i 0(-D) 0 0{-D') -+ 0,
and if D is defined locally by a and D' by a', then F* <g> F* is locally the Koszul
complex on (a, a'). Using the notation ch(D) = D — \D2 + • • • as above, we
must show that
ch(F* ® F*)[Y) =ch(D)ch(D,)[y].

LOCAL CHERN CHARACTERS 399
It would be possible to show this directly from the graph construction, but
we prove it here by blowing up D D D' to produce a filtration of the above type
on E+ ®F*.
Let /: Y —► Y be the blow-up of Y along D C\D'. Let Spec A be part of an
affine cover of Y, and let a generate D and a' generate D' on Spec A The part
of Y over Spec A is covered by affine pieces
Ui = Spec A[a/a'\ and t/2 = SpecA[a'/a\.
The image of d:0{-D - D') -+ Q(-D) 0 Q(-D') is generated in Spec A by
(—a', a). Thus the pull-back to Y is contained in the invertible subsheaf
generated in U\ by (—l,a/a;) and in f/2 by (—a'/a, 1). Let Z£ be the exceptional
fiber in Y, generated in f/i by a' and in f/2 by a. By comparing the transition
functions one checks that this invertible subsheaf of 0(—D) 0 Q(-D') is locally
isomorphic to 0(E); by comparing the generators for different affine pieces of Y,
it is seen to be globally isomorphic to 0(—D — D' + E). Here we use D and D'
also to denote the pull-backs of D and D' to Y.
A similar computation shows that the quotient of 0(—Z?) 0 O(-D')
modulo this invertible subsheaf is isomorphic to Q(—E), and we have a short exact
sequence of complexes.
0 0
1 1
0 -+ Q(-D-D') -+ Q(-D-D' + E) -+ 0
1 1 1
0 -+ Q(-D-D') -♦ O(-D)0O(-D/) -+ 0
1 1 1
0 > 0(-E) > 0
I I
0 0
The top row can be written Q(-D-D'+E)®(Q(-E) -* 0), and, by additivity,
we have
ch(£* ® F*) = ch(F) - ch(0(-Z? - Z?; + ^)) ch(F)
= ch(£)[l - ch(0(-D -D' + E))}
= ch(E)ch(D + D' -E).
Let F = D + £>; - F. We must show that
/*(ch(£;)ch(F)) = ch(D)ch(D').
We show, in fact, that if P(E, F) is any homogeneous polynomial with rational
coefficients which is symmetric in E and F, then
0
0
f*(P(E,F))=P(D,D').

400
PAUL C. ROBERTS
To do this, it suffices to prove:
(a) /*((£ + F)n) = {D + D')n for any integer n.
(b) f*(EFP(E,F)) = DD'f*P(E,F).
Statement (a) is straightforward, since
f*((E + F)n) = U((E + D + D' -E)n) = f*((D + D')n) = (D + D')n,
since /*(£>) = D and /*(£>') = £>'.
To show (b), we note first that D — E and D' — E do not intersect (they are
the "proper transforms" of D and D'), so that
(£> - E){D' -E) = DD' -DE- D'E + E2 = 0.
Thus, £F = E(D+D'-E) = DE+D'E-E2 = DD'. Hence U(EFP(E,F))
= f*(DD'P(E,F)) = DD'f*P{E,F), as was to be shown. The general formula
above now follows by induction on the degree of P(E, F).
References
1. P. Baum, W. Fulton, and R. MacPherson, Riemann-Roch for singular varieties, Inst.
Hautes Études Sci. Publ. Math. 45 (1975), 101-145.
2. S. Dutta, Generalized intersection multiplicities of modules, Trans. Amer. Math. Soc.
276 (1983), 657-669.
3. S. Dutta, M. Hochster, and J. E. McLaughlin, Modules of finite projective dimension
with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291.
4. H.-B. Foxby, The MacRae invariant, Commutative Algebra Durham 1981, London
Math. Society Lecture Notes Series, 72, 121-128 (1982).
5. W. Fulton, Intersection theory, Springer-Verlag, 1984.
6. H. Gillet and C. Soulé, K théorie et nullité des multiplicités d'intersection, C. R. Acad.
Sci. Paris Ser. I. Math. 300 (1985), 71-74.
7. C. Peskine and L. Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris, Ser. A-B,
278 (1974), 1421-1424.
8. P. Roberts, The MacRae invariant and the first local Chern character, Trans. Amer.
Math. Soc. (to appear).
9. , The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math.
Soc. (N.S.) 13 (1985), 127-130.
10. J.-P. Serre, Algèbre locale-multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag,
1957/58.
University of Utah

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Enumerating Contacts
ROBERT SPEISER
A contact takes place when two varieties in Pn meet, but not transversally.
The counting of contacts has led, over the past century or so, to the discovery
of some very beautiful and fundamental geometry, whose importance to older
generations was highlighted by Hilbert's challenge, in his fifteenth problem, to
develop this, and some related areas, rigorously.
Recently, the study of contacts has been especially active, due to new
conceptual approaches and more powerful techniques.
For example, suppose given a projective variety Vt, moving in a 1-parameter
algebraic family, and another projective variety X, fixed, but such that the
general Vt does not contact X. Then the number of V* which do make contact with
X is finite, and we can determine it by the Contact Formula, a classical result
recently made rigorous [FKM].
To formulate the underlying idea precisely, one uses the conormal schemes of
Vt and X. For a variety V, the conormal scheme CV, by definition, parametrizes
all possible pairs (x, H), where x is a point of V and H is a hyperplane tangent
to V at x. Then one counts contacts by counting the intersection points of the
conormal schemes. This process is considered in greater detail in §1.
These results on simple contacts provide background for investigating higher-
order contacts. New work here has led to some exciting possibilities, whose
common point of departure is the classical enumerative study of triangles, begun
by Schubert [S2].
For example, suppose that two reduced plane curves C and Z?, in general
position, move in 1-parameter families. Then at most finitely many C meet
some D at least triply at some common point; denote by N the number of these
contacts. To find TV, we strengthen the machinery of §1.
To grasp the main idea consider a triple contact between two plane curves as
a limit of transverse intersections. Following three intersection points as they
converge to a triple intersection, we can form the triangle they span, including
the sides along with the vertices. (For certain applications, as in §4 below, it will
help to order the vertices and sides, but never mind for now.) In the limit, the
1980 Mathematics Subject Classification (1985 Revision). Primary 14C17, 14D05, 14M15.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
401

402
ROBERT SPEISER
three sides will approach the tangent line common to both curves at the point
of contact, so we obtain, as before, the information needed to detect contacts,
in this case ordinary tangencies.
To distinguish the triple contacts, however, this plainly won't be enough. But
what extra structure, exactly, should we attach, in order to capture the second-
order differential picture in the limit? Based on a key idea suggested by Schubert
[S2], Semple gave an explicit construction [Se] which provides the information
necessary to strengthen the conormal scheme.
Here is the gist of Semple's construction. In addition to the configuration
of vertices and sides that we have already introduced, include the 2-parameter
family of conies through the 3 intersection points as above. Then, in the limit,
each general enough conic in the 2-parameter family will have a triple contact
with each of the original intersecting curves, at the given point of intersection.
The space of such ordered triangles, equipped with the extra structure we have
described, together with their limits, is denoted by W*. While the construction
goes back to [Se] its detailed study, as continued recently in [RSI, RS2, and
RS3] has led to new discoveries.
Here, as in the case of simple contacts, the parameter variety, now VF*, has
an intersection calculus which can be worked out in detail. (A good beginning
was made by Tyrrell [T], while the full determination will appear in [RS3].
Another version has just been found by Collino and Fulton, using a torus action.)
Compared to the study of simple contacts, the triangle theory is deeper. For
example, the space W* is no longer a natural blowup, and its geometry, while
impressive, needs much more insight to uncover.
We shall outline the triangle theory, along with several alternatives, in §2.
Each alternative, roughly speaking, carries a different kind of extra structure to
the limit, as a given triangle degenerates, and each contributes crucial
information about the others. Much recent work has been devoted to these alternative
theories, and to their respective applications.
Section 3 presents, with proofs, some new results of mine. (It is heartily
suggested that a reader new to the subject pass directly to §4, which
continues the survey.) Here the preceding triangle varieties are compared with two
other natural choices: the Hilbert scheme Hilb3 P2 and Kleiman's third
iteration scheme [K], which we denote by X3. These results provide, among other
things, an intrinsic definition of VF* via the Hilbert scheme, independent of any
particular families of auxiliary curves, and they help to clarify the connection
with multiple-point theory.
One of the most interesting applications of the triangle theory is to the
classical problem of enumerating cuspidal plane cubics. (Compare [Z, M, S, and
KS].)
Each cuspidal cubic carries a natural triangle, formed by the cusp, the flex,
their join, together with the cuspidal tangent, the inflectional tangent, and their

ENUMERATING CONTACTS
403
intersection. In general families, this triangle degenerates exactly when the
cuspidal cubic does, so we can reduce questions about degenerating cuspidal cubics
to questions about degenerating triangles. Schubert, in [S], counted these
degenerations by determining how often each pair of vertices (resp. sides) come
infinitely close.
A challenging key step in Schubert's discussion, however, provides the stimulus
for some current work, which is described in §4.
Enumerative geometry has been, of course, one of the main historic sources
of contemporary algebraic geometry. Its recent development has given us not
only a deeper appreciation of the phenomena connected with tangency and with
projective duality, especially as a variety degenerates; we are also, I hope, at the
point of building a satisfactory higher-order theory as well.
One aim of this paper is to survey recent progress in this area. Another is
to suggest that the classical enumerative viewpoint, centering as it does upon
the interplay between the intrinsic geometry of a projective variety, the way that
variety embeds in its projective space, and the way the given embedding deforms,
should be very pertinent to us now. Although this survey offers worthwhile
evidence, I think, to back this claim, I'm looking forward to the efforts of the
next few years to give us more.
1. The conormal scheme and its descendants. The counting of
ordinary tangencies provides an elegant case study of how one solves a nontrivial
enumerative problem. Schubert treats it in his first chapter of his book [S].
Indeed, by 1879, when he wrote, it was already a classic. So many basic ideas make
their appearance: completed objects (compare Laksov's report in this Seminar),
the explicit determination of cycle classes by means of degenerations, the use
of dual cycle bases, and so on. Each idea, moreover, appears in an especially
transparent, simple way.
The general setting. Suppose given closed subvarieties V,W, and a point p in
PN. We shall say that V and W have a contact at p if (1) p is a smooth point
on both V and W', and (2) V and W are not transversal at p. More precisely,
(2) requires that the embedded tangent spaces TPV and TPW are contained in
some hyperplane of PN. (For p singular on V, W', or both, we shall extend this
definition of contact by continuity, using the conormal scheme, to be introduced
below.)
For example, two plane curves have contact at a common smooth point
precisely when they are tangent there, while two curves in PN, for N > 2, make
contact anywhere they meet. As the dimensions of V and W grow in comparison
to TV, the condition of contact becomes stronger. At the upper extreme,
generalizing the case of plane curves, two divisors have contact at a common smooth
point p only when their tangent hyperplanes at p coincide.
By introducing the conormal schemes of V and W (for more details, compare
[K2]) we can always reduce the counting of contacts to the counting of ordinary
intersections.

404
ROBERT SPEISER
Here, briefly, is the definition. Denote by / the incidence correspondence of
points and hyperplanes on PN. (Hence / is also the incidence correspondence of
points and hyperplanes on the dual projective space PNv.) Denote by Vsin the
smooth set of V. Assigning to each p E Vsm the set of all (p, H), where H is a
hyperplane containing TPV, we obtain CVsm, a subscheme of /. Its closure in /
is defined to be the conormal scheme, denoted CV, of V. The projection of CV
to PN is obviously V, while the projection of CV to PNv is, by definition, the
dual variety of V, denoted by Vv.
If CV is also the conormal scheme of Vv, we say that V is reflexive. (In
positive characteristics, this can fail. For a survey of this fascinating and rapidly
developing theory, see Kleiman's excellent survey [Kl].)
Return now to our two closed subvarieties V, W C PN. Clearly V and W
are tangent at a common smooth point p E PN exactly when CV and CW
have a nonempty intersection in the fiber of / over p. This reformulation of the
definition of a contact extends directly to arbitrary points p E V D W, so, from
now on, we shall work with contacts in this extended sense.
To discuss contacts with a moving V C PN, consider now a family {Vt},
parametrized by a scheme T, with total space V C PN x T. We shall assume
that T is smooth and that the general Vt is reduced. Working as before, we
obtain a conormal scheme CV, this time in / x T. Again, CV projects to V in
PN x T, while the projection to PNv x T maps CV onto a subscheme, denoted
by Vv, which is the total space of a family, denoted {Vtv} and called the dual
family, in which the dual of the general Vt moves.
These preliminaries aside, suppose given a family {Vt}, and a closed point
t E T. What might the special fiber (CV)r look like, if VT happened to be
singular, say, or if VT were nonreduced? In characteristic 0, the picture is quite
clear.
THEOREM 1.1 (Kleiman [K2]). Assume the characteristic is 0. Then the
irreducible components of (CV)r are all of the form CW, for irreducible, closed
subschemes W ofVT. If such a W is not an irreducible component ofVT, then
W is contained in the singular locus ofVT.
Instead of giving the proof, we shall examine the main idea in an important
special case.
Plane curves. Now we have
7 = {(p,L)eP2xP2v|PeL},
and, for a reduced curve X in P2, the conormal scheme CX is the closure of
cxsm = {(p,Tpx)\Pexsin},
which is the graph of the rational map from X to P2v assigning to each p E Xsm
the tangent line TPX.
EXAMPLE. Suppose X is a smooth curve, whose degree is d and class (defined
to be the degree of Xv) is dv. Choose a point c E P2, off X, and a line L off c.

ENUMERATING CONTACTS
405
Denote by T the affine line with coordinate t. Choose homogeneous coordinates
x, 2/, z such that c = (0,0,1) and such that L is the line where z = 0, and define
Xt to be the image of X under the map sending (x, y, z) to (x, ?/, tz), for tET.
We call such a family {Xt} a homolography with center c and image L.
Hence, in a homolography, we have X\ = X, while X0 is a d-fold copy of
L. Roughly speaking, we form Xt by "pushing X down along the fibers" of the
projection, denoted 7r, from c to L. Clearly, as t varies, the dv lines through
c carrying ramifications of n\Xt are independent of t. Further, each point q at
which one of these lines meets L is such that every line through q is a limit of a
tangent of Xt, for t approaching 0.
It follows that the fiber of CX at t = 0 contains the conormal scheme of the
point q. We shall call such a point of a special fiber VT a vertex (= sommet) as
Zeuthen did in [Z].
Thus, in the example, writing in cycle notation, we have
[(CX)o] =dL + dv • vertices.
In the intersection ring A*/, denote by A (resp. Av) the class of the pullback of
the hyperplane sheaf 0(1) on P2 (resp. P2V). By conservation of cycle classes,
we obtain the identity
(1) [CX]=d{\v)2+dv\2
in A*L As one should expect, this formula for [CX] is self-dual if X is reflexive.
Continuing in P2, let's look at A*I in more detail. (Further reasoning in this
vein can be found in [G], based on ideas in Schubert's book [S]. More general
flag varieties are discussed in [F] and in its references.) Denote by B the blowup
of P2 x P2 along its diagonal, A. A moment's thought will verify that we have
B = {(p,q,L)<=(P2)*xP™\p,qGL}.
Here we have a space of completed objects: pairs of points, each equipped
with a joining line. Further we have a diagram with fiber square
I > B >P2V
1 1
A ! >P2 xP2,
where i and j are the inclusions, and 0 sends (p, g, L) to L.
Now /, as a bundle over A, is just the projectivized normal bundle of the
embedding i. Hence we know its intersection ring, via the Chern classes of P2,
as soon as we can identify its tautological divisor class. For this we can use the
map (j) to relate it to the exceptional class [/] on B and the pullback, Av, of the
class of a line in P2V. After a brief effort, we arrive at the following result:
(2) As a Z-algebra, we have
A*I = Z[A, AV]/{A3, (Av)3, A(AV) = A2 + (Av)2},

406
ROBERT SPEISER
and, under the intersection pairing, the subsets
{\,\v}cA1I and {(Av)2, A2} c A21
are dual bases.
Again, the statements are projectively self-dual.
Counting tangents. Now we shall use the conormal class (1) and the duality
result (2) to count the number of members of a one-parameter family of plane
curves which are tangent to a given, fixed curve.
So suppose given a 1-parameter family {V*}, parametrized by a smooth,
complete curve T. Denote by V C P2 x T the total space of the family, and by p the
projection from I xT to I. Denote by ji and /iv the characteristic numbers of
the family: the number of Vt through a general point, and the number tangent
to a general line, respectively. More precisely, /i (resp. /iv) is defined to be the
degree of the intersection cycle A2 -p*[CV] (resp. (Av)2 -p*[CV]). Then we have
(3) p*[CV] = /iAv + /ivA.
Indeed, (3) holds for some integer coefficients, by (2), and we can use duality
to compute them immediately.
Finally, denote by e the number of Vt tangent to X. If X is general for the
action of PGL(2), it is easy to see that N is the degree of the intersection cycle
p*[CV] • [CX]. (In nonzero characteristic, however, each intersection may appear
with a nontrivial multiplicity.) Using (2) again, we obtain the identity
£ = dy jJL + djJLy
a special case of the Contact Formula.
The proof above generalizes directly [FKM]. (One can check, by the way, that
each such contact takes place at a smooth point on both intersecting curves, if
the data are in general position. For details, compare [FKM] for the general
case, or [F, pp. 187-193] for curves.)
2. Triangles and second-order data.
An old problem. Given two 1-parameter families of plane curves, in general
position relative to the action of PGL(2), find the number of triple (or higher)
contacts between curves in one family and curves in the other.
The solution was found by Schubert [S] in 1880, based on a penetrating,
general approach. The first rigorous treatment of this work appears in [RSI,
RS2, and RS3]. The theory works for any families whose general members, and
their duals, are reduced, contain no lines, and have, at worst, ordinary cusps and
nodes for singularities. (Related but weaker versions can be based, for example,
on the results of [EL]. An alternate approach to Schubert's results, via a torus
action, has just been found by Collino and Fulton.)
The number of triple contacts depends on two new invariants, k and fc', as
well as the previous invariants ji and // of the two given families.

ENUMERATING CONTACTS
407
To define k and k' for any one-parameter family of plane curves whose general
member and its dual are as above, denote by % (resp. %') the cycle on P2 (resp.
P2V) traced by the cusps (resp. the inflectional tangents) of the curves in the
family. Then k (resp. k') is defined to be the degree of % (resp. X'), as cycle on
P2 (resp. P2V).
(Note: since ordinary cusps are dual to inflectional tangents, it follows that
X and X', hence k and fc', are symmetric under projective duality, as are /i and
Returning to the original problem, suppose the two families are given by
morphisms Vi —► T\ and V2 —► T2, respectively, and denote by N the number of
triple contacts between their members. Then Schubert's solution is the formula
N = /iifcj + Mife + fc'lM2 + fcl/4 + 3(/ii/i2 + Ml/4)>
where each subscript indicates the family whose invariant is being computed.
While there are strong heuristic grounds to feel that there ought to be such a
number TV, we must begin by defining it, explicitly, as an intersection number.
The construction. To define TV, the first step is to construct a space, denoted
by X*, to parametrize the "second-order data" on P2, and, simultaneously, on
P2V. This construction (due essentially to Semple [Se]) begins the description,
in explicit, modern terms, of the geometry behind Schubert's result.
Here is some motivation. It is natural to expect that X* should be an
algebraic P1-bundle over the incidence correspondence / of points and lines in P2.
Indeed, / parametrizes the "first-order data" on P2, that is, points in P2 plus
assigned tangent directions. Further, above a given point of /, each point of
the corresponding fiber of X*/I should measure, in effect, a "curvature value",
giving 0 at flexes, and, dually, 00 at cusps. (We need the value 00, for example,
to handle flexes in the dual plane, since these are dual to cusps in P2.)
Now for the construction. First we define a suitable base space, parametrizing
the triangular configurations in P2. Denote by W the incidence correspondence
{all (x, y, z, L, M, N) € (P2)3 x (P2)3 | x, y € L, x,z,€ M, and y, z € N} .
Clearly no second-order information appears on VF; we just have points and
"secants" joining them.
To provide the necessary extra structure, denote by G the Grassmannian
of 2-planes in the P5 of all plane conies. Hence a point of G parametrizes a
2-parameter linear family of conies in P2. We define a rational map
r0: W —► G by assigning to each general point (x, y, z, L, M, N) of W the point
of G parametrizing the 2-parameter family of conies through x, y, and z.
The variety W* of Schubert triangles is defined to be the closure, in W x G,
of the graph of tq. The effect of adding the conies is to provide curves which
contact each other triply as the vertices x, î/, z approach a common limit.
As Semple observed in [Se], W* is smooth, while W is singular. The
singular locus of W is the subvariety, denoted by X, parametrizing the degenerate
triangles of the form (x, x, x, L, L, L).

408
ROBERT SPEISER
Finally, we define X* to be the inverse image of X in W*. Since X c W is a
copy of the incidence correspondence /, we have the Cartesian square
X* —
I
I —
—► w*
I
—► w.
Here the arrow W* —► W is birational, and there is a regular map r: W* —► G
compatible with the rational map r0: W —► G.
To determine A • X*, it is important to note that the projection X* —► /
has two disjoint sections, denoted by sq and s^. To each given point of X,
that is, to each point and assigned tangent direction, the S{ attach the curvature
values indicated by the subscripts. (Other curvature values depend on the local
coordinates, hence do not give sections defined over all of X.)
We define s0 (resp. Sqq) by equipping each point (x,L) of / with the two-
parameter family of conies having L as a component (resp. having x as a singular
point).
Two further triangle varieties. Denote by W the blowup of W along its
singular set X. As Semple points out [Se], the singularity of W along X is conical,
so W is smooth. Further, W dominates W* by a birational morphism. In fact,
W is the blowup of W* along X*, as shown in [RSI].
There is another triangle variety dominated by W', denoted by 5, which first
appears in [RSI]. To construct 5, we first blow up (P2)3 along its diagonal P2,
obtaining a smooth variety, denoted by A. Then B is defined to be the blowup
of A along the strict transforms of the three 4-fold diagonals of (P2)3. (By the
way, B is self-dual [RSI]: we could have begun with P2v instead.)
It is easy to verify that B dominates VF by a birational morphism. Indeed, one
obtains the three lines joining the vertices of a triangle (x, y, z) E (P2)3 because
the three diagonals are defined by the vanishing of the Pliicker coordinates of
the given lines, as computed from the homogeneous coordinates of x, y, and z.
Denote by XB the pullback of X from W to B. A fundamental argument in
[RSI] shows that W dominates 5, and is, in fact, the blowup of B along XB'.
Denote by X the pullback of X to W. The projection XB —► X, like the
projection X* —► X, is a P1-bundle, but its fibers measure something new:
the relative rates at which the vertices x, y, and z approach each other. (This
information, of course, is completely independent of any given curvature.)
Putting all our varieties together, we obtain two commutative diagrams,
and

ENUMERATING CONTACTS
409
where the arrows of the first diagram are dominant birational morphisms, and
the arrows of the second can all be shown [RS2] to be P^bundles. Further
[RS2], the second diagram is Cartesian!
The intersection ring of X*. Now X*/X is of the form P{E) for a locally free
E of rank 2 on X. Since the sections s0 and s^ are disjoint, they correspond
to a splitting of E into a direct sum of line bundles, so, if we understand this
splitting, it will be easy to determine A*X*.
To accomplish this, lift the sections s0 and Soo to sections of X —► XB. These,
too, are disjoint. We also know that X/XB is of the form P(£')> where E' is
the pullback of E, and the lifted sections correspond to the pullbacks of the
summands of E.
But X —► XB is also the projection of the normal bundle of XB in £,
because W is the blowup of B along XB. This leads to another natural splitting
of E'. Denote by C the divisor on B parametrizing triangles with collinear
vertices. (One can project to W to interpret elements of B as triangles.) Similarly,
we define the divisor D of points of B parametrizing triangles with concurrent
sides. Then a detailed local inspection [RSI] shows that XB is the transverse
intersection of C and D. This gives the second splitting of E'.
By the Krull-Schmidt theorem, the two splittings have summands which are
isomorphic in pairs. After identifying the corresponding summands, it is
straightforward to compute their Chern classes on XB. From this, the determination
[RS2] of the intersection ring of X* follows directly.
In fact, denote by rj (resp. ç) the classes in AXX* representing the image of
sq (resp. Sqo), and let a\ (resp. 0:2) represent the pullback of a hyperplane in P2
(resp. P2V). Then we have the following result.
THEOREM 2.1 [RS2]. (1) We have
A*X*=A*/[»?]/(»?2+3(ai-a2h).
(2) The class of a point is
a\a2r) = aialri = a\a2Ç = ai^.
(3) Under the intersection pairing of AXX* and A3X*, we have dual bases
{ai,a2,rj + 3ai = ç + 20:2} and {a\r), a\r], a\a2 — otio^}-
(4) Under the intersection pairing of A2X* with itself we have dual bases
{a2, ol\, aif, a2r)} and {a2 + 3a2, ai + Za\, a\, a\}.
Parametrizing the contacts. To complete the solution of the problem at the
beginning of this section, we need to associate with each given 1-parameter family
{Vt} of curves the subvariety of X* parametrizing the second-order data carried
by the curves Vt- In characteristic 0, one can do this directly, by referring to a
local parametrization of W* given by Semple [Se].

410
ROBERT SPEISER
In this way each curve Vt in the family determines a proper transform
V* C X*, and the Vt* sweep out the proper transform V* c X* of the
family. These dominate the analogous conormal schemes, and augment them with
the second-order information which they lack.
To understand the subvarieties of X* which result, we need to have some
information about how a general Vt* degenerates, just as we did before, when
studying the conormal scheme it dominates. This second main result tells what
happens on a possibly nonreduced special fiber Vt.
THEOREM 2.2 [RS2]. Assume dim(T) = 1, fix a closed point t e T, and
choose a smooth point p E Vt, not a vertex and not the limit of a moving
singularity. Then the only point of Vt* over p E Vt is the one corresponding to the
second-order data of (Vi)red at p.
The proof is based on representing Vt locally by means of a power series
y = /(x, £), which is regular in t as well as x. (This representation is possible by
an old result of Jung.)
To solve the problem posed at the opening of this section, we can now proceed
as in the previous section, first using duality to write the classes of the proper
transforms in terms of a pair of dual bases, and then using duality again to find
the product.
The intersection ring ofW*. The determination of A*W* in [RS3] is based
on the diagram of triangle varieties VF, VF*, £, and VF above. Using the standard
results about intersection rings of blowups, one shows, in this situation, (1) that
A*W* is generated as a ring by Pic VF*, and (2) that the dimension of each AW*
agrees with that predicted by Schubert.
For (2), it is actually shown that A*W* and A*B are isomorphic as graded
abelian groups (although they are not isomorphic as rings). Next, the dimensions
of the graded pieces of A* B are found, via the definition of B by repeated blowups
of (P2)3. Finally, one checks directly that the relations predicted by Schubert
among the generators of AW* hold. Then Schubert's triangle calculus follows
immediately. (Collino and Fulton follow a similar outline; the main new feature
is the use of a torus action to handle (2).)
3. Comparison theorems. This section contains new results of mine, which
compare the triangle variety W* with the Hilbert scheme Hilb3P2, and with
Kleiman's iterated blowup scheme, to be described below. I want to thank Steve
Kleiman for his comments on some earlier versions of this work.
The Hilbert scheme. We define a rational map
VF ^ Hilb3 P2
by the assignment
(x, y, z, L, M, N) H-+ {x} U {y} U {z}.
Our main result is the following.

ENUMERATING CONTACTS
411
THEOREM 3.1. The triangle variety W* is the closure, in W x Hilb3P2, of
the graph ofr.
PROOF. By definition, W* is the closure of the graph of the rational map
VK AG(2,5)
{2-parameter linear
families of conies in P'
i
which sends a general point (x, t/, 2, L, M, N) of W to the linear system of conies
through x, t/, and z. (Here, for a general point, we can take one where x, t/, and
z are distinct.)
The theorem would follow immediately if we had a closed embedding of
Hilb3P2 in G(2,5) compatible with r and s, but, as we shall see below, this is
not the case. We do, however, have a canonical morphism i: Hilb3 P2 —► G(2,5),
such that s = ior, with closed image.
That i exists will follow from Grothendieck's construction of the Hilbert
scheme [Mu]. Although presented there in the case of divisors, the method
works in general. Since we shall need to take advantage of our special situation
in order to obtain some sharper estimates, here are the relevant details.
Let Z be a subscheme of P2 whose Hilbert polynomial is the constant function
3, and let / be its defining ideal sheaf. We show first that / is 3-regular, in the
terminology of [Mu, p. 99].
To show that / is 3-regular, we reason as follows. The short exact sequence
of coherent sheaves
0 - /(-l) - 0(-l) - Oz(-l) = Oz - 0
on P2 gives the long exact cohomology sequence
0 = #°(0(-l)) - H°{Oz) - ^(/(-l)) - ^(©(-l)) = 0;
hence dim /f1(/(—1)) = dim H°(Oz) = 3. Now consider the short exact sequence
of coherent sheaves
(*) 0-+J(-l) -+J-+J|L-+0,
where L is a line in P2 which misses Z. Since I\L = Ol by the choice of L, we
find that I\L is 0-regular. By the estimate [Mu, p. 102, last display] it follows
that / is 3-regular, since dim/f1(/(—1)) = 3 as we have shown. In particular,
by Castelnuovo's result [Mu, p. 99], we have ^1(/(2)) = #2(/(2)) = 0.
Now for the construction of the morphism i. Start with the natural map
tf°(P2,0(2)) ± H°{Z,Gz{2)) = H°(Z,Oz).
Since the cokernel of q is H1(I(2)) = 0, and since the dimension of the source
is 6, the assignment Z *-> q defines a morphism from Hilb3 P2 to G(2,5), as one
sees immediately by passing to the direct image sheaves (cf. [Mu, p. 106, III])
whose fibers at Z E Hilb3 P2 are the source and target of q.
Indeed, because 3-regularity gives H1(I(2)) = 0, it follows that the relevant
R1 vanishes. In fact, beginning with [M, p. 106, assertion I], one can decrease

412
ROBERT SPEISER
the number mo there by 1 throughout and still obtain a morphism. In our case,
of course, this means that we can take tuq = 2. (Only the argument that i is
an immersion fails for the new mo: in the key step [Mu, p. 107, V] it no longer
follows that the twisted ideal sheaf is spanned by its sections.)
Clearly i satisfies the required compatibility with r and s. That i is not an
embedding follows because any subscheme Z contained in a line L in P2 maps
to the point in G(2,5) corresponding to the 2-parameter family of conies which
contain L as an irreducible component, so i is not even injective. To show that
the image of i is closed, one reduces as in [Mu, p. 109, bottom] to the case of
a family parametrized by a smooth curve, and then applies the specialization
construction [H, Proposition 9.8, p. 258].
REMARK 1. For any integer d > 0, denote by PN the space of plane curves of
degree d. Hence we have N = dimH°(P2,0(d)) - 1. For any d > 2, the method
above embeds Hilb3P2 in the Grassmannian G(N — 3, AT), by the map which
associates to each subscheme Z the (N — 3)-parameter family of plane curves of
degree d through Z. For later use, we denote this embedding by id-
To prove Theorem 3.1, we shall map W* to the iteration scheme X3, using a
variant of the case d = 2 above, and then map X3 to Hilb3 P2, using a variant
of the case d > 2 of Remark 1. We shall carry this out after Corollary 3.3 below.
REMARK 2. Instead of using 2-parameter families of conies in the definition of
W*, Theorem 3.1 shows that we could just as well have used (N — 3)-parameter
families of curves of any degree d > 2, where N is the dimension of the
projective space of plane curves of degree d. But now, using the Hilbert scheme, we
can generalize W* directly from configurations involving 3 points and lines to
configurations involving k > 3 points. These ideas will be developed elsewhere.
The iteration scheme. Here is the definition [K] of Kleiman's iterated blowup
scheme, which also parametrizes triangles. Denote the blowup of P2 x P2 along
its diagonal by X<i and view it as a P2-scheme, via the composite of the first
projection with the blowup map. Then X3, the iterated blowup scheme, is
defined to be the fiber product of Xi with itself over P2, blown up along its
diagonal.
To investigate how points of X3 are separated by plane curves of any degree
d > 2, choose a point [x,t/,2] of X3. In other words, x is a point of P2, y is a
point of the blowup B[x] of P2 at x, and z is a point of the blowup B[XiV] at y of
the previous blowup. We denote by E\ and E<i the exceptional divisors on B[XiV]
corresponding to x and t/, and by £ = Ob(—Ei — ^2) the standard invertible
sheaf on -B[x,y]- Finally, denote by C(d) the tensor product of C and the pullback
of Op2(d). The global sections of £(d) corresponds to the curves in P2 through
the perhaps infinitely near points x and y.
PROPOSITION 3.2. For all d > 2, the global sections of the invertible sheaf
C(d) separate points on B[XiV].
PROOF. First of all, Op2(l) is very ample. So choose 2 sections which vanish
simultaneously just at x. These sections determine an embedding of the blowup

