Conference Board of the Mathematical Sciences
REGIONAL CONFERENCE SERIES IN MA THEMA TICS
supported by the
National Science Foundation
Number 54
INTRODUCTION TO INTERSECTION THEORY
IN ALGEBRAIC GEOMETRY
by
WILLIAM FULTON
Published for the
Conference Board of the Mathematical Sciences
by the
American Mathematical Society
Providence, Rhode Island

Expository Lectures
from the CBMS Regional Conference
held at George Mason University
June 27-July 1, 1983
1980 Mathematics Subject Classifications. Primary 14C17, 14C15, 14C40, 14M15, 14N10,
13H55.
Library of Congress Cataloging in Publication Data
Fulton, William, 1939-
Introduction to intersection theory in algebraic geometry.
(Regional conference series in mathematics, ISSN 0160-7642; no. 54)
"Expository lectures from the CBMS regional conference held at George Mason
University, June 27-July 1, 1983"-T. p. verso.
Bibliography: p.
1. Intersection theory. 2. Geometry, Algebraic. I. Conference Board of the
Mathematical Sciences. II. Title. HI. Series.
QA1.R33 no. 54 [QA564] 510s [512'.33] 83-25841
ISBN 0-8218-0704-8
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Contents
Preface v
1. Intersections of hypersurfaces 1
1.1 Early history (Bezout, Poncelet) 1
1.2 Class of a curve (Plucker) 2
1.3 Degree of a dual surface (Salmon) 2
1.4 The problem of five conies 4
1.5 A dynamic formula (Severi, Lazarsfeld) 5
1.6 Algebraic multiplicity, resultants 6
2. Multiplicity and normal cones 9
2.1 Geometric multiplicity 9
2.2 Hilbert polynomials 9
2.3 A refinement of Bezout's theorem 10
2.4 Samuel's intersection multiplicity 11
2.5 Normal cones 13
2.6 Deformation to the normal cone 15
2.7 Intersection products: a preview 17
3. Divisors and rational equivalence 19
3.1 Homology and cohomology 19
3.2 Divisors 21
3.3 Rational equivalence 22
3.4 Intersecting with divisors 24
3.5 Applications 26
4. Chern classes and Segre classes 29
4.1 Chern classes of vector bundles 29
4.2 Segre classes of cones and subvarieties 32
4.3 Intersection formulas 34
5. Gysin maps and intersection rings 37
5.1 Gysin homomorphisms 37
5.2 The intersection ring of a nonsingular variety 39
5.3 Grassmannians and flag varieties 41
5.4 Enumerating tangents 44
iii

CONTENTS
b.
7.
8.
9.
10.
Degeneracy loci
6.1 A degeneracy class
6.2 Schur polynomials
6.3 The determinantal formula
6.4 Symmetric and skew-symmetric loci
Refinements
7.1 Dynamic intersections
7.2 Rationality of solutions
7.3 Residual intersections
7.4 Multiple point formulas
Positivity
8.1 Positivity of intersection products
8.2 Positive polynomials and degeneracy loci
8.3 Intersection multiplicities
Riemann-Roch
9.1 The Grothendieck-Riemann-Roch theorem
9.2 The singular case
Miscellany
10.1 Topology
10.2 Local complete intersection morphisms
10.3 Contravariant and bivariant theories
10.4 Serre's intersection multiplicity
References
47
47
49
50
51
53
53
54
55
56
59
59
60
62
65
65
69
73
73
74
76
78
81

Preface
These lectures are designed to provide a survey of modern intersection theory
in algebraic geometry. This theory is the result of many mathematicians' work
over many decades; the form espoused here was developed with R. MacPherson.
In the first two chapters a few episodes are selected from the long history of
intersection theory which illustrate some of the ideas which will be of most
concern to us here. The basic construction of intersection products and Chern
classes is described in the following two chapters. The remaining chapters contain
a sampling of applications and refinements, including theorems of Verdier,
Lazarsfeld, Kempf, Laksov, Gillet, and others.
No attempt is made here to state theorems in their natural generality, to
provide complete proofs, or to cite the literature carefully. We have tried to
indicate the essential points of many of the arguments. Details may be found in
[16].
I would like to thank R, Ephraim for organizing the conference, and C.
Ferreira and the AMS staff for expert help with preparation of the manuscript.

1. Intersections of Hypersurfaces
1.1. Early history (Bezout, Poncelet). A most basic question in intersection
theory is to describe the intersection of several algebraic hypersurfaces in «-space,
i.e., the common solutions of several polynomials in n variables. The ancients
certainly knew about the possible intersections of lines and conies in the plane,
and they also knew that rational solutions of two quadric equations in three
variables behaved like solutions of one cubic equation in two variables [61].
We do not know who first observed that two plane curves of degrees p and q
should intersect in pq points. By 1680 Newton [48] had developed an elimination
theory for two such equations. This produced a resultant, which was a polynomial
in one variable of degree pq whose solutions gave an abscissa of the intersection
points of the two curves. The corresponding construction and assertion for n
equations in n variables were made in 1764 by Bezout [5, 6]. Bezout's treatment
was entirely algebraic, although he briefly interpreted his result for n = 2 and
n = 3: the number of intersections of two plane curves (or three surfaces in space)
is at most the products of their degrees.
By referring to the resultants, which are polynomials in one variable, one can
also discuss the possibilities of nonreal solutions, asymptotic solutions, and multi-
multiple solutions. As geometry developed, the first two of these situations were
subsumed by considering intersections of hypersurfaces #,,..., Hn in complex
projective space P?. Now we assign an intersection multiplicity
i(P)-i{PtHx ¦¦-tfj
to a point P of the intersection П #,; if the #, do not meet transversally at P, this
multiplicity will be greater than one.
Although there was little early discussion of this multiplicity, the governing
principle of continuity was well understood, at least since Poncelet [51]. If the //,
vary in families #,(f), with #,@) = #,, and Р,(г),..., Pr(t) are the points of the
general intersection П#,(г) which approach P as t -> 0, then
/(/>,#, "¦ Hn)= ii{Pj(t),Hx(t)-..HH(t)).
Varying the #, so that the #,(r) meet transversally, this determines the multiplic-
multiplicity /(?,#, •••#„).
In all the above discussion, it is assumed that the intersection of the hyper-
hypersurfaces is a finite set, or at least that P is an isolated point of C\Hr