ENUMERATING CONTACTS
413
B[x] in P2 x P1. Now the composite of this embedding of B[x] with the Segre
embedding of P2 x P1 is given by certain sections of the tensor product of the
pullback of Op2(2) to B[x] with the invertible sheaf on B[x] corresponding to
the divisor —E\. This tensor product is very ample, so it separates points and
tangent directions on B\xy Hence, when we impose the additional basepoint t/,
the resulting linear system will separate points on B[XiV] as was to be shown.
COROLLARY 3.3. For each point [x,y,z] of X3, the linear system of plane
curves of degree d>2 through x, y, and z has dimension N — 3, where
N = dimtf°(P2,Op2(d)) - 1
is the dimension of the complete linear system of all plane curves of degree d.
PROOF. In the iterative process above, the linear system obtained on any
given blowup by subtracting the exceptional divisor E\ from the members of the
linear system given by the pullback of Op2 (d) will be very ample, of dimension
N — 1. Imposing two new basepoints, even infinitely near, will therefore reduce
the dimension by exactly 2. This completes the proof.
We now construct a canonical morphism, denoted by /, from X3 to Hilb3 P2.
Denote by G the Grassmannian G(N — 3, N) of linear systems of codimension
3 in the PN of plane curves of some degree d > 2. It follows from the last
corollary that each point of X3 maps to the point of G corresponding to the
linear system of curves of degree d through x, t/, and z. Since X3/X2 is the total
space of the universal family {B[XiV]} of twofold blowups, reasoning similar in
style to that for the Hilbert scheme, based on the last proposition, verifies that
we have a morphism. To see that X3 maps to the Hilb3 P2, the latter identified
with its image via id in G (see Remark 1), consider three noncollinear points,
x, t/, z E P2. Then the corresponding point Z of Hilb3 P2 obviously has the same
image in G as the point [x, y,z] of X3: the (N — 3)-dimensional linear system
of plane curves of degree d with basepoints x, t/, and z. Since the set of all such
[x, t/, z] is dense in X3, it follows that / maps X3 to the Hilbert scheme.
We now construct a dominant birational morphism, denoted 0, from W* to
X$. Denote by Y3 the fiber product which we blow up to obtain X$. There is
a natural birational map p from W onto Y3, since Y3 parametrizes the triangles
in P2 with one side (TV, in our notation) missing. Indeed, each factor of Y3, the
blowup X2, parametrizes lines, each equipped with two ordered points, and the
structural map selects the first point.
Consider the diagonal, denoted by A, in Y3: by definition, X3 is the blowup of
Y3 along A. Identifying the complement of the pullback of A in X3 with Y3 — A,
we obtain a birational correspondence g from W* to X3, via the composite
dominant birational map from W* to Y3.
It remains to show that g is a morphism. Since W* and X3 are both smooth,
it suffices, by Zariski's Main Theorem, to show that for each a E W* there
corresponds a unique b E X3. Denote by p the image of a in Y3. If p is off A,

414
ROBERT SPEISER
clearly we have a unique b. So consider a p E A. Thus we have p = (x, t/, t/, L, L),
in our usual notation.
Now consider a point b = [g, r, s] of X3 corresponding to a. Clearly we have
q = x. I now claim that r is also determined by p, hence by a. Indeed, if r is
off the exceptional curve of £[x], this is clear: in effect we have r = y. If r is on
the exceptional curve of B^, then, by definition, r corresponds to the tangent
direction at x given by L.
Now the 3-parameter family of conies through [x, r] separates points on B[x^,
by the last proposition, taking d = 2. Hence the map from X3 to G(2,5) is
injective on £[x,r]. It follows that there is exactly one point [x, r, s] of -B[x,r]
having the same image in G(2,5) as does a E VF*. Clearly we must have b =
[x,r, s], since the maps from VF* and X3 to G(2,5) are compatible with #. It
follows that g is a morphism.
PROOF OF 3.1. Since W and VF* are canonically isomorphic over dense open
sets, it follows that the composite map fg from VF* to Hilb3P2 extends the
rational map from VF* to Hilb3 P2 induced by r. Hence VF* is the closure of the
graph of r, as was to be shown.
To state a second key result, denote by D the closure in X3 of the set of points
[x, y,z], where y and z are both on the exceptional fiber at x, but z is not on
the exceptional fiber at y.
THEOREM 3.4. The birational map g from W* to X3 above is an
isomorphism only on W* — <7_1D.
PROOF. We know that W* and X3 are both smooth, so, by Zariski's Main
Theorem, it is enough to show that g is injective only on W* — g~xT>. Choose
a point p of X3. Then p maps to a point q of the base space Y3. By the
construction, q is a configuration consisting of points x, y, z and lines L, M,
where L joins x and !/, and M joins x and z.
When the images of y and z in P2 are distinct and different from x, they
determine a unique join N, which, together with the original configuration g,
determines a point of W. This point of VF, combined with the value
f(p) E Hilb3P2, gives a unique preimage of p in VF*. The same holds if y
and z are infinitely near, with images distinct from x.
Now suppose p is such that y and z map to the same point x of P2. If y and
z determine different tangent directions at x, the 2-parameter family of conies
through [x, y, z] is given by the square of the maximal ideal of P2 at x. Hence
the point in VF corresponding to a preimage of p is not uniquely determined,
since we have no way of recovering the third line N from the family of conies.
On the other hand, suppose y and z do determine the same tangent direction
at x. (This happens if y is on the exceptional curve of B^, and z is on the
exceptional curve of B[x^/B[xy) Denote by F the 2-parameter family of conies
through [x, i/,;z]. Then the general member of F is tangent at x to the line,
denoted L, through x with tangent direction given by y. The image of [x, !/, z]

ENUMERATING CONTACTS
415
in y3 is (x, x, x, L,L), so its preimage in W is of the form (x, x,x, L,L, TV), for
some line N through x.
By the elementary result [RSI, (2.6.1), p. 1237], if L and N were distinct, F
would be the 2-parameter family of conies given by the square of the maximal
ideal of P2 at x. As we have seen, this is impossible for F; therefore N = L,
so [x,t/,;z] has a unique preimage inW*. It follows that / fails to be biregular
exactly on <7_1D, and the proof is complete.
4. Cuspidal cubics. Consider a cuspidal cubic curve C in P2. Denote the
cusp and flex by c and v, the cuspidal and inflectional tangents by q and w. Then
the dual curve Cv is also a cuspidal cubic, with cusp w, flex g, and cuspidal and
inflectional tangents v and c. Let x denote the intersection point of the lines q
and w, and, dually, let z denote the line joining c and v. We obtain a triangular
configuration of points and lines, the singularity triangle of C, denoted d(C),
whose dual configuration is the singularity triangle of Cv.
Following Zeuthen [Z], the union of a nonsingular conic Q with one of its
tangent lines L will be called a curve of type cr. It is easy to degenerate a
cuspidal cubic to a curve of type cr; when this happens, the dual cuspidal cubic
also degenerates to a curve of type cr, consisting of the union of the dual conic
Qv with the line dual to the intersection point of Q and L. (For proofs of these
and later assertions, compare [KS]. A different treatment, based on double-point
theory, has been found independently by Sacchiero [Sa].)
The cuspidal cubics C are parametrized by a locally closed 7-fold, denoted by
DC, of the P9 of plane cubics, Denote by G the group of linear automorphisms
of P2; then DC is a G-orbit. So is the locally closed 6-fold parametrizing the
curves of type cr, denoted by S. In fact, S is the only 6-dimensional orbit on the
boundary of DC.
Now let's consider the singularity triangle d(C) of a cuspidal cubic C as it
degenerates to a curve S of type cr. It is not difficult to show that d(C) limits
to a degenerate triangle d(S), depending on S alone, consisting of 3 coincident
vertices and 3 coincident sides.
The enumeration of cuspidal plane cubics begins with a study of 1-parameter
families {Ct} of plane cubics, for t on a complete, nonsingular parameter curve
T, such that the general Ct is cuspidal, and the only noncuspidal Ct are of type
cr. For such a family, we have the invariants m, m' of §1. We define further
invariants c, v, x, g, w, and z (abusing notation, as in [S]) as follows: c is the
degree in P2 of the cycle traced by the cusp of Ct, while v and x are defined
similarly; then g, w and z are defined analogously in the dual plane. Hence, for
example, c is the number of cusps of Ct on a given line, and q is the number of
cuspidal tangents of Ct through a given point.
We would like to write a formula for the number, denoted (again, we abuse
the notation) by cr, of t E T such that the curve Ct is of type cr. To do so, of
course, we must define a as an intersection number, and here the triangle theory
of §2 comes into play. As t varies, the singularity triangle d(Ct) ranges over the

416
ROBERT SPEISER
triangle variety W', and the curves of type a are parametrized by precisely those
t such that d(Ct) lies in the copy X of the incidence correspondence of points
and lines. If X had codimension 1 in VF, we could define a as the degree of the
pullback of X to T under the map which sends t to d(Ct); sadly, however, X has
codimension 3.
Hence we do the next best thing: we blow up X in W, obtaining the divisor X
in the triangle variety VF of §2. Then the invariant a is defined to be the degree
of the pullback of X under the natural lift to W of the previous map from T to
W.
The main result for enumerating cuspidal cubics is the following simple-
looking statement.
Theorem 4.1 (Zeuthen [Z], Maillard [M], Schubert [S]). For
a family {Ct} as above, whose general member is a cuspidal plane cubic, the
following identities hold:
(1) 2cr = c + w = v + g,
(2) 2<t = ia + ia'.
The proof, however, is far from simple. One obtains (2) from (1) by counting
the number of t for which a general projection from P2 to a line ramifies on C*,
along a fixed line through the projection center. (This count follows from a new
and quite general ramification result, which also contains Plucker's formula for
the class; compare [KS].)
Finally, to prove (1), we need to count the degenerate singularity triangles.
Here we take the same basic approach that Schubert did in [S]: we count the
number of times each pair of vertices (resp. sides) of Ct coincide, as the triangle
d(Ct) degenerates. In other words, we shall reduce the question to enumerating
pairs of coincident points.
So we need to consider the following general situation: a curve T and a map
/ from T to P2 x P2. Denote by A the diagonal, and blow up P2 x P2 along
A to obtain a blowup B with exceptional divisor D. Assume that f(T) is not
contained in A. Then lift / to B: the degree, denoted £, of the pullback of D
to T is well defined. Write f(t) = (x(t),y(t)), and, abusing notation as before,
write x (resp. y) for the degree of the cycle on P2 traced by x(t) (resp. by y{t)).
It is not difficult to show that we have
e = x + y - L,
where L denotes the degree, in the dual plane P2v, of the cycle traced by the line
L(t) joining x(t) and y(t). And clearly, e counts, in a natural way, the number
of t € T for which x(t) and y(t) coincide. (Since T is a nonsingular curve,
L(t) extends to all t by continuity.) The result above is Schubert's Coincidence
Formula; compare [S, G, and KS].
We apply Schubert's formula to the 3 pairs of vertices of the singularity
triangle; we can also apply it, over the dual P2, to the 3 pairs of lines. We obtain 6

ENUMERATING CONTACTS
417
coincidence formulas, each counting, in a natural way, the number of Ct of type
a. Among them are the following three:
&i = c + v — z, &2 = z + w — v, as = z + q = c.
Adding the first two, the right side is c + w. Adding the first and third, the
right side is v + q. Hence assertion (1) holds, provided that we can identify, as
Schubert did, each ai with a.
Here, however, is a fascinating leap in Schubert's reasoning. Discrepancies
among a and the ai can only occur if different pairs of vertices (resp. sides)
approach their common limits at different rates, as the general Ct limits to some
special curve S of type a. If the rates differ, the multiplicities with which the
degeneration S is counted in the different coincidence formulas will not agree,
and the proof of (1) will fail. In his text, Schubert simply identifies the ai [S,
p. 110], writing a for each [S, p. 110], and passes on.
Justifying Schubert's identification of the ai (the first rigorous treatment of
this approach appears in [KS]) has led to an important application of the triangle
theory of §2 to the enumeration of plane cubics, based on new results which
deepen and advance the classical program.
Having initiated the triangle theory, it may seem paradoxical that Schubert,
who had foreseen so much, did not seem to be aware of this particular possibility.
Easy examples, indeed, support Schubert's claims: in [KS], in fact, the general
case is eventually reduced to such examples. Perhaps Schubert suspected as
much.
Also important, however, is that Schubert's original triangle calculus,
designed to study triple contacts, simply does not apply in the current situation,
for reasons which I think he understood. Indeed, on the parameter space W*
which models his theory, the degenerate triangles correspond to the points of a
subvariety of codimension 2, while the degenerate members of a family of
cuspidal cubics, on the other hand, will typically appear in codimension 1. Judging
by the discussion on p. 157 of [S2], it appears that Schubert had every reason
to be aware that his own triangle calculus would not be valid here.
The necessity of working on W in place of VF* takes us, however, entirely
outside the nineteenth-century context. The result comparing the ai and a
depends upon a completely new identity, in a triangle calculus fundamentally
different from Schubert's: the intersection ring of W'.
When Semple gave the original construction of W in [Se], suggesting the study
of its intersection theory as a topic for future investigation, he made no reference
to the problem of enumerating cubics. Semple's suggestion has been taken up
recently, as we have seen, from several viewpoints, some surprising, and here the
new developments begin.
The reader is referred to the original papers [RSI, RS2, KS, RS3], and to
work in progress soon to appear, for further information.

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REFERENCES
[EL] G. Elencwajg and P. Le Barz, Détermination de l'anneau de Chow de Hilb3 P2,
C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 635-638.
[F] W. Fulton, Intersection theory, Springer-Verlag, 1984.
[FKM] W. Fulton, S. Kleiman, and R. MacPherson, About the enumeration of contacts,
Algebraic Geometry—Open Problems (Ravello, 1982), Lecture Notes in Math., vol. 997, Springer-
Verlag, 1983, pp. 146-155.
[G] D. Grayson, Coincidence formulas in algebraic geometry, Comm. Algebra 7 (1979), 1685-
1711.
[H] R. Hartshorne, Algebraic geometry, Graduate Texts in Math., no 52, Springer-Verlag,
1977.
[K] S. Kleiman, Multiple-point formulas. I: Iteration, Acta Math. 147 (1981), 13-49.
[Kl] , Tangency and duality, (Proc. Conf. Algebraic Geom., Vancouver, 1984), CMS
Conf. Proc, vol. 6, Amer. Math. Soc, Providence, R.I., 1986, pp. 163-225.
[K2] , About the conormal scheme, Preprint, 1984.
[KS] S. Kleiman and R. Speiser, Enumerative geometry of cuspidal plane cubics, (Proc. Conf.
Algebraic Geom., Vancouver, 1984), CMS Conf. Proc, vol. 6, Amer. Math. Soc, Providence,
R.I., 1986, pp. 227-268.
[M] S. Maillard, Thèse, Cusset, Paris, 1872.
[Mu] D. Mumford, Lectures on curves on an algebraic surface, Ann. of Math. Stud., vol. 59,
Princeton Univ. Press, Princeton, N.J., 1966, lectures 14-15.
[RSI] J. Roberts and R. Speiser, Enumerative geometry of triangles. I, Comm. Algebra 12
(1984), 1213-1255.
[RS2] , Enumerative geometry of triangles. II, Comm. Algebra 14 (1986), 155-191.
[RS3] , Enumerative geometry of triangles. Ill, Comm. Algebra (to appear).
[S] H. Schubert, Kalkul der abzdhlenden Géométrie, Teubner, Leipzig, 1879; reprinted by
Springer-Verlag, 1979.
[S2] , Anzahlgeometrische Behandlung des Dreiecks, Math. Ann. 17 (1880), 153-212.
[Sa] G. Sacchiero, Numeri charatteristici delle cubiche piani cuspidali, Preprint, 1985.
[Se] J. Semple, The triangle as a geometric variable, Mathematika 1 (1954), 80-88.
[T] J. Tyrrell, On the enumerative geometry of triangles, Mathematika 6 (1961), 159-164.
[Z] H. Zeuthen, Détermination des charactéristiques des systèmes élémentaires des cubiques, C.
R. Acad. Sci. Paris 74 (1872), 521-525.
Brigham Young University

Algebraic Cycles

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Cycles on Arithmetic Schemes
and Euler Characteristics of Curves
SPENCER BLOCH
0. Introduction. This paper, which considers in detail the subject of one of
my lectures at the Bowdoin conference, evokes the germ of a theory of algebraic
cycles and intersection numbers on regular arithmetic schemes (i.e., schemes of
finite type over a Dedekind ring) and uses these tools to establish a long-expected
formula for the change in Euler characteristic in a degenerating family of curves
in mixed characteristic. The formula has some number-theoretic interest, but
the focus here will be on the geometric ideas involved in the proof, most
particularly the idea of intersection numbers and Lefschetz numbers associated to
endomorphisms of arithmetic schemes.
I cannot claim to have a completely general theory. All too often in the sequel
we will fall back on the crutch of working in low dimensions. Indeed, as Steve
Kleiman has pointed out to me, we don't even know how to put a ring structure
on the Chow group of an arithmetic scheme. However it should become clear to
the reader at least what one can expect in general, and what would be useful.
Certainly, the work of Faltings has brought home to us the importance of doing
geometry on these schemes.
The main ideas for this work came first from the work of K. Kato and
S. Saito in Tokyo and the opportunity I had to talk and correspond with them,
and second from the powerful intersection theory of Fulton and MacPherson, as
exposed in the book of Fulton [3]. This is a revised version of an earlier preprint
which contained an error in the proof of Lemma (7.17). I am very grateful to
T. Saito for pointing out the mistake and to K. Kato for helping to repair it.
The whole treatment of §7 in this version is much simpler. I should also like
to acknowledge helpful conversations with H. Gillet, L. Illusie, S. Kleiman, G.
Laumon, M. P. Murthy, and V. Srinivas.
Let S = Spec (A) be the spectrum of a complete discrete valuation ring with
algebraically closed residue field. Write s (resp. 77, resp. fj) for the closed (resp.
generic, resp. geometric generic) point of S. Let /: X —► S be flat and proper.
1980 Mathematics Subject Classification (1985 Revision). Primary 11G25, 14H25, 14C17,
14C25.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
421

422
SPENCER BLOCH
We assume X is regular and the generic fibre Xv —► r\ is smooth. For z either s
or 77, let
X(X)Z = ^(-îrdimtfl^QO,
the étale Euler characteristic of the corresponding fibre. We ask for a formula
calculating x{X3) - XpG?)-
In characteristic zero (A = C [[£]]) the result is understood (cf. [3, example
14.1.5]); dt gives a section of the sheaf n^/C of Kâhler differentials, to which
one can associate a localized chern class Z(sf) E CH0(XS). (I will follow the
notation in Fulton's book [3] except that I prefer to denote the Chow group of
dim n cycles by CHn.) If Xs is smooth outside a single point x, then
Z(sf) = (Milnor number) • [x],
where the Milnor number is the local invariant of the map / at x studied in [7].
In characteristic 0, for X a degenerating family of varieties of any dimension,
degZ(sy) = {-l)dimX(x{Xs) - x{Xfj))- Note that one can (and we will) think
of (—l)dlmXZ(s/) as a local contribution to the cycle-theoretic self-intersection
of the diagonal
(0.1) (Ax • AX)S = (-l)dim*Z(S/) = X(*.) " x{Xn).
We will be most interested in the mixed characteristic and pure characteristic
p analogues of this result. There are two problems with (0.1) in these cases.
First, in mixed characteristic, our construction of Z(sf) doesn't make sense.
(What is fi^yC?) Second, even in the pure charp case, the formula is wrong!
It is clear, however, from the global Grothendieck-Ogg-Shafarevich formula [8]
that the appropriate correction factor is the Swan conductor (cf. [8] as well as
the discussion below)
sw(X/S) = dimHomgai(sw5,iT(X^)).
(Notations like Rom(A, H* (Xf})) or tra\H*(Xfj) are used as shorthand for the
familiar alternating sums.) In the geometric (i.e., pure charp) case, the
conjectural local formula for a regular scheme X flat and proper over S reads
(0.2) (Ax • Ax)s - - sw(X/S) + x(Xs) - xM-
When the points of nonsmoothness for X over S are isolated, this was proved
by Deligne [2, exp. XVI].
In the arithmetic (i.e., mixed characteristic) case we must first define
(Ax-Ax)s- For this, we modify slightly the graph technique in Chapter 18
of [3] to prove
THEOREM (VAGUE). Let £ be a coherent sheaf of finite homological
dimension on a scheme X of finite type over a regular noetherian base. Let Y C X
be a closed subscheme, and assume £ is locally free of rank d on X — Y. Then
there exists a "reasonable" definition for chern classes c»(£) E CHdimX-i(^) for
i >d+l.

CYCLES ON ARITHMETIC SCHEMES
423
Suppose now X is a regular scheme, flat, proper, with smooth generic fibre
over S = Sp(A) as above, and d = fibre dim(X/S). We apply the theorem with
£ = fi^/5, the sheaf of Kàhler differentials, and Y = Xs, defining
(Ax.Ax), = (-l)d+1cd+1(fi^/5) e CH0(XS).
We will also frequently use the same notation for the degree of this cycle, so
(Ax.Ax)sEZ.
CONJECTURE. For any such X, we have with this definition
(Ax • Ax)s = - sw(X/S) + x(Xs) ~ xi**)-
THEOREM. The conjecture is true when fibre dim(X/S) = 1.
Here is a sketch of the proof. The localized chern class construction and
the definition of (Ax • Ax)s are discussed in §1. In §2, we discuss the Fulton-
MacPherson intersection theory and use it to define Lefschetz numbers or Lef-
schetz 0-cycles
(0.3) (Ax.r,)ECH0(Xs),
when a: X —► X is an automorphism lifting a nontrivial automorphism of the
base S. By the stable reduction theorem [1], there exists a finite totally ramified
T —► S with generic point rj' —► T and closed point £, such that XV' admits a
model V over T whose special fibre Vt is reduced with normal crossings. We may
assume rj' /rj is galois, and the action of the group G = gal^/rç) lifts to an action
on V. In §3 and §4 we calculate the Lefschetz numbers (0.3) on V (Theorem
4.1). Let W be obtained from V by a sequence of blowings-up of closed points so
there exists a regular map h: W —► X [11]. In §5, we introduce (without proof)
a projection formula
(0.4) \G\ - (Ax • Ax)s - (h x h)*(Aw . [W x W))s.
Using the results of §4, we show how the main theorem follows from (0.4) together
with a formula relating x{(W/G)3) — x{X3) to the intersection (Aw R), where
R is the "part" of [W x W] supported on fibres of h x h of dimension 2. This
formula for (Aw • R) is established in §6, and then the projection formula is
proven in §7.
1. The localized self-intersection (Ax • Ax)s. The purpose of this section
is to define the localized self-intersection class (Ax • Ax)s for X a regular scheme
of finite type over S = Spectrum of a discrete valuation ring. We assume X has
Krull dimension d and the generic fibre of the structure map /: X —► S is
smooth. In the case S is smooth over a field, this class coincides with the class
(—l)dZ(sf) discussed in [3, example 14.1.5]. In general, however, we cannot
use this approach because we do not have n^/fc. Instead we will generalize
slightly the construction of localized chern classes in Chapter 18 of [3], in order
to define (Ax. Ax)s = (-l)dCd(fi^/5) E CH0(XS) directly. Fulton emphasizes

424
SPENCER BLOCH
the definition of the localized chern character, but gives the localized chern class
as an example [3, 18.1.3].
We work in the category of schemes of finite type over a regular noetherian
base. Let Y be such a scheme, and let X c Y be a closed subscheme. Let £ be
a bounded complex of locally free Oy-modules. We assume (i) £* = 0, i > 0;
(ii) WP(£)\y-x is locally free of rank e for some e > 0; (iii) !K*(£) is supported
on X for i < 0. Note these properties are stable under pullback. Our objective
is to associate bivariant chern classes c%x(E) E CHn(X —► Y) in the sense of
[3, Chapter 17] for all n > e + 1. By definition, such a bivariant class is a
family of maps CEL(Y') -+ CH*_n(X') for any Y' -+ Y and X' = X xY Y',
satisfying certain functorial compatibilities. The reader who is unfamiliar with
these objects may prefer to think of the fundamental classes c%x(£) D [Y] E
CHdimY-n{X). In our application, Y will be the regular scheme denoted X in
the previous paragraph, X = Y9 the closed fibre, £ will be a resolution of fiy/5,
and e = dim Y/S = d — 1.
The construction of these classes follows, mutatis mutandis, the construction
of [3]. I will give the details for a two-term complex E\ —► E0 (the only case we
will use) and let the reader work out the more general situation. Let ei = rkE{.
Define
G = Grassei(£i 0£o),
the Grassmannian of rank e\ subbundles of E\ 0 E0. Let pr0: G —► G0 =
Grasseo(^o) = Y be the structure map, and let p^ = identity: G —► G. Define
£ = pi-Q £0 — Prî £i € Ko(G)i where £0 — ^o and £i are the canonical rank e;
subbundles. Let H = Grassei {E0) and let i: H —► G be the map associating
to a rank e\ subbundle P C E0 the subbundle (0,P) C^i04 Note that
z*(£) = 0 g Kq(H) where 6 is the canonical rank e = e0 — ei quotient of £^0 on
As in [3] we consider the (noncommutative) diagram
y x A1 ► G x A1 c G x P1 <-G x oo = G
U U
(Y - X) x A1 c (Y - X) x P1 i^x^c^xP1
where 0(y,A) = (graph(Adi(y)),A) and ip{y) = {di(y)(£i(y)) C £0(y)}- Given
a cycle a on Y, let a0 = a\y-x- Choose a' on G x P1 restricting to 0*(a x A1)
on G x A1 and a" on H x P1 restricting to </>*(a°) x P1 on ff° x P1. Set
1 — ilo(a' ~ a") ^ %>*{G) = free abelian group generated by cycles on G. The
same argument as in [3] shows 7 is well defined independent of the choice of a',
a!' up to a cycle supported on H Xy X. I claim 7 is supported on G Xy X.
To see this, it suffices to consider the case X = 0, so di(E\) C E0 is & rank ei
subbundle, and to show we can choose a', a" such that 7 = 0.
Define a map 0: Y x P1 —► G x P1 by associating to (y, (A0, Ai)) the pair
(B, (A0, Ai)), with £ c E\(y) x E0{y) the subbundle
B = {(vi, v0) G £i(y) x di(^i(y))|A0t;o = Aidi(vi)}.

CYCLES ON ARITHMETIC SCHEMES
425
Note that over oo = (0,1), the bundle B is (0,di(Ei(y))). Given a on Y, let
a' = 0+(a x P1), a" = i/)*(a) x P1. One sees as in Fulton that i^ot' - a") = 0.
We may now define
Cn,x(£)na = »?.(cn(0n7),
where rj: Gx —► X is the projection. Since 7 is well defined up to a cycle on
i: Hx C Gx and since i*(f) = 0 has vanishing chern classes above degree e, this
is well defined as a cycle class for n > e + 1.
Many of the properties verified by Fulton go through in this more general
context. For example, c^x(£) depends only on the quasi-isomorphism class of
£. In particular, if F is a coherent sheaf of finite homological dimension on Y
such that F\y-x is locally free of rank e, then c%x(F) is defined for n > e + 1.
The following proposition is proved in the same way as [3, 18.1(b)] replacing the
chern character with the chern class. Many of the details are left for the reader.
PROPOSITION ( l. I ). (i) Let £ and 7 be coherent sheaves of finite
homological dimension on Y, and assume 7 is locally free of rank donY — X for some
closed X C Y. Then for n> d and any m, the bivariant classes c%x(7) ' cm(£ )
and cm(£ ) • c%x(7) are equal.
(ii) //, furthermore, £ is locally free of rank e < m on X — Y, then c^x(7) '
cm(E) = cn(7)-c^x(E).
(iii) Let 0 —► 7^ —► 7^ —► 7^ —► 0 be an exact sequence of coherent
sheaves of finite homological dimension on Y. Assume jW is locally free of
rank dW on Y -X and let n > d^. Then
cixi?{2))= £ 4(*(1)K(y(3)).
p+q=n
where c'p(7^) denotes the usual chern class if p < dW and the localized class
clxifpXiW.
PROOF. With notation as above,
(cm(£ ) • <x(?)) H a = r?.(r?*(cm(£ )) D c„(0 D 7)
- v*{cn(0 n v*(cm(e. )) n 7) = <£iJC(y) n cTO(£ ) n a,
proving (i).
Assertion (ii) is a consequence of the projection formula for bivariant classes
[3, p. 323]. Indeed, suppose given a diagram
X' £ Y'
9'i 9[
X £ Y £ Z
with cartesian square and g proper. Let c G CH*(X —► Y) and d € CH*(Y' —► Z)
be bivariant classes. Then the projection formula says
c ■ g.(d) = gU9*(c) ■ d) e CH*(X - Z).

426
SPENCER BLOCH
In our case, let i: X —► Y be the inclusion and consider the diagram with
Z = y, Y' = X, and f — g — i. It follows as in [3, Proposition 18.1(a)] that
z*(cm x(£ )) = cm(£ )> so the projection formula gives
<x(9r)-cm(£)-i*«x(9r))-^,x(£)-
The reader should check that the somewhat implausible-looking formula
i*c = z*z*c
holds for bivariant operators, proving (ii).
Finally, for (iii), the argument is exactly parallel to [3, Proposition 18.1(b)]
and is left for the reader. Q.E.D.
COROLLARY (1.2). Let T —► S be a finite, totally ramified extension of
discrete valuation rings. Write s and t for the closed points of these schemes,
and let dx/s denote the order on S of the discriminant of T/S. Let Y be a
regular scheme which is flat and proper over T with smooth generic fibre. Then
(Ay . Ay )9 - (Ay . Ay)t = ~dT/s • x(l^),
where x(Yfj) denotes the Euler characteristic of the geometric generic fibre.
PROOF. Let /: Y —► T be the structure map. Suppose for a moment the
sequence
(*) 0 "~* f*^T/S ~^ ^Y/S ~^ fty/T ~^ 0
is exact. Since dT/S = length(fiy /5), it follows from (l.l)(iii) that
cd\mY,x(^Y/s) = Cdim Y,x(^Y/t) + dT/S ' ? (^dim Y-\ (Oy/r))'
Here X = Yt and i: X —► Y is the inclusion. Note that restricted to the geometric
generic fibre Y^, Cdimy-i(^y/T) is a zero cycle of degree = (—l)dlmy~1x(^)-
By specialization, the same is true of z*(cdimy-i(^y/T))- To obtain the desired
formula, multiply all terms by (—l)dlmy.
It remains to show (*) is exact, i.e., f*fl^,s C fly/S' Consider a diagram
X >• P' ^P
y\ Vsmooth
/smooth

CYCLES ON ARITHMETIC SCHEMES
427
with T embedded in Q, X embedded in P, P' = P xQ T, and P/Q/S smooth.
There is a corresponding diagram of sheaves (N* = conormal bundle)
0
I
Npi/P\X
1
h*^Q/S\X
1
f*^T/S
I
0
-
-
-
0
I
TV*
1
Çlp/s\X
1
n1
ilx/s
I
0
-
-
-
0
I
AT*
1
n^/g|x
1
o1
lLX/T
I
0
Since all schemes in the diagram are regular, the two top rows of sheaves are
locally free. It follows that the two top rows and all columns are exact. Indeed,
the only question is injectivity on the left, which may be checked over the generic
point of S. The fact that /*n^ ,s c ^x/s *s now a diagram chase. Q.E.D.
2. Fulton-MacPherson intersection theory. In this section I will
summarize results from the book of Fulton [3] in the form that will be convenient
for us.
To begin, consider a cartesian diagram of algebraic schemes over a field
W -+ V
I I
X -Ù Y
with i a regular immersion of codimension d. Let g: W —► X and write N =
g* {Nx/y ) f°r the rank d vector bundle on W obtained by pulling back the normal
bundle of X in Y. We abuse notation by writing N also for the scheme
N = V(A0 = Sp(Sym(7V*)).
(Quite generally, we follow Fulton's convention rather than Grothendieck's for
affine and projective schemes, e.g., we write P{N) = proj(Sym(7V*)), not
proj(Sym(iV)).)
Now let / C Oy be the ideal of W. Since the square is cartesian, there is a
surjection N -» 7//2, and hence a closed embedding of the tangent cone
C{W/V)à7n. SP O 7"//n + 1) - N.
Quite generally, one has dimC(W/V) = dimV [3, Appendix B.6.6].
There is a map on Chow groups
s*: CH*+d(7V)^CH*(W0
which can be thought of as "pullback along the zero section" (although it is
technically simpler to show the structure map p: N —► W induces an
isomorphism p*: CH*(W0 -+ CH*+d(7V) and define s* = (p*)"1). One then has

428
SPENCER BLOCH
[C{W/V)] € CHdimV(A0, and one defines
(2.1) X • V = 8*[C{W/V)] € CHdimV-d{W).
Fulton shows this definition is equivalent to another, given in terms of the
Segre classes of C(W/V). He considers a projective scheme
P = P(C 0 l)d= Proj ((©/n//n+1) [T]) yW
with T an indeterminate. The total Segre class is defined by
(2.2) s(W/V) = s(C) = U (5>(0P(1))*) e CH.(W).
One gets
(2.3) X • V = {c(N) n s(W/V)}dimv-d.
Here c(N) is the total Chern class of N = 1 + ci(N) + • • •. Since W may
have singularities, there is no multiplicative structure on CH*(W), but Fulton
defines chern classes of vector bundles as operators on cycles, the operation being
denoted (~1.
A variant of (2.3) [3, Corollary 9.2.2] will be useful. Suppose the ideal / of
W in V can be written / = Id • Ir with Id the ideal of a Cartier divisor in V.
Assume the residual scheme R: Ir = 0 has pure codimension d in V. Formula
(2.3) yields in this case
(2.4) X • V = {c(N) n s(D/V)}dimy.d + £ el[Ri].
Here Ri are the components of i2, and e; are their algebraic multiplicities in V.
(These are the usual multiplicities when V is regular at the generic points of the
Ri.) A)so,8{D/V) = c{ND/v)-1.
Note that generalizations are possible. For example, we do not need X, F, V,
W to be algebraic schemes over a field; as discussed in [3, Chapter 20] it suffices
to work with schemes of finite type over a regular noetherian base. (In fact,
in our applications, W will be defined over an Artin ring. Since all algebraic
manipulations of cycles are done in CH*(W) = CH*(W/rred), we can use the
formalism developed for cycles over a field.)
Another generalization is to the case X —► Y not necessarily a regular
embedding. We can work with the conormal sheaf
N^/Y = J/J2, X: J = 0.
(The normal sheaf is *Kom(N*, Ox), but it is important not to lose information
by dualizing.) If, for example, N^,Y has finite homological dimension we can
make sense of (3) by defining
(2.5) c(N) = (g*(c(N*x/YW,
where the * outside the parentheses means to multiply the term in codimension
* by (-!)*.

CYCLES ON ARITHMETIC SCHEMES
429
On the other hand, even if N^,Y is not of finite homological dimension, it may
happen that g*{N^c,Y) *s l°caUy free °f rank d, so formula (2.1) makes sense.
Note, however, that in cases where (2.1) and (2.3) both make sense, but ^x/y ^
not a vector bundle, it is not clear they give the same answer! (I am indebted to
K. Kato for this remark.) We will see that in our case both constructions apply
and they give the same result.
EXAMPLE (2.6). Let X be a regular scheme which is flat and of finite type
over a regular noetherian base S. Assume the structure morphism /: X —► S
is smooth over the generic points of S. Take Y = X Xs X, and let i: X —► Y
be the diagonal. Note N^,Y ~ ^x/S' ^he sheaf of Kàhler differentials. Locally,
X can be embedded as a complete intersection in a smooth scheme P/S. The
sequence
(2.7) 0 - Nx/P - QP/S \x - tfx/s - 0
is exact (injectivity on the left can be checked on the general fibre of X/S), so
fi^/5 has homological dimension < 1. We can, therefore, use equations (2.3)
and (2.5) (with d = dimX £ codim X/Yl) to define
(2.8) AxTECH^r^Ax))
for any h: V ^ X xs X.
One could hope to use this technique with V = A Xs B to put a product
structure on CH*(X), but there are technical problems I am uncertain about.
One case where the above construction works well, however, is when V = Ta is
the graph of an automorphism of X lifting a nontrivial automorphism of S. The
remainder of this section will be devoted to the precise situation of interest to
us. A number of the hypotheses could be weakened.
EXAMPLE (2.9). Let A be a complete discrete valuation ring with uniformiz-
ing element -k and algebraically closed residue field k = À/7TÀ. Assume as given
an automorphism a =^ id of A such that j = j(a) = ord(cr(7r) — 7r) < oo. Suppose
further that a is of finite order and acts trivially on k. Let A0 C A be the subring
of elements fixed under cr, and let S = Sp(A0), T = Sp(A). Let X be a regular
T-scheme of Krull dimension d which is flat and proper over T. We assume the
reduced special fibre Xt has at worst normal crossings. We assume as given a
lifting of g to an automorphism a of X. Let Ta clx§ X be the graph of a
(note the fibre product is over S, not T). We want to define the Lefschetz cycle
(Ax-r,)ECH0(Axnr,).
Note that Ax H Ta is a scheme over the artinian ring Sp(A/7r-7A), so there is a
degree map CH0(Ax HIV) —► Z. For the most part we will abuse notation and
write (Ax • IV) for the Lefschetz number obtained by applying the degree map
to the Lefschetz cycle.
LEMMA (2.10). Let X3■ C X be defined by the ideal (ir3"). Let i: Z -+ X be a
morphism of schemes, and assume i factors through Xj. Then i*fi>x/S is locally

430
SPENCER BLOCH
free of rank d. Moreover, in the derived category there is an exact triangle
Oz[l}^ Li*(Qx/s)^i*Qx/s
PROOF. It suffices to consider the case Z = Xj. By [10], we can embed
T —► Ag. Build a diagram
X ^p
smooth
L5
^smooth
s '
and hence a diagram of sheaves (locally free in the center and left columns)
o - W/a - r^A/s\r - mbs - °
lb I I
o _> N*x/P - nP/s\x - nx/s - o.
The two left-hand sheaves in the top row are isomorphic to Ox, and the map
labelled a is divisible by 7rJ. Moreover, the cokernel of b is N^/v with U =
<7_1(T), and this sheaf is locally free. We now get a complex
(2.10.1) 0 - Oz - i*(W£/P) - i'i&p/slx) - i^x/s - 0
which is exact everywhere except possibly at N*. Since i*(N^,p)/Oz is locally
free, the lemma will follow if we show i*^x/S is locally free of rank d.
Since X is regular and the special fibre has normal crossings, we can find local
coordinates £i,... ,td at a given closed point x, an integer m < d, and integers
fti,... ,Wm such that
Ox,x = a[[«i, ... ,*d]]/(n*iJ" - ™); * G A[[^i- • •>*<*]]*•
Thus Hx/5 nas ^oca^ generators dt\,..., d^, d7r, and relations of the form
m
2_, aidti + fccfor = 0 = E'^d-K,
i=i
where £ is the Eisenstein polynomial with coefficients in Aq satisfied by 7r. Since
E'(-k) is divisible by cr(7r) — 7r, this second relation dies on Xj. Also since X is
regular, one at least of the functions ai,..., am, 6 is invertible at x. This proves
the lemma. Q.E.D.
We return now to the problem of defining (Ax -IV). Since cr(7r) — -k vanishes
on fix(cr) = Ax H rff, we have ^x/5|fix(<j) locally free of rank d — Krull dim X
by Lemma (2.10). We can therefore use formula (2.1) and define
(Ax • rff) - a*[C(fix(*)/rff)] = **[C(fix(a)/X)].
We then get
(Ax • rff) = {c((n^/5|fix(CT))*) • 5(fix(a)/x)}dim0
= {^^x/s)* ^(nx(a)/X)}dim0,

CYCLES ON ARITHMETIC SCHEMES
431
where the first equality is the same calculation as [3, Proposition 6.1], and
the second is from Lemma (2.10).
We are particularly interested in the case dimX = 2. Write (cf. [3, Chapter
9]) fix(cr) = D U R, where Delis the Cartier divisor defined locally by the
G.C.D. of the defining equations for fix(cr), and R is the 0-dimensional residual
scheme obtained by factoring out the G.C.D. from these equations. Formula
(2.4) applies in our case and yields
(Ax ■ IV) = {c((n]t/s|Dn • c(iVD/x)-1)}dimo + [R].
Note ci(fi^/5) = Kx/Si the class of the relative canonical bundle. (To see this,
embed X —► P smooth over S and look at the exact sequence 0 —» N^,p —►
^p/sl* ~~¥ ^x/s ~^ 0- Both KX/s and ci(^x/s) are represented by the line
bundle det(fî},/5|x) ® det(Nx/p).) Thus
(2.11) (Qx ■ Ta) = -Kx/s D-(DD) + [R].
This formula will play a basic role in our calculations.
One important ingredient we don't know how to establish is any sort of
projection formula. To see how such a thing would be useful, consider the following
THEOREM (2.12). Let X/S be a family of curves as above, and let p: X' —►
X be the blow-up of a closed point x E X. Let a be an automorphism of X as
above, and assume x is fixed by a, so there is an induced automorphism a' of
X'. Then
(A.r) = (A'.r')-i.
REMARK. If we had a projection formula, we would use the fact that T =
(PxP)*(r') and the hope (more difficult to justify since A is not a Cartier divisor)
that (p x p)*(A) = A' + E x E to try to write
(A.r) = (A.(pxp)*r') = ((pxPyA.r') = (A'.r') + (ExE.r')
= (A'. r') + (E. E)T> = (A'. r') - 1.
PROOF OF THEOREM. Lacking such a projection formula, we give a direct
argument. Choose local parameters t, u at x such that on X', E is covered by
open sets U\ and [/2? where E:u — {) onU\ and u/o~(u) is a unit there, while
E: t = 0 and t/a(t) is a unit on [/2- Write
/ = a(t) -t; t' = t/u; f = a(t') - t';
g = a(u) - u; u" — u/t; g" — <r(u") - u".
Using the identities t — t'u and u = u"t we find
/' = {uf - tg)/ua{u), g" = {gt - uf)/ta(t).
Write fix(cr) = V(/, g) = D\JR, and let F = 0 be a local defining equation for D
near x. Let f/F and g/F vanish at x to orders m and n respectively. We may
assume m > n > 0. Suppose first that ord£;((^/ - tg)/F) = n + 1. One finds
nx(a') = p*D + (n - 1) • E + V{{uf - tg)/Funa(u), gu/Fun) on Ux,
nx(a') = p*D + (n - 1) • E + V{{uf - tg)/Ftna{t), ft/Ftn} on U2.

432
SPENCER BLOCH
These equations can be rewritten
fix(a') = p*D + (n - 1) • E + V{{uf - tg)/Funa{u), u}
+ V(//FWn-V(W),(//W) ontfi,
fix(a') - p*L> + (n - 1) • E + V{(tx/ - tg)/Ftna{t), t}
+ V(g/Ftn-1a(t)J/Ftn) onU2.
More geometrically, this says
fix(a') = p*D + (n - 1) • £ + {strict transform of V(u/ - ty/F)} • E
+ {(strict transf. of V(f/F)) + (m - n)E} - {strict transf. of V(^/F)}.
Since
deg{strict transf. V(f/F)} • {strict transf. V(^/F)}
= degV(f/F) • Vfa/F) - n • m = c (defining c)
and p*KX/s + E = KX'/s-> we can compute by formula (2.11) above
(r . A') = -D2 + (n - l)2 - i^x/5 -I> + n-l+n + l + c + (m-n)n
= (r . A) + (n - l)2 + n - 1 + n + 1 - n2 = (r . A) + 1.
Finally, we must consider the case ordjg;((w/ — tg)/F) > n -f-1. Necessarily in
this case n = ra, and
fix(a') = p*D + n£ + V((tx/ - ^)/Ftxn+1a(Tx), 0/Ftxn)
= p*L> + n£ - V(tx, g/Fun) + V(f/Fun'lG(u), Fun) on tfi.
Similarly, on [/2?
fix(a') - p*L> + n£ - V(t, f/Ftn) + F(//Ftn'1a{t)J g/Ftn).
Note that 1 — (t/u)(g/f) vanishes on E to order > 1. Since f/g is defined and
nonzero on E, we get f/g\E — t/u\E- This implies that the strict transform of
g/F meets E at the point t/u = oo with multiplicity one greater than the local
degree of V{f/Ftn, t) at t/u = oo. Thus
fix(a') = p*£> + n£ - {strict transf. V(^/F)} • E + 1
+ {strict transf. V{f/F)} • {strict transf. Vfa/F)}.
Again we compute using (2.11) and the fact that the local degree at the point x
is
degxV{f/F,g/F) = n2 + {strict transf. V{f/F)} • {strict transf. Vfa/F)},
to get (T'. A') = (r. A) +1. This completes the proof of the theorem. Q.E.D.
3. Lemmas on fixed point sets. Let A be a complete discrete valuation
ring with algebraically closed residue field k = /t, and uniformizing element 7r.
Write T = Sp(A) with rj (resp. rj) and t £T the generic (resp. geometric generic)
and closed points. Let a: A —► A be an automorphism of finite order, and let
j = j(p) — ord(cr(7r) — 7r). We assume a ^ id.

CYCLES ON ARITHMETIC SCHEMES
433
In this section we consider a T-scheme /: W —► T with / proper, flat, and
fibre dimension 1. We assume W is regular, the generic fibre Wv —► rj is smooth,
and the special fibre Wt is reduced, with normal crossings (semistable case). We
further suppose given a lifting of a to an automorphism (also denoted a) ofW.
Let fix(cr) denote the fixed point scheme of a in W. We remark
(1) Since a ^ id on T, fix(cr) c Wt (set-theoretically). Scheme-theoretically,
fix(a) CjWt.
(2) Let #i and 02 be local parameters on VF at a closed point w E fix(cr). Then
the scheme fix(cr) is defined locally by cr(#i) - 6\ and a{02) - #2- This follows
from the "derivation-like" identity
(3.0) <r(ab) - ab = <r{a){<r{b) - b) + b[a[a) - a).
(3) We can write fix(cr) as a union of schemes D U i2, where D is a Cartier
divisor on W defined locally by the G.C.D. of all functions in the ideal defining
fix(cr) and R is the zero-dimensional residual scheme (cf. [3, Chapter 9]).
Our first goal in this section is to analyze the multiplicities of the closed points
in R. The next three lemmas compute these multiplicities at points in R which
are isolated in fix(cr), i.e., do not lie on D. If A is a ring and / c A is an ideal
such that A/1 has finite length, we write
l(A/I) = length(v4//) = col(J).
LEMMA (3.1). Let x €W be an isolated fixed point of a, and assume the
structure map f:W^>T is smooth at x. Then the multiplicity of x in R is
given by
j(a){mult, of x as fixed point of a acting on fibre Wt}.
PROOF. Write Qw,x = A[[*]]. We have
ûx(a)x S* Sp(A[[*]]/(<7(tt) - 7r,a(t) - t)).
The fact that x is isolated in fix(cr) implies fix(cr)x is flat over Sp(A/(7r-7)), and
one can use, e.g., [3, Lemma A.4.1] to conclude
/(fix(a)x) = /(A/(^))-/(A;[[<]]/(a(0-0),
which is the desired identity. Q.E.D.
LEMMA (3.2). Let x €W be a double point ofWt which is fixed by a but
isolated in fix(cr). Assume a stabilizes (i.e., does not switch) the branches ofWt
through x. Then
(3.2.1) /(M*)*)=J-(*i+*2-2) + l,
where the S{ are the multiplicities of x as a fixed point of a on the two branches
ofWt through x.
Proof. Let
(3.2.2) A = 0^x = A[[t,e}}/(te-7r),

434
SPENCER BLOCH
with the two branches defined by t and e. Since a stabilizes the branches, we
have a(t) = ut, o~(e) = ve for u,v E A*. Let a — u— 1, b — v — 1. We want
l{A/{at,be)) = j - {sx + s2 - 2) + 1.
We have
Z(A/(at,£)) = *i; Z(A/(t,&£)) = a2; Z(A/(*>0) = 1>
so we must show
Z(i4/(a, 6)) = j ■ («i + «2 - 2) + 1 - «i - «2 + 1 = (j - l)(«i + «2 - 2).
Note
unit • 7T7 = cr(7r) — 7T = ££(a& + a + 6)
and hence
l(A/(a, b)) = l(A/{a, ab + a + b)) = l{A/{a, ^-V"1))
= (j-l)(l(A/(a,t))+l(A/(a,e))).
I claim
(3.2.3) /(v4/(a, 0) = 52-l; Z(^/(a, e)) = *i - 1.
Indeed, the second claim is clear. For the first, note that for some integers m, n
and a, /?, 7, d E v4*, we have
a = a*Sl-1+i8cn+0(*Sl,£n+1,7r),
6 = 76S2-14-a^m-fO(r+1,£S2,^).
Since (a +1)(6 + 1) = 1 mod tJ~1eJ~1 we must have n = 52 — 1, m = si — 1,
and we can take 7 = — /?, 9 = — a. Thus
a = a^i-1 + j3e92~l + 0(*Sl, eS2, tt),
and the first equality in (3.2.3) follows. This proves the lemma. Q.E.D.
LEMMA (3.3). Let hypotheses be as in Lemma (3.2), but now assume o~
switches the branches ofWt at x. Then
i(fixWx) = 2j - 1.
PROOF. Write a(t) = ue, o~(e) — vt; u,v € A*. We have uv7r - -k — unit • 7T7,
so uv — 1 + wk3~1 with it; E A* and v = u-1 + hit3"1, /i € v4*. We compute
Z(fix(a)x) = l{A/{t -ue,e- vTlt - ^3~li)) = Z(A/(* - tx£,tt*"1*))
= l + (j-l).l(A/(t-U€,1c))
= (j - l)(/(A/(* - tX£, 0) + /(A/(* - tX£, £))) + 1
= 2j - 1. Q.E.D.
We continue to write fix(cr) = D U R. Our goal now is to compute the
multiplicities of points in R lying on D.

CYCLES ON ARITHMETIC SCHEMES
435
LEMMA (3.4). All components of D occur with multiplicity < j.
Moreover, no two components with multiplicity j in D meet, and R is contained (set-
theoretically) in the union of components of Wt occurring in D with multiplicity
i such that 0 < i < j — 1.
PROOF. Since cr(7r) — 7r E ideal defining fix(cr), the first assertion is clear. For
the second, suppose components defined by t and e meet, with te = 7r. Then
a(7r) - 7T = unit • 7T7 = a(e)(a(t) - t) + t(a(e) - e).
If both components occur with multiplicity j\ the right-hand side of this equality
lies in ir3 times the maximal ideal at the point, a contradiction.
Finally, let y E Wt be a smooth point, and let 7r, 0 be local coordinates at
y. If the component of Wt through y occurs in D with multiplicity j\ then 7T7
divides a(0) — 0, so the ideal defining fix(cr)2/ is generated by irJ, and there is no
embedded component. Q.E.D.
LEMMA (3.5). Let Y C D be an irreducible component occurring with
multiplicity i < j — \, and let y E Y. Then there exists a local defining equation t
for Y at y with ordt(o~(t) — i) — i + 1.
PROOF. The ideal of fix(cr) at y is generated by elements a(u)—u for u a unit
at y since a acts trivially on the residue field at y. We can find such a u with
ordt{o-(u) — u) = i. Replacing t by ut if necessary, we may assume therefore that
ord*(cr(£) — i) < i +1. (Calculate a(ut) —ut using the "derivation-like" identity.)
Suppose ord*(cr(£) — i) <i < j, and write -k — vt. Then
(3.5.1) j = ordt(<r(7r) - ir) = OTdt{t(a{v) - v) + a(v){a(t) - t)) < i,
since a(v) is a £-adic unit. This contradiction proves the lemma. Q.E.D.
Let Y C D be a reduced and irreducible curve, and assume Y appears in D
with multiplicity i < j — 1. We can define a derivation
(3.6) Da,Y: 0Y -Oy(-tY)
as follows. For any n > 1 we have a — 1: Ony —► Ony, and this endomorphism
is zero if and only if n < i. From the exact sequence
0 - Oy(-zT) - 0(î+1)y - OlY - 0
we deduce a nonzero map V — 1": O^y —► Oy(—zT). It follows from Lemma
(3.5) and (3.0) that V — 1" is zero on the ideal defining Y c iY, so there is an
induced map Da,y as in (3.6).
LEMMA (3.7). Let y E D and assume y is a smooth point ofWt. Let Y
be the irreducible component of D on which y lies, and assume Y appears in D
with multiplicity i. Then the multiplicity of y in R is
l{Ry) = {j-i).OTdy(D(rtY)-
PROOF. We have already seen this when i" = j, so we may assume i < j. Let
7T, 6 be local coordinates at y. Note
ordy{Da,Y) = OTdy{{<j{6) - 6)/ttj) = co1(tt, {a{0) - 0)/**)

436
SPENCER BLOCH
while
(<7(tt) - 7T,a(0) - 0) = 7T* • ((a(?r) - tt)/^, (<t(0) - 0)yV)
= 7ri-(7H'-1,(a(ff)-ff)/Ô,
so Ry\ 7T-7-1 = (<r(0) - 0)/7r* = 0 has multiplicity (j - i) ordy (Daty)• (Note that
VW-W«)-«)A^) Q.e.d.
LEMMA (3.8). Let Y,Zc fix(cr) 6e distinct irreducible reduced curves
appearing with multiplicities h, i respectively in D. Let y £ Y D Z and assume
h < j. Then the multiplicity of R at y is given by
(3.8.1) {j - h) • {ordyiD^y) ~ % ~ 1) + (j ~ %) • {0Tdy{Da,z) - h - 1) + 1.
(Strictly speaking, Daz is not defined when i — j. However, the coefficient
j — i = 0 in that case, so the formula yields
{j-h)-{0Tdy{Da,Y)-i-l) + l-)
PROOF. Let m = min(z +1, j). Let A be the formal completion of the local
ring at t/, and take e, t E A with Y : £ = 0, Z: e = 0, e£ = 7r, and
a(0 - « = /?e^+1, a(c) - e = bemth, P,beA.
Suppose first i — j~m. Apply (3.0) to 7r = et, getting
(3.8.2) <y{e)P + b = unit •^~fc-1.
The defining ideal for i2 at t/ is (/?£, 6), so we must compute
col(/ft, 6) = col(/3,6) + col(*, 6).
Since Da,Y{e) = {<T{e)-6)/th,
col(6, £) = ord^Z^y) — m.
On the other hand, e = cr(e) • unit, so from (3.8.2)
col(/3,6) = col(<r(e)/?,&) - col(a(c),6)
= (j -h- l)(ordy (D^y ) - m) - col(<r(e), 6)
= (j-h- l){ordy{Da,Y) -m)-(j-h-l) col(e, 0
= (j-h- l){ovdy{D^Y) - m - 1).
Finally,
/(iîy) - col(/fc,6) = (j - ^(ord^D^y) -j - 1) + 1
as claimed.
We now consider the case i < j. Equation (3.8.2) is replaced by
(3.8.3) 0{(r{e)/e) + b = unit • *'-*-V"'-1.
The same argument as before yields
col(6, i) = ordy(Z>ffjy) — i — 1,
col(/3, c) = ordy(Z?a,z) - h - 1.

CYCLES ON ARITHMETIC SCHEMES
437
If by chance y = ft + l = i + l, the right-hand side of (3.8.3) is a unit, so
l{Ry) = col(/?i, eb) = col(/3, e) + col(i, b) + col(i, c)
= ord^Z^z) - h - 1 + ord^Z^y) - i - 1 + 1,
which is the desired formula in this case. To finish, since the situation is
symmetric, we need only consider the case j — 1 > i > h. The right-hand side of
(3.8.3) is divisible by i, so col(/?,i) = col(M),
col(/3,6) = col^^-^-V-*-1)
= (j - fc - 1) col(6,0 + (j - i - 1) col(/?, e),
/(iîy) = col(/?i, e6) = {j - h) col(6,0 + (j - t) col(/3, c) + 1
= (j - i){ordy{D^z) -h-l) + {j-h) ordy{Da,Y) - t - 1) + 1.
This proves the lemma. Q.E.D.
One final case remains to be considered.
LEMMA (3.9). Let Y, Z be irreducible components ofWt meeting at a point
y. Assume Y c D with multiplicity i < j but Z (/L D. Let s = order of fixed
point at y of a acting on Z. Then
l{Ry) = {j - i)(oTdy(DaiY ~ 1) + i ■ {s - i - 1) + 1.
{Again, as in Lemma (3.8), the first term on the right is to be ignored if i = j.)
PROOF. The calculations are similar to those of Lemma (3.8) so I will run
through them quickly. Suppose first i < j. Fix coordinates i, e as in Lemma
(3.5), with Y: t = 0. Write a(t) - t = ai*+1, <r(e) - e = aet\ The usual
calculation gives
(3.9.1) a + aat1 +a = unit • e3'1 • ^'"*"1.
Hence
l{Ry) — col(ai, ae) = col(ai, e) + col(ai, a) = s — i + col(ai, a)
= 5 - t + col(i, a) + col(a, e3'1^'1)
= j • {s - i - 1) + 1-f col(i, a) + {j -i-l) col(a, i).
Note col(a, i) = ordy Day — 1. If y = z + 1, we get
/(i2y) = j (5 - j) + 1 + ordy D^y ~ 1.
If i < j — 2, (a, i) = (a, i) by (3.9.1), so the lemma follows in that case also.
Finally, we must consider the case i = j. The relevant formulas are
a(t) - t — at3\ <r(e) - e = aet3, unit • e-7-1 = a + oaiJ + ai.
We have /(i2y) = col(a, ae). Also
col(a, e) = s — j, col(a, i) — col(a + aat3 + ai, t) — j — 1,
col(a, a) = col(a, ai) — col(a, i)
= coUa,^"-1) - col(a,i) = (j - l){s - j) - j + 1.
The desired formula, /(i^) = j • (5 - z - 1) + 1 follows. Q.E.D.

438 SPENCER BLOCH
4. Lefschetz fixed point formula (semistable case). Our objective in
this section is to prove
THEOREM (4.1). With hypotheses as at the beginning o/§3 (semistable case)
we have
(4.1.1) (Aw • IV) = (j(g) - 1) • tvalH^Wfj) + tT*\H*{Wt).
PROOF. The first step is to reinterpret the desired formula.
LEMMA (4.2). Let W be the normalization of'Wt. Let S = {xi,... , xr} C Wt
be the double points, and define for x E §
{— 1, g fixes branches through x,
+ 1, a switches branches through x,
0, g(x) j=- x.
Then formula (4.1.1) is equivalent to (W = normalization ofWt)
(4.2.1) (IV ■ Avy) - j(g) • (IV ■ Aw) + (2j - 1) £ lx.
PROOF. Let M be the Q/ vector space with generators b\x and &2x for x E §,
and relations b\x + b<ix = 0. Note cr acts on M, and trg\M — — ^lx.
Write p: W —► W* for the normalization map. One has r(p*Q/5w/Qz,wt) — ^
so
(4.2.2) tr<r|iT(W) = tr <r\H*{Wt) - ]£/*.
On the other hand, writing T for the spectrum of the integral closure of A in
k(fj) and a: Wn -+ ÏF = W xT T, one has (cf. [2])
{M, m = 0, n = 1,
Hm(Wt,Qi), « = 0,
0 otherwise.
The Leray spectral sequence for a implies
(4.2.3) tTv\H*(Wri) = tTv\H*{Wt)+^lx.
Since (IV • Aw) = trcr|if*(W) by the Lefschetz fixed point formula, we get
that (4.1.1) is equivalent to
(IV ■ Avy) = J(g) • (tTG\H*(Wt) + £/*) - £/*
(4.2.4) = j . (tra|JT(W) +2 • J>) - J>
= y-(r,.Aw) + (2y-i)-^/,.
This is the desired formula (4.2.1). Q.E.D.
We turn now to the proof of Theorem (4.1). Substituting the formula for
(IV • Avy) from (2.11), the equation to be checked becomes
(4.3.1) -(Kx/S D)-(DD) + [R}=j • (IV ■ Aw) + (2j - 1) • £ lx.

CYCLES ON ARITHMETIC SCHEMES 439
We rewrite the formulas for l(Ry), yeS, from Lemmas (3.2), (3.3), (3.8), (3.9).
(i) If y belongs to two components Y, Z of D appearing in D with multiplicities
h and z, then
KRy) = U ~ h) ' OTdDa,Y + U ~ %) ' ordDa,z - U - l){h + i) + 2hi + {2j - 1) • ly.
(ii) If y belongs to a single component Y of D appearing in D with multiplicity
i, and if the multiplicity of the fixed point of a at y on the nonpointwise-fixed
component of Wt containing y is s(y), then
l(Ry) = {j - i) - oxdy D^y + j • s{y) + (2j - 1) • ly - i • (j - 1).
(iii) If y does not belong to D, and if the multiplicities of the fixed point of a
at y on the two components are denoted s\(y) and S2{y) (if cr switches branches,
we take S{(y) = 0), then
l{Ry) = j'{si{y) + 82{y)) + {2j-l)-ly.
For Y C D with multiplicity z'y, we have since 0 i=- D^Y € r(F,Ty(-zY)),
(4.3.2) (j - iy) ■ ^ ord, Z^y = (X(y) - iy^ . Y))(j - iy).
y€Y
Let Be Wbe the normalization of D (with reduced structure), and let V =
W — D. Using (i)-(iii) above together with Lemmas (3.1) and (3.7) and formula
(4.3.2), we calculate
^l{Ry)-^2{2j-l)-lv= J2U-iy)(x(Y)-iy(Y.Y))
ally yeS Y CD
(4.3.3) - ^2(j - l)iY +J-J2 s(y) + £*V • *"z.
(1) y€V (2)
The sum labelled (1) is over all pairs y E Y such that y E § and FcD. Sum
(2) is over triples y, F, Z with Y,ZcD and ?/ € Y fï Z. Substitute now in
(4.3.1), using
(IVAv) = 2>(y)
2/GV
to get
- Kx/S D-(D.D)+J2{(J- iv){x{Y) - iY(Y . Y)) + (j - l)iY(Y . Y)}
YCD
+ ^ «V • iz
(2)
k £ x(Y).
YCD
Using the formula x(^) = —Kx/s ' Y - (Y . Y) for the reduced curves Y, this
equation is straightforward to check so Theorem (4.1) is proved. Q.E.D.
5. The plan. In this section we will outline the proof of (0.2) for a family
of curves. Let S = Sp(A) be the spectrum of a complete discrete valuation ring

440
SPENCER BLOCH
w
i
T
—
_►
X
ï
S.
with quotient field K and algebraically closed residue field k = A/7rA. Let X
be a regular scheme flat and proper over S with fibre dimension 1 and smooth
generic fibre. Let T = Sp(A') be a sufficiently large totally ramified extension of
S so X Xs T has a semistable regular model V; i.e., the fibre Vt is reduced with
normal crossings [9]. Write K' for the quotient field of A'. One knows that the
galois representation of K/K' on H*(Xfj) is tame. The rational map V —► X
is not necessarily everywhere defined, but after a succession of blowings-up of
closed points, we get a scheme W —► V fitting into a diagram
(5.1) V
Unfortunately, it is not true in general that the special fibre Wt is reduced, but
we can suppose Wt, red is a union of smooth components with normal crossings.
PROPOSITION (5.2). Formula (0.2) holds for V/T.
PROOF. There is no Swan term, and a standard geometric argument shows
x{Vt) — x(^) is the number of double points in the special fibre. We must show
that this is equal to c^v (QyiT). Fix an embedding V C P with P smooth over
T, and consider the exact sequence
0 -♦ nv/p "♦ np/r\v -♦ nv/r -♦ 0.
Let ujy/T — JCom(det(Ny,p), det(fip/T|V)) denote the dualizing sheaf. Given
a local section r of fi^/T, let f denote a lifting to fip/T|V. Consider the map
(5.2.1) p: VtyjT ^uy/T
defined by r \-> {n \-> n A f}. Note that Ïïy/T has homological dimension 1
and is locally free outside the double points, hence is torsionfree. It follows
that p is injective. One checks easily that the cokernel of p is the skyscraper
sheaf isomorphic to a copy of the residue field kx at each double point x. We
apply (1.1)(iii) to the resulting sequence noting that C2{kx) = — [x] to prove the
proposition. Q.E.D.
PROPOSITION (5.3). Formula (0.2) holds for W/T.
PROOF. This amounts to showing if Y = BL(x/X) for x eX, then (Ay . Ay )s
= (Ax • Ax)s + 1. Let /: Y —► X. Consider the exact sequence
0 —► / ^x/s ~^ ^Y/s ~^ ^y/x -^ 0.
The arrow on the left is injective by the same argument as in Corollary (1.2).
Write E c Y for the exceptional fibre. There is a natural map fly ix ~^ ^k whidi
is easily seen to be an isomorphism. Indeed, if t and u are local coordinates at
x, then the local coordinates upstairs at some point can be taken to be u and
t/u. It follows that fiy/X is generated locally by d(t/u). Since E: u = 0 and
u - d(t, u) = dt — {t/u) • du E f*ftx/si

CYCLES ON ARITHMETIC SCHEMES
441
the result follows. Again we may apply (l.l)(iii). Since fi^/s has homological
dimension 1 and / is faithful, in the derived category we find Lf*Vllx,s = /*fi^/5,
so c\(f*VLlx,s) = /*ci(n^/5). This class clearly meets ci(n^) trivially, so
(Ay . Ay )s = (AX . AX)s + cIyb We)-
We have an exact sequence
0 -+ uy/s -+ wy/s{E) -+ n^ -+ 0,
whence again by (l.l)(iii)
cIy.&e) = -ci(wy/s) • £ - -ci(wy/s(£)) .£7 + ^-^ = 2-1 = 1.
This proves (5.3). Q.E.D.
We continue with the outline of the proof of (0.2). With notation as in (5.1),
let a be an automorphism of V lifting a nontrivial automorphism a of T over S.
Then by (4.1)
(5.4) (Av .IV) = (j(a) - 1) • tT<T\H*{Vri) +tr<7|JT(Vi).
It follows, moreover, from (2.12) that (5.4) continues to hold with V replaced by
W.
The scheme Wxx W has pure dimension 2 and is a local complete intersection.
As a cycle on W x s W, [W Xx W] = XVeG ^ + ^ where i2 is a sum with
suitable multiplicities of components E x E' with E, E' C.W collapsing to the
same point of X. We will prove in §6 the formula
(5.5) \G\-1-(Aw.R) = X(Xs)-x(Wtf,
where x{Wt)G denotes the alternating sum of the dimensions of the G-fixed part
of the cohomology of Wt.
It is natural to think of
[W xx W] = {hx /i)*Ax.
Since (h x h)*Aw = \G\ • Ax, one might expect a projection formula
(5.6)
\G\-(Ax.Ax)a = (Aw.[WxxW})s = (Aw.Aw)s + y}2(Aw.r(,) + (Aw.R).
Let us check that such a formula is what we need. Substituting from the previous
steps (using also (1.2)), we get
(Ax • Ax)s = IGp1 I x(Wt) - x{Xn) - (swr/s(l) + \G\ - 1)X{X„)
+ £(i(<r)-l)tr<7|ff*(X„)
+ ]Ttx<T\H*(Wt) + \G\X(XS) - \G\X(Wtf
= -sw(X/S)-X(Xfi)+x(Xs).