1.2. Class of a curve (Plucker). An important early application of Bezout's
theorem was for the calculation of the class of a plane curve C, i.e., the number of
tangents to С through a given general point Q:
Equivalently, the class of С is the degree of the dual curve С v. If F(x, y. z) is the
homogeneous polynomial defining С and Q = (a: b: c), then the polar curve CQ
is defined by
FQ(xtytz) = aFx + bFy + cF.t
where Fx = dF(x, y, z)/dX, Fv, F, are partial derivatives. This is defined so that
a nonsingular point P of С is on CQ exactly when the tangent line to С at P
(defined by XFX(P) + YFV(P) + ZFZ(P) = 0) passes through Q. One checks that
С meets CQ transversally at P if P is not a flex on C, so
class(C) = #CC\CQ = degCdegq, = n(n - 1),
if n is the degree of C, and С is nonsingular.
If С has singular points, however, they are always on С П CQ, so they must
contribute. For example, if P is an ordinary node (resp. cusp) and Q is general,
then
/(ЛС-Св)-2 (resp./(Л С C^) = 3).
This gives the first Plucker formula [50]
n(n - 1) = class(C) + 28 + Зк,
if С has degree и, 8 ordinary nodes, к ordinary cusps, and no other singularities
1.3. Degree of a dual surface (Salmon). In 1847 Salmon [53] made a similar
study of surfaces. If 5 с P3 is a surface, the degree of the dual (or "reciprocal")
surface S v is the number of points P e S such that the tangent plane to S at P

contains a given general line /. (This number is one of the projective characters o>f
5, now called the second class of S.)
For a point Q e P3, let SQ be the polar surface of S with respect to Q: if
F(jc, >>, z, w) defines 5 and Q = (a: b: c:d\ then a/; + Z?/^ + cF, + rffw de-
defines Sq. Taking two points Qlt Q2 on /, one sees as before that a nonsingulax
point P of 5 is on SQi П 5^2 if and only if the tangent plane to S at P contains Л
Thus for S nonsingular of degree л, and (),, Q2 general,
deg(Sv) - #5 П Sg, П SQj - л(л - IJ.
As before, all singular points of 5 are contained in S П SQi О Sq2. If P is an
isolated singular point of 5, its contribution to the total n(n — IJ is the
intersection multiplicity i(P, S • SQi- SQ2). For example, the contribution of an
ordinary double point is two, so degEv) = л(л— IJ-26 if 5 has 5 ordinary
double points.
If S is singular along a curve C, however, a new phenomenon occurs, a problem
of excess intersection: how to compute the contribution of С to the total
intersection n(n — IJ, so that n(n — IJ diminished by this contribution, and by
contributions of other singular points, yields degE v). Salmon initiates a study of
the contribution of a curve С to the intersection of three surfaces in space when С
is a component of their intersection. For example, if С is a line, he gives its
contribution as m + n + p - 2, where m, л, р are the degrees of the surfaces.
Salmon justifies this by saying that the answer must be independent of the choice
of surfaces of given degrees, and then he calculates it directly in the degenerate
case when the first is the union of a plane containing С and a general surface of
degree m - 1. This surface meets the other two surfaces in (m - \)np points,
m - 1 of which are on the line C. The plane meets the other two surfaces in
curves of degrees л - 1 and p - 1 in addition to C; these curves meet in
(л - IXp - 1) points. The total number of points of intersection outside С is
therefore
(m - \)np - (m - 1) + (л - \)(p - 1) = mnp - (m + л Л- р - 2),
as asserted. In case С is a double line on the first surface, he calculates its
contribution азт + 2л + 2/>-4Ьу working out the case where this surface is
the union of two surfaces containing С

If C is a double line on a surface 5 of degree л, this analysis predicts 5 л - 8 as
the contribution of С to the intersection of S with Sq{ and Sq2. However, as
Salmon points out, there are special points on C, called pinch points (or "cuspidal"
points), where the two tangent planes to S coincide.
pinch point
If С is the line x = у = 0, and S is the surface Uy2 + Vxy + Wy2 = 0, then
these pinch points are the intersections of С with the surface V2 = 4UW, so there
are 2 л - 4 pinch points on S. Thus C, together with its pinch points, diminishes
the degree of S v by Eл - 8) + B л - 4) = 7л - 12. For example, a cubic with a
double line (e.g.y2 - zx1 + x3) has a dual surface of degree three.
Salmon also considers more general curves. If С is a complete intersection of
surfaces of degrees a and b, and С is a component of intersection of three surfaces
of degrees m, л, and p, then he finds that the contribution of С to the total
number of mnp is ab(m + n + p — (a + b)). Concluding this remarkable paper,
he deduces that if such С is an r-fold curve on a surface S> then it diminishes the
degree of the dual by
ab[(r - l)Cr + 1)л - r2(r - \){a + b) - 2r(r - 1)].
1.4. The problem of five conies. Problems of excess intersection arise frequently
in enumerative problems. The famous problem of the number of plane conies
tangent to five given conies in general position is a typical example of this. A
plane conic is defined by a quadratic polynomial ax2 + by2 + cxy + dx + ey 4- /,
unique up to multiplication by a nonzero scalar, so the space of conies can be
identified with P5. One checks that the condition to be tangent to a fixed
nonsingular conic is described by a hypersurface of degree six in P5. The desired
conies are then represented by the points in the intersection of five such
hypersurfaces Я, П • • • П H5. There are not 65 = 7776 such conies, however, as
originally thought by Steiner and others. Indeed, the Veronese surface V s P2 of
conies which are double lines is contained in П#,, and one can show (cf. §4
below) that the contribution of V to the intersection is actually 4512, which leaves
3264, the actual number of (nonsingular) conies tangent to five given conies in
general position.

Note that the conies tangent to a fixed line form a quadric hypersurface in P6.
Given five general lines, the Veronese contributes 31 to the predicted intersection
number 25 for the five quadrics. Since everyone knew that there is only one
nonsingular conic tangent to five general lines (by duality, for example), it is
curious that these false answers were proposed when the lines are replaced by
curves of higher degree.
In spite of the clear exposition of the importance of excess intersections in
enumerative geometry by Salmon and Cayley, such considerations played little
role in the great development of enumerative geometry at the hands of Chasles, de
Jonquieres, Schubert, Halphen, Zeuthen, and others. For one thing, they avoided
writing equations for varieties and, especially, for parameter spaces. In general,
however, their work can be interpreted as calculating intersections on appropriate
spaces so that the intersections become proper. Often these spaces are blow-ups of
the naive spaces, which amounts to adding structure to degenerate figures. For
example, a classical approach to the space of conies amounts to working on the
space of complete conies, which is the blow-up P5 of P5 along the Veronese; in
this model a point in the exceptional divisor corresponds to a double line together
with a pair of points on the line. The proper transforms of the hypersurfaces Ht
then meet properly on P5 outside the exceptional divisor, and once one knows an
appropriate "intersection ring" for P5 one may calculate their intersection.
The same approach works for quadrics of arbitrary dimension. The beautiful
study of complete quadrics was initiated by Schubert, who found many enumer-
enumerative formulas. The rigorous construction of these parameter spaces and their
intersection rings has been carried out by Semple and Tyrell, with modern
re-examination by Vainsencher, Laksov, and Lazarsfeld. Realizing the spaces as
orbit spaces of suitable group actions, by Demazure and by De Concini and
Procesi, has led to a clearer understanding of their structure.
1.5. A dynamic formula (Seven, Lazarsfeld). In general, if #t,...,#/f are
arbitrary hypersurfaces in P", with d, = deg(#,), Severi [58] proposed to assigji
numbers i(Z) to certain distinguished subvarieties Z of the intersection locus
Я, П • • • П #„, so that
Each irreducible component of П #, should be distinguished, and each isolated
point should be assigned its intersection multiplicity. In general, as in Salmon's
examples, there may also be imbedded distinguished varieties. Severi's dynamic
procedure, corrected and completed by Lazarsfeld [40], can be summarized as
follows. If f, is a homogeneous equation for #,, consider deformations H,(t) of H{
defined by homogeneous polynomials Ft + tGt + t2G- + • • •. For a given sub-
variety Z of П #,, lety(Z) be the number of points of П #,(/) which approach Z
as t -* 0, for a generic deformation; in fact,y(Z) of the points will approach Z for