442
SPENCER BLOCH
The proof of the main theorem is now complete modulo verification of formulas
(5.5) and (5.6).
6. Exceptional fibres. Let X/S be our usual regular family of curves over
a complete discrete valuation ring with algebraically closed residue field. Let
T —► S be a finite ramified cover, which we assume galois with group G. Finally,
let W —► T be a birational desingularization ofXx§ T. Let /: W —► X be the
natural map. We assume G acts on VF, so / factors through W/G.
We consider the fibre product W Xx VF as a cycle on W x s W. We can write
[W Xx W\ — Q + i2, where the components of Q are of finite degree over X,
while those of R are all of the form E\ x Ei with E\ c Wa smooth irreducible
curves collapsing to a point in Xa. Our purpose in this section is to prove
THEOREM (6.1). With hypotheses as above
(deg f)-1 (Aw . R) = x(Xs) - X(Wtf.
Here x{X9) is the étale Euler characteristic of the special fibre and x{Wt)G is
shorthand for ][^(—l)*(dimi7L (VF*)G). The superscript G means invariants.
PROOF. We begin by reducing the theorem to a question about birational
maps. By hypothesis, G acts on VF, and the quotient W/G is a normal scheme,
flat and proper over S with fibre dimension 1. The schemes (given reduced
scheme structure) {W/G)t and Wt/G coincide.
LEMMA (6.2). X(Wt)G = x(Wt/G) = x((W/G)t).
PROOF OF Lemma. It suffices to show H*{Wt)G = H*{Wt/G). This
follows in a standard way from the existence of a transfer map H*(Wt) —►
H*{Wt/G). Q.E.D.
The intersection (Ajy.i?) is not mysterious. Indeed (Aw-Ei x ^2) =
(E\. E2)w is the usual intersection of divisors on a local arithmetic surface
[2], exp. X]. A procedure of Mumford [12] enables one to define intersection
numbers in Q (not necessarily in Z) for Weil divisors on a normal surface.
Given such a surface Z, one chooses a birational desingularization 7r: V —» Z.
For a divisor D on Z, define 7r*D to be the unique Q-divisor on V satisfying
-k±-k*D — D and ir*D .E = 0for every E c V with 7r(E) =point. By definition
(D.Di) = {ir*D.ir*Di).
LEMMA (6.3). Let p: Z' —» Z be a finite map of degree d. Then
{p*D.p*D,) = d-{D.D').
PROOF. Define a birational desingularization 7r': V —► Z' fitting in a square
V -* V
I I
Z' -> Z
Now 6 : V' —» V has degree d also, so
{D . D')z = [>k*D . k*D')v = d~x {6*tt*D . e*^D')v>.

CYCLES ON ARITHMETIC SCHEMES
443
Note <(0*7T*£>) = p*D. Also if E c V and tt'(E) = point, then tt(0(£)) =
point, so {0*7T*D.E) = (tt*D.6*E) = 0. It follows that 6*tt*D = 7r'*p*D.
Hence
(D. D')z = d'1 ((9*tt*D . 0*tt*£>') = d^ip'D. p*£>'). Q.E.D.
We can now reformulate the theorem.
THEOREM (6.4). Let Y —► X be a birational morphism of arithmetic
surfaces, with X regular and Y normal. Define a cycle R of dimension 2 on Y Xs Y
as before, by taking the components of Y Xx Y not of finite degree over X (i.e.,
all components except Ay). Then
(AY.R) = X(X3)-x(Ys).
PROOF. Write R = Y^tijEi x Ej- The assertion is
J2eij(Ei.Ej)Y = x(X9)-x(Ys).
It is convenient to fix a resolution of singularities Z —► Y such that the
composition Z —► X is obtained by a succession of blowings-up of smooth points [11].
Let T denote the associated graph. That is, T is the labeled graph with one node
for each irreducible curve in the exceptional fibre of Z/X and one edge between
intersecting curves. Each node is labeled with the self-intersection number of
the corresponding curve.
Note the following stability property of the numbers e^ above. If E[ is the
strict transform of E\ on Z, the coefficient of E[ x E'- in Rz is still e^. Indeed,
these coefficients depend only on the structure of the morphism Z —y X at the
generic point of E[ x £', and this is the same on Y.
LEMMA (6.5). The integers eij depend only on the graph T (and not, e.g.,
on whether or not we are in a mixed characteristic situation.)
PROOF. It follows from the above remarks that we can ignore Y and check
this simply for Z/X. We blow down a — 1 curve on Z corresponding to a node
of T to get
Z = BL(z/Z') -+ Z' -+ X.
v
Write E'i, i < n, for the curves on Zf collapsing on X. Let Ei be the strict
transform of E^ and let En = p~x(z) be the exceptional curve for ZjZ'.
Note Rz = (pxpy(Rz>+&z')-&z, whereby definition (pxp)*(E[ xE'2) -
p*E[ x p*E'2 and (p x p)*(Az>) = Az + En x En. By induction on the number
of vertices, the coefficients in Rz> are computable from the graph T' for Z'/X.
Note p*E[ = Ei if the nodes corresponding to Ei and En are not joined by an
edge, and p*E[ = Ei + En otherwise. Clearly, T determines the values for the
e^ as well. Q.E.D.
REMARKS, (i) Given T and Z/X, any set / of nodes of T can be blown down
to get a normal surface Y.

444
SPENCER BLOCH
(ii) x{Xs) - x{Ys) equals -#{E c Y \ E —► pt. on X}. In particular, this
number is also dependent only on T and /.
Returning to the proof of the theorem, the above discussion has shown
everything to depend only on T and /. We may therefore take X = P^H This is the
main point. The rest of the proof, which we leave the reader to check in detail,
consists of verifying that Y satisfies Poincaré duality over Q so one can define a
class [Ay] € H*(Y x Y, Q), [Ay] = £a* <g> a* with {a*} a basis of H*(Y) and
{a*} the dual basis. One defines [Y xx Y] G H*{Y x Y, Q) by pulling back [Ax]
and one verifies that
[Ay] = [yxx Y)-y}2eijEixEj,
with the same coefficients e^ as before. It is now a formality to check that
[Ay] • [Ay] = X(Y) and [Ay] • [Y xx Y) = [Ax]2 = x(X). Thus X(X)-X(Y) =
Yleij(Ei -Ej)- Even though the geometric situation is somewhat different than
what we started with, the numbers are exactly the same on the right and the
left, and the theorem follows. Q.E.D.
7. The projection formula. We keep the notation of the previous sections;
S is the spectrum of a complete discrete valuation ring with algebraically closed
residue field, W and X are degenerating families of curves over S, and f:W—*X
is an S morphism of finite degree d. (We do not assume / finite.) At one point
we will assume W —► S factors through a nontrivial totally ramified extension
T —► S. This section will be devoted to a proof of the projection formula:
THEOREM (7.1). r(Ax.Ax)s = (Avy.[Wxx w])st
PROOF. To begin, we substitute the definitions, so the desired formula
becomes
(7.1.1) fctirtx/s) = M^w/sT ■ s(&w/W xx W)}deg0.
All cycles are taken in the chow group of the closed fibre Ws. More precisely,
the Si and c<i are on Ws and we want
(7.1.2)
rc2(nxx/s) = s2(Aw/W xx W) - s1(Aw/W xx W) • c1(Q1w/s) + c2(Q1w/s).
LEMMA (7.2). (i) /"fiws = Z//*n^y5 (in the derived category.)
(ii) The sheaves fix/s> ^w/s> an^ ^w/x a^ ^ave homological dimension 1.
(iii) The sequence
(7.2.1) 0 —► / ftX/s ~^ ^w/s ~^ ^w/x ~^ 0
is exact.
PROOF OF LEMMA. We have already seen (ii) for fi^/s and fiJyyS. Part
(i) follows from the general remark that if £ has homological dimension < 1
on X and / : W —► X is a morphism of schemes such that W is reduced and
every irreducible component of W dominates X, then L/*£ = /*£. One simply
remarks that /* preserves injections of locally free sheaves.

CYCLES ON ARITHMETIC SCHEMES 445
For (ii) and (iii) consider the diagram
W ► XxP Q > Q
smooth
if/^smooth
s
One has a corresponding diagram of sheaves
0
0
~*■
~~*
0
i
NW/P
I
iyW/XxP Q
1
0
~~*
~^
b
0
1
f*ttp/s\w
1
oi i
lLQ/S\W
1
^Q/plw
1
0
~~*
~*
"*
f*^x/s
la
n1
lLW/S
1
n1
lLW/X
1
0
~~¥
~*
-*
0
0
0
A diagram chase shows Kera = Kerfc. But N^,XXp q is locally free since W
and X Xp Q are regular, and a is injective at the generic point of VF, so these
kernels are zero. Assertions (ii) and (iii) follow. Q.E.D.
By Lemma l(i),
fciVx/s) = CiWVx/s) = CiWVx/s)-
Using (7.2.1) and (l.l)(iii), we find
C2(rtw/S) = /*c2(n^/5) +c2(n^/x) + /*ci(n^/5) .Cl(n^/X),
ai^w/s) = /*ci(n]r/s) -hci(n^/x).
Substituting these identities, the desired equation (7.1.2) becomes
(7.3) -52 +ci$i =c2H-ci(ci -ci),
where Si = Si(Aw/W x W), C{ = Ci(fijyy5), and C{ — Ci(fiJyyX). The next step
will be to replace (7.3) by a formula involving only invariants of the embedding
Aw —► W xx W, i.e., to eliminate ci(fijyy5).
LEMMA 7.4. Let i: D —► W be a reduced, irreducible Cartier divisor
supported on the closed fibre. Then
i*a(ttw/s) = ci(i*nw/s)-
PROOF OF LEMMA. The assertion can be rewritten
ci(Lf*n^/5) = ci(f*nj^/s).
Recall we have W —► T —► S with T/S a nontrivial totally ramified extension.
It follows that ftw/S has rank 2 at every point of Z?, so i*Q^,s is locally free

446
SPENCER BLOCH
of rank 2. Fix a global embedding W —► P with P smooth over S. There is an
exact sequence
0 —► NWfP —► tïp/s\w -^ ^w/s ~^ 0-
This is a resolution of H^/5 by locally free sheaves, and rkN^,p = rkfip/5 — 1.
Restrict this sequence to D, and define 9 to be the kernel
(7.4.1) 0^9^ i*Nb/P - fVp/slw - i^w/s - 0.
It follows that 9 is a line bundle on Z?, which we must show = Od- Let J c K c
Op be the ideals of VF and Z>. I claim JK = JnK2. Clearly JK C JnK2, and
the question of equality is local. Since VF is regular, it is defined locally in P by
/i = • • • = fn = 0 with the /» forming part of a system of parameters. Since
D c W is Cartier, we can choose g (locally) with D: g = f\ = • • • = fn = 0. An
element ft E JnK2 can be written /i = Y^^ifi-> h ~ YlKjfifj + Y1 h[ fig + h"g2.
Going mod J, we see h" G J, so fc E JK as claimed.
Let 7T G O5 be a uniformizing parameter. We have -k E if2 H- J so there is a
diagram
K/(K2 + J)
The bottom row is (7.4.1) together with the remark that K/(K2 + (71-)) injects
into i*fip/5. (Indeed, K/(K2 + (71*)) is the conormal bundle of Z? in the closed
fibre Ps.) The lemma follows by a diagram chase. Q.E.D.
We turn now to an outline of the proof of (7.1). Let A denote the formal
completion of Aw in VF Xx VF. In Lemma (7.5) below we will construct a
formal scheme P locally isomorphic to VF x A2 and an embedding of A in P as
a codimension 2 local complete intersection extending the 0-section imbedding
of VF in P. We then apply a variant of the basic commutativity result from
Fulton-MacPherson intersection theory to the diagram
W = W
1 1
A -* P
Since both A and VF are l.c.i. in P, we can do the intersection in either order.
The usual formula A-W = W -A would yield
{c(NA/P\W)}dim0 = {c(NW/p) - s{W/A)}dim0.

CYCLES ON ARITHMETIC SCHEMES 447
This is an equality in CH*(W) which is not very useful to us. In order to have
a formula in CH*(H/rs), we remark that, with notation as in [3], the
projective completion of the normal cone of W in A can be written (as a cycle in
P(NW/P 0 1))
P{CWA® 1) = W + Ploc(CWl0 1),
where W c P(NW/P 0 1) via the subbundle 1 and Ploc is supported over the
closed point s. It follows that s(W/A) = 1 + sloc{W/A), where sloc is a well-
defined bivariant class in CR*(WS —► W) [3, Chapter 18]. We have already
seen in §1 that c(N^/A) = 1 + cloc(A^/yl), where cloc has a similar bivariant
interpretation. The identity on chern classes
«M) • c(NX/P\W) = c(N^/P)
enables us to rewrite the commutativity formula in two equivalent ways:
(*) {c(NA/P\W)[(l + cloWM)*)(l + sloc(W/A)) - l]}dim0 = 0;
(**) [c(Nw/P) • sloc(W/A)]d[m0 + [c(Nw/P) - c(NA/P\W)]dim0 = 0.
Note the expressions inside the square brackets [...] make sense as bivariant
classes and hence both (*) and (**) (more precisely, both (*) and (**) cap
product with the fundamental class [W]) can be viewed as classes in CHo(Ws).
These classes are equal, and we will verify in Lemma (7.6) that they vanish.
Formula (*) yields
c{NA/P\W).d1+d2=0,
where the di are the classes inside the [...]. In particular,
Now the completed normal cone P(C\yA 0 1) coincides with
Proj(Sym(7V^M0l))
over codim. 1 points in W where the fibre dim. of N^,A < 1, so d\ can be taken
to be supported on those components of W9 where the rank of N^,A is 2. The
key remark, which is due to K. Kato, is that this implies
-^(rn^.d^ci^/pi^-di.
Namely, the exact sequences (recall fij^ ,x — N^,A)
0 - N1/P\W - N^/p - Vw/X - 0,
n —► /""O1 —► o1 —► o1 —► n
u —> J lLx/s ^ lLw/s ^ lLw/x ^ u
give
ci (NA/P\W) + ci (/*n^/s) = d (Nw/P) + ci (fi^/s).
For any D C.W & reduced divisor along which Q^ ,x has rank 2,
Nw/p\d — ^w/x\D — ^w/s\D'

448
SPENCER BLOCH
By Lemma (7.4) this implies
-c1(f*n1x/s)-D = c1(NA/P\W)-D.
Since d\ is a sum of such Z?, we get Kato's remark. Finally, substituting in the
formula (*) above yields
{c(rnx/sy ■ Hnlw/Xy ■ ,(w/w xx w) - i]}dim0 = o.
This completes the proof of Theorem (7.1).
It remains to construct the scheme P and verify the localized commutativity
formula (*).
LEMMA (7.5). There exists a formal scheme P locally [on W) isomorphic
to W x A2, and an imbedding of A (= formal completion of W Xx W along
Aw) in P as a local complete intersection of codimension 2, extending the zero
section imbedding W —► P.
PROOF. Locally on W such a P exists. Indeed writing J for the ideal of Aw
in .A, we have a split exact sequence
0 -+ J -+ (Da -+ Qw -+ 0.
Since J/J2 = ^w/X 1S locally generated by two elements, we get local surjections
Ow 1^1,^1 —► Qa by Nakayama. More precisely, since the maximal spectrum
of W is one-dimensional, we can write VF as a union of two affine open sets,
W = W\ U W% such that J /J2 is generated by two global sections on each Wi.
(Choose W\ meeting each component of Ws nontrivially. The complement of
W\ will then meet the maximal spectrum in only finitely many points. J /J2
will be generated by two sections over the corresponding semilocal ring, and it
suffices to lift these to some neighborhood.) By lifting generators of J /J2 to
generators of J, we get surjections <\>{\ CV^i,^] —► OyilH^. Note that these
will be local complete intersections, e.g., by (E.G.A. IV, 19.3.2, Publ. Math.
I.H.E.S. No. 32). Moreover we can find an automorphism <p of Ov^nv^I^i^I
such that <p = id mod(£) and 02 • <p = <t>\ over W\ fl WV Indeed, we can clearly
find a homomorphism (p. The only point is to check that we can take <p to be an
automorphism. For this it suffices to take <p to be an isomorphism mod(£)2, so
the problem is purely module-theoretic. Note that a <p can certainly be chosen
to be an isomorphism mod(£)2 at all generic points of (H^i fl W^)a, so it will fail
to be an isomorphism mod(^)2 at only finitely many points of Max(Wi fl ^2)-
Let D C W\ fl W2 be a principal divisor finite over S containing these points,
and replace Wi by W2 — D. This proves the lemma. Q.E.D.
The final point to be checked is the commutation formula:
LEMMA (7.6). With notation as above, the 0-cycle
[c(Nw/P) ■ sioc(W/A)]dim0 + \c{Nw/P) - c(NA/P\W))dim0
is trivial in CHo(H^a).
PROOF. The argument is a variant on [3, Chapter 6]. Let
q: P{NW/p 0 1) -► W

CYCLES ON ARITHMETIC SCHEMES 449
be the projection, and consider the diagram
0 - 0(-l) - q*(Nw/P®l) - £ - 0
II 1 1
0 - 0(-l) - q*{NA/P\W®l) - v - 0.
As in [3, Proposition 6.1(b)], the assertion of the lemma will result from applying
q* to the equality
(1) c2(0 • \P{CWA®1) - [W]} = (cafa) - c2(0)[W]
in CH+(P(ATvy/p e i)s). (Note ci(0(-l)) • [W] = 0.) Consider now a diagram
of schemes
w
i
Wi
Ï
w
->
—
-t.
A'
Ï
Ai
ï
A
->
-►
-••
P'
i
Pi
i
P
where P\ is obtained from P by blowing up A and P' is the blow up of the
inverse image W\ of W on Pi. The analogous identity
(2) Cl(f) • [P(CwA' e i) - [W]] = MV) - c^aw'}
is easy to check because W and .A' are Cartier divisors on P'. As in [3, Theorem
6.3], the bundles £ and rj admit subbundles £' and 7/ when pulled back over
W. In fact, r\' is already defined over W\. Write J = r\jrf', £ = £/£', so
ci(5) • ci^) = c2fa) and ci(£) • citf') = c2(£).
The idea now is to multiply the divisor identity (2) above by c\ (3) • c\ (£) and
project to CH*(P(7VV/p 0 l)a). Let
tt: P(iVV/P 0 1) x W -* P(AW/p 0 1)
be the projection. One checks (e.g., using [3, Theorem 6.3] that
Mci(3) ' [W]) = tt*(ci(£) • [W]) - [W] € CK.{P{NW/P 0 1)).
The right-hand side of (2) multiplied by c\(3) • ci(£) yields
ci(£) • (c3(f?) - c2(0)[W] - (Cl(5) - Cl(£)) • c3(0 • [W"].
When we apply 7r* to this expression, using the projection formula we get the
right-hand side of (1).
For the left-hand side we must compute
(3) Mci(J) ' c2(0 • [P{CW'A' 0 1)- [W])).
The map A' —► ^4i is birational, so [3, Proposition 4.2(a)] the cycle
P{CWlA1 0 1) - [^1]

450
SPENCER BLOCH
is the direct image of the corresponding cycle on the "prime level." As remarked
above, jF is already defined on the "1-level," so we can apply the projection
formula and rewrite (3)
tfi*(ci(y) • c2(0 • [P{CWlA1 0 1)- [Wi]]),
where w\ is the projection from the "1-level." Now A\ —► A is flat, so [3,
Proposition 4.2(b)], P{CwxA\ 0 1) is the pullback of P(CWU 0 1). We may
therefore apply the projection formula again, getting
(c3fa)-c2(0)-[W].
This is the left side of (1), so the proof is complete.
Bibliography
1. M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology 10
(1971), 373-383.
2. P. Deligne et al., Groupes de monodromie en géométrie algébrique. II. Séminaire de
Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 II), Lecture Notes in Math., vol.
340, Springer-Verlag, 1973.
3. W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.
4. Luc Illusie (Editor), Cohomologie l-adique et fonctions L, Séminaire de Géométrie
Algébrique du Bois-Marie 1965-1966 (SGA 5), Lecture Notes in Math., vol. 589, Springer-
Verlag, 1977.
5. K. Kato, Vanishing cycles, ramification of valuations, and class field theory, Preprint,
Univ. of Tokyo.
6. G. Laumon, Caractéristique dEuler-Poincaré des faisceaux constructibles sur une surface,
Preprint.
7. J. Milnor, Singular points of complex hyper surfaces, Ann. of Math. Stud., no. 61,
Princeton Univ. Press, Princeton, N.J., 1968.
8. M. Raynaud, Caractéristique dEuler-Poincaré d'un faisceau et cohomologie des variétés
abéliennes, Dix Exposés sur le Cohomologie des Schémas, North-Holland, 1968.
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representations for two-dimensional local rings, Preprint, Univ. of Tokyo.
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University of Chicago

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Zero-Cycles and if-theory on Singular Varieties
MARC LEVINE
Introduction. One of the few definitive results known on the structure of
the group of zero-cycles mod rational equivalence on a variety X,Aq(X), is the
theorem of Roitman [8]:
THEOREM. Let X be a smooth projective variety over an algebraically closed
field. Then the map ax- Aq(X) ~► Alb(X) induced by the Albanese morphism
is an isomorphism on the torsion subgroups.
(Actually, Roitman proves the above except for the problem of p-torsion in
characteristic p > 0; this portion was filled in by Milne [6].) This has implications
for the affine case; in [7] and [2] this is used to show that the group A0(X)
is torsion free if X is a smooth affine threefold. The same argument shows
that Ao(X) is torsion free for X smooth and affine in characteristic zero. The
technique is limited by resolution of singularities.
In [3], we have extended Roitman's theorem to the case of a projective variety
smooth in codimension one (modulo p-torsion in characteristic p), where A0{X) is
replaced by the group CH0(X, XSing). This is defined as follows: let Zo(X, XSing)
be the free abelian group on the smooth points of X, and let Ro{X,Xs\ng) be
the subgroup generated by cycles of the form ic* ((/)), where C is a (closed)
reduced irreducible curve on X with CCiXS[ng = 0, and / is a rational function
on C. CH0(X,X3ing) is the quotient Z0{X,Xsing)/Ro(X,XSing).
If X has singularities in codimension one, we must allow curves which meet
the singular locus in a nice way, together with special rational functions on these
curves. This led to the definition in [5] of the relative Chow group CH$(X,Y),
where y is a closed subset of X containing XSing. In this paper, we show that
CH0(X, Y) is torsion free (mod p-torsion in characteristic p > 0) if X is an affine
variety over an algebraically closed field. We then connect this cycle theory with
the if-theory of X, to show that the map CH0{X, Y) —► Kq(X) is injective, and
that CH0(X,Y) is isomorphic to the subgroup F0K0{X) of K0(X) generated
by the residue fields of smooth points of X (with the same restriction on the
characteristic).
1980 Mathematics Subject Classification (1985 Revision). Primary 14-02, 14C25.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
451

452
MARC LEVINE
In the last section of this work, we show that CH0{X,Y) and FqKq{X) are
isomorphic (mod p-torsion as above) if X is projective, and Y has codimension
at least two, using a different argument.
1. Zero-cycles on a singular variety. In this section, we recall various
definitions of the group of zero-cycles modulo rational equivalence on a singular
variety, and show that they all agree. We also prove some results of Bertini type
on singular varieties.
LEMMA 1.1. Let C be a curve on a variety Z, and A a closed subset of
Z, not containing C. Then there is a sequence of monoidal transformations,
u: Z —► Z with point centers lying over C fl A, such that the proper transforms
u~x[C] and u~1[A] are disjoint.
PROOF. First, we may replace Z with a variety Z' containing Z, and similarly
replace A with a closed subset of Z' not containing C. Thus we may assume that
Z is an affine space An and A is a hypersurface. We then resolve the singularities
of C which lie on C fl A by monoidal transformations with point centers lying
over Cf)A; changing notation we may assume that C is smooth. Taking suitable
local analytic coordinates xi,..., xn at a point p of C D A, we may assume that
C is defined by x\ — • • • = xn_i = 0.
Let / = 5^a/M/(x) be a local defining equation for A at p. Since C is not
contained in A, / contains a term of the form ax™ with a ^ 0, m ^ 0. If we blow
up p, the proper transform [C] of C lies in the coordinate patch with coordinates
xi/xn = xij.-.jXn-i/xn = x'n_1, xn, and [C] is defined by x\ = •• • = x'n_x — 0.
The proper transform of A has defining equation f\{x\,..., x,n_1,xn) = x~dl •
/(xi,... ,xn), in this patch, where d\ is the degree of the monomial of least
degree in /. Thus we see that f\ has a term ax™~dl. Repeating this procedure,
we find that fk has a nonzero constant term for some k <m. Doing this at all
points of Cn A gives the desired sequence of monoidal transformations. Q.E.D.
COROLLARY 1.2. Let X be a quasiprojective variety, Y a closed subset of
X containing XSing, and C a curve on X, no component of which is contained
in Y. Then there exists a projective closure X of X such that, letting Y and C
denote the closures of Y and C in X, we have
(i)FnC_=ynC, and
(ii) X — Y is normal.
PROOF. Let X* be a projective closure of X, and let Y*,C* be the closures
of Y and C in X*. By the previous lemma, there is a sequence of monoidal
transformations u: X* —► X* with point centers lying over C* D (Y* — Y) such
that the proper transforms of C* and Y* have no intersection away from CdY.
Replacing X* with X*, and changing notation, we have shown that X* satisfies
(i).
TV
Let u: X* —► X* be the normalization of X*. Let A be the sheaf of Ox*
algebras defined by
Ax = (ix)*(Ox)xnu*(Ox.iv)x, xeX\

ZERO-CYCLES AND if-THEORY ON SINGULAR VARIETIES 453
where the intersection takes place in k(X*). As A is a sub-Ox*-algebra of
u*(Qx+n), A is finite as an Ox* module, hence a finitely generated Ox*
algebra. In addition, if x is in X, then Ax = Ox,x- If z is in X* — Y*, then there
is a neighborhood U of x in X* with XC\U smooth; hence (ix)*(Ox)x contains
u*(Qx+n )x, and thus Ax — u*(Qx+n )x. If we let X be the variety Spec0x„ (.A),
then X is a projective closure of X, and satisfies (i) and (ii). Q.E.D.
LEMMA 1.3. Let X be a quasiprojective variety of dimension n > 2, Y a
closed subset of X containing XSing, and W a subscheme of X, of pure codi-
mension i, such that Y fl supp(W) is a finite set, and W is defined by a regular
sequence at each point of YD supp(W). Then for all d sufficiently large, a general
choice of global sections ft,..., ft of 3w{d) satisfies
(i) ft,..., ft is a regular sequence of forms on X;
(ii) if Wj is the subscheme of X defined by (ft,..., gj), then Wj is reduced
and irreducible for j = 1,..., i — 1;
(iii) if E is the smallest extension ofWi — W to a closed subscheme of X, then
E is reduced and irreducible and E fl supp(W) fl Y = 0.
PROOF. By the Chinese Remainder Theorem, there is for all d sufficiently
large, a choice of forms ft,..., ft in H°(X,3w{d)) such that 3w,x is generated
by ft, • • •, ft at each x in Y fl supp(W), and ft,..., ft form a regular sequence
at each such x. As both conditions are open conditions, a general choice of
ft,...,ft in H°(X,3w(d)) is a regular sequence at x and generates 3w,x>> for
each x in Y fl supp(W).
For d sufficiently large, a basis of H°(X, 3w{d)) defines a separable map
0: X —► PN, N = h°(X,3w{d)) — 1, which is a morphism outside of supp(W).
Let 0o- X — supp(W) —^ PN be the induced morphism. By Lemma 1.2 of [4],
if H is a general hyperplane in PN, the subscheme ())q1(H) of X — supp(W)
is reduced. Taking d sufficiently large so that 0 is a birational map, Bertini's
Theorem implies that (J)q1(H) is irreducible. Thus if i > 1, and ft is a general
element of H°(X, 3\y{d)), the subscheme W\ defined by ft is irreducible, and is
reduced outside of supp(W). On the other hand, each point of supp(W) — Y is
a smooth point of X: hence W\ is reduced at each point of supp(W) — Y. As a
general choice of ft is part of a regular sequence of length i > 1 at each point of
Y flsupp(W), W\ can have no embedded component on Y flsupp(W). Thus W\
is reduced and irreducible. The assertions (i) and (ii) then follow by a simple
induction.
For (iii), the argument above applied to X = Wi-\ shows that Wi — W is
reduced and irreducible for a general choice of ft,..., ft. We have already seen
that a general choice of ft,..., ft generates 3w at each point of y fl supp(W),
which finishes the proof of (iii). Q.E.D.
Let X be a quasiprojective variety of dimension n, and Y a closed subset of
X containing XSing. We let Z0{X, Y) be the free abelian group on the closed
points of X — Y. Let Z be a closed subscheme of X, of pure dimension one,
with no component of Z contained in Y. We let k(Z,Y)* be the subgroup of

454
MARC LEVINE
k{Z)* consisting of those / which are units in 0z,x at each point x of Z fl Y.
If / is in fc(Z)*, and Zi,...,Za are the irreducible components of Z, then /
determines elements /2 of k(Zi)* by restriction. If the associated cycle of Z is
the sum Z — J2\ n%Zi then we define (/) to the the zero-cycle
l
As in [5], we define Ro(X, Y) to be the subgroup of Z0{X, Y) generated by cycles
of the form iz+ ((/)), where
(i) Z is a pure dimension one closed subscheme of X, with no component
contained in Y,
(ii) Z is defined by a regular sequence at each point of supp(Z) fl Y,
(iii) /isinfc(Z,Y)*.
We call a subscheme Z of X satisfying (i) and (ii) a Cartier curve on X, relative
to Y.
We recall from [4, §1] that a filter S on X is a finite collection Si of irreducible
closed subsets of X, including each irreducible component of Xsing. In [4, §2] we
have defined groups Z*(X, 5), and R%(X, S), and we have shown in §2.8 of [4]
that
CH0{X,Y) = Zn{X,S)/Rn{X,S) = CHn{X,S)
def.
if Y = \J{ Si. Zn(X, S) is the same as the group Z0(X, Y) defined above; it is not
necessary to give the definition of Rn(X, S) here. The map Zn(X, S) —► K0(X)
defined by sending a smooth point x to the class of the residue field k{x) in
Kq(X) descends to a homomorphism 7: CHn(X, S) —► K0(X).
One can also define a subgroup Rq(X, Y)red of i2o(X, Y), which is generated
by zero cycles iz* ((/)), where Z and / satisfy (i)-(iii) above, and in addition
(iv) Z is reduced and irreducible.
In fact, these two equivalence relations are the same:
LEMMA 1.4. i2o(X,Y)=i2o(X,Y)red.
PROOF. Let Z be a subscheme of X, and / an element of fc(Z)*, satisfying
(i), (ii), and (iii) above. If Z has a component Zi disjoint from Y, we may, after
suitably modifying /, assume that Zi is reduced. Since Z is a complete
intersection at each point of Z fl Y, it then follows that Z is a complete intersection at
each of its generic points.
By Lemma 1.3, we can find a reduced, irreducible surface T containing Z such
that Z is principal in a neighborhood of the finite set Z fl Y fl T. In addition,
since T is constructed as a complete intersection of forms of large degree, there
is a locally principal subscheme D of T such that
(a) D contains Z,
(b) D equals Z at each generic point of Z, and
(c) {D-z)nznY = 0.
In particular, the function / in k(Z,Y)* extends to a function g in fc(Z},Y)*

ZERO-CYCLES AND if-THEORY ON SINGULAR VARIETIES 455
by setting g = / on Z, and g = 1 on D — Z. Since iz*((f)) = *d* ((#)), we
may assume that Z is locally principal on T. Similarly, we may assume that
the invertible sheaf M := Ot{Z) is very ample. Let t0 be a section of M with
Z = (Jo), and let Jqo be a section of M with divisor Zoo satisfying
(a') Zoo is reduced and irreducible,
(b') Zoo contains no component of Z,
(c') Zoo H Y is finite and disjoint from Z fl Y.
Let L be a very ample line bundle on T so that L(—Z) is also very ample. Let
ft be the rational function on ZUZqo defined by ft = f on Z, ft = 1 on Zoo. Then
ft is in k(Z U Zoo, 50*. Thus there is a finite set S contained in (Z U Zoo) — Y
such that ft is a regular function on (Z U Zoo) — S. Let Soo be a section of L such
that
(i) (soo) is reduced and irreducible,
(ii) (so©) contains S,
(iii) (soo) intersects Z U Zoo properly,
(iv) {soc)n{ZuZoo)nY = 0.
(we may need to replace L with a high tensor power to accomplish this). We
may also assume that T — (sqq) is affine, so the function h lifts to a regular
function H on T — (soo)î for N sufficiently large, the section s0 := Hs^ of Z/N
on T — (soo) extends to a section s0 of LN on T with s0/s^ = ft on Z U Zoo-
Altering so by any section of Ln <g> I(zuZoo) preserves this equation, so we may
assume that
(v) (sq) is reduced and irreducible,
(vi) (so) H Y is finite and disjoint from Z U Zoo-
We have
iz*((f)) = (s0)-Z-N(soo)-Z,
0 = iZ. ((1)) - (S0) • Zoo " JV(Soo) • ^oo.
Let 0o be the restriction of to/too to (s0), and let <7oo be the restriction to (sqo)-
Then #o is in fc((s0),50* by (iv), and 0oo is in k((soo),Y)* by (vi). In addition,
iz- ((/)) = (so) • (Z - Zoo) - JV(soo) • (Z - Zoo)
= *(«>)• ((0o)) - W • HSooy ((9oo)).
As (so) and (soo) are both reduced and irreducible, the right-hand side of the
above is in Rq(X, Y)red, completing the proof. Q.E.D.
2. Torsion in CH0(X,Y) for affine X. Using essentially the technique [1,
3, and 8], we show that CH0(X, Y) is torsion free, mod p-torsion, for X an affine
variety.
Let X be a quasiprojective variety, and Y a closed subset containing XSing.
Suppose Z is a subvariety of X, not contained in Y, and locally a complete
intersection along Y. Then the inclusion of Z in X induces a homomorphism
%z* : CHo(Z,ZHY)^ CH0(X, Y).

456
MARC LEVINE
In particular, if C is a reduced irreducible curve on X, not contained in Y, and
a complete intersection at each point of C C\ Y, we get a homomorphism
ic* : CH0(C, CHY)^ CH0(X, Y).
We will refer to a curve C satisfying the above as a good curve on X, relative to
y.
If C is a curve, and S a finite set containing CSing, then the first Chern class
gives an isomorphism from Pic(C) to CH0{C,S). Thus, if C is a good curve
on X, relative to y, we get a homomorphism, also denoted z"c*, from Pic(C) to
CH0(X,Y).
If C is a good curve on X, relative to y, we let C be a complete model of
C, containing C as an open subset, with each point of C — C smooth on C.
This uniquely determines C. The inclusion of C in C induces a homomorphism
i*c : Pic(C) —► Pic(C). We let Kq denote the kernel of i*c. From the localization
sequence
Z0(C - C) -* Pic(C) -* Pic(C) -* 0
we see that Kq is the subgroup of Pic(C) generated by the point of C — C.
LEMMA 2.1. Suppose X is an affine variety of dimension at least two, Y a
closed subset containing XSing, and C a good curve on X relative to Y. Let n be
an integer prime to the characteristic, z an n-torsion cycle on C (cl(nZ) = 0 in
Pic(C)). Then ic+{z) = 0 in CH0(X,Y).
PROOF. By Corollary 1.2, there exists a projective model X for X such that,
letting C denote the closure of C, XSing has codimension at least two on X at
each point of (C — C) D XSing. As C is a complete intersection at each point of
CnY, Lemma 1.3 shows that there is a reduced irreducible surface S containing
C such that S = S H X is a complete intersection on X, and C is defined by a
single equation in a neighborhood of C C\ Y on S. As XSing has codimension at
least two at each point of (C — C) fl XSing, we may choose S as above so that
each point of C — C is either smooth on S, or is an isolated point of SSing-
We resolve the singularities of S which are on C — C, and resolve the
singularities of C which are on C — C, by monoidal transformations with centers
lying over C — C. This gives a projective surface S, with a projective birational
morphism u: S —► S, such that
(a) u is an isomorphism over S,
(b)FHC) = c, _
(c) each point of C — C is smooth on S.
Let F = S — S. As S is affine, F supports a very ample divisor D. Replacing
D with a sufficiently high multiple, we may assume there are sections so, Sqo of
Og{D) such that
(i) sq is a section of 3§(D);
(ii) the subscheme Wo defined by s0 is reduced and equal to C in a
neighborhood of C H u-1 (Y) ;

ZERO-CYCLES AND if-THEORY ON SINGULAR VARIETIES 457
(iii) letting E be the closure of Wo - C, E is irreducible, E intersects u~l(Y)
properly, and E n C fl u~1(Y) = 0;
(iv) (5qo) is contained in F.
We may also assume that sq + tSoo — st defines a reduced irreducible subscheme
Wt of S for general t.
Let W Ç S x P1 be the subscheme defined by X0s0 + Xis^ ((X0 : X\)
homogeneous cordinates for P1). Since Wq is reduced we have
h\WuOWt) = h\W0,0Wo)
if Wt is reduced and irreducible. Thus there is a neighborhood U of 0 on P1
such that the relative Picard scheme 7r: Picu{Wu) —► U of WV = pj1^) exists
[10]. Let z be an n-torsion cycle on C. We lift cl(z) to an element 5 of Pic(C)
such that nz is in Kc> After a base extension q: U' —> U, there is a cycle Z\ on
S xU', supported on (W xPi [/') fl (F x U') and flat over [/', and a point 0' over
0, such that Z\ • Wo> is contained in the smooth locus of C — E, and represents
nz in Pic(C). We denote W xPi U' by Wv> and let tt;: Pic(^/) -► C/; be the
relative Picard scheme.
Let o\\U' —► Picf/'(H^t//) be the section determined by Zi, where we may
have to shrink U' to define c\. Pic(W0') is an extension of Pic(C) x Pic(^) by
an n-divisible group; hence there is an element b0 in Pic(W/o/) lifting (i,0), with
nbo = cri(O'). Let B be an irreducible component of (n x id)-1 {g\(U1)) passing
through 60. B is étale over U'. Let p: B —► P1 be the morphism q o 7r;. Let
Z Ç W^ xPi 5 be a cycle representing 5, i.e., the linear equivalence class of
Z(6) = pr^ (Z • (VKpjfc) x b)) on VFp(5) is 6. (We may have to shrink B to get
such a Z.) Shrinking B again we may assume that Z is supported on (S — Y)x B.
Replacing B with its normalization, we assume that B is smooth.
Let B be a complete smooth model of 5, and let 600 be a point of B lying
over 00 in P1. Let B* = B U {boo}. Since Woo is contained in S — S, it follows
that Z is closed in S x 5*, and ZCiS xboo is empty. As the cycle Z(6) satisfies
nZ(b) ~j Zi(6) on Wp(6) for 6 in 5
and Zi(6) is supported on F, the cycle Z(b) on S is n-torsion in C/foCS'î 5 fl Y)
for each 6 in B. By [5, Proposition 4.1], the class of Z(b) in CH0(S, S fl Y) is
constant over 5*; as Z(&oo) = 0, we have Z(6) = 0 in CH0(S,S fl Y) for all 6
in B. Since * = Z(60) in CH0(S, S fl Y), z is therefore zero in C/f0(S, S fl Y):
hence 2 is zero in CH0(X, Y) as desired. Q.E.D.
COROLLARY 2.2. Le£ X be an affine variety, and Y a closed subset
containing XSing. Let u: X* —► X be a sequence of monoidal transformations with
point centers lying over X — Y. Suppose C is a good curve on X*, relative to
Y, n an integer prime to the characteristic, and z an element ofPic(C)n. Then
ic.(z) = 0 inCH0(X*,Y).
PROOF. The map u induces an isomorphism rz* : CH0{X*, Y) —► CH0(X, Y).
If C is contained in the exceptional locus of u, then C is complete, and z has

458
MARC LEVINE
degree zero. Thus u*(z) = 0, which proves the corollary in this case. If C is not
contained in the exceptional locus, then, letting D = u(C), the map
zD*: Pic{D) ^ CH0{X,Y)
factors as
Pic(D) —+ Pic(C) > CH0{X*,Y) —► CH0(X,Y).
u* ic+ tx*
As the kernel of u* : Pic(Z>) —► Pic(C) is n-divisible, we can find an element u
of Pic(£>)n with z = u*{oj). Thus
0 = iD*(u) = u*(ic*(z))\
hence ic*(z) = 0. Q.E.D.
Let X, y, X* be as in Corollary 2.2. If C is a good curve on X*, n an integer
prime to the characteristic, and z a zero-cycle supported on Creg — Y such that
cl(z) is in nPic(C), pick a zero cycle y on Creg — Y such that c\(ny) = c\(z) in
Pic(C), and define ri^}(z) to be ic*{y) in CH0{X*,Y). By the above corollary,
tiq1(z) is well defined.
LEMMA 2.3. Let C,C be good curves on X*, and z a zero cycle on
(Creg — Y) C\ {C'Teg — Y) that is n divisible in Pic(C) and Pic(C') (n prime
to the characteristic). Then
nç\z)=nç}{z).
PROOF. By symmetry, we may replace C with a general complete intersection
of hypersurface sections containing supp(^). We note that such a curve is affine;
hence cl(^) is automatically n-divisible. In particular, we may assume that
C flC C\Y is empty, that C and C are smooth at each point of C C\ C, and
that C and C have distinct tangents at each point of C C\ C.
Arguing as in Lemma 2.2, there is a reduced, irreducible surface S on X,
which is a complete intersection of hypersurface sections, such that
(i) S contains CuC and both C and C are Cartier on S in a neighborhood
of S H Y ;
(ii) S has a projective model S such that C, C' are the closures of C, C on 5;
(iii) each point of C — C and C' — C is smooth on S.
In addition, as C and C intersect nicely, we may assume
(iv) S is smooth at each point of C D C'.
Again by symmetry, we may replace C' with a general hypersurface section of S
containing supp(^).
By Lemma 1.3, there is a line bundle L on S, and sections so, Sqq of L such
that, letting Wt be the subscheme of S defined by s0 H- ^œ, we have
(a) Wt is reduced and irreducible for general t;
(b) Wt contains supp(^) for all £, and is smooth at each point of supp(^) for
general t and for t = 0;
(c) Wo is equal to C in a neighborhood of C fl (S D Y); Wo is reduced;
(d) if E is the closure of Wq — C, E is irreducible and E fl supp(^) = 0.

ZERO-CYCLES AND if-THEORY ON SINGULAR VARIETIES 459
Let W® — S C\Wt. By the remark above concerning C, it is enough to show
that n^1(z) = n^0(z) for general t.
Let U be a neighborhood of 0 in P1 such that Wt is reduced and irreducible
for all t in U — {0}. We then have
h}{WuOWt) = hl{W0,0Wo) for all t in U.
Thus, letting W be the subscheme of S x U defined by so + ^œ, the relative
Picard scheme 7r: Picu{W) —► U exists. Shrinking [/, we may assume that
supp(^) is smooth on Wt for all t in U. Let cr: £/ —► Picu{W) be the section
determined by the cycle z. As the map
Pic(VKo) -► Pic(C) x Pic(£)
is surjective with n-divisible kernel, there is an element b0 in Pic (Wo) lifting
(?/,0), where y is in Pic(C) with ni*c(y) = cl(^) in Pic(C). In addition, there is
an irreducible component B of (n x id)-1(cr([/)), étale over [/, passing through
b0. Take a cycle Z Ç S Xu B representing B; shrinking U we may assume that Z
is supported on (X — Y)Teg xv B, and that ZnWn(b) x b is smooth on Wn(b) x &
for each b in 5. Let Z(b) denote the cycle ps{Z • (Wn^) x &)) f°r b m £• We
have
nZ{b) = z in CH0{S,SHY).
Thus
n(Z(6) - Z(6o)) = 0 inCflo(5,Sny).
The family of cycles Z(b) — Z(b0) parametrized by B thus defines a family of
n-torsion cycles in CHo(S,S fl Y), which by [5, Proposition 4.1] is a constant
family. Thus Z(b) — Z(b0) in CH0(S,Sr)Y) for each b in B; hence we also have
equality in CH0(X*,Y). Thus
"^(sHnj^s) inCH0(X*,Y)
for general £, as desired. Q.E.D.
The maps n^1 thus define a map
n~x\: Z0{X\Y) ^ CH0{X\Y).
n^\ is a homomorphism, for if 21,22 are cycles in Z0(X*, Y), we may take a
good affine curve C containing z\ and z2 in Creg — V\ and then
n^l(zi + 22) = riçl(zi + z2) = n^1 {z^ + n^1 {z2)
= n*l(*i) + n*l(*2).
In addition, we have
LEMMA 2.4. Le£ X 6e affine, Y a closed subset of X containing XSing, and
u : X* —► X a sequence of monoidal transformations with point centers lying over
X — Y. Then the following diagram commutes:
Z0{X\Y) -^-> CH0(X*,Y)
Z0(X,Y) -^ CH0(X,Y)

460
MARC LEVINE
PROOF. By induction, we may assume that X* is the blow-up of X at a point
p of X — Y. Let E = u~1{p). Let z be in Z0(X*, Y). We can write z as a sum
z = z\ + 22, where supp(^i) is disjoint from E and supp(^2) is contained in E.
Let C be a good curve on X, containing supp(u*(2i)) U {p} in its smooth locus.
Let C* be the proper transform of C to X*, and let q = C* fl £. Let Dbea
rational curve contained in E, and passing through supp^) U {q}. Let d be the
degree of z<i. We have
= u*(n^l (*i + d{q))) + u*(n^1 (*2 - d ■ (9)))
= u* (n^î (^i H- d($))) = n^1 (ti* (*i + d{q)))
= n51(Tx*(^)) = n^1(ix*(^))
as desired. Q.E.D.
LEMMA 2.5. n^(i2o(X,y)) = 0 in CH0(X,Y).
PROOF. As #0(X, Y) = Ro{X,Y)Ted, we need only show that n£(z) = 0 if
2 is a cycle of the form z'c* ((/)), where C is a good curve on X, relative to Y,
and / is in fc(C, Y)*. Let u : X* —► X be a sequence of monoidal transformations
with point centers lying over X — Y such that the proper transform [C] of C is
smooth away from Y. We have
«.(nx1-(*"[C]-((/)))) = »i1(*"c-((/)))
by Lemma 2.4. On the other hand,
«xl(*"[C]-((/))) = ng((/)) = 0.
which proves the lemma. Q.E.D.
Thus n^1 : Z0(X, Y) —► CHq(X,Y) descends to a homomorphism n^1 :
CHq(X)Y) —► Ci?o(-X\50 which is inverse to multiplication by n. We have
just shown
THEOREM 2.6. Let X be an affine variety, and Y a closed subset containing
^sing- Then CHq(X,Y) is torsion free, except possibly for p-torsion in
characteristic p > 0.
COROLLARY 2.7. Let X be an affine variety of dimension n, and S a filter
on X. Then
1:CHn{X,S)^FnK0{X)
is an isomorphism, modulo p-torsion in characteristic p > 0. If n <p, then 7 is
an isomorphism.
PROOF. In [4, Corollary 5.4] we show that the top Chern class cn defines a
homomorphism
cn:FnK0{X)^CHn{X,S)
satisfying
cn o 7 = ±(n - 1)! x id, 7 o cn = ±(n - 1)! x id.

ZERO-CYCLES AND if-THEORY ON SINGULAR VARIETIES 461
The corollary is now immediate from Theorem 2.6, and the equality CH0(X, Y)
= CHn(X, S) if Y = \JSi€S Si. Q.E.D.
In particular, the map 7: CH0(X, XSing) —► FnK0(X) is an isomorphism mod
p-torsion, and is an isomorphism if in addition n < p.
3. The projective case. Using arguments similar to those of §2, we have
shown in [3, Theorem 1] that, if X is projective and Y a closed subset of X
containing XSing such that codimx(^) > 2, then the Albanese map
ax: CH0{X,Y) -► Alb(X)
defines an isomorphism
ax: CH0(X,Y)tOT ^ A\b(X)tOT
modulo p-torsion in characteristic p. Roitman [8] and Bloch [1] had previously
proved this result in the smooth case, and Milne [6] has shown that ax is an
isomorphism on the p-torsion part, when X is smooth. Using these results, we
will show that 7: CH0(X,Y) —► FqK0(X) is an isomorphism, mod p-torsion
for X projective, Y as above, and is an isomorphism if either X is smooth and
Y = 0, or if n < p.
We first note that if X is projective, then ax : X —► Alb(X) = A induces a
homomorphism ax\ : K0(X) —y Ko(A). Indeed, let X* Ç X x A be the closure
of the graph of ax • X —► A, and define ax\ to be p<i\ o p\. As ax is a morphism
at each smooth point of X [9, II, Theorem 6], we have
aXi([k(x)}) = [k(ax(x))}
for each smooth closed point of X. Indeed, p\ is an isomorphism over a
neighborhood of x, and hence, letting y = p^1^)
aX\[k{x)] =pnp5(fc(x))
= ^2(-lY[RW(k(y))}
= [pi*k(y)] = [k(ax(x))}.
Thus, if x = Yl n%xi is a zero cycle on X — Y, we have
lA{ax*{z)) = ax\{lx{z))
where lx: CH0(X,Y) -► F0K0{X), ^A: CH0{A) -► F0K0(A) are the cycle to
if-theory maps.
Let P be the Poincaré line bundle on A x Â, A = Pic0(A). P defines a map
~P:K0(A)^K0(Â) by
~P{z) = P2i(p[{z)®P)
for z in K0(A). If x is a closed point of A, then
~ P{[k{x)]) = P2! (P ® 0xxi) = p2* (P 0 Oxxa)
asxxi has relative dimension zero over A. Thus P restricts to a map
- P: F0K0(A) - Pic°(i) Ç ffo(i),

462
MARC LEVINE
and as cl(p2* (P ® OxXa)) = z under the isomorphism Pic0(À) = A, we see that
w P (H n»[fc(x»)l) = ^2 UiXi'
Then ^ P o oxi defines a map
-PoaX!: /T0(X)->j4
and
^PoaX\ \^2rii[k(xi)}j = ^2niax(xi).
We have therefore factored ax : CH0(X, Y) —► A as
ax —^Po ax\ ° 7x-
Since ax is an isomorphism on the torsion subgroups if X is smooth and Y = 0,
and an isomorphism mod p-torsion if y has codimension at least two, we have
shown
LEMMA 3.1. Let X be a projective variety. Then 7x- CHq(X,Y) —►
FqKq(X) is an injection on CH0(X,Y)tor if X is smooth and Y is empty, and
is an injection modulo p-torsion if codimx Y > 2, and Y D XSing.
As mentioned in Corollary 2.7, if S is a filter on X, the top Chern class
cn: FnK0{S) -► CHn{X,S) satisfies
7x°cn = ±(n- 1)! xid,
n = dim A.
cn°lx = ±(n- 1)! x id,
Letting Y = \JSies s»> we have CHo{X, Y) = CHn(X, S). Hence we have shown
THEOREM 3.2. Let X be a projective variety, and Y a closed subset
containing XSing. If X is smooth, then 7x: CHq(X) —► FqKo(X) is an isomorphism. If
codimx(y) > 2, then 7x- CHq(X,Y) —► F0K0(X) is an isomorphism modulo
p-torsion; z/dim(X) < p, then 7x- CHq(X,Y) —► F0K0(X) is an isomorphism.
References
1. S. Bloch, Torsion zero cycles and a theorem of Roitman, Compositio Math. 39 (1979),
107-127.
2. S. Bloch, M. P. Murthy, and L. Szpiro, Algebraic cycles and number of generators of
ideals in affine rings, Preprint.
3. M. Levine, Torsion zero cycles on singular varieties, Amer. J. Math. 107 (1985), 737-
758.
4. , A geometric theory of the Chow ring on a singular variety, Preprint.
5. M. Levine and C. Weibel, Zero cycles and complete intersections on singular varieties, J.
Reine Angew. Math. 359 (1985), 106-120.
6. J. S. Milne, Zero cycles on algebraic varieties in non-zero characteristic: Roitman's
theorem, Compositio Math. 47 (1982), 271-288.
7. N. Mohan Kumar and M. P. Murthy, Algebraic cycles and vector bundles over affine
threefolds, Ann. of Math. (2)116 (1982), 579-591.
8. A. A. Roitman, The torsion in the group of zero-cycles modulo rational equivalence, Ann.
of Math. (2)111 (1980), 553-570.
9. A. Weil, Courbes algébrique et variétés abeliennes, Hermann, Paris, 1971.
10. M. Artin, Algebraization of formal moduli. I, Global Analysis (papers in honor of
K. Kodaira), Univ. of Tokyo Press, Tokyo, 1969, pp. 21-71.
Northeastern University

Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Zero Cycles Modulo Rational Equivalence
for Some Varieties Over Fields
of Transcendence Degree One
CHAD SCHOEN
Let X denote a smooth, geometrically irreducible, projective variety over an
infinite field k of finite transcendence degree over the prime field. Let K be an
algebraically closed extension field of fc, and let Dq(Xk) denote the group of
degree-zero zero cycles modulo rational equivalence. It is natural to ask whether
the albanese map <pk : D0{Xk) —► A\bx{K) is an isomorphism. We shall mainly
consider this question in the case that dimX = 2 and the cycle class map N.S.
(Xk) <8> Qi —► H2(Xk,Qi{1)) is not surjective. In this case Mumford [9] has
shown that ker <pk is large, if K — C. Later Bloch [3] extended this result to
universal domains of arbitrary characteristic. The main step in his argument
establishes that for any fc-morphism from a smooth projective curve, j : C —► X,
the induced map, j* : D0(Ck) —► D0{Xk) is not surjective, provided K contains
the function field k(X). Now if C is taken to be a general hyperplane section of
X, the natural map D0{Ck) —► Albx(-^) is surjective, which implies <pk is not
injective.
Each of the above arguments relies on the hypothesis trans, deg. (K/k) > 2.
In this paper we shall be concerned with the case trans.deg. {K/k) = 1. In
order to put the results in the proper perspective, two things should be said
concerning the case trans, deg. {K/k) = 0. First, a theorem of Roitman, as
extended by Milne [8], shows that <pk is an isomorphism if K is the algebraic
closure of a finite field. Secondly, there is a conjecture (or at least a "recurring
fantasy" [4]) of Beilinson and Bloch concerning the order of vanishing of L-series
and the ranks of Chow groups which would imply that <pk is an isomorphism
when K = Q [1]. However there is presently no known example to support this
conclusion, and so far, only very little evidence for the conjecture.
In the first part of this paper we consider the surface X = E x E, where E is
the Fermât cubic curve, over the field K = Q(£). It is shown that D0{Xk) is not
"finite-dimensional" in the sense of Bloch [3]. More precisely, given any smooth
1980 Mathematics Subject Classification (1985 Revision). Primary 14-02, 14C25.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
463

464
CHAD SCHOEN
projective curve over K and any if-morphism j : Ck —► Xk, the cokernel of
the induced map y* : D0(Ck) —► Dq(Xk) is shown to have infinite rank. This
implies that <pk is not an isomorphism.
The method of proof is based on the fact ([12] or [11]) that the rank of
the group of one cycles modulo algebraic equivalence on Wq is infinite, when
W is a threefold constructed from the elliptic modular surface for level three
structure. The main step involves constructing a correspondence between W
and E x E x E. The existence of such a correspondence is predicted by the
Tate conjecture. In fact the Tate and Hodge conjectures imply the existence
of correspondences between several elliptic modular varieties and self-products
of elliptic curves. However, it is not apparent that the methods used here will
produce correspondences in other cases.
In §3 we consider k = algebraic closure of a finite field and X a smooth
projective k-variety dominated by a product of projective curves. If K is the algebraic
closure of the function field of a curve over fc, then <pk is an isomorphism. This
is a consequence of work of Soulé [14].
After the first sections of this paper were written, I learned from Madhav Nori
that he had developed a way of showing that <pk is not an isomorphism, based
on the study of one cycles algebraically equivalent to zero on an appropriate
product of curve and surface. In §4 I have attempted to explain Nori's interesting
technique and to illustrate its relatively wide range of application by an example.
The result is similar to that in §2 except with the Fermât cubic replaced by
an arbitrary curve of positive genus! With more work one could undoubtedly
construct other examples. Nonetheless, the method has a fundamental limitation
due to the fact that a certain Hodge structure has to split off a piece of pure type
(2,1),(1,2). When this fails to occur, one should probably return to studying
the Abel-Jacobi homomorphism on null-homologous one cycles modulo cycles
algebraically equivalent to zero.
I wish to thank M. Nori for explaining his ideas and letting me include a brief
description in this paper. I also wish to thank V. Srinivas for helpful discussions.
1. The correspondence. Write X for Pq and Y for the subvariety of
Pq x Pq defined by the bihomogeneous equation
v0(x£ + x? + x£) - vix0xix2 = 0.
Let 7T be the restriction to Y of projection onto the second factor. It is known
that 7T : Y —► X is isomorphic to the natural compactification of the universal
elliptic curve with level-3-structure [13, Remark 5.6]. More precisely, -k : Y —► X
may be taken to be the morphism Y(3) -+ X{3) of [12, (1.12)] (cf. [11, §2]).
Set W = Y xn Y and let W denote the blow up of the singular locus of W.
The rational Hodge structure H3(W*n,Q) has dimension-two pure type (3,0),
(0,3) and complex multiplication from the field, N = Q(^), of cube roots of
unity ([12, (1.5) and (1.7)] or [11]). The complex multiplication is induced by
an order-three modular automorphism of W^n. Let E denote the Fermât cubic

ZERO CYCLES MODULO RATIONAL EQUIVALENCE
465
curve. Then the Hodge conjecture implies the existence of a correspondence
between W^n and (i£3)an which induces an injection on H3. This section is
devoted to constructing such a correspondence defined over N.
The first step is to recognize that W is covered by a two-parameter family of
elliptic curves with j-invariant zero. This is most easily seen by working with
the birational model W C P2 x P2 which is defined by
(1.1) detfX° + X? + xi yo + y? + y23l 0.
L x0xix2 yoyiy2 J
Introduce inhomogeneous coordinates
Xi = Xi/x2 and & = y;/y2, i € {0,1}.
Now the curves we wish to study are cut out by the two-parameter family of
rational surfaces x0 — cy0,x\ = ey\. Then W is birational to
Spec Q[c, e, y0, yi]/(yl(cz - ce) + 2/i(e3 - ce) + 1 - ce).
There is an obvious map to Spec Q[c, e], whose geometric generic fiber is
birational to the Fermât cubic curve. It is not evident (at least to me) whether or not
this map has a rational section. However, a rational section may be introduced
by taking the quotient with respect to an appropriate automorphism. Define 7
by
Vi°l= Q^Vi, co-y = c, eo-y = e.
If the base field is 7V,then we may take c, e, w0, wi, where
wo = (&(c3 - ce)yl + £2(e3 - ce)y\ + (1 - ce))/(-3y0!/i),
wi = (fs(c3 " ™)yl + Çs{e3 - ce)y\ + (1 - ce))/{-3y0yi)
as generators for the field of invariant functions (cf. [5, (24.16)]). One checks
that
(1.2) wl + w\ + (c3 - ce)(e3 - ce)(l - ce) = 0.
This hypersurface is birational to W/ (7).
The automorphism 7 is actually very natural. It is induced by an element of
Aut(P^ x P^) also denoted by 7.
^°7= £3 Xii Vi°l= C3
Villi the terminology of [11, §2], 7 is the biregular automorphism of Wn
corresponding to translation by the order-three section (—s2) x^ (—82). The
fixed points of this action are the nine singular points in the fiber of W above
v\/vq = 00 in X. These correspond to the points
x^ = Xi+i = Yj = yj+1 = 0, z,y mod 3,
on W. Clearly the action the group (7) extends to a biregular action on W. By
blowing up a sequence of smooth curves and points (defined over Q) in the fiber
over 00, one arrives at a threefold W on which 7 acts biregularly and W/(i) —V
is smooth. In the case at hand, these blow ups may be done explicitly thanks to

466
CHAD SCHOEN
the explicit equation for W. Quite generally, given any ^ action on a smooth
threefold (over a field of characteristic ^ 3) one can blow up points and smooth
curves to get a birational model whose quotient is smooth. In fact, one reduces
to enumerating the linear actions of fis up to isomorphism on the ring of formal
power series in three variables over the algebraic closure. In each case it is easy
to see which blow ups are necessary.
Recall that H°(W, ft3) is one-dimensional [11]. A basis element may be given
as the residue
uj = Resdxo dxi dy0 dyi/{x% + x\ + 1)î/o2/i - {Vo + 2/i + l)«o«i-
Evidently u is 7-invariant so the trace is a basis for H°(V, fi3).
As we have seen there is a rational map from V to SpecQ[c, e]. If we base
change with the obvious map
T := SpecQ[c,e,z]/{z3 - (c3 - ce){e3 - ce){I - ce)) -+ SpecQ[c,e],
we obtain by (1.2) a constant family of Fermât cubic curves over T. Thus it is
of interest to establish
LEMMA 1.3. The elliptic curve E\ : w2 — 1/4 - x3 = 0 is Q-isogeneous to
E and T is birational to E\ x E\.
PROOF. We may think of T as a degree-three cyclic cover of P1 x P1 with
inhomogeneous coordinates c and e on the latter. We shall produce a pencil of
elliptic curves on P1 x P1 which is the image of a constant family of elliptic
curves on T. For this consider the pencil of (2,2) curves on P1 x P1 generated
by c(c — e2) =0 and e(ec — 1) = 0. The equation for a general member of this
pencil
(1.4) c2 + (t- l)e2c-te = 0
may be rewritten using u = c + e2(t — l)/2 as
u2 - (t - l)V/4 - te = 0,
which becomes
w2 - 1/4 - t(t - l)v3 = 0
when w = u/(t — l)e2 and v = l/(t — l)e. From (1.4)
t(t - 1) = (c3 - ce)(e3 - ce)(l - ce)/e3(ce - l)3,
and the function field of T is obtained from Q(c,e) by adjoining an element r
which satisfies
r3 = t{t - 1) = (t - 1/2)2 - 1/4.
This yields the pair of equations
w2 - 1/4 - [rv)3 = 0, (t- 1/2)2 - 1/4 - r3 = 0
for a birational model of T.

ZERO CYCLES MODULO RATIONAL EQUIVALENCE
467
To see that E\ is Q-isogeneous to E consider the /i3 action on
Spec Q[*i,£2]/(^+ 4 + 1)
given by Z{ —► £3^. Let x — —z\z<i and w — z\-\-\. Then Q(w,x) is the fraction
field of the quotient and w2 = x3 + \.
Resolution of singularities gives dominant, proper Q-morphisms
V J- U Z E3
with <p a composition of blow ups with smooth centers followed by an isogeny.
Putting everything together gives a sequence of morphisms
WNtwN^VN^UN^E%
for which
ip+T*K+il>* : H°{WN, fi3) - H°{E3N,Q3)
is evidently injective. Since H3(W^n^C) is of pure type (3,0),(0,3), the induced
map on singular cohomology with Q coefficients or étale cohomology with Q/
coefficients is injective. Of course, the induced map
i: J2(W™)^ J2((£3)an)
on intermediate Jacobians is nonzero. In fact, i is an isogeny onto its image
J2((£3)an)'. _
REMARKS, (a) The geometry of the variety Wn is very interesting. It has 117
isolated ordinary double points and no other singularities. 81 of these nodes arise
from collapsing the three torsion sections of Wn • The remaining 36 come from
the triple points in the singular fibers. The general determinantal hypersurface
for pr* Op2(3) 0 pr£ Op2(3) has only 81 isolated double points.
(b) The Tate conjecture predicts that our correspondence should be defined
over Q. Using [6] one shows that the L-series for the Gal(Q/Q) representation
on H3{W-q, Qi) is (up to finitely many Euler factors) the Mellin transform of
the unique weight-four normalized cusp form for T(3). But this L-series and its
twists by Dirichlet characters satisfy the same functional equations as L(*,s)
and its twists, where \I> is the Hecke character defined on fractional ideals of
Z[£3] prime to \/^3 by the rule
\P(a) = a3, where (a) = a and a = 1 mod>/—3. From this the two L-series may
be seen to coincide.
It is known that L(E, s) = L(x, s), where
X : /(3) - AT*
is defined by x(a) = a, where (a) = a and a = lmod3. A consequence of this
is that the submotive H3{E3)' of Hodge type (3,0),(0,3) has L-series L(*,s).
Thus the Gal(Q/Q) representations are ismorphic [10].
(c) In our construction, the only map which is not defined over Q is /c. V
is defined over Q and W/(i) is too, but they become isomorphic only over N.