WILLIAM FULTON
any deformation for which the first order parts (Gh..., G,?) belong to a certain
open set Uz of the space of л-tuples of polynomials of degrees </,,...,</„. For any
point P set i(P) -JiP). Only finitely many points will have i(P) * 0. For an
irreducible curve C, set
Лес
so i(C) is the number of points that generically approach C, but not any
particular point on C. Inductively,
the sum over all proper irreducible subvarieties К of Z. Then E/(Z) = d{ • • • dn,
which achieves the desired decomposition.
We will later see a static construction of this decomposition, which is also valid
in contexts where such deformations are unavailable. It should be emphasized,
however, that in spite of the existence of a rigorous general theory, and some
explicit formulas, the actual computation of the contributions i(Z) remains a
difficult problem.
For plane curves, following Segre [55], Lazarsfeld gives the following answer. If
H, = Dt + ?, where D, and D2 meet properly, and P is a point in E, let G, be
generic as above, let A, be equations for Dt, and let F be the curve defined by
Afi2 - ^2G,.Then
i{P) = i{P,EF) + i(P,DrD2).
For example, if #, = 2L, + L2, #2 = L, + 2L2, with Lu L2 lines meeting at a
point P, then the Segre-Lazarsfeld formula shows that
/(/>)-/(I,)-i(L2)-3.
1.6. Algebraic multiplicity, resultants. For an isolated point P in the intersection
of hypersurfaces #,,..., Hn in P", a modern static definition of the intersection
multiplicity is
i(P,Hx •••#J = dimc0/)/(/1,...,/J,
where 0^ is the local ring of P" at /\ and /, is a local equation for #, in вР. If P is
the origin in С" с P", 6P is the localization of C[A",,..., Xn] at the maximal ideal
(A",,..., Xn). Or one may replace ®P by its completion С[[Я",,..., Xn]], or by the
ring С{ЛГ,,..., Хп) of convergent power series. This algebraic construction of
intersection multiplicity dates from Macaulay [42].
Let us verify the agreement of this definition with that obtained from elimina-
elimination theory, at least for plane curves. Suppose the curves are defined by poly-
polynomials f(xy y) and g(xt y), and the two curves do not meet at infinity on the
^-axis. Thus we may assume
/(*. У) = ao(x)ya + a,{x)yn-[ + • • • + an(x)

INTRODUCTION TO INTERSECTION THEORY 7
with <яо@) * 0. Let A = C[x](jc) be the local ring of the *-axis at the origin. Then
A[y]/(f) is an A -algebra which is a free Л-module of rank n, and one may
construct the resultant r = R(f, g) in A by
(It is a formal exercise, left to the reader, to show that this agrees with the usual
definition, as in [60].)
We must show that the order of vanishing of г at x = 0 is equal to the sum of
the intersection numbers of the two curves at all points P on the >>-axis:
Now A[y]/(f, g) is finite dimensional over C, so it is a direct sum of its
localizations ©/>/(/, g), where P varies over the points on the y-axis on both
curves. Therefore
Since the order of vanishing of r at x = 0 is dimc^/(r), the equation to be
proved is
dimcA[y]/(fyg) = dimcA/(r).
This is a special case of an important algebraic fact:
Lemma. Let A be a one-dimensional Noetherian local domain, M a finitely
generated free A-module and ф: M -* M an A-homomorphism. Then
The length of an Л-module N is d if there is a chain of submodules N = Noz>
N} D • • • D Nd = 0, where ЛГ/ЛГ + , is isomorphic to the residue field of A. In
case A contains a subfield К which maps isomorphically to its residue field, then
lengthy N - dimKN.
When A is a discrete valuation ring, the lemma is an exercise in elementary
divisors. For the general case see [16, A2.6].

2. Multiplicity and Normal Cones
2.1. Geometric multiplicity. A subvariety X of СN is defined by a prime ideal
/(X) in C[Xx,..., XN]. The coordinate ring Г(X) is the residue ring
A (closed) subscheme Z of X is determined by an ideal I = I(Z) of Г( A'), which
is a subvariety if /(Z) is prime. In this case the local ring of X at Z is the
localization of T(X) at /(Z), and is denoted 6Z x.
If Z is a subscheme of X, the irreducible components of Z are the sub varieties of
X corresponding to the minimal prime ideals of T(X) which contain /(Z). If V is
such a component, the geometric multiplicity of V in Z is defined to be the length
of the Artinian ring
The cycle of Z, denoted [Z], is defined to be the formal sum
where K,,..., Fr are the irreducible components of Z, and m, is the geometric
multiplicity of V, in Z. For example, if X = С and Z is the scheme-theoretic
intersection of n hypersurfaces which meet properly, then
the sum over the points P in Z, with i(P) the intersection number described in
§1.6.
For an arbitrary variety Xy subschemes Z are defined by ideal sheaves # = #(Z).
On any affine open U с X which meets Z, 3 is given by an ideal in the coordinate
ring of U, which is prime if Z is a subvariety. The local ring of X along Vy and the
geometric multiplicity of a component V of Z can be defined using any such U.
2.2. Hilbert polynomials. A subscheme Z of P* is defined by a homogeneous
ideal I = I(Z) in QA'q,..., Xn]. If QAJ,,..., A^], denotes the homogeneous
polynomials of degree /, such an ideal / is the direct sum of its intersections It
with C[XQ,..., XN]r Two homogeneous ideals define the same subscheme when
their homogeneous pieces are the same for all but finitely many /. The Hilbert
polynomial of Z is the polynomial Pz(t) such that

10 WILLIAM FULTON
for all sufficiently large t. Indeed, one shows (cf. [30, §1.7; or 57]) that the right
side is a polynomial of degree equal to the dimension of Z, for t » 0. If
n = dim(Z), one may define the degree of Z, deg(Z), to be the coefficient of
t"/nl in Pz(t), i.e.
(i) Pz{t) = deg(Z)/"/rt! + lower terms.
It also follows that if [Z] = Lm^V,] is the cycle of Z, then
(ii) deg(Z)= ? m,deg(!0.
If К is a sub variety of P^, and Я is a hypersurface of P* not containing V, then
(Hi) deg(KPl tf) = mdeg(F).
It will later become clear that this definition of deg(K) agrees with the
geometric notion of counting intersections of V with complementary linear
spaces. In fact, we shall have no need for Hilbert polynomials, although they have
played an important role in the modern algebraic development of multiplicity.
2.3. A refinement of Bezouf s theorem. The elementary facts about degree in the
preceding section, together with an important join construction, allow a simple
proof of the following proposition. A stronger result will appear later when more
intersection theory is available.
Proposition. Let K,,..., Vs be subvarieties of P*, and let Z,,..., Zr be the
irreducible components of V] О • • • П Vs. Then
t deg(Z,)<fldeg(K7).
Proof. By a simple induction, one may assume 5 = 2. Construct the ruled join
J = J(VltV2) in P2N+' as follows. Let X09..., XN% YOt...,YN be homogeneous
coordinates on P2"+'. Let P," (resp. P2") be the linear subspace of P2"+1 defined
by the vanishing of all Y, (resp. all Xt). Identifying P," with P", one has Vx с РД
Let J be the union of all lines from points of Vx to points of V2. Algebraically, the
homogeneous coordinate ring of J is simply the tensor product of the homoge-
homogeneous coordinate rings of K, and V2. One verifies that
@ degG) = deg(K1)deg(K2).
Let L be the linear subspace of P2yv+I defined by Xf = Yt, 0 < / < N. Then
L = P" and
(ii) Lnj=VlC)V2.
Thus we are reduced to the case where one of the varieties being intersected is a
linear subspace.
Since a linear subspace is an intersection of hyperplanes, one is further reduced
inductively to the case where one of the varieties, say Vx, is a hyperplane. In this

INTRODUCTION TO INTERSECTION THEORY 11
d V2 and the proposition holds with equality, or [К, П V2] =
E'., m,[Z,], where the Z, are the irreducible components of V] П K2, and by (ii)
and (iii) of §2.2 (for any hypersurface K, not containing V2\
2.4. Samuel's intersection multiplicity. Suppose Я,,..., Hn are hypersurfaces in
an л-dimensional variety K, and P is an isolated point of ПНГ Let /4 = вР v be
the local ring of К along P, and assume each Я, is defined by one element/ in A.
Let /«(/,,...,/„). Then Л// is finite dimensional over C, and if P is a
nonsingular point of V, one may use dimcA/I to give a workable definition of
the intersection multiplicity i(P, Я, • • • Hn) as in §1. The following is a standard
example of the failure of this definition in general.
Example. Let V be the image of the mapping ф: С2 -> С4 defined by
фE, t) = (s4, s3t, st3, t4\ let P be the origin, and let Я, and Я2 be the hyper-
hypersurfaces of V defined by the coordinates xx and x4 respectively. By varying Hx
and Я2, the principle of continuity requires that the intersection multiplicity is 4.
However, one calculates that the ideal of V is generated by хххл - x2x3,
xfjc3 - *2, x2xj - x], and xjx4 - x]xx, from which it follows that
dimcA/(xlt x4) = 5.
Samuel [54] defines the multiplicity i(P) = i(P, Я, • • • Hn) to be the coeffi-
coefficient of tn/n\ in the Hilbert-Samuelpolynomial
(i) P(t) = dimc(^//r) = i(P)t"/n\ + lower terms
for t » 0. To see that dim(A/I') is a polynomial of degree n in r, for t » 0, one
may proceed as follows. Let A — A/I and consider the surjection of graded rings
(ii) A[Xlt...tX,]-+ ®I'/I'+l
which maps Xt to the image of/j in I/I2. The kernel of this homomorphism is a
homogeneous ideal which defines a subscheme P(C) of projective (n - l)-space
P" over Л. (Those who feel uncomfortable with projective space over a ring
such as Л may realize P(C) in P" X F, since Л is a residue ring of A.) This
scheme P(C) is the projective normal cone to Г)Н( in V. We shall discuss normal
cones in succeeding sections. Here we shall use the fact that P(C) has pure
dimension n — 1, so its Hilbert polynomial has the form
(iii) iV,(/) = dimc/'//'+1 = i(P)t-x/(n - 1)! + -.-
for / » 0. A simple calculation shows that this definition of i(P) is the same as
that in (i). However, since P(C) cl^l.1, the only component of P(C) is the
underlying variety P? ~ * of Рд "' and, therefore,
(iv)

defines the multiplicity i(P) without reference to Hilbert functions. In addition,
sinceP(C)c PA"-',and
it follows that
(v) i(P) < dimc(A) - dim
We see also that equality holds in (v) if the morphism (ii) is an isomorphism.
This is related to the important notion of a regular sequence.
Definition. A sequence of elements/,,..., fd in the maximal ideal of a local
ring A is a regular sequence if/, is a non-zero-divisor in A, and if, for / = 2,..., d,
the image of/, in Л/(/,,...,/_,) is a non-zero-divisor. (This is equivalent to
asserting that the Koszul complex
Q-+Ad(Ad)->Ad-l(Ad)-+ ••• -+Ad-+A
defined by/,,...,/, is exact, giving a resolution of A/I. In fact, the multiplicity
i(P) may also be defined to be the alternating sum of the dimensions of the
homology groups of this complex, cf. [57].)
The dimension of a local ring A is the length л of a maximal chain of prime
ideals PQ с Р] с • • • с Рп с A. If A is the local ring of a variety V along a
subvariety Wy the dimension of A is the codimension of W in V. The ring A is
Cohn-Macaulay if its maximal ideal contains a regular sequence of dim(y4)
elements. For example if P is a nonsingular point of K, then QP v is Cohen-
Macaulay.
The following lemma contains the main facts from commutative algebra that
we will need. For proofs, see [38 or 57].
Lemma, (a) If A is Cohen-Macaulay, a sequence f{,...,fd of elements in the
maximal ideal of A is a regular sequence if and only if
(b) Let f",,..., fd be a regular sequence in a local ring A, and let I = (/h..., /rf).
Then the canonical homomorphism
A/I[Xx,...,Xd]^ 0/•//•+',
which takes Xt to the image off in I/I2, is an isomorphism. Moreover, the kernel of
the canonical surjection
А[ХХ,...,ХЛ\^ 0/'
r = 0
is generated by the elements fXj - fjXn 1 < / <j < d.
For example, with notation as at the beginning of this section, if вР v is
Cohen-Macaulay, it follows that
i.e., Samuel's sophisticated multiplicity agrees with the naive multiplicity of §1.