468
CHAD SCHOEN
However there is a degree 3 Q-rational map W/(i) —► V. The resulting degree-9
Q-rational map W —► V is described by identifying Q(V) with Q(c,e,vo,vi),
where
^o = (Cs^o + ûwl + (c3 - ce)(e3 " ce)(! " ce))/(-3w0wi),
vi = (Cl^o + few? + (c3 - ce)(e3 - cc)(l - ce))/(-3w0wi).
Note that the equation
^o + vi + U - ce)(e3 ~ ce)(cS - ce) = 0
holds. Using this one may define the correspondence over Q.
2. Infinite-dimensionality of the Chow group. Unless the contrary is
specifically indicated, all schemes in this section are understood to be varieties
(finite type, reduced, separated schemes) over Q. Given such a variety V we let
Ak{V) (resp. BkiV)) denote the group of fc-dimensional algebraic cycles modulo
rational (resp. algebraic) equivalence. See Fulton [7] for the definitions. The
proof of [12, Theorem 4.7] shows that rank£?i(i£ x E x E) — oo. An immediate
consequence of this is
PROPOSITION 2.1. Let T be a smooth projective curve which dominates E.
Then rank Bx(E x E xT) = oo.
PROPOSITION 2.2. Let L be a finite extension of Q(E). Suppose given a
smooth projective curve C/L and an L-morphism j : C —► (E x E)l. Then the
cokernel of
i. : A0(CT) -+ A0((E x E)T)
has infinite rank.
PROOF. Let T be a smooth projective model for L over Q. Let S C ExExT
denote the closure of the image j(C). Consider the commutative diagram of exact
sequences,
Bi(S)
a
UmBi(ExExD) -^ Bi(ExExT) —+ lim BUE x E x (T - D)) — 0
D D
limBi(£x E x D - SD) —+ BX{E xExT - S) —+ WmBiiEx E x (T - D) - ST-d) -* 0
D D
0 0 0
The direct limits are taken over finite sets of closed points D c T. Sd and
St-d denote the base change of S with respect to inclusions of subschemes D
(resp. T — D) in T. Let p : S —► S be a resolution of singularities of S. Now
B\(S) is finitely generated, and it follows easily from standard exact sequences
relating B\(S) and B\{U), where U — complement of the exceptional divisor in

ZERO CYCLES MODULO RATIONAL EQUIVALENCE
469
S = complement of singular locus in S, that B\(S) is finitely generated. From
(2.1) and the fact that the image of b has rank 4, a diagram chase shows that
(2.3) limBX{E x E x {T - D) - ST-d)
has infinite rank. But
lim A^E xEx(T-D)- ST_D) ~ A0((E xE- j(C))L)
surjects onto (2.3). Finally note that the kernel of the pullback
A0((E xE- j(C))L) - A0((E xE- j(C))T)
in torsion [3]. Since j+ factors through A0{j{Cj;)), the desired result follows from
the exact sequence
A)(i(C)r) - A0((E x E)T) -, A0((E xE- j(C))T) - 0.
3. Fields of transcendence degree one over a finite field. Let X be
a smooth, projective, geometrically irreducible variety over a field L. Fix an
algebraic closure, L, of L. The following argument is implicit in Beilinson [1,
(5.2)].
LEMMA 3.1. Let £ be a set of intermediate extensions in the tower L c L
such that every M € £ is finite-dimensional over L and every finite extension of
L is contained in some M E £. Suppose that rank Albx(M) = rank(Do(Xm)) <
oo for all M E £. Then the natural map <p-£ : Dq{Xj^) —► A\bx{L) is an
isomorphism.
PROOF. The surjectivity of <p-jr is clear. Let z denote a degree-zero zero
cycle, rational over some M E £, with [z] E ker^. By Roitman's theorem as
extended by Milne [8], [z] is not torsion. To show that [z] = 0, it suffices to
show that (Cm • Dq(Xm)/torsion —► Albx(M)/torsion is injective. Since these
are free, finitely generated Z-modules of the same rank, we need only check that
the cokernel is torsion. Fix an integer m for which the difference morphism
tfM:^x*M^AlbxM,
#M((zi,.--,Zm),(2/i,..-,2/m)) = class of ÇJTXi -^Tyi)
is dominant. Given an element of infinite order p E Albx(M), the image of an
effective M-rational zero cycle in the fiber \P-1(p) is a nonzero multiple of p.
Thus coker <pm is torsion.
PROPOSITION 3.2. Let X be a smooth, projective, geometrically irreducible
variety of dimension n over the finite field Fq. Assume furthermore that there is
a finite, dominant Fq-morphism from a product of projective curves to X. Let L
be a finitely generated extension field of Fq of transcendence degree one. Then
<p-j- : Dq(X-j-) —► A\bx{L) is an isomorphism.
PROOF. Let M be a finite extension of L, k the algebraic closure of Fq in
M, Ta smooth projective curve over k with function field M. Let £ denote the

470
CHAD SCHOEN
collection of those M for which (i) T has a fc-rational point and (ii) the pullback
map Ai(Xk)®Q —► Ai(X^ )<8>Q is surjective (such fc exist by [14, (3.2)]). (It is
not really necessary to make such restrictions on M, but it makes the following
arguments slightly easier.) For any M E £ consider the exact sequence
(3.3) limMXk xD)-^ A%(Xk x T) - At^(XM) - 0,
where the direct limit is taken over effective divisors on T. Since Pic°T(fc)
is a torsion group, rank(imfri) = ra,nkAi(Xk). Another application of Soulé's
result [14, (3.2)], together with Poincaré duality, yields rankAi(Xfc x T) =
rank An(Xk x T) < oo and rank Ai(X^) = rank An-\(Xk) < oo. From (3.3)
rankZ?0(XM) = rank Ai(Xfc x T) - rank Ai(Xk) - 1
(3.4)
= rank An(Xk x T) - rank An-i(Xk) - rank An(Xfc).
Assume for a moment the equality
(3.5) rankAn_i(Xfc) = rank N.S. (XM).
Now use (3.3) with i = n to rewrite (3.4) as
rankDo(XM) = rank An_i(XM) — rank N.S. (XM)
= rank Pic0(XM) = rank Albx(M).
By (3.1) we are done, modulo the proof of (3.5).
Since An-i(Xk) <g> Q = N.S. (Xk) <8> Q an easy intersection theory argument
shows that the composition v : Xm —y Xk x T —► Xk induces an injection
i/* : An-i(Xk) <S> Q —► N.S. (Xm) <8) Q. To verify surjectivity, note that an
invertible sheaf F on Xm may be extended to an invertible sheaf F on X x T
which we restrict to the fiber X xt over a fc-rational point of T. By modifying F
by an element of the image of is*, we may assume that the cohomology class of
F\xxt in H2(X^Qi(l)) is 0. It follows that the class of F in #2(X^-Q/(1)) is
also zero, so a multiple of F lies in Pic0 Xjj. Since the pullback N.S. (Xm)®Q —►
N.S. (Xjj) <8) Q is injective, the class of F in N.S. (XM) ® Q is zero and (3.5)
follows.
4. Nori's method. Let T be a smooth projective irreducible curve over an
algebraically closed subfield fc C C. Write L for the function field of T. For
certain irreducible, projective surfaces X, smooth over fc, we wish to produce
nontorsion elements of Ker^L : Dq(Xl) —► Albx(^). For a variety V, it is
convenient to denote by -Di(V) the group of 1-dimensional cycles algebraically
equivalent to zero modulo rational equivalence. From the exact sequence
linu4i(X xd)^ii(Ixr)-^ A0(XL) -► 0,
d
where the direct limit is taken over finite sets of closed points in T, it follows
that
DX(X x T) :=f Di(X x T)/[limAi(X x d) D DX(X x T))

ZERO CYCLES MODULO RATIONAL EQUIVALENCE
471
may be viewed as a subgroup of Dq(Xl). Nori uses an Abel-Jacobi homomor-
phism to try to detect nontrivial elements of D\(X x T). For this, define the
following sub-Hodge structures
H2(X™)t := ( N.S. (X))x c tf2(Xan),
M = image {H2{X*n)t ® H^T™) c if3(Xan x Tan)).
The complex torus
J = #3(Xan x Tan, C)/{Mè + F2H3(Xan x Tan) + # 3(Xan x Tan, Z))
is a quotient of the usual intermediate Jacobian of Xan x Tan. Since Mo PlM^ =
0, J is isogenous to the intermediate Jacobian associated to the Hodge structure
M.
The composition of the usual Abel-Jacobi map with the canonical quotient
map gives rise to a homomorphism
a\Dx{XxT)-^ J.
It is not difficult to see that a induces a well-defined map
â:2?i(XxT) ->J.
This map is zero unless J contains an abelian subvariety of positive dimension.
Assuming that this condition is fulfilled, the problem is to find z £ D\(X xT)D
kenpL with ct(z) not torsion in J.
EXAMPLE 4.1 Suppose that T has positive genus and X = T xT. Giving an
abelian subvariety of J amounts to giving a sub-Hodge structure of M of pure
type (2,1),(1,2). To produce such, use the inclusion from the Kunneth formula
£ : H1 (Tan)®2 — tf2(Xan), f(i/ ® i/) = prj i/ U pr£ i/.
to define N.S.(X)' := ^(N.S.pf)). The polarization on H!(Tan)®3 restricts to
a nondegenerate pairing on £_1 (H2(Xan)t) ®H1 (Tan). The image of H1 (Tan)®
N.S.(X)' under the orthogonal projection
will give rise to the desired nonzero sub-Hodge structure. One need only check
that the kernel of the orthogonal projection, which is N.S.(X)' <g> if^T3,11), does
not coincide with if^T*11)® N.S.(X)'. Since N.S.(X)' is a nonzero, proper
subspace of #1(j*an)®2 wnicn is stable under the operation of switching the
factors in the tensor product, this will follow from
LEMMA 4.2. Let W be a finite-dimensional vector space and V C.W ®W
a proper subspace. If V ®W C W®3 is stable under the symmetric group of
permutations of the three factors in the tensor product, then V = 0.
PROOF. Let (wi)\<i<n be a basis for W. Suppose Y^%u% ® w% ^ ^ w^n
some u\ ± 0. For each j E {l,...,n}, Yliui ® w% ® wj € V ®W. This
implies ^2{Ui <g> Wj <g> w% E V <8>W, and hence ui <8> w3- E V. Now for any

472
CHAD SCHOEN
m E {1,..., n}, ui<g>Wj® wm E V <g> W, from which wm <g> Wj <g> u* E V <g> VF, and
hence wm 0^GV follows. Thus F = W <g> VF.
Let P2 : T x T —> T denote projection onto the second factor. The generic
fiber of this projection is written Tl. If C C T x T is a cycle, write Cl for
the pullback to the generic fiber. Fix a point Jo £ T(k) and use (t0 x T)L as
a base point of Tl to get an embedding Tl —► Alb^L. There is also a product
embedding XL —► AlbxL • Given a zero cycle z on T and a divisor D on T x T,
the composition T3 = X x T —► Xl —► AlbxL sends 2 x D to a zero cycle on
AlbxL- This zero cycle may be written as the Pontrjagin product of two zero
cycles on (AlbTL)2 = AlbxL, namely as
(4.3) \{z x T)L x (t0 x T)L] * [(t0 x T)L x DL\.
Now choose the divisor D such that its cohomology class lies in im(£) and
such that ^(T3,11) <g> [D] is not orthogonal to M. The first hypothesis implies
that the degree of the zero cycle Dl is zero. By the second assumption there is
a nonzero homomorphism of complex tori
P : Pic°(Tc) - J, P(z) = a{z x D).
Of course (3 induces an isogeny from an abelian subvariety, Ac C Pic°(Tc),
onto an abelian subvariety of J. Necessarily, Ac is obtained by extension of
scalars from an abelian subvariety A c Pic°(T) defined over the algebraically
closed field k. Select a degree-zero zero cycle z on T, whose divisor class is a
point of infinite order in A(k). Then ct(z x D) has infinite order in J. On the
other hand, the cycle z x D on T3 = X x T maps to a zero cycle on AlbxL,
which is the Pontrjagin product of degree-zero zero cycles (4.3). Such a cycle
is in the kernel of the map Z?0(AlbxL) —► AlbAibx(^) — Albx(^). It follows
that the class of^xDinDiflxT) c Dq(Xl) is an element (of infinite order)
in ker<£>L : D0(Xl) —► Albx(^)- Finally, if K = L, then ker^/c is not zero
[3, Lemma 3].
It is interesting to observe that if the zero cycle z on T were chosen so that its
divisor class was an element of A(k) having sufficiently large finite order, then
z x D would give rise to a nontrivial torsion class in Ker <pl .
References
1. A. Beilinson, Height pairing between algebraic cycles, Preprint.
2. S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39
(1979), 107-127.
3. , On an argument of Mumford in the theory of algebraic cycles, Algebraic Geometry
Angers 1979, A. Beauville, editor, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980, pp.
217-221.
4. , Algebraic cycles and values of L-functions, Crelles Journal 350 (1984), 94-108.
5. J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London
Math. Soc. 41 (1966), 193-291.
6. P. Deligne, Formes modulaires et representations l-adiques, Séminaire Bourbaki 355,
Lecture Notes in Math., vol. 179, Springer-Verlag, 1971, pp. 139-172.
7. W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3) 2 (1984).

ZERO CYCLES MODULO RATIONAL EQUIVALENCE
473
8. J. S. Milne, Zero cycles on algebraic varieties in non-zero characteristic: Rojtman's
theorem, Compositio Math. 47 (1982), 271-287.
9. D. Mumford, Rational equivalence ofO-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968),
195-204.
10. K. Ribet, Galois representations attached to eigenforms with nebentypus, Lecture Notes
in Math., vol. 601, Springer-Verlag, 1976, pp. 17-52.
11. C. Schoen, Null-homologous 1-cycles on a family of abelian surfaces. I, manuscript (1984).
12. , Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53
(1986), 771-794.
13. T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972) 20-59.
14. C. Soulé, Groupes de Chow et K-théorie de varieties sur un corps fini, Math. Ann. 268
(1984), 317-345.
Duke University

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Rational Equivalence of O-Cycles
on Normal Varieties over C
V. SRINIVAS
In the basic paper [M], Mumford defined the notion of infinite dimensionality
for the Chow group of 0-cycles on a smooth projective variety, and showed that
if X is a smooth projective surface over C with T(X, fi^) ^ 0, then A0(X) is
infinite-dimensional. Mumford's results were refined and extended to arbitrary
dimension by Roitman [RI, R2], who defined suitable invariants measuring the
"growth" of the Chow group, and in particular showed that if X/C is smooth and
projective, and T(X, Qqx) ^ 0, for some q > 2, then A0(X) is infinite-dimensional
in Mumford's sense. Mumford's result (for surfaces) has been extended to
characteristic p > 0 by Bloch [Bl].
In [SI] the author constructed numerous examples of affine cones over curves
(over C) with nonzero Chow group of 0-cycles; this gave examples of normal
graded rings of dimension 2 with nontrivial projective modules, and hence non-
trivial NK0. The proof used relative if-theory, defined in terms of Quillen's
higher if-groups. Next, in [S2] we showed that if X/C is a normal projective
surface with H2(X, Ox) ^ 0, then A0(X) is infinite-dimensional, in a suitable
sense generalizing Mumford's definition. This contains the previous result on
cones. We also gave a suitable result in charp > 0. Again the proofs use the
methods of higher K -theory and relative if-groups, and are partly motivated
by a conjectural if-theoretic formula for ker(A0(X) —► A0(Y)), where Y —► X
is a resolution of singularities. This point of view has also proved important
in related work about modules of finite length and finite projective dimension
over 2-dimensional local rings, and in the study of NK0 and K-\ for surfaces.
Finally, the relationship of H2(X, Ox) with 0-cycles is directly seen in terms of
deformations of the functor if2-
However, as far as the infinite-dimensionality theorem for normal projective
surfaces over C is concerned, the author recently realized the following. Let X
be a normal projective surface over C, U C X the set of smooth points. If u/^
1980 Mathematics Subject Classification (1985 Revision). Primary 14C25; Secondary 13C10,
14E15.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
475

476
V. SRINIVAS
is the dualizing sheaf for X (X is Cohen-Macaulay), then by duality,
dime #2(*,0x) = dimcr(A>£) - dimcr(£/,fi^)
since oJxlu — &u an(^ wx nas depth 2 everywhere (by local duality, for example).
If we now carefully reformulate Mumford's proof, replacing X by U everywhere
and making a few other modifications, then the infinite-dimensionality theorem
follows, without the use of higher if-theory. Similarly, Roitman's results on
the growth of the Chow group, suitably reformulated, remain valid for normal
projective varieties, provided we define the Chow group suitably and replace
X by U in various constructions. In particular, we have (see §1 for precise
definitions)
THEOREM. Let X/C be a normal projective variety, U c X the set of
smooth points. Suppose T(U, fi^) ^ 0 for some q > 2. Then Ao(X) is infinite-
dimensional. Further, if Y C X is a closed subset of dimension < q — 1 such
that Y c U, then A0(X - Y) £ 0.
COROLLARY 1. Let X/C be a normal projective surface with H2(X, Ox) i=- 0.
Then Aq(X) is infinite-dimensional.
We easily obtain the generalization of our result on cones over curves to the
case of cones over arbitrary smooth varieties.
COROLLARY 2. Let Z c P£ be smooth of dimension d > I, and let
Ac@H°(Z,Oz(m))
m>0
be its homogeneous coordinate ring. Suppose H°(Z,ujz{—1)) ¥" 0 (equivalently
Hd(Z,Qz{l)) ^ 0). Then F0K0(A) ^ 0. In particular there exist nontrivial
projective A-modules of rank (d + 1) which do not have unimodular elements.
(Here Foifo(^) is the subgroup of the Grothendieck group of projective modules
generated by the classes of smooth points of Spec A.)
The reader can consult [S2] for examples related to Corollary 1. The
interested reader can easily work out the analogue of Roitman's theorem on the
growth of the Chow group (Theorem 1 of [Rl]) along similar lines; we obtain a
more general result in [S3], where we give an application of the Theorem to the
construction of indecomposable projective modules on affine 3-folds.
1. Definitions and preliminaries. For any normal quasiprojective variety
X over an algebraically closed field fc, let U C X be the set of smooth points,
and define the Chow group of 0-cycles
( . _ Free abelian group on points of U
Rational equivalence
where the group of cycles rationally equivalent to 0 is defined to be the group
generated by cycles of the form (/)c> where C C X is a curve such that
-^sing — 0 and / E fc(C)*. This is the "correct" Chow group (at least

RATIONAL EQUIVALENCE OF 0-CYCLES
477
up to torsion), for the following reasons. If X has a unique singular point, then
CH0(X) = i7n(X,3Cn,x), n = dimX, where %n,x is the Zariski sheaf associated
to the presheaf V h-> Kn(T(V,0^)), where Kn is Quillen's nth if-group (see
[C]). For normal X, if F0K0(X) is the subgroup of the Grothendieck group
generated by points of [/, then CR0(X) -» F0Ko(X) and the kernel is annihilated
by (n - 1)! by results of Levine [LI]. If dimX = 2, K0{X) C K0{X), the kernel
of the rank map on vector bundles, and F0K0(X) = ker(K0(X)^»P\cX), then
det
F0K0{X) is indeed the subgroup generated by points of U (and is isomorphic to
CH0(X)by [LI], or by [C]).
If X is projective, let A0(X) = ker(CH0(X) -» Z) be the subgroup of CH0(X)
generated by cycles of degree 0. If X is noncomplete, let A0(X) = CH0(X).
Now let X be a normal projective variety over a universal domain, U C X the
set of smooth points.
DEFINITION. A0{X) is finite-dimensional if the natural map
ln:Sn(U)xSn(U)^A0(X)
is surjective for some n > 0. If no such n exists, A0(X) is infinite-dimensional.
In [S2], we showed that A0(X) is finite-dimensional iff the Abel map A0(X) -»
AlbX is an isomorphism. If Y —► X is a resolution of singularities (e.g., if
k = C), then AlbX = Alb Y, so that A0{X) s A0{Y) s Alb y.
From now onwards we let fc = C and let X, [/ be as in the statement of the
Theorem. We recall a lemma of Mumford.
LEMMA 1. Let V be a nonsingular variety over C, G a finite group of
automorphisms of V, W = V/G, n: V —► W the quotient map [we assume the
quotient exists and is a normal variety). Let u E r(V, fly) be G-invariant. Then
for any morphism f: S —► W from a smooth variety S, there exists a unique
q-form r\s G T(S, Qqs) such that if S = (S xw V)Ted, p: S -+ S, /: S -+V, then
P*Vf — f*w is torsion in fi|.
S
pI
S
PROOF. See Mumford [M, §1].
LEMMA 2. Letu e r({/,n^) be nonzero. Then for each map f: Z -+ Sn(U),
where Z is a smooth variety, there exists ojf E T(Z,Qqz) such that
(i) iff: Z^Sn{U), g: Z -* Sm{U), and{f + g): Z ^ Sm+n(U) is induced
by addition ofO-cycles, then Uf +wg = Wf+g,
(ii) z/y —► Z -^ Sn(U) where Y, Z are smooth, then f*ug = wgof, and
(iii) if f: Z —► £/, Men a;/ = /*cj.
PROOF. For each n > 0 let u/n) = X>*w, where p*: £/n -+ {/ is the zth
projection. Then for G = Sn, the symmetric group acting on Un by permuting
W

478
V. SRINIVAS
the factors, u/n) is G-invariant. Hence Lemma 1 above applies, to yield a form
Uf on Z associated to a morphism /: Z —► Sn(U) = Un/G and the form u/n)
on Un. The properties (i), (ii), (iii) are clear from [M, §2].
LEMMA 3. Sn(U) x Sn(U) contains a countable union of constructible sets
ZUZ2,... such thatif(A,B)eSn{U) x Sn{U) then
oo
A-B = 0in A0{X) o (A,B) e \J Z%.
Further for each i, there is an integer m > 0, a reduced scheme Wi, and a set of
morphisms e{\ W% -+ Z%, fn W% -+ Sm(U), gi: ^xPU Sm+n{U) such that
we have equations between 0-cycles
gi{x,0) = pi(e,-(x)) + fi(x), #(x, oo) = p2(^(x)) + /»(x),
for all x e Wiy where pj : Sn{U) x Sn{U) -+ Sn{U), j = 1,2, are the projections.
PROOF. It is shown in [S2, §1] that if A, B are zero cycles on X of equal
degree then [A] - [B] = 0 in A0{X) ^3a 0-cycle C such that deg(A + C) =
deg(£ + C) = n > 0, and 3 a morphism /: P1 -+ Sn{U) such that /(0) = A + C,
/(oo) = B + C. Now consider T c Sn(X) x Sn(X) given by (A,5) <E T *>
3C e Sm(X) for some m > 0, and a morphism /: P1 -+ Sn+m(X) such that
/(0) = A + C, /(oo) = 5 + 0. (We can think of T as the graph of rational
equivalence of zero cycles in the sense of Fulton [F].) By [M, Lemma 3] there is
a countable set of varieties Zi C Sn{X) x Sn(X) such that T = |j£i Z%\ further
for each i there is an integer m > 0, a reduced scheme W^, and morphisms
^i * ** i ► Z/i,
flft-:^xP1-^Sm+n(X),
such that we obtain equations between 0-cycles
&(x,0) = Pi(e»(x)) + /i(x), &(x,oo) = P2(^(x)) + /,-(x).
(Note that though Mumford is working in a situation where X is nonsingular, the
proof of Lemma 3 cited above used only the projectivity of X.) Now Sm+n(X) —
Sm+n(£/) = Y is a closed subset of Sm+n(X). LetWt = Wl~p1{g-\Y)). Then
if Zi = ëi(Wi), and /», ^, e^ are the obvious restrictions, then W^, Zi, e^, /i, ^
have the required properties.
The next lemma (generalizing the "Theorem (essentially due to Severi)" [M])
is not needed as such in the proof of our theorem, but it will motivate our proof,
and has some independent interest.
LEMMA 4. Let f: Z -+ Sn(U) such that all the 0-cycles f(s), s e Z, are
rationally equivalent (Z is smooth). Then ujf = 0.
PROOF. Fix a base point A e f{Z). Then f(Z) x {A} c Uï î zù so that if
Ti = (Wi xeij Z)ied,

RATIONAL EQUIVALENCE OF 0-CYCLES
479
then Z is the union of the images of the Ti under the second projections. Hence
there exist a nonsingular variety T (an open set in some TJ), a dominant mor-
phism e: T -+ Z, and morphisms g: T -+ Sm{U), h: T x P1 -+ Sm+n({7) such
that we have equations between 0-cycles
/*(*, 0) - ?(*) + / o e(0, fc(«, oo) = ?(*) + A
Let /: T —► Sn(U) be the constant map with image A. Thus A| Tx{o} = ? + /oei
^1 Tx{oo} = 0 + I, where # + / o e, # + / are morphisms T —► Sm+n(£/) obtained
by addition of 0-cycles. Now Uh is a g-form on T x P1, so^ = p*ry for some
g-form 77 on T. In particular, Uh\ tx{o} = <*7i| rx{oo}> i-e-> by Lemma 2 above,
since / is a constant map. Hence ujfoe = e*^/ = 0- Since e: T —► Z is a dominant
morphism of smooth varieties over C, ujf = 0, proving the lemma.
2. Proof of the theorem. As remarked earlier, A0(X) is finite-dimensional
iff Aq(X) = AlbX. Since X is normal, we can find a smooth complete
intersection curve C C X such that C is disjoint from the singular locus. Then we have a
homomorphism J(C) —► A0(X), where J(C) is the jacobian, such that the
composite J{C) -» AlbX. Thus iîAo{X) is finite-dimensional, A0{X-C) = 0. Thus
it suffices to prove the result that if T(U, f^) ^ 0, q > 2, then A0(X - Y) ^ 0
for any (complete) algebraic set 7 C 17 of dimension < q.
Let w€T([/, fi^) be nonzero, where g > 2. If A0(X — Y) = 0, then every zero
cycle on X is rationally equivalent to a cycle supported on Y, since YnXSing = 0.
In particular if
Tn = {(x,A,B) eUx Sn{Y) x Sn+1(Y)|x + [A] - [fl] = 0 in A0(X)}
and pi : Tn —► U is the projection, then U = U^Li Pi(rn)- By Lemma 3 above,
each Tn is a countable union of constructible sets. Thus for some smooth variety
Z, there is a dominant morphism /: Z —► U, together with morphisms g: Z —►
Sn{Y), h: Z -+ Sn+1(Y) such that we have an equation
[/(*)] + W)\ ~ [H*)] = 0 inAo(X).
Thus the morphism (f + g,h):Z-+ Sn+1 (17) x Sn+1 (17) has its image contained
in U~i ^ where Z% C Sn+1(J7) x Sn+1(f7) are the constructible sets given by
Lemma 3 (applied to Sn+1 (£/)). If W^ are the associated reduced schemes, then
some reduced fiber product of W{ and Z over Sn+1(£/) x Sn+1(£7) dominates Z
under the projection. Replacing Z by an open subset of such a fiber product,
we may assume given a smooth variety Z, a dominant morphism f:Z—+U,
morphisms?: Z ^ Sn(Y), h: Z^Sn+1(Y), e: Z -* Sm{U), and k: ZxPU
Sm+n(U) such that we have equations between 0-cycles (instead of classes in
MX))
k(z,0) = f(z) + g(z) + e(z), k(z,œ) = h(z) + e(z), z € Z.

480
V. SRINIVAS
If a;/, 0Jg, oJh, ue are the associated g-forms on Z, and ujk the g-form onZxP1,
then
Uk\ Zx{0} = ^fc|zx{oo}
as g-forms on Z, i.e., by Lemma 2,
Hence Uf = Uh — wg. But we claim c^, cjg are 0, giving ojf = f*oj = 0, i.e.,
u = 0 (since /: Z —► £/ is dominant) which is a contradiction.
Thus it suffices to prove that if g: Z -+ Sn([/) factors through Sn(Y) c
Sn(U), then cjg = 0. By definition, cjg is the unique g-form on Z such that if
Z = (Z Xsn(u) Un)Ted, and we have the diagram
Z -^ Un
pi 1
Z -+ Sn{U)
9
then ijf*(u/n)) — p*(ug) is a torsion g-form on Z. In particular if Z0 c Z is a
smooth open set dominating Z, then g*(u^)\ z0 = P*(^g)| z0- But g: Z0 ^> Un
factors through Yn C Un, and u/n) restricts to a torsion g-form on Yn (since
dim y < q). By Lemma 1 of Mumford [M], any map of reduced schemes over
C takes torsion differentials to torsion differentials, i.e., g*(uj^)\ z0 = 0. Hence
P*{ug)\ z0 = 0, and since p: Z0 —► Z is dominant, uj9 = 0.
3. Projective modules on graded rings. Here we prove Corollary 2 stated
in the introduction. We recall the situation: let Z c P^ be a smooth variety,
A C 0n>o H°(Z, Gz(n)) its homogeneous coordinate ring. Then we must show
that if r(Z,wz(-l)) 7e 0, then F0K0{A) ^ 0. We first consider the Chow group
Ao(Spec A) (defined as in §1 in the normal case). If A is the normalization of A,
then Spec A —► Spec A is a homeomorphism, and an isomorphism on the sets of
smooth points. Thus by definition of the Chow group, A0(SpecA) = A0(SpecA),
and we will consider the latter group below.
Let X c Pn+1 be the projective cone over Z, and g: X —► X the
normalization. Then X has a unique singular point P E X, resolved by a single blow up
7r: Y —► X. If E = 7r_1(P) is the exceptional divisor, then there is a morphism
/: X —► Z which is a P1 bundle such that E is a section. Then
u = x - {P} s x - {<?(P)} sy-£,
so that U is a geometric line bundle over Z whose 0-section is the oo-section of
X (the complement of Spec A c X). If h : £/ —► Z, then one has exact sequences
for g > 1
o - 0 t4(-n) - /,n«, - 0 nr x(-«) - o,
n>0 n>\
by a simple computation, using the fact that the line bundle h: U —► Z is
associated to the invertible sheaf 0^(1). By a result of Bogomolov [Bo],
r(Z,n|_1(-n)) = 0 for n > 0

RATIONAL EQUIVALENCE OF 0-CYCLES
481
unless q - 1 = dim Z, i.e., fi|_1 = wz. Thus if T(Z,uz{-l)) = 0, then
r(tf,n^) = r(z,n£) foraiig.
In any case
T{U,u>u) = ®T(Z,u>z(-n)).
In particular if T(Z,ujz(—1)) ^ 0, then j40(X) is infinite-dimensional, and
A0{X - Y) ^ 0 for any proper closed subset Y C X such that P 0 Y,
i.e., Yet/. This holds, therefore, with Y equal to the 0-section of [/, i.e.,
A0 (Spec i) = A0 (Spec A)^0.
To obtain results about projective modules on A, let F0K0{A) be the
subgroup of Kq(A) generated by the classes of smooth points. Then A0(Spec A) -»
FqKq(A). By a result of Levine [L2] this map is an isomorphism. (This follows
if dim A = 2, i.e., dimZ = 1 from the Murthy-Swan cancellation theorem [MS].)
There is a simple argument due to Collino [C] that gives the result up to
torsion. Let m be the maximal ideal of the vertex, S = A — m, DJls the category of
S-torsion finitely generated A-modules. We have a localization sequence (Bass
[Ba])
KX{A) - Ki{An) £ K0{ms) - K0(A) - K0(Aa) - 0.
Now Am is local, so that K0(Am) = Z, Ki(Am) = Am, and the map d has the
explicit description d(f/g) = [A/fA] — [A/gA] for f,g G S. Further
K0(ms) = \mifesG0(A/fA),
where Go is the Grothendieck group of coherent sheaves. If Aq(A//A) is the
Chow group of Fulton [F], and F0Go(A/fA) the subgroup of Go generated by
modules of finite length, then Aq(A//A) -» F0G0{A/fA) with torsion kernel,
by the Riemann-Roch theorem [BFM]. But clearly
limfc A%(A/fA) = A0(SpecA),
by definition of the latter.
Relative if-theory as in [S2] suggests that
A)(Spec A) Ï 0 <£> r(Z, wz(-l)) Ï 0
<*> r(y, nqY) § r(u, nqv) for some q > 2.
This is consistent with Bloch's general philosophy [Bl] that differentials should
"control" the size of the Chow group of 0-cycles. In this vein, we give a simple
example of an affine cone with trivial Chow group, such that the projective cone
has infinite-dimensional Chow group.
Let Z c Pq be a smooth quartic surface. If Q E Z is a general point, 3R ^ Q
such that if L is the line in P3 through Q and i2, then L <£_ Z and (L, Z) = Q+3R
as cycles.1 If X c P4 is the projective cone, P £ X the vertex, and S a point
1Let R be a point of triple ramification of the projection Z —► P2 with center Q.