INTRODUCTION TO INTERSECTION THEORY 13
2.5. Normal cones. If W is a subscheme of an affine variety V, defined by an
ideal / in the coordinate ring A of V, the normal cone С = CWV to W in К is
defined to be
C=Spec( 0/'//'+l).
The isomorphism of the coordinate ring of W with A/I = I°/Il determines a
morphism/?c: С ~* W, called the projection, and a closed imbedding sc\ W -* C,
called the zero section, withpc°sc= idw. U fd,..., fd generate /, the canonical
surjection of A/I[XU..., Xd] onto ®I'/Il+' determines a closed imbedding of
С in W X C*:
pr
If /,,...,X/ is a regular sequence, it follows from the preceding lemma that
С = Ж X C^. In general, since С is defined by a homogeneous ideal, it is a
subcone of Ж X C^, i.e., С is invariant under multiplication by C* on the fibres
C.
In spite of the marvelous brevity of this algebraic definition of normal cone, i ts
geometry is not so simple. Considerable study, beginning with [32], has been
devoted to the case where W is a nonsingular subvariety. For example, if W = P
is a point, then CPV is the tangent cone to V at P; if V is a hypersurface in Cd, aad
P is the origin, one may check that CFV is the hypersurf ace in Cd defined by the
leading homogeneous term of an equation for V. However, as is evident from the
preceding section, the normal cones of interest for intersection theory are usually
defined by ideals which are not prime ideals, i.e. Ж is a subscheme of K, but not
usually a subvariety. There has been extensive recent study of associated graded
rings 0/'//f+1 in commutative algebra; one hopes that useful criteria for
identifying the irreducible components of C, with their multiplicities, may emerge.
The projective normal cone P(C) = V(CWV) is defined by
In concrete terms, if generators for / are chosen as above, P(C) is the subscheme
of W X Pd~' defined by the same equations that define С in Ж X Cd.
A closely related and equally important construction is that of the blow-up of a
variety V along a subscheme W. This is a variety V — Bl w V> together with a
proper morphism it: V -* V, satisfying:
(i) The inverse image scheme E = ir~'(W) is a Carrier divisor on V, called the
exceptional divisor: at each Q e E, the ideal IQQ у has one generator.
(ii) E is isomorphic to V(CWV), and the mapping from E to Ж induced by it is
the projection from P(C) to W\
i
W

(iii) The induced mapping from V — E to V — W is an isomorphism.
A quick definition of Bl w V is
@0
0 i
/ = 0
the mapping it determined by the isomorphism of A with /°. If /h..., fd generate
/, B\WV is the subvariety of К X P^ defined by the kernel of the canonical
homomorphism from A[Xx,...t Xd] onto 0/'. In case /,,..., fd is a regular
sequence, Bl^ V is defined by the equations/Ду - fjX,t i <j, by the lemma of
§2.4. In general, Bl w V is the closure of the graph of the morphism from V - W
to Pd~] defined by (/,:... :fd).
Note that, since A is a domain, 0/' is also a domain, so Bl^ V is a variety.
The identification of E — ^'{{W) with P(C) follows from the canonical isomor-
isomorphism
Over the subvariety of Prf"' where the coordinate X, is not zero, W is defined by
the equation/,, since^ = (XJ/Xl)]r
One important consequence of this construction is that each irreducible compo-
component of E = P(C) has dimension d ~ 1. Indeed, Я is locally defined by one
equation in the J-dimensional variety V, and any such subscheme has pure
codimension one.
The above constructions globalize to the case of an arbitrary proper closed
subscheme W of an arbitrary variety V. If 9 is the ideal sheaf of W in K, they are
written
0 J'/3'+I),
BlH,K=Proj( 0 }').
They may be constructed by covering V by affine neighborhoods, over which the
preceding constructions apply, and gluing over the overlaps.
In case the imbedding of W in V is a regular imbedding, i.e., local equations for
the ideal of W in V form a regular sequence in local rings of K, then CWV is a
vector bundle, called the normal bundle to W in K, and also denoted NWV. If К
and W are nonsingular, this agrees with the definition of NWV as the quotient of
tangent bundles:
When D is an effective Cartier divisor, on a variety X, NDX is the restriction to D
of the associated line bundle ®X(D) on X. If E = P(C) is the exceptional divisor
on the blow-up F of a variety V along a subscheme W, then
is also the dual line bundle to the canonical line bundle 0CA) on P(C).

It is a useful exercise to examine a normal cone which is not a vector bundle.
For example if W is the intersection of two curves which have common compo-
components in the plane V, then CWV will have irreducible components which lie over
each irreducible component of W, and other varieties as well. If the curves are
written in the form Dx + E, D2 + E, where D{ and D2 meet properly, then CWV
has components over each component of E and over each point in D{ П Dz,
including those points which are in E. To see this, let du d2, e be polynomials in
R = C[X, Y] defining Z),, D2, ?, and set / = (dxe, d2e\ the ideal of W. One
verifies that the kernel of the homomorphism A/I[UX, U2] -» ®1"/1п+\ which
takes Ц to d,e mod /2, is generated by d2Ux - dxU2. Therefore CWV is the
subscheme of W X C2 defined by d2Ux — dxU2, from which one may read off the
components of С Note that these components lie over precisely the subvarieties
of W which contribute to the intersection product by the Severi-Segre-Lazarsfeld
prescription (§1.5).
2.6. Deformation to the normal cone. In light of the principle of continuity, a
reason why one can expect to use normal cones to compute intersection products
is because there is a deformation from the given imbedding of a subscheme W of
V to the zero section imbedding of W in the normal cone С = CWV. The affine
version of this is a closed imbedding
WxC -* M°
pr \ / p
С
with M° = M°WV a variety of dimension one greater than dim(F), such that over
t * 0, the imbedding of W X {/} in M,° is isomorphic to the given imbedding of
W in K, while the imbedding of Ж X {0} in Л/о° is isomorphic to the zero section
imbedding of W in CWV.
Suppose V is affine with coordinate ring A, and W is defined by the ideal
/ = (/,,..., /rf). Let Af be the closure of the graph of the morphism
(V- W) X C* - P*
by (P, t) -> (/,(/»):... :fd(p)'- 0, in V X С X Pd. Note that W X С is imbedded
in A/by
W X С - W X С X {@:... :0: 1)} с А/.
Over / = 0, one sees that the fibre A/o of A/ contains the blow-up V = B\w V, but
this is disjoint from W X @). We shall see that the complement to V in A/o is the
normal cone С = CWV, so that A/° = A/ - К is the desired deformation space.
An algebraic version of this deformation was studied by Gerstenhaber [24]
using the graded ring В defined by
5= ••• e ГТ-" e ••• ® IT~1 ®a®at®--- ®ATn® •••,
i.e., В = е00 Г"Т", with /m = Л for m < 0, and Г an indeterminate. One
may define M to be the affine variety whose coordinate ring is B. The projection