482
V. SRINIVAS
on the ruling of X through Q such that S ^ P, Q, let L' c P4 be the line
through S and R. Then 1/ £ X, and (X, Z/) = S + 3# as cycles. Thus if m c A
is the maximal ideal of S on the affine cone Spec A, then m is generated by 3
linear functions (equations for Z/fl A4), and [S] = 0 in A0(Spec A). Since Q was
any general point, this proves A0(SpecA) = 0. Since T(Z,ojz) = C, A0(X) is
infinite-dimensional.
References
[Ba] H. Bass, Algebraic K-theory, Benjamin, New York-Amsterdam, 1968.
[Bl] S. Bloch, Lectures on algebraic cycles, Duke Univ. Math. Ser. IV, Duke University,
Durham, N. C, 1980.
[Bo] F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math.
USSR-Izv. 13 (1979), 499-555.
[BFM] P. Baum, W. Fulton, and R. Macpherson, Riemann-Roch for singular varieties, Inst.
Hautes Études Sci. Publ. Math. 45 (1975), 101-146.
[C] A. Collino, Quillen's %-theory and algebraic cycles on almost non-singular varieties, Illinois
J. Math. 25 (1981), 654-666.
[F] W. Fulton, Rational equivalence on singular varieties, Inst. Hautes Etudes Sci. Publ. Math.
45 (1975), 147-167.
[LI] M. Levine, A geometric theory of the Chow ring for singular varieties, Preprint.
[L2] , Zero cycles and K-theory on singular varieties, these Proceedings, Part 2, pp.
451-462.
[M] D. Mumford, Rational equivalence ofO-cycles on algebraic surfaces, J. Math. Kyoto Univ.
9 (1969), 196-204.
[MS] M. P. Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36
(1976), 125-165.
[Rl] A. A. Roitman, Rational equivalence ofO-cycles, Math. USSR-Sb. 18 (1972), 571-588.
[R2] , r-equivalence of zero dimension cycles, Math. USSR-Sb. 15 (1971), 555-567.
[SI] V. Srinivas, Vector bundles on the cone over a curve, Compositio Math. 47 (1982),
249-269.
[S2] , Zero cycles on a singular surface. II, J. Reine Angew. Math. 362 (1985), 4-27.
[S3] , Indecomposable projective modules over affine domains, Compositio Math. 60
(1986), 115-132.
Tata Institute of Fundamental Research, India

Commutative Algebra

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Syzygies: The Codimension of Zeros
of a Nonzero Section
E. GRAHAM EVANS, JR. AND PHILLIP GRIFFITH
If M is a vector bundle of rank k on the punctured spectrum of a regular local
ring having dimension greater than k + 1 and if m is a section of M, then one
expects that the zeros of m have codimension fc. Of course, the dimension of
the subscheme of zeros could be much smaller (for example, m could fail to have
any zeros at all), or the dimension could be much larger (for example, m could
be the zero section). If M is merely a reflexive sheaf, then the situation becomes
more complicated. This article examines the subscheme of zeros of a section of
a sheaf and presents a connection between the vanishing of the cohomology of
the sheaf and the codimension of the zeros of a section.
Throughout we will assume that R is a regular local ring which contains a
field and that m is its maximal ideal. In addition we assume that M is a finitely
generated reflexive module over R. We present our arguments from the point of
view of commutative ring theory since this is the area with which we are most
familiar and since we are not aware of a proof of these results from a purely
geometric approach.
A fuller treatment of this material can be found in our book [5]. The beautiful
article [7] by Horrocks has served us well as a bridge from commutative algebra to
algebraic geometry, particularly for its treatment of cohomology. We recommend
it as a bridge from algebraic geometry back to the commutative ring theory in
this material.
Definition. Let m e M. Then 0M(m) = {/(m) | / e Hom(M,i2)} is
called the order ideal of m in M. If we view M as a sheaf on Speciî, then
m determines a homomorphism M* —► R via / —► /(m). The image of this
homomorphism is Om(j^) and Spec(i2/OA/(wi)) is the closed subscheme of zeros
of m.
We say that the i2-module M is a fcth syzygy if M fits into an exact sequence
(*) 0 -+ M -+ Rnk~i -+ ► iT1 -+ Rn° -+ N -+ 0.
1980 Mathematics Subject Classification (1985 Revision). Primary 14F05; Secondary 13C15,
13D05.
©1987 American Mathematical Society
0082-0717/87 $1.00+ $.25 per page
485

486
E. GRAHAM EVANS, JR. AND PHILLIP GRIFFITH
If M does fit into an exact sequence such as this, we may extend the
coordinate maps /i,..., /nfc_! (if necessary) to a generating set /i,..., fnk-x > • • •, fq of
Hom(M, R). We then obtain an embedding
0^M^Rq^C^0
in which the dual sequence
0 -* C* -* (Ri)* -* M* -* 0
is exact. In this situation the order ideal Om{^) will be generated by the
coordinates of the image of m. We may now repeat this process for the cokernel
C of M —► Rq, provided C is torsion free. In fact, if we begin as above with M a
fcth syzygy, we are guaranteed that this process will continue until M is realized
as a fcth syzygy in a free resolution which has an exact dual (see [5, pp. 49-63]
for more details). Thus, under the assumption that M is a fcth syzygy, we may
assume our original exact sequence (*) has the property that its dual
0 <- M* <- (iT*-1)* <-•••<- (Rny <- (iT0)* <- N* <- 0
is exact and, in particular, is the first fc terms in a free resolution of M*. One
may properly view the preceding construction as a "universal" representation of
M as a fcth syzygy. Furthermore, a closer look at this construction yields that
(i) M is a first syzygy if and only if M is torsion free,
(ii) M is a second syzygy if and only if M is reflexive, and
(iii) M is a fcth syzygy, for fc > 3, if and only if M is reflexive and Ext*(M*, R)
= 0for i= l,...,fc-2.
Assuming that the dimension of R is n + 1, the Ext*(M*, R) are dual to the
sheaf cohomology Hn~l(X, M*), where X denotes Spec R — {m}, the punctured
spectrum of i2, and M* denotes the associated sheaf of M* on X. Thus one can
rephrase the fact that M is a fcth syzygy in terms of the cohomology of M*. In
the situation where M is a vector bundle on X, one can use the duality between
Hl(X,M) and Hn~l(M*) to phrase the condition in terms of M alone.
The connection between the "syzyginess" of a module and the vanishing of
the Ext1 (M*,R) was first discussed in Auslander and Bridger's memoir [1].
We now derive the fundamental connection between fcth syzygies and order
ideals. We remark that "pd" denotes projective dimension.
THEOREM A. Let M be a reflexive R-module without free summands and
assume M is a kth syzygy. Let I ^ 0 be an ideal such that pdR/I < fc and let
me M. Then
(1) the order ideal Om{^) cannot equal I, and
(2) if me M — mM, then Om{^) is not contained in I.
PROOF. (1) Suppose that 0M{m) = 1 and let
0 -+ M -+ iT*-1 -+ ► iT1 -+ Rn° -♦ N -+ 0

SYZYGIES
487
be a "universal" resolution as described above which realizes M as a fcth syzygy.
By splicing this exact sequence together with a free resolution of M we achieve
a free resolution
(**) 0 -+ Rnd -+ ► Rnk -+ Rnk-> -+ ► Rn° -+ N -+ 0
of TV in which M sits as a fcth syzygy.
Since pd R/I < fc, we have that
Tor£(i2//,7V) = 0.
Using the resolution (**) to compute Tor we see that, if ]T) rjej in i2nfc has image
m in M, then after tensoring with R/I we obtain the result that Yl^jëj goes
to zero in Rnk~l (here the notation "x" indicates "x modulo /"). Thus J2^j^j
is the image of an element from i2nfc+1, say Yl^ifi 1S sent to Yl^j^j- Then the
image ]T) sj/i in the original resolution (**) must be ]T) rjej+w, where it; E IRnk.
Thus the image of Ylrjej + w in M is zero and hence m is the image of —w.
That is, m must be in IM. Hence we can express m as m = ^Ylbvmv with the
b„ € I and mv E M. Then for each / E Hom(M, R) we have
We remark that /(M) C m since M has no free summands. Thus
OM{rn) — I Ç m/;
but this is impossible by Nakayama's Lemma unless 7 = 0. However, this
contradicts our assumption that 7^0.
(2) We may assume that the resolution (**) is minimal. Since m E M — mM
we have that m is a part of a minimal generating set m = mi, m2,..., mnk of M.
We may also assume that this generating set was the one used for constructing
the basis of Rnk; thus we have e\ in Rnk is sent to m. As in (1) above we
can conclude that e\ + w is in the image of i?nfc+1, when w E 7i2nfc. But this
contradicts the assumption that the resolution is minimal.
These results have their clearest implication in the situation where R/I is
Cohen-Macaulay of projective dimension I < k. In this context we may restate
them to say, if Y c X is a Cohen-Macaulay subvariety of codimension /, then
no section of a fcth syzygy has zeros which are ideal-theoretically Y, and further
that no section which is a minimal generator has zeros completely inside Y.
Needless to say the Cohen-Macaulay hypothesis, while technical, cannot
simply be omitted. However one may call upon Hochster's [6] maximal Cohen-
Macaulay modules in order to obtain a substantial improvement on (2).
This theorem is found as Theorem 3.14 in our book [5]. It was proved in [2]
originally for the case that M is a vector bundle on the punctured spectrum of a
local ring. Later we used the result without the bundle assumption. The removal
of that assumption is new. Unfortunately the requirement that R contain a field
(in order to have maximal Cohen-Macaulay modules on all images of R) is still
needed.

488 E. GRAHAM EVANS, JR. AND PHILLIP GRIFFITH
THEOREM B. Let M be a kth syzygy without free summands and let m E
M — mM. Then the height ofOm{™) must be at least k.
PROOF. Our starting point is now standard. Let
0 -+ M -+ Rnk~i -+ ► flni -+ Rn° -+ N -+ 0
be a universal sequence which realizes M as a fcth syzygy and extend this
sequence to
(* * *) 0 -+ Rn* H Rn*-i -+ >Rnk f-X Rn*-^ -+ ► Rn° -+ N -+ 0,
a free resolution of N. We may assume the maps /<*,..., fk have their entries in
the maximal ideal m. Note that if M were to have a (nonzero) free summand,
then the map Rnk —► Rnk~1 would have some entries consisting of units.
Let m £ M — mM have Om(™) = I and suppose that the height of / is less
than k. Further we assume that the map Rnk —► M C Rnk~l carries e\ to m.
By [5, Theorem 1.11], there is a Cohen-Macaulay i2-module C over R/I so
that pdC = height I < k. Then as before, we have that Tor£(C,7V) = 0.
However, after tensoring (* * *) with C we obtain the complex
0 —► cnd —►•••—► Cnk —► Cnk~l —►-.•—► Cn°
which must be exact at Cnk. But elements of the form e\ <8> c are sent to
ICnk~l = 0; thus the e\ <g> c must be in the image of Cnk+1. However, this
implies that C = mC since the entries of fk are in m. Hence this contradicts the
fact that C is a Cohen-Macaulay module. We remark that this corrects a flaw
in [4] which relied on [2, Proposition 1.6]. The latter reference is a version of
Theorem B which was verified only in the case of vector bundles on the punctured
spectrum.
Let M be a reflexive module of rank r. We would expect that a section of M
corresponding to a minimal generator should have zeros of codimension r. Alas,
some such sections have their zeros of higher codimension. However, if R/m is
algebraically closed and if M is not free, then some m E M — mM will have its
zeros of codimension at most r. This fact is proven in [3]. We require R/m to
be algebraically closed in order to ensure that some equations have solutions.
These equations are nearly linear in the sense that we have equations /» = 0
in i?[xi,... ,xn] all of degree one in the Xj. But then we must include other
equations by taking a prime ideal minimal over the ideal generated by the fa and
in the end "solve" them all. We suspect that requiring R/m to be algebraically
closed is "overkill" and asking that R/m be infinite should suffice.
Regardless of certain technicalities involving i2/m, the preceding discussion
provides a clean argument of the fact: if R is a regular local ring containing a
field and if M is a nonfree fcth syzygy, then M must have rank at least fc. This
theorem is the main result in [2].

SYZYGIES
489
References
1. M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94
(1969).
2. E. G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), 323-
333.
3. , Order ideals of minimal generators, Proc. Amer. Math. Soc. 86 (1982), 375-378.
4. , The syzygy problem: a new proof and historical perspective, Commutative Algebra:
Durham 1981 (Durham, 1981), London Math. Soc. Lecture Note Ser., no. 72, Cambridge
University Press, Cambridge, 1982, pp. 2-11.
5. , Syzygies, London Math Soc. Lecture Note Ser., no. 106, Cambridge University
Press, Cambridge, 1985.
6. M. Hochster, Topics in the homological theory of modules over commutative rings, CBMS
Regional Conf. Ser. in Math., no. 24, Amer. Math. Soc, Providence, R. I., 1976.
7. G. Horrocks, Vector bundles on the punctured spectrum of a regular local ring, Proc.
London Math. Soc. (3) 14 (1964), 689-713.
University of Illinois, Urbana

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Intersection Problems and Cohen-Macaulay Modules
MELVIN HOCHSTER
The object of this note is to describe the current state of knowledge of several
problems in local algebra on which there has been recent progress, and a couple
on which there has not, but which should be reexamined in the light of that
progress.
1. Multiplicities. One of these problems is Serre's conjecture on intersection
multiplicities and its generalizations. We shall henceforth assume that all rings
are commutative, with identity, that modules are unital, and that local rings
are Noetherian. Let M, N be finitely generated modules over a local ring R
and suppose that pd^M = n < oo and that l(M ®r N) < oo, where "/"
denotes length. In this situation the intersection multiplicity in the sense of
Serre, \r{M,N) (or, simply, x(A^AO), is defined to be
i=0
In [S] Serre proved that if (i2,m) is a regular local ring (in which case, of
course, pd^M < dimR automatically), then if M, N are finitely generated
modules such that l(M <g> N) < oo (all tensor products, Tor's, etc. are assumed
to be taken over R unless otherwise specified) then
(0) dim M + dim TV < dimR (where dimQ = dim(i2/AnnQ)) and, provided
that the completion R of R is isomorphic to a formal power series ring over
a discrete valuation ring or field (e.g., if R is equicharacteristic or of mixed
characteristic p but unramified in the sense that p^m2), Serre also proved that:
(1) If dim M + dim N < dim R, then x(M, N) = 0 and
(2) If dim M -h dim N = dim R, then x(M, N) > 0.
Serre said: "Il est naturel de conjecturer que le théorème 1 [théorème 1 is
the result described by (0), (1), and (2) above] est vrai pour tous les anneaux
réguliers—" [S, p. V-14]. In fact, until recently, it seemed possible that the
conjecture might be valid for an arbitrary pair of finitely generated modules
1980 Mathematics Subject Classification (1985 Revision). Primary 13E05, 13C13, 13D05.
The author was supported in part by a grant from the National Science Foundation.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
491

492
MELVIN HOCHSTER
M, N simply under the assumption that one of the modules, say M, have finite
projective dimension.
Recently, Paul Roberts (see [R3, R4, R5]) and, independently, H. Gillet and
C. Soulé (see [GS]) have established that the vanishing part of Serre's conjecture,
that is, the statement in (1) above, is true for arbitrary regular local rings R.
(The argument given by Roberts depends on a theory of local Chern classes and
a local Riemann-Roch theorem: the theory given, for example, in [F] suffices.)
The arguments actually work for complexes but we shall stick to the module case
here. Moreover, it is not necessary that R be regular. The arguments of either
Roberts or Gillet-Soulé show that if R is a local complete intersection, then
(1°) Let M, N be finitely generated i2-modules such that dim M + dim TV <
dimiî, and such that both M and N have finite projective dimension over R.
Suppose also that l(M <g> N) < 00. Then x(M, N) = 0.
Roberts has also shown that (1°) holds for a local ring R of finite type over a
regular local ring and such that the singular locus is of dimension at most one.
While there was some evidence that the conclusion of (1°) might hold under
the weaker hypothesis that just one of the modules M, N, say M, have finite
projective dimension (for example, the graded case is done in [PS2], where the
strategy that led to the later proofs was suggested, and more evidence may be
found in [D1-D5]), a counterexample has recently been obtained in [DHM]. In
this counterexample the local ring R is
K[XuX2,X3,X4]m/{X1X2-X3X4), where m - (Xt).
It is shown that i2, which is three-dimensional, possesses a module M of length
15 such that pd^M = 3 (dim M = 0), and yet if P denotes the prime ideal
(xi,X2)i2 [xi denotes the image of X{ in i2), then x{M,R/P) = -1. Since
dim M + dim(i?/P) = 0 + 2 = 2<3 = dimiî, the stronger conjecture is not
true, even for hypersurfaces with an isolated singularity.
Many questions remain. The problem of establishing the positivity part of
Serre's conjecture, statement (2), is still open. It is possible, so far as I know,
that the conclusion of (1°) is valid over every local ring i2, provided that both M
and N have finite projective dimension, and it is also possible that (2) holds over
every local ring i2, provided that both M and N have finite projective dimension,
even though it has not been proved even for regular R.
2. Cohen-Macaulay modules. We shall comment briefly on one idea for
settling the positivity conjecture (2) in the regular case because it shows the
connection between Serre's conjecture on multiplicities and the existence
question for Cohen-Macaulay modules, which is one of our central concerns. The
point is that the proofs of the vanishing part of Serre's conjecture automatically
focus attention on the existence of small Cohen-Macaulay modules, since the
existence of small Cohen-Macaulay modules implies the positivity conjecture in
the regular case.

INTERSECTION PROBLEMS AND COHEN-MACAULAY MODULES 493
Let (i2,m) be a local ring. We recall that an i2-module M (not necessarily
finitely generated) is called a big Cohen-Macaulay module if there is a system of
parameters x = x\,..., xn for R such that
(i) (x)M ^ M (where (x) is the ideal generated by the X{) and
(ii) for 1 < i < n, X{ is not a zerodivisor on M/(xi,..., Xi-\)M.
If M is finitely generated and satisfies these conditions, then it is simply
called a Cohen-Macaulay module (the adjective small is sometimes included for
emphasis). Condition (i) is equivalent to the condition
(i°) mM^M
since m is nilpotent modulo (x), and so if M is finitely generated it is simply
equivalent to the condition that M ^ 0. (In the literature a finitely generated
module M over R is sometimes called Cohen-Macaulay if it is Cohen-Macaulay
in our sense over R/AimM. The reader should note that we are requiring here
that depth M = dimR rather than depth M = dim M.)
The main result of [H3] asserts:
THEOREM. Let (i2,m) be a local ring such that i2red contains a field. Then
R possesses a big Cohen-Macaulay module.
This is also known, and, in fact, rather trivial, if dimR < 2 (take the module
M to be the integral closure of fi/p, where p is a minimal prime of R such that
dimfi/p = dimiî). However, the general case (i.e., the mixed characteristic
case) remains open, even in dimension three. It is worth noting that it suffices
to consider the case of complete normal local domains. (Oddly, the only known
proof of the existence of big Cohen-Macaulay modules in the equicharacteristic
case depends on first using Artin approximation to reduce to the case of local
rings essentially of finite type over a field, then reducing to characteristic p > 0,
and finally making use of the Frobenius endomorphism. See [H3].) Small Cohen-
Macaulay modules are known not to exist for "pathological" local rings. (If M is
a finitely generated Cohen-Macaulay module over i2, R' — R/AimM is catenary
and the Cohen-Macaulay locus {P E Spec(i2/): R'P is Cohen-Macaulay} is open.)
On the other hand, the question of whether small Cohen-Macaulay modules exist
for "reasonably well-behaved" local rings, e.g., excellent local rings, is completely
open in dimension three and higher. It is not known in the complete case nor
for local rings essentially of finite type over a field. (In dimension two or less one
may, of course, kill a suitable minimal prime and take the integral closure.)
Suppose that a local ring R is a finite module over a regular local ring A. Then
there is a very simple way to explain what it means to give a finitely generated
Cohen-Macaulay module over R: it is precisely a nonzero finitely generated R-
module which is free as an A-module. In the case where R is a domain, giving
such a module is then equivalent to embedding R as a subring of size r square
matrices over A for a suitable positive integer r so as to extend the embedding
of A as the scalar matrices.
While the analogous statements are not true for big C-M modules, P. Griffith
[Gr] has shown that if R is a complete local domain module finite over a regular

494
MELVIN HOCHSTER
local ring A and if R possesses a big Cohen-Macaulay module, then there exists
a big Cohen-Macaulay module which is a nonzero A-free (not necessarily finitely
generated) A-module. In particular, this is indeed the situation for equichar-
acteristic complete local domains, for in this case big Cohen-Macaulay modules
are known to exist.
We shall discuss a couple of positive results about existence of small Cohen-
Macaulay modules below. Before doing this, however, we want to indicate the
connection with the multiplicities conjecture.
PROPOSITION. Let (i2,m) be a regular local ring and suppose that for every
prime ideal P of R (or of R) that the domain R/P (respectively, the domain
R/P) has a small Cohen-Macaulay module. Then Serre's conjecture on
multiplicities holds for R.
Since the conjecture for R implies the conjecture for i2, the parenthetical
statement follows once we have proved the result for R itself. The argument is
given, for example, in [H2], but since it is very brief we repeat it here. Because
of the recent results of Roberts and Gillet-Soulé it remains only to prove the
"positivity" part of the conjecture. Because x is biadditive (see [S]) we can
reduce to the case where M = R/P = S and N = R/Q = T are prime cyclic
modules (and, hence, also local rings in their own right), such that dim S +
dimT = dimR. Let M', N' be small Cohen-Macaulay modules over S, T
respectively. M' will be torsion-free over S of some rank s > 0 and will have
a prime filtration in which s of the factors are copies of R/P = S and the
other factors have the form R/P1 for some prime P' strictly larger than P. A
similar statement holds for N': let t > 0 be its torsion-free rank over T. The
biadditivity of x then implies that xr(M,N) is the sum of st terms each of
which is Xr(S,T) and other terms of the form xr(R/P', S/Q'), where P Ç P',
Q ÇQ', and at least one of the two inclusions is strict. Hence, these other terms
all vanish, for in every case we shall have dim R/P' + dim R/Q' < dim R, and we
conclude that x(M', N') = stx(S, T). Thus, to prove that x(S, T) > 0 it suffices
to prove that x(M',N') > 0. But it follows from [S, Théorème 4, p. V-19] that
Tori(M',N') = 0 for i > 1 and hence that x(M', N') = l(M'®N') > 0. Q.E.D.
Unfortunately, while big C-M (we henceforth use this abbreviation for "Cohen-
Macaulay") modules are known to exist at least in the equicharacteristic case,
very little is known about the existence of small Cohen-Macaulay modules. It
is shown in [M] that the homogeneous coordinate ring of a finite product of
smooth curves has a small C-M module. The situation is somewhat better in
characteristic p > 0, for then it is known that the homogeneous coordinate ring of
any smooth variety has a small C-M module. More precisely and more generally:
PROPOSITION. Let R be a finitely generated nonnegatively graded algebra,
where Rq is a perfect field K. Assume that R is unmixed in the sense that, for
all P E Assiî, dim(i2/P) = dimiZ. Let M be a finitely generated graded module
such that all of the primes in Ass M are minimal primes of R, and suppose that

INTERSECTION PROBLEMS AND COHEN-MACAULAY MODULES 495
Mp is C-M for all prime ideals P of R except possibly when P is the irrelevant
ideal m = ]Ci>o^* Then R has a graded small C-M module D {i.e., DP is
C-M for all primes P of R, including P = ra).
This result was discovered by Hartshorne and Peskine-Szpiro and then
rediscovered by the present writer several years later. See [H4].
The argument is extremely simple. Let d = dimiî. The hypotheses imply
that H(M) = @i<dHlm(M) (local cohomology with support in ra) has finite
dimension as a vector space over the perfect field K. Call this dimension N. View
R as a module over itself via the eth iteration of the Frobenius endomorphism
and then give M a new i2-module structure by restriction of scalars. Call this
module eM. Then H^M) = e/f^(M), whence H(eM) has dimension N as a
vector space over K for all e. But eM splits into a great many direct summands
when e is sufficiently large. In fact, for each residue j modulo q — pe, we may
let
i=j mod q
where the sum may be thought of as a direct sum of vector spaces over K.
Obviously, M — @- Dj as a if-vector space. While the Mj are not i2-submodules
of M, they are JR-submodules of eM. Thus, eM splits into an arbitrarily large
number of nonzero summands for large e, as claimed. As soon as the number
of summands exceeds TV, one of them, call it D, must satisfy H (D) = 0:
otherwise, H(eM) will have dimension greater than N. But the vanishing of H(D) is
precisely the condition needed for D to be C-M.
This argument does not appear to be helpful in the nongraded case nor does
it seem that the graded equicharacteristic 0 case can be deduced from the
characteristic p case. The problem is that one cannot control the "size" (e.g., the
number of generators or the rank) of the small C-M modules being constructed
by this method without referring to p. If one knew not only that C-M modules
exist in positive characteristic but that their sizes could be bounded by
numerical data that are constant on the closed fibers of "sufficiently flat" families of
algebras (where the base is a finitely generated algebra over the integers), one
could deduce the equicharacteristic 0 case from the positive characteristic case.
3. The direct summand and canonical element conjectures. While
the existence of big C-M modules has remained intractable in mixed
characteristic and the existence of small C-M modules has resisted proof even in dimension
3 in the equicharacteristic case, it appears that there is hope of settling a weaker
form of these conjectures. In order to explain why it is worthwhile to have the
weaker form, we shall first give a number of statements. Each of these statements
has the property that it is implied by the existence of big C-M (and, hence, of
small C-M) modules. Each of these statements is a theorem for rings containing
a field. Each of these statements is a conjecture in the mixed characteristic case.
We shall refer to each of them as a "conjecture."

496
MELVIN HOCHSTER
(i) DIRECT Summand CONJECTURE. If RÇ S is an extension of
Noetherian rings and R is regular, then R is a direct summand of S as an R-module.
(ii) Peskine-SzpiRO-Roberts New Intersection Conjecture. If a
finite complex of finitely generated free modules over a local ring R has nonzero
finite length homology, then the length of the complex is at least the dimension
ofR.
(iii) Evans-Griffith Syzygy Conjecture. A finitely generated kth
module of syzygies of a finitely generated module over a regular local ring which
is not free has torsion-free rank at least k.
(iv) Peskine-Szpiro Intersection Conjecture, first form. If M,
N are finitely generated nonzero modules over a local ring R such that l(M®N)
is finite, then dim N < pdR M.
(v) Intersection Conjecture, second form. If R ^ S is a homo-
morphism of Noetherian rings, M is any R-module, I = Ann#M, and Q is a
minimal prime of IS, then height Q < pdRM.
(Note that this statement is not interesting unless IS is a proper ideal of S
and pdfl M is finite.)
(vi) BASS'S QUESTION. // a local ring R possesses a finitely generated
nonzero module of finite injective dimension, then R is Cohen-Macaulay.
(The converse is true. See [B].)
(vii) M. AUSLANDER'S ZERODIVISOR CONJECTURE. Let M be a nonzero
finitely generated module over a Noetherian ring R and let x be an element of R
which is not a zerodivisor on M. Suppose that pd# M is finite. Then x is not a
zerodivisor on R.
These conjectures are related as follows: the existence of small C-M modules
in the complete case => the existence of big C-M modules => [(i) and (ii) and
(iii)] while (ii) => (v) => (iv) => [(vi) and (vii)].
The questions raised in (vi) and (vii) arose in [B] and in [Ai,Aa],
respectively. A tremendous breakthrough was made in [PSi] where it was shown
that (iv) implies both (vi) and (vii) and where (iv) was proved in characteristic
p (by clever use of the Frobenius endomorphism) and in the most important
equal characteristic 0 cases as well (using Artin approximation and reduction
to characteristic p). The new intersection theorem was proved independently in
characteristic p (and, hence, in many cases in equal characteristic 0) in [PS3]
and [Ri]. (Later, Roberts gave an analytic proof of the new intersection
theorem using the Grauert-Riemenschneider vanishing theorem: see [R3].) That the
existence of big C-M modules implies (v) and that (v) implies (iv) were noted
in [Ha]. That the existence of big C-M modules => (ii) is easy, as is (ii) => (v).
The idea used to prove the existence of big C-M modules in [H3] can also be

INTERSECTION PROBLEMS AND COHEN-MACAULAY MODULES 497
used to prove directly that the equal characteristic 0 cases of (iv), (v), (vi), and
(vii) can be deduced from the positive characteristic p case.
(i) was first proved in positive characteristic in [Hi]—the equal characteristic
0 case is trivial. (The result may be proved when R is a normal ring containing
the rationals by first killing a minimal prime P of S disjoint from R — {0} and
then using (l/d)Tr to retract S/P to i2, where Tr = Tr^/x is the field trace from
the fraction field L of S/P to the fraction field K of R and d = [L : K].) That
the existence of big C-M modules => (i) is obtained in [Hi] (see also [H3,H6]).
It is quite easy to reduce to the case where R is a complete local regular ring
and S is a domain module-finite over R. If one has a small C-M module for S,
then, as noted earlier, one has an embedding of S in a ring of square matrices
over i2, and the required retraction can then be obtained by projection on a fixed
diagonal entry: e.g., map each matrix to its 1,1 entry. If one has only a big C-M
module one may use the representation theorem of [Gr] to give an analogous
argument or one may proceed a bit differently as follows: let x\,..., xn be a
regular s.o.p. (system of parameters) for R\ i.e., the x's are a minimal set of
generators of the maximal ideal of R. Then, as observed in [Hi], the inclusion
map of R into S splits if and only if for all integers t > 0 the equation
(*) *i-"4 = Êvi*«+1
has no solution in S. It is not hard to see that if a local ring S with s.o.p.
xi,..., xn has a big C-M module then (*) has no solution. In fact, we have
(i°) MONOMIAL CONJECTURE. // x\,..., xn is a system of parameters for
a local ring S, then for all integers t > 0 (*) has no solution in S.
It is possible to show that (i) and (i°) are equivalent. Thus, (i°) is known for
local rings containing a field but is not known, even in dimension 3, in mixed
characteristic.
That the existence of big C-M modules => (iii) was proved in [EGi]. See
[EGa] for a more detailed exposition which contains more refined results.
To explain why (v) is referred to as an "intersection" conjecture, we examine
the case where R = Z[x\,... ,xn] is a polynomial ring and M = i2/7, where
/ is the ideal generated by the X{. To give a homomorphism R —► S is the
same as to specify values for the Xi, call them s^, in S. Then (v) becomes
the assertion that a minimal prime of an ideal J of a Noetherian ring S such
that J has at most n generators has height at most n (for n = pd^M). This
is, of course, the Krull height theorem, which controls the codimension of an
intersection with a subscheme defined locally by at most n equations and so is,
indeed, an intersection theorem. The author contends that all of the conjectures
discussed in this section are intersection theorems in that they are equivalent
to statements about the behavior of the codimension of an intersection under
certain conditions. We shall return to this point later.