from M° to С corresponds to the canonical inclusion of C[T] in 2?, and the
imbedding of W X С in M° to the canonical surjection of В onto A/I[T]. Since
the canonical homomorphism from A[T] to В becomes an isomorphism after
inverting T:
BTsA[T]T,
the imbedding ^xCcM°-p"'@)is isomorphic to the trivial imbedding of
^xCinFxC. Over T = 0, since
ОС
в/тв= ф in/in+\
we see that Л/о° = CWV, with W imbedded as the zero section.
An equivalent construction of this deformation is to define M to be the
blow-up of К X С along W X {0}. The normal cone to this imbedding is the cone
С 0 1 = Specj 0 /y/"+1 *A/fA/I[T]),
so the exceptional divisor is the protective completion P(C Ф 1) of C, where
С = CWV. The blow-up V = B\w V is also contained in M as a divisor, and if p:
M -* С is the projection to C, then the scheme Mo = p~l@) is the sum of two
Cartier divisors, P(C Ф 1) and K, which meet in P(C). We have a commutative
diagram:
'*! '0
W -> WXC <- Ж
Г Г Г
к = л/, ^ м ^ л/0 = Р(се1) + к
I 1р I
{1} - С - {0}
This last construction of the deformation works equally well for arbitrary
varieties V and arbitrary closed subschemes W of V.
The deformation to the normal cone often functions as an algebro-geometric
analogue of the topologists' tubular neighborhood. Note, however, that even if V
and W are nonsingular, there is usually no neighborhood of W in V which is, even
complex analytically, isomorphic to a neighborhood in the normal bundle. In the

singular case, the normal cone may be a bundle over W even when no neighbor-
neighborhood of Ж in V is topologically a product of W with a disc.
2.7. Intersection products: a preview. Normal cones will be basic to our general
construction of intersection products, even when the intersection is not proper. To
intersect hypersurfaces (Cartier divisors) #,,..., Hd on a variety V, consider the
set-up:
Я, П • • • П Hd -» V
Г fs
HxX--XHd -* VX--XV
To intersect subvarieties K,,..., Vs of a nonsingular variety ЛГ, consider (" reduc-
reduction to the diagonal"):
К, П • • • П И, -> F, X • • • X Vs
Г Г
X - Л'Х ••• X Л'
s
Неге б denotes the diagonal imbedding.
Each of these is a special case of the situation:
W ^ V
Г Г
x -> у
/
Неге К is an л-dimensional subvariety of У, and /: X -» У is a regular imbedding
of codimension d, i.e. A' is locally defined in У by a regular sequence of d
elements; and W is the intersection scheme X П V, the subscheme of У defined
locally by the equations for X and for V. Our goal is to construct and compute an
intersection product X • K, which will be an equivalence class of cycles of dimen-
dimension n — d on X (in fact on W).
Since/is a regular imbedding, the normal cone to # in У is a vector bundle of
rank d on X, denoted NXY. There is a canonical imbedding of CwV'm NXY:
cwv -
i
w
i
X
In fact, if V and У are affine with coordinate rings A and B, and W and X are
defined by ideals / and /, respectively, then JA = /, so there is a surjection
which corresponds to the imbedding of CwVin NXY.
We saw in the previous section that С has pure dimension n = dim(K). By the
procedure of §2.1, С determines a cycle [C] on NXY. The class X- V will be
constructed by "intersecting the cycle [C] with the zero section of NXY".
Explicitly, X • V will be represented by a cycle of the form Ел,[^], where Vt are

18 WILLIAM FULTON
(at - d)-dimensional subvarieties of X, and [C] is (rationally) equivalent to the
pull-back cycle Y/ni[NxY\yy (The next few sections will explain these terms.)
In fact if [C] = Em, [C,], with C, the irreducible components of C, and Z, is the
support of the cone C,, then the intersection of C, with the zero section is a
well-defined cycle class a, on Z,, and
Х- К = Х>,а,.
We shall see that in the case of hypersurfaces considered in §1.5, these Z, are the
distinguished varieties found by Seven and Lazarsfeld, and the contribution /(Z,)
is simply the degree of m,a;.
In case the intersection is proper, i.e. dim W = n — d, then X • V is a well-
defined cycle on W. If W{,..., Wr are the irreducible components of W, then
so X • V = LnijlWj]. The coefficients m, agree with the multiplicities discussed in
§2.4 in case n = d.
In the opposite extreme, when V = X, so W — X, then CWV = A' is the zero
section of NXY. The intersection of the zero section with itself will be the "top
Chern class" of NXY, and we will have the self-intersection formula
X-X=cd(NxY)n[X].

3. Divisors and Rational Equivalence
3.1. Homology and cohomology. Before beginning to develop a theory of
rational equivalence, Chern classes, intersection products, etc., let us look to
topology for a model.
A complex projective variety X has homology groups HqX and cohomology
groups HPX. Homology is со variant, cohomology contra variant. There are cup
products
HpX в
and cap products
which make H*X= фНрХ a skew-commutative graded ring, and H*X =
ШЯ^ап Я*^-module. If /: X - Y is a mapping, /*: H*Y-> H*X is a
homomorphism of rings, and one has the important projection formula,
/•(/•« П*)- an f*b,
Any /c-dimensional subvariety V ol X determines a class denoted cl(K) in
H2kX. This follows for example from the fact that К can be triangulated to be an
oriented 2/r-circuit. If/: X -» У is a morphism, and/(K) = W, then
where deg(V/W) is defined as follows. If dim(W) < dim(K), then deg(V/W) =
0. If dim(W) = dim(K), there is an open W° с FFsuch that Kn/^')-» W°
is a finite sheeted topological covering, and &q%(V/W) is the number of sheets of
this covering; algebraically, deg(V/W) is the degree of the function field R(V) of
К as a field extension of R(W).
If X is nonsingular of dimension л, capping with d(X) gives the Poincare
duality isomorphism
ncl(X)
HX
19