498
MELVIN HOCHSTER
Of the conjectures discussed above, it is perhaps the direct summand
conjecture (i) which seems most accessible and, perhaps, least surprising. It is therefore
of some interest that in the only case where these conjectures are open, i.e., mixed
characteristic, conjecture (i) implies all the others. Thus, the direct summand
conjecture seems to have more or less the same "observable" consequences as the
existence of big C-M modules. This is one of the main results of [He]. However,
so far as I know, it does not imply the existence of big C-M modules.
The reader should note that while the existence of small C-M modules (as
was shown earlier) suffices to complete the proof of Serre's conjecture on
multiplicities in the regular case, it has not been possible so far to give an argument
using instead either the existence of big C-M modules or the direct summand
conjecture. Nonetheless, part of our interest in these latter two conjectures stems
from the possibility of making such a substitution.
We next want to discuss two more recent "conjectures." The first is the
canonical element conjecture. Let (i2, m, K) be a local ring of dimension n. A
truncated resolution of K by finitely generated free i2-modules, say
0 - syznK - Fn_i ► Fi - F0 - K - 0,
represents an element in Extn(if, syznK). One way of defining the local coho-
mology Hlm(M) of a module M is
HUM)=\\mtExtiR(R/m\M),
and so there is a natural map Om- Ext*(if,M) —► Hlm(M). Applying this
when M — syznK and i = n we obtain an element tjr = 0m(^) which is
called the canonical element for R. Although tjr depends on various choices, the
different ry^'s can be identified. In particular, the question of whether tjr = 0 is
independent of these choices. Then we have:
(viii) Canonical Element Conjecture. For every local ring R, r)R is
not 0.
The discussion which follows summarizes some of the results of [He]. It turns
out that t]r is not 0 if and only if for some i2-module M, Qm*> Ext^(if, M) —►
H^(R) is not zero, where n = dim R. If R has a big C-M module, then t]r is not
0. The issue is the same for R as it is for R. If R has a canonical module fi, then
rjn is not 0 if and only if 6q is not 0. Moreover, it is shown in [H6] that tjr is not
0 if and only if for every system of parameters x\,..., xn of R and every map
<p* of the Koszul complex i£*(xi,..., xn\ R) to a free resolution F* of K such
that the map of augmentations is the natural surjection i2/(xi,..., xn) —► if,
the map <pn is not 0. It is in this form that the fact rjn is not 0 is used in [He]
to deduce the other conjectures (i)-(vii).
To summarize: (viii) follows from the existence of big C-M modules, is true in

INTERSECTION PROBLEMS AND COHEN-MACAULAY MODULES 499
the equicharacteristic case, and implies (i)-(vii). Another key point (see [He])
is:
THEOREM. The fact that t}r is not 0 for a given local ring R follows from
an infinite family of cases of the direct summand conjecture.
Thus, the direct summand conjecture for all local rings is equivalent to the
canonical element conjecture. Some progress has been made on the canonical
element conjecture for special classes of local rings: see, for example, [D6] and
[HuK]. A somewhat different point of view introduced by P. Roberts in an
unpublished manuscript (but see [HuK]) concerning the canonical element has
been useful: we viewed
r,R € H%(SyznK) S syznK ® H£(R),
and from this one can see that tjr represents an element of Torn(if, H^R)).
(This was pointed out to the author by J. Lipman.) The canonical element
conjecture is also equivalent to the assertion that this element is not 0. Now,
H^R) = Hm R/(x\,... ,x£J, where the X{ are a system of parameters. From
this one can see that tjr is not 0 if and only if for every s.o.p. y = y\,..., yn
and map ip+: if*(y; R) —► G*, where G* is a free resolution of R/(y) by finitely
generated free modules, the element of Torn(if, K) represented by ^n(l) ® 1 in
Gn®K (where we have written 1 for the generator of the nth term of the Koszul
complex) is not 0.
We also note that there is a strengthening of the new intersection conjecture
which is (by a recent result: see [De]) equivalent to the canonical element
conjecture. We first observe that it is not difficult to show that (ii) is equivalent to
the same assertion with the apparently stronger hypothesis that H0{F*) is not
0, where F* is the free complex under discussion. One then has:
(ii°) Improved New Intersection Conjecture. // a finite complex
F* of finitely generated free modules over a local ring R has the properties that
H0(F*) has a minimal generator which is killed by a power of m and that Hi(F*)
has finite length for i > 1, then the length of F* is at least the dimension of R.
The various proofs of the new intersection conjecture in the cases where it is
known ( [Ri, PS2, R2] and via big C-M modules) all work just as well to show the
improved version. It is observed in [He] that the canonical element conjecture
implies (ii°) and that the argument of Evans and Griffith actually shows that
(ii°) implies (iii). As mentioned above, it is shown in [De] that (ii°) implies the
canonical element conjecture as well.
One point worth emphasizing is that the direct summand conjecture does
imply all the others given with Roman numerals. We therefore conclude with a
few more remarks about it. One is that, by an argument given in [He], it suffices
to prove the case where the regular ring is a formal power series ring over an
unramified, discrete valuation ring.

500
MELVIN HOCHSTER
A second point we want to mention is that there is a stronger conjecture, due
to Koh [K], which moves in a different direction. This conjecture asserts that if
S is a module-finite extension of a Noetherian ring R and pd^ S is finite, then
R is a direct summand of S. This is true (as was pointed out to the author by
J. Lipman) if R contains the rationals, by a trace argument. It is not known,
however, in characteristic p nor in mixed characteristic, even if R is a local ring
of a plane curve (at a singular point).
Finally, we want to justify the assertion, made earlier, that even the direct
summand conjecture is an intersection theorem. For simplicity we work in the
equicharacteristic case over an algebraically closed field if, and limit the
discussion to finitely generated if-algebras. (One can, in any case, reduce to this
situation using Artin approximation.) We work with the monomial conjecture,
which is equivalent: of course, both are theorems in the equicharacteristic case.
The "conjecture" then asserts that if the equation x\ ■ ■ -x^ = Y%=i Vi^1 nas
a solution in a finitely generated if-algebra i2, then the ideal generated by
the x's has height at most n — 1. (If it has height n, localizing at a
minimal prime will produce a local ring of dimension n in which the x's are an
s.o.p.) But one can recast this statement as follows: If X = Spec(T/(F)), where
T = if[xi,...,xn,2/i,...,2/n, zi,...,zr) and F denotes x\ ■ ■ -x^ - YTi=\ 2/izJ+1,
and V is the subvariety of X defined by the vanishing of the £;, then an n-
dimensional subvariety W of X cannot have the origin as an isolated point of its
intersection with V. This viewpoint is discussed in detail in [H5].
Added in proof. Paul Roberts (Le théorème d'intersection, preprint) has
recently established (ii) in general, and (iv)-(vii) then follow, (i), (ii°), and (iii)
remain open questions.
References
[Ai] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961),
6631-6645.
[A2] , Modules over unramified regular local rings, Proc. Internat. Cong. Math.
(Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 230-233.
[B] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28.
[Di] S. P. Dutta, Generalized intersection multiplicities of modules, Trans. Amer. Math. Soc.
276 (1983), 657-669.
[D2] , Weak linking and multiplicities, J. Pure Appl. Algebra 27 (1983), 111-130.
[D3] , Symbolic powers, intersection multiplicity, and asymptotic behavior of Tor, J.
London Math. Soc. (2) 28 (1983), 261-281.
[D4] , Frobenius and multiplicities, J. Algebra 85 (1983), 424-448.
[D5] , Generalized intersection multiplicities of modules. II, Proc. Amer. Math. Soc. 93
(1985), 203-204.
[De] , On the canonical element conjecture, Preprint, University of Pennsylvania, 1985.
[DHM] S. P. Dutta, M. Hochster, and J. E. McLaughlin, Modules of finite projective
dimension with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291.
[EGi] E. G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981),
323-353.
[EG2] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Series 106,
Cambridge Univ. Press, Cambridge, 1985.

INTERSECTION PROBLEMS AND COHEN-MACAULAY MODULES 501
[F] W. Fulton, Intersection theory, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo,
1984.
[Gr] P. Griffith, A representation theorem for complete local rings, J. Pure Appl. Algebra 7
(1976), 303-315.
[GS] H. Gillet and C. Soulé, K-théorie et nullité des multiplicités d'intersection, C. R. Acad.
Sci. Paris. Ser. I Math. 300 (1985), 71-74.
[Hi] M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math.
J. 51 (1973), 25-43.
[H2] , Cohen-Macaulay modules, Conference on Commutative Algebra, Lecture
Notes in Math., vol. 311, Springer-Verlag, Berlin/Heidelberg/New York, 1973, 120-152.
[H3] , Topics in the homological theory of modules over commutative rings,
CBMS Regional Conf. Ser. in Math., no. 24, Amer. Math. Soc, Providence, R.I., 1975.
[H4] , Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors,
Queen's Papers in Pure and Appl. Math. 42 (1975), 106-195.
[H5] , The dimension of an intersection in an ambient hypersurface, Proceedings of the
First Midwest Algebraic Geometry Seminar, Lecture Notes in Math., vol. 862, Springer-Verlag,
Berlin/Heidelberg/New York, 1981, pp. 93-106.
[He] , Canonical elements in local cohomology modules and the direct summand conjecture,
J. Algebra 84 (1983), 503-553.
[HuK] C. Huneke and J. H. Koh, Some dimension 3 cases of the canonical element conjecture,
Proc. Amer. Math. Soc. (to appear).
[K] J. H. Koh, The direct summand conjecture and behavior of codimension in graded extensions,
Thesis, University of Michigan, Ann Arbor, 1983.
[M] F. Ma, Splitting in module finite extensions and Cohen-Macaulay modules and algebras,
Thesis, University of Michigan, Ann Arbor, 1983.
[PSi] C. Peskine and L. Szpiro, Dimension projective finie et cohomology locale, Inst. Hautes
Études Sci. Publ. Math. 42 (1973), 323-395.
[PS2] , Syzgies et multiplicités, C. R. Acad. Sci. Paris Ser. A-B 278 (1974), 1421-1424.
[Ri] P. Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. Ecole
Norm. Sup. (4) 9 (1976), 103-106.
[R2] , Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture,
J. Algebra 66 (1980), 220-225.
[R3] , The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math.
Soc. (N.S.) 13 (1985), 127-130.
[R4] , Local Chern characters and intersection multiplicities, these Proceedings, Part 2,
pp. 389-400.
[R5] , The MacRae invariant and the first local Chern character, Preprint, University of
Utah, 1985.
[S] J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag,
Berlin/Heidelberg/New York, 1965.
University of Michigan

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Proceedings of Symposia in Pure Mathematics
Volume 46 (1987)
Decompositions of Torsionfree Modules
over Affine Curves
ROGER WIEGAND AND SYLVIA WIEGAND
The two main problems in the study of direct-sum decompositions are
existence and uniqueness. The first problem is to decompose objects into direct
sums of simpler objects, and the second is to make sense out of existing direct-
sum decompositions. Most of our work in this paper deals with the uniqueness
question. The strongest possible uniqueness result is the Krull-Schmidt theorem,
which holds for coherent sheaves over any projective variety. (See [17, p. 86].)
Krull-Schmidt fails for sheaves over affine varieties; even the weaker uniqueness
property, direct-sum cancellation, usually fails.
Our main results, in §2 and §3, give a precise picture of the obstruction to
direct-sum cancellation for torsionfree coherent sheaves on a reduced affine curve.
As an application we show that over an irreducible curve in characteristic zero,
stably isomorphic torsionfree (finitely generated) modules are actually
isomorphic. Our primary tool in this study is the conductor square (or pullback) for
the coordinate ring R:
R ~ R
i in
R/ç ~ R/c
where R is the integral closure of R in its classical quotient ring and ç is the
conductor ideal of R in R. Most of the information we need is contained in the
bottom line of the pullback, an inclusion of Artinian rings. This phenomenon
is explained in the appendix, where we show that curves whose pullbacks have
isomorphic bottom lines have analytically isomorphic singularities. In §4 we
discuss the existence problem and its relationship to the multiplicity^ the curve.
1. Notation and conventions. Throughout this paper R denotes a one-
dimensional reduced Noetherian ring with module-finite normalization R. If
1980 Mathematics Subject Classification (1985 Revision). Primary 13C05, 14H20; Secondary
13E15.
Both authors thank the National Science Foundation for partial support for this research.
©1987 American Mathematical Society
0082-0717/87 $1.00 + $.25 per page
503

504
ROGER WIEGAND AND SYLVIA WIEGAND
Pi,..., Pt are the distinct minimal prime ideals of R, then R is the direct
product of the Dedekind domains (R/Pi)~. All modules are assumed to be finitely
generated. If rM is torsionfree (meaning M can be embedded in a free module),
the projective ^-module (R <8>r M)/torsion is denoted by RM. The rank of a
torsionfree i?-module M is the £-tuple (ri,..., rt), where r{ is the dimension of
Mpi as a vector space over Rpi. The group of units of a ring S is denoted by
Every torsionfree i2-module M arises as a pullback (or Cartesian square)
M ~ RM
(l.i) I W
M/çM i RM/cM.
Let £m be the ring of endomorphisms of the bottom line of (1.1); that is, Em
is the i?/ç-algebra of i?/ç-endomorphisms of RM/cM that carry M/cM into
itself. (Em is a module-finite algebra over the Artinian ring R/c.) Every R/c-
endomorphism of RM/cM has a well-defined determinant in R/c (computed
componentwise for the decomposition of R/c induced by that of /£), and we let
Am be the subgroup of (R/c)* consisting of determinants of units of PEm>
Following [19], we define an action of (R/c)* on the set J of isomorphism
classes of faithful torsionfree i?-modules: Given M E 7 and u E (R/c)*, we
choose any i?/ç-automorphism <p of RM/cM with determinant u, and define
Mu by the pullback
Mu >^ RM
(1.2) I Um
M/çM i RM/cM >%> RM/cM.
The isomorphism class of Mu is independent of the choice of <p, and we have a
well-defined group action. (See [19].) To distinguish Mu, u E (R/c)*, from a
direct sum of n copies of M, we write M^ for the latter.
2. The obstruction to direct-sum cancellation. The orbits of the action
defined in §1 delineate the failure of direct-sum cancellation for torsionfree R-
modules:
2.1. THEOREM. Let M, N, and X be faithful, torsionfree R-modules.
(2.1.1) M®X^N®XoN^Mufor some u e Ax.
(2.1.2) Ifue (R/c)*, then Mu ^ M o u € Am(tt(£*)).
(2.1.3) (M 0 N)u ^ Mu 0 N ^ M 0 Nu for each u € (R/c)*.
(2.1.4) Am®n = AMA;v.
PROOF. Property (2.1.3) is clear from the construction of (M0iV)u, since one
can produce the required automorphism with determinant u by splitting a rank-
one summand from either RM/cM or RN/cN, and letting <p be multiplication
by u on that component. The other assertions are proved in [21; 1.6, 1.7, 1.9].

DECOMPOSITIONS OF TORSIONFREE MODULES
505
Let Gx(M) denote the set of isomorphism classes of modules TV such that
M®X^N®X.
2.2. COROLLARY. Let M and X be faithful, torsionfree R-modules. Then
Mu <-► û is a one-to-one correspondence between Gx(M) and the group
Ax/(Axn(AMMir)))).
2.3. PROPOSITION. Let M and N be faithful torsionfree R-modules.
(2.3.1) If M is isomorphic to a direct summand of a direct sum of copies of
N, then Am Q Ajy.
(2.3.2) If M and N are locally isomorphic, then AM = Ajv-
(2.3.3) If M is projective, AM = &r = (R/c)*-
(2.3.4) If M has rank (ri,... , r*) and u E (R/c)*, then (rzri,... ,rzrt) E Am-
( The coordinates correspond to the direct decomposition of R/c induced by that
ofR.)
PROOF. (2.3.1) is an obvious consequence of (2.1.4). One can prove (2.3.2)
directly from the definition of Am; or one can use (2.3.1) and the fact [10, 3.1]
that "is a direct summand of a direct sum of copies of" is a local property.
Clearly AR = (R/c)*, and the other equality in (2.3.3) follows from (2.3.1). To
prove (2.3.4), observe that multiplication by u is in £m> and its determinant is
The genus of M is the set of isomorphism classes of modules locally isomorphic
to M. Using (2.3.2) and (2.2), we see that cancellation holds for modules in the
same genus:
2.4. COROLLARY. Let M andX be faithful torsionfree modules in the same
genus. Then Gx(M) = {M}.
Since A^ = (R/c)*, every class £x(M) is contained in C^(M), which, by
[19, 2.3], consists of those TV in genus (M) such that RN = RM. (This class is
usually referred to as the restricted genus of M [13, 2.9].)
Also Qr(M) is the stable isomorphism class of M, consisting of those TV such
that N 0 P = M 0 P for some projective module P. In general £r(M) is not
always the smallest of the classes Cx(M), since Ax may be properly contained
in Ar. To illustrate this, we borrow a construction from [6]. Let v be the nxn
nilpotent matrix with l's on the superdiagonal, O's elsewhere.
2.5. THEOREM. Suppose R/c contains elements a and (3 such that
{l,a,a2,/?} is linearly independent over R/c. Then, for each n > 1, there is
a torsionfree R-module M of constant rank n such that Em = (R/ç)W], whence
AM={un:ue(R/ç)*}.

506
ROGER WIEGAND AND SYLVIA WIEGAND
PROOF. Let W = {x + yct + vyf):x,y G (i?/ç)(n)}, an i?/ç-submodule of
(R/ç)(n\ Define M by the pullback diagram:
(2.5.1) i i
Then M is torsionfree of rank n, and, by [19, 2.1], the square above is the
standard pullback (1.1) for M. The computations of Drozd and Roiter in case
(A) of [6, 2.2] show that Em — {R/ç)[v], the ring of matrices of the form
u
0
w
V
u
w
V
0
w
u
0
0
V
u
0
w
V
u
u,v,w,... E R/c.
3. Torsionfree modules over curves. Let k be an algebraically closed
field and suppose C = specie is a reduced affine curve over k. Recall that R (or
C) is called seminormal provided R/c is reduced. Here we need only the weaker
condition that R/c is reduced, that is, R/c = k^s\ where s is the number of
singular points of C.
3.1. THEOREM. Let M and N be stably isomorphic torsionfree R-modules
of constant rank n. Suppose either that R/c is reduced or that char(fc)f n. Then
PROOF. We have to show that Qr{M) is trivial. By (2.2) and (2.3.4) it is
enough to show that A# = {R/c)* has nth roots. If R/c is reduced this is clear.
In the other case, one uses the following simple lemma:
3.2. LEMMA [21, 2.3]. Let (;4,M) be an Artinian local ring, let v € A*,
and suppose the coset v + M is an nth power in (A/M)*. If char(A/M)\n, then
v is an nth power in A*.
If C is reducible, modules with nonconstant rank may have more interesting
stable isomorphism classes. In [21, 3.3], there is an example of a curve over C
with two irreducible components and a torsionfree module M of rank (1,2) such
that Qr(M) is parametrized by C*. Also, a construction in [21, 3.4] shows that in
characteristic 2, (3.1) is false without the assumption that n is odd. The example
below is an adaptation of this construction to any nonzero characteristic, using
(2.5).
3.3. EXAMPLE. Over any algebraically closed field k of characteristic p^O,
there exist an irreducible curve C = speciî and a torsionfree i2-module M of

DECOMPOSITIONS OF TORSIONFREE MODULES
507
rank p whose stable isomorphism class Qr{M) is parametrized by the additive
group (fc,+).
PROOF. Let 01,02,03,04 be distinct elements of fc, consider k[x] where x2 =
0, and define n to be the natural homomorphism from the polynomial ring k[t]
onto (fc[x])(4) with kernel Ilt=i(* "" ai)2- Define R by the pullback
R ^ k[t]
(3.3.1) I i*
k[x] ~ (M*D(4)
where j is the diagonal embedding. Then R is a finitely generated fc-algebra
and (3.3.1) is the conductor square for i2, by [21, 3.1]. (Note that R is the ring
of polynomials / satisfying /(aj = /(a?) and /'(a^) = f'{a,j) for all i,j.) Let
a = 7r(£), set n = p, /? = a3, and let M be the module of (2.5). Then Am =
{up:u€ {k[x})*} = fc*. By (2.2) GR(M) is parametrized by (fc[x])*/fc* = (fc,+).
The equations defining the curve C = speciî are messy, but it is easy to
see, using the result in the appendix of this paper, that the singularity of C is
analytically isomorphic to that of the curve in 4-space defined by the ideal
{Y-x2,z,w)n{z-x2,Y,w)n{w-x2,Y,z)n(Y,z,w).
In [19] it was shown that direct-sum cancellation for torsionfree modules fails
over any irreducible singular curve. A simpler proof was given in [20], using the
following two lemmas:
3.4. LEMMA. Let C = speciî be a connected curve over fc. Then R*/k* is
a finitely generated group.
PROOF. Let Ci be the irreducible components of C and look at the map
i i i
Any function in the kernel is constant on each Ci and hence constant, since C is
connected. Therefore R*/k* embeds in Y[i(R*/k*), which is finitely generated
by [20, 1.8].
3.5. LEMMA [20, 1.7]. Let A C B be an integral extension of Artinian
rings with infinite residue fields. If B*/A* is finitely generated then A = B.
Next we use these two lemmas to classify those reduced affine curves C over fc
such that the torsionfree modules satisfy direct-sum cancellation. The irreducible
components of such a curve must be nonsingular. Also we will see that C has to
be seminormal. Geometrically, this means C has normal crossings: Whenever
two or more irreducible components meet at a point x, their tangent lines at x are
linearly independent. (See [3] or [5].) The final requirement is that the following
graph (which represents the configuration of the irreducible components) be
acyclic: Let graph (C) be the bipartite graph with vertex set {xi} U {C3}, where
the Cj are the irreducible components of C and the Xi are the points of C that

508
ROGER WIEGAND AND SYLVIA WIEGAND
2 3 4
/MW
Figure l
lie on more than one irreducible component. There is an edge {i,j) from Xi to
Cj if and only if Xi E Cj. Three curves and their graphs are sketched in Figure
1. The first (intended to be a union of lines in 3-space) is acyclic. The other two
are not.
3.6. THEOREM. Let C = speciî be a reduced affine curve over an
algebraically closed field k, and let C\,...,Ct be the irreducible components. The
following conditions are equivalent:
(1) Torsionfree R-modules satisfy direct-sum cancellation.
(2) The natural map Pic R —► Pic R is an isomorphism.
(3) (£/£)* =A«0r(Â*)).
(4) Each Cj is nonsingular, C has normal crossings, and graph (C) is acyclic.
(5) Each Cj is nonsingular, C is seminormal, and graph (C) is acyclic.
PROOF. The equivalence of (2) and (3) follows from the Mayer-Vietoris
sequence of the conductor square. (See [1, IX, 5.3].) For (1) implies (3), suppose
u e {R/c)* - Ar{tt{R*)). Then Ru 0 R ^ R 0 R by (2.1.1), but Ru ¥ R by
(2.1.2).
We have already remarked that (4) and (5) are equivalent. It remains to be
shown that (3) => (5) and (5) => (1). We may assume C is connected.
For (3) => (5), note that conditions (2) and (3) are inherited by the Cj, so (3)
implies that each Cj is nonsingular, by [19, 3.2]. Let R be the seminormalization
of R. Since speciî —► speciî is a bijection [18], speciî is also connected, and
R /k* is finitely generated by (3.4). If R ^ R then R/c is a proper extension of
R/c, and we see from (3.5) that {R/c)* is not finitely generated over {R/c)* =
Ar, whereas A#(7r(i? )) is. This contradicts (3) and shows that (3) implies C
is seminormal.
To show that (3) also implies that graph (C) is acyclic, we let {x\,... ,xs}
be the singular points of C and let T be the set of ordered pairs {i,j) such that
Xi E Cj; that is, T is the set of edges of graph (C). We examine the natural
map $: A# x R* —► {R/c)*, which is surjective by (3). Since R/c is reduced,

DECOMPOSITIONS OF TORSIONFREE MODULES
509
Figure 2
we can regard an element of Ar = (R/ç)* as an s-tuple a = (ai,..., as), where
ai E fc*, and an element of (R/ç)* as an assignment of a label ^ G k* to
each edge (i, j) E T. Similarly, an element of R* is a £-tuple / = (/i,..., /*),
with fj E fi*. The map $ takes (a,/) to the element with aifj{xi) on edge
(i,j). Let # = (fc*)^^ be the subgroup of fi* consisting of functions that are
constant on each component of specfi*, and suppose £ E $(A# x H). If graph
(C) contains a cycle traversing edges ei,e2,...,em with corresponding labels
fi,...,fm, then the alternating product fif^&f^1 • • • fm1 must be 1. This
relation shows that the cokernel of $ restricted to Ar x H maps onto fc*, which
is not finitely generated. Since R*/H is finitely generated by (3.4), coker^ is
not finitely generated, and condition (3) fails. This completes the proof that
(3) => (5).
Supposing now that (5) holds, we show (1). Let M, TV, X be torsionfree R-
modules such that M 0 X = N 0 X. We may assume all three modules are
faithful, by replacing X by X 0 R and discarding the components Cj where M
vanishes. To prove that M = N it suffices to show that {R/c)* = Am{k{R*)),
by (2.2). Let rank(M) = (ri,..., rt), and consider the modified map #: Ar x
fi* -► (fi/ç)* taking (a,/) to the labelling Ç with ^ = a~rj fj{xi). By (2.3.4),
Im\I> Ç AAf(7r(i2*))- Let d E fc* be arbitrary, let (p, g) be any edge of graph
(C), and let 6 be the element of {R/ç}* with d on edge (p, g) and l's on all the
other edges. Since {R/c)* is generated by elements of this form, it will suffice to
produce a pair (a, /) in Ar x H such that \P(a, /) = 6.
The graph 9 obtained from graph (C) by removing edge (p, q) has two
connected components: 9'> containing xp, and 9", containing Cg. Choose b £ k*
such that 6~r9 = d, and build (a, /) as follows: On 9', o,i — b and /y = 6r>; on
9", fl< = /y = 1. (See Figure 2, where (p,g) = (2,3).)
In the general setting of §1, condition (2) of the theorem is not sufficient for
condition (1). In [19, 2.8], an example is given in which Pic R — 0 but direct-sum
cancellation fails for torsionfree i2-modules.

510
ROGER WIEGAND AND SYLVIA WIEGAND
4. Multiplicity. We now return to the more general setting in §1. We
denote the multiplicity of i2, the maximum of the multiplicities of the local rings
of i2, by n(R). Obviously ji{R) — 1 if and only if R = R. In fact, /j,(R) is
always the minimum number of generators required for R as an iî-module, or,
equivalently, the minimum number of generators for R/ç as an R/ç-modu\e.
Further, if R ^ i2, then /j,(R) is the sharp bound on the number of generators
needed for the ideals of R. (See [9, 2.1] for proofs of these assertions.) In this
section we survey results relating the multiplicity to the existence of direct-sum
decompositions.
Many characterizations of the rings R of multiplicity < 2 have appeared in
the literature. (See [2, 9, 12, 14, 19].) One characterization is that every ring
between R and R is Gorenstein [2, §7]; the following proposition, essentially due
to Bass, is another:
4.1. PROPOSITION [19, 2.6]. /j,{R) < 2 if and only if every faithful
torisonfree R-module has a faithful ideal as a direct summand.
Thus /j,(R) < 2 implies that every torsionfree module is a direct sum of ideals,
and the converse holds if R is a domain. (In the terminology of Levy and Haefner,
rings for which every torsionfree module is a direct sum of ideals are called £/
rings; they have been completely classified in [11].) In multiplicity 2 there is
actually a canonical form for torsionfree modules of constant rank, discovered
by Borevich and Faddeev [4] in the case of classical orders:
4.2. PROPOSITION [14, 7.1]. Assume fi(R) = 2, and let M be a
torsionfree module of contant rank r. Then there exist rings R{, 1 < i < r,
and an invertible ideal I of Rr, such that R Ç Rx Ç • C Rr C R and
M = R\ 0 • • • 0 Rr-i 0 /. Moreover, the Ri are unique, and I is unique up
to isomorphism.
As an example, suppose that R is the coordinate ring of the cusp or the node.
Then there are no rings properly between R and #, and since Pic R is trivial, we
conclude that for each r, there are only r nonisomorphic nonprojective modules
of rank r, namely, R(r~^ 0 i?W, 1 < i < r. This is somewhat surprising, since
the projectives of a given rank are parametrized by (fc, +) for the cusp and fc*
for the node.
If fi(R) > 4 and speciî is connected, there are indecomposable torsionfree
modules of constant rank n for every n > 1. For local rings, this is almost
proved in [6], using constructions similar to (2.5). A slight modification of the
approach in [6] works for a general local ring of the sort we have been considering.
Then a simple gluing argument gives the global result.
Multiplicity 3 is the most interesting: There are a few £/ rings; for
example, the coordinate ring of 3 lines (coplanar or not) meeting at a point. (See
[11].) The other rings can be categorized depending on the size of rad(i2/ç), the
nilradical of R/c.

DECOMPOSITIONS OF TORSIONFREE MODULES
511
4.3. ASSERTION. Let /i(i2) = 3. Then there is a bound on the ranks of
indecomposable torsionfree R-modules if and only if (jdid(R/ç))(R/R) is a cyclic
R/c-module.
This assertion is proved in [6] and [8] under strong finiteness assumptions.
The "only if" direction is true in general, and it seems likely that the matrix
reductions for the "if" direction in [6] and [8] can be made to work in general.
The ring R = k[t3,t4,tb] has conductor t3k[t] so rad(#/ç) = 0. Thus,
presumably, R has a bound on the ranks of its indécomposables. On the other hand
S = k[t3,t7] has conductor t12k[t]. It is easily checked that {t4,t6} is a
minimal generating set for (rad(S/ç))(5/S), which implies S has indécomposables
of every rank.
Appendix. Analytically isomorphic singularities. Let (i2, M) be a one-
dimensional reduced local ring with finite normalization R and conductor ç. Our
goal in this section is to show that if R contains a field and R ^ R, then the
completion R is determined by the bottom line of the conductor square for R.
The first thing we need to know is that R is reduced. We are grateful to L. S.
Levy for showing us the following:
A.l. LEMMA. The completion of R is reduced, and its conductor square is
the M-adic completion of the conductor square for R.
PROOF (LEVY). The bottom line R/c —► R/c of the conductor square is
unaffected by completion, since R/c has finite length. Therefore the M-adic
completion of the conductor square is
R ~ R
(A.l.l) I l*
R/c » R/c
This is still a pullback, since completion is exact. Now ç is M-primary, so R is
the ç-adic completion of R. But this is the same as the J-adic completion, where
J is the Jacobson radical of R. Since R is a direct product of discrete valuation
rings (none of which is a field), so is its completion R. This shows that R is
reduced. The rest of the proof proceeds as in [21, 3.1]: No nonzero ideal of R/c
is contained in i?/c, and since (A.l.l) is a pullback, kerir is the largest ideal of
R contained in R. Finally, ker 7r contains a non-zerodivisor; otherwise R/c would
be one-dimensional. It follows that (A.l.l) is the conductor square for R.
A.2. THEOREM. Let k be a perfect field, and let R\ and R2 be one-
dimensional reduced local k-algebras with finite normalizations Ri ^ Ri and
conductor ideals c^. The completions R\ and R2 are k-isomorphic if and only if
there are compatible k-isomorphisms ot'.Ri/^ —► Ri/ç^ and 0:Ri/ç—> Ri/ç<i>
PROOF. By the lemma we may assume R\ and R2 are complete. If R\ = i22
we can obviously find compatible isomorphisms R\ —► R2 and R\ —► Ri inducing

512
ROGER WIEGAND AND SYLVIA WIEGAND
the required a, /?. Conversely, suppose a, /? are given. If we can lift /? to an
isomorphism 7: R\ —► #2, the pullback of a and 7 over ft will be an isomorphism
from R\ onto i?2-
By Cohen's structure theorem [15, Corollary 2, p. 206] R\ is a direct product
of rings of the form ^[[x]], where the Kj are the residue fields of R\. Now
çx Ç rad(^i), so the Kj are also the residue fields of R/c, which is the direct
product of the rings Kj[[x]]/(xnJ). Applying the same argument to #2, we see
that R\ = ^2, and the only problem is to find a fc-isomorphism that lifts /?. Now
(3 has to carry each direct factor Kj[[x]]/(xnj) of Ri/çx isomorphically onto
one of the indecomposable factors ^[[x]]/(xn/) of #2/^2' and clearly Kj = K\
and rij = n\. Therefore it will suffice to prove that every fc-automorphism of
if [[x]]/(:rn) lifts to a fc-automorphism of -K"[[x]].
If n = 1 this is clear, so we assume from now on that n > 2. Let <pn
be a fc-automorphism of fc[[x]]/(xn). If we can lift <pn to a fc-automorphism
<pn+i of if [[x]]/(xn+1), and so on, the limit of <pn, <pn+i,... will be the required
automorphism of K[[x}\.
Let 7rm:lf[[z]] -► K[[x]]/{xm) and q: K[[x]]/(xn+1) -> K[[x]]/{xn) be the
natural maps, and consider the following diagram of fc-homomorphisms:
K[[x)) ^n > K[[x]]/{x»)
(A'2'2) î "^-„ h
k ^K[[x]\/{xn+1).
We want to fill in the homomorphism w so that both triangles commute. To
do this it is enough to show that K[[x]\ is formally smooth over fc. (See [15,
28.3, pp. 198-199].) Since fc is perfect, K is separable over fc by [15, Corollary,
p. 194], hence formally smooth over fc by [15, 28.1, 28.D]. Also, K[[x]] is formally
smooth over K [15, Example 3, pp. 200-201], so K[[x\] is formally smooth over
fc by transitivity [15, 28.E].
We now have the map w in (A.2.2). It is a fc-homomorphism because the
lower left triangle commutes. We claim w is surjective. Since qw is surjective, it
suffices to show that no proper subring of K[[x]]/(xn+1) maps onto K[[x]]/(xn).
The verification (which uses our assumption that n > 2) is left to the reader.
Thus w is surjective, and it is continuous by [15, Remark, p. 199]. Since ck —► 0
in K[[x]], w(x) is a nonunit of K[[x]]/(xn+1). Therefore w(xn+1) = 0, so w
factors through 7rn+i, say w = <£>n+i7rn+i. The map <£>n+i is the desired lifting,
and the proof is complete.
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