In the nonsingular case, ЯД = Я* X has an intersection product. If A and В are
subvarieties of X of dimensions a and 6, one may therefore define A • В in
H2a+2h_2n(X). This product may be refined by using relative groups:
and similarly clE) e Я2л~26(^, # - В), so
(Я4л-2а-26(*, *- (Л П
and thus A - В lives in H2a+2b_2n(A П 5). Note that if the intersection is
proper, this last group is free on the classes of irreducible components of Л П В,
so this determines A • В as a cycle; in other words, this gives a topological
construction of intersection multiplicities.
Complex vector bundles Ein X have Chern classes c,(?) in Я2'*, satisfying:
(i)co(?)= l;c,(?) = 0if/> rank(?).
(ii) If/: Y -+X,c,{f*E) = /%(?).
(iii) If 0 -* E' -+ E -* E" -» 0 is an exact sequence, then
(iv) If L is an algebraic line bundle on X and s is a section of L with
zero-scheme Ds * X, then
where cl([DJ) = Em,.01B),) if [Ds] = Ет,.Д. is the cycle of Ds.
For a line bundle L e Я^Л',©^), c,(L) e H2(X,Z) may be constructed as
the coboundary of L from the exact sequence
2 it/ exp
0 -> Z -> 0л -> 0* -> 0.
For a vector bundle E of rank r, let P(?) be the projective bundle of lines in E,
with projection/?: P(?) -> X and universal (tautological) exact sequence
0 -> LE -> /?*? -> Q? -> 0
with L? a line bundle. The line bundle 0?A) is dfined to be the dual of L?. Let
I = c,@(l)). Following Grothendieck [28], one may then define the Chern classes
of E by the identity
(Such an equation exists by the structure of Я*Р(?) as an Я*Х-а^еЬга.)
The total Chern class c(E) is defined to be 1 + Cj(?) + • • • + cr(?). The total
Segre class of E is the formal inverse:
s(E) = c(E)-\
so sQ(E) = l,.s,(is) — —cl(E)ys2(E) = c,(?J — c2(E), etc. A calculation using
(*) shows that
*,(?) П cl(X) = р*(Г-|+1 П cl(P(?))).

This gives an alternate construction of Chern classes—or at least of their images
in homology: define s({E) П cl(A') by this last formula, and invert formally to
obtain с(Е)Пс\(Х).
For complete (compact) varieties, singular homology is satisfactory. To extend
to arbitrary varieties, Borel-Moore homology, constructed from locally finite
chains, is more appropriate. With this homology, every variety К has a fundamen-
fundamental homology class cl(K).
3.2, Divisors. A Weil divisor on an л-dimensional variety X is an {n — l)-cycle
on X, i.e. a finite formal combination E/iJFJ] of subvarieties of codimension 1. A
Cartier divisor on X is determined by local data consisting of a covering {U,} of X,
and rational functions/ e R(U,)* = R(X)*, such that on overlaps Ц П UjJ/fj
is a nowhere vanishing regular function, i.e., /// G T(U, C\ Ujy 0*). Local data
(Ut\ fl) define the same Cartier divisor if//// e T(U, П UJ9 0*) for all ij.
If D is a Cartier divisor on X given by local data {Un /}, and V is any
subvariety of X, then the functions / for Ц П V =*= 0 are unique up to units in
the local ring &y x. Such a rational function / is called a /oca/ equation for 2) at F.
If V is of codimension one in X, and we write/ = a/6 with a, Ь е 0KJr, we may
define the order of 2) at V by
where / denotes the length; note that since 0K x has dimension one, §y x/(a) and
^V,,v/W nave dimension zero, so finite length. It is not hard to verify that this is
independent of the choice of a and b.
Each Cartier divisor D on X determines an associated Weil divisor, denoted
ID], by
[D]-T,<«dy(D)[V],
the sum over the codimension one subvarieties V of X. The Cartier divisors on X
form a group Div(Ar), the sum D + E of two Cartier divisors being defined by
multiplying local equations for D and E. The mapping D -* [D] defines a
homomorphism
from Div(Ar) to the group Zn_x(X)oi Weil divisors on X
A Cartier divisor D on X determines a line bundle 0B)) - 0x{D). If {[/,, /} are
local data for ?>, the transition functions for 0(I>) from the Ц neighborhood to
the Ut neighborhood are the units///,; thus a section of 0(Z)) is given by a
collection of regular functions s{ on Ц such that
s, = (f,/fJ)-sJ
on U, О Uj. A Cartier divisor D is effective if it is defined by local equations/
which are regular; in this case s, =/ determines a canonical section of 0B)).
Equivalently an effective Cartier divisor is a subscheme of X whose ideal is locally
defined by one equation; this subscheme is the zero-scheme of the canonical
section of 6B)).

Any / G R(X)* defines a principal Cartier divisor div(/). Two Cartier divisors
D and E determine isomorphic line bundles on X if and only if they differ by a
principal divisor. It is not hard to show that any line bundle on a variety comes
from some divisor, so
Div(X)/Principal divisors = ?\z{X).
Note that in topology a Cartier divisor D determines a cohomology class
c,((9(Z))) in H2X, while [D] determines a hornology class d[D] in H2n_2X. One
can show that
Cl(e(D))nd(X) = c\[D].
We will develop a rational equivalence theory with analogous properties. For this
last formula to hold, note that it is necessary that the class of [div(/)] must be
zero for any principal divisor div(/).
3.3. Rational equivalence. For any variety (or scheme) X over any field K, let
ZkX be the group of /c-cycles ЕиД^] on X, i.e. the free abelian group on the
A:-dimensional subvarieties of X. Two /c-cycles are rationally equivalent if they
differ by a sum of cycles of the form
where /| б Л(^)*, with Wt subvarieties of X of dimension к + \. (Strictly
speaking, [div(/)] was defined in the preceding section to be a /c-cycle on Wx\ we
freely use the same notation for the cycles they define on any larger variety.) The
group of /г-cycles modulo rational equivalence on Х'\ъ denoted AkX, and we write
A*X= ф AkX=Z*X/~ ,
where - denotes rational equivalence.
Although the preceding definition is usually simplest to work with, it may be
shown to be equivalent to the following more classical one. Two /г-cycles are
rationally equivalent if they differ by a sum
L»,([W)] - M»)])
with и, integers, Vt subvarieties of X X P1 whose projections to P1 are dominant,
and Vt@) and J^(oo) are the scheme-theoretic fibres of V, over 0 and oo, regarded
as subschemes of X = X X {0} and X X {oo}.
Note that AnX = Z[X] — Z if X is an и-dimensional variety. More generally, if
# is a scheme of dimension n, then AnX is the free abelian group on the
/i-dimensional irreducible components of X.
Iff: X -* У is a proper morphism, the formula
/,[K] = deg(K//(K))[/(K)]
determines a homomorphism/*: ZkX -> ZkY. To have a covariant ("homology")
theory, the following fact is basic:
Theorem. ///: X -* Y is proper, and a and a' are rationally equivalent cycles on
X, then ft0L and /* a' are rationally equivalent cycles on Y.

Thus there is an induced homomorphism, the push-forward
ft:AkX^AkY,
making Ak a covariant functor for proper morphisms.
For example, if A' is a projective curve, and Y = Spec(iC) is a point, the
theorem asserts the familiar fact that, for a rational function r on X,
i.e. r "has as many zeros as poles". Another important case is when X and У are
/i-dimensional varieties, and/is surjective: if r e R(X)*, then
/*[div(r)] = [div(tf (/•))].
Here N(r) e R(Y)* is the norm of r, the determinant of multiplication by r on
the finite dimensional Z?(Y)-space R(X). This formula is a consequence of the
basic lemma in §1.6. The theorem, in fact, can be deduced from these two cases,
cf.[16,§1.4].
In particular, if A'is complete, i.e. the projection p: X -» Spec(A") is proper, the
degree of a zero cycle is well defined on rational equivalence classes. We set
a = deg(a)=/?J)c(a),
x
identifying^0(Spec(iC)) with Z.
There is an important class of morphisms /: X -> У for which there is a
contravariant/?w//-6ac/c/*: AkY -» Ak+nXt where n is the relative dimension of/".
For any /c-dimensional sub variety V of Y, f~\V) will be a subscheme of X of
pure dimension к + и, and we will define
r\v\-[r\v)\.
(Note that f~\V) denotes the inverse image scheme, defined by pulling back
equations for V in Y; its cycle is defined as in §2.1.) This class of morphisms
includes:
A) projections p: Y X T -» У, Г an и-dimensional variety; here p*[V] =
IV XT];
B) projections p: E -^ Y (resp. P(?) -» Y) from a bundle to its base; here
C) open imbeddings/: G -* У, with л = 0, and/*[K] = [К П ?/]. If ?/ is the
complement of a closed subscheme X of У, and / is the inclusion of X in У, the
sequences
are exact;
D) any dominant (nonconstant) morphism from an (n + l)-dimensional variety
to a nonsingular curve.
A class of mappings including these for which this pull-back is well defined on
rational equivalence classes is the class of flat morphisms; f: X -* У is flat if each

24 WILLIAM FULTON
local ring в у x is flat as a module over 0^y, with W ~f{V). This includes all
smooth morphisms. For most applications here, the above examples suffice.
The following proposition is needed to complete the construction of intersec-
intersection product outlined in §2.7.
Proposition. Let E be a vector bundle of rank r on X, pE: E -» X the projection.
Then the pull-back homomorphisms
p*E: AkX -. Ak + rE
are all isomorphisms.
UsE: X -* E is the zero section imbedding, and a is any k-cycle or cycle class on
E, define the intersection of a. by the zero section, denoted ^(a), to be the class in
Ak_r(E) that pulls back to a:
Note that by the proposition, a is equivalent to a cycle of the form ?л,[?|Д and
clearly the intersection of such a cycle with the zero section should be ЕлД^].
In particular, in the situation of §2.7, the intersection class X • V is a well-
defined class in An_d(W), with W = X П V. Indeed, the normal cone С to W in
V determines an л-cycle [C] on the restriction N of NXY to W, and we may set
X-V=s%[C].
As for the proof of the proposition, the surjectivity of p*E follows by a
Noetherian induction argument, using the exact sequence of C) above. The
injectivity, and in fact a formula for the inverse sEi uses Chern classes (§4).
Another elementary operation on rational equivalence is the exterior product
AkX®A,Y^Ak+l(Xx Y)
defined by [V] X [W] = [V X W].
3.4. Intersecting with divisors. If D is a Cartier divisor on X, and a a /c-cycle on
Xy we define an intersection class
D ¦ a e Ak_l(Z)y
where Z is the intersection of the support of D (the union of varieties at which
local equations are not units) and the support of a (the union of varieties
appearing in a with nonzero coefficients). By linearity it suffices to define D • [V]
if К is a subvariety of X. Let / be the inclusion of V in X. There are two cases:
(i) If V is not contained in the support of Z), then by restricting local equations,
D determines a Cartier divisor, denoted /*Z), on V. In this case, set
/>-[K]-[fD],
the associated Weil divisor of i*D on V. In this case D • [V] is a well-defined
cycle.

INTRODUCTION TO INTERSECTION THEORY 2 5
(ii) If К с Supp(-D), then the line bundle 0x(D) restricts to a line bundle
i*6x(D) on V. Choose a Carrier divisor С on К whose line bundle is isomorphic
to this line bundle: 0K(C) = i*®x(D), and set
the associated Weil divisor of C. Since С is well defined up to a principal divisor
on К, [С] is well defined in Ak_x(V).
In case D is an effective Carrier divisor on X, this class D • [V] agrees with the
class D • V constructed in §2.7. In case (i) this is immediate, while in case (ii) it
amounts to the fact that for a Carrier divisor С on a variety V, the cycle of the
zero section [V] is rationally equivalent to the cycle [^"'(C)] in the line bundle
L = 6И(С), with it: L -» V the projection. When С is effective, corresponding to
a section s of L, an explicit rational equivalence may be constructed as follows
(cf. §2.6): let
Z = {(Л (Xo: X,)) e L x P'|X0^(P) = X,P}.
Then Z@) = ir~ '(C), and Z(oo) is the zero section.
In general, because of the ambiguity in case (ii), D • a is only defined up to
rational equivalence. If the restriction of the line bundle 0х(О) to D is trivial,
however, D • a can always be defined as a cycle. Namely, if К с Supp(Z)), s«t
D • [V] = 0. This applies when D is the fibre of a morphism from X to a
nonsingular curve; the cycle D • a is then called the specialization of a.
This intersection product satisfies the formal properties one would expect for a
"cap product". For example:
A) If a ~ a', then D • a = D • a' inA*(Supp(D)).
B) If D - D' is principal, then D ¦ a = D' • a in i4+(Supp(a)).
C) {Projection formula) If /: У -> I is a proper surjective morphism of
varieties, D a Carrier divisor on X, and а а /c-cycle on У, then
/;(/•!>-a)-/>-/,a
in i^.jfZ), with Z = Supp(D) П /(Suppa), and/': /~'(Z) -+ Z the morphism
induced by /. There is a similar compatibility with flat pull-backs.
From A) and B) it follows that the operation product D • a determines
products
?\c(X) ® AkX -+ Ak_x(X).
For a line bundle L and cycle class a we shall write c,(L) П a for this product:
cx(Qx(D)) na = D-a.
This will be the basis for the study of Chern classes in the next chapter.
If D is an effective divisor on X and/is the inclusion of D in X, it follows from
A) that [V] -* D • [V] determines a "Gysin" homomorphism

This is the key to showing that the general intersection product is well defined on
rational equivalence classes. Note that this is a strong form of the principle of
continuity: all such intersection operations, applied to rationally equivalent
cycles, will give classes of the same degree. It also includes the statement that
specialization respects rational equivalence.
In fact properties B) and C) are straightforward to prove. Property A) then
follows from B) and the following basic commutativity law, on which most of the
subsequent theory depends.
Lemma. Let D and E be Cartier divisors on an n-dimensional variety X. Then
D[E] = E-[D]
in An_2(Supp(D) П Supp(?)).
Consider the case where X is a surface, and it: X ~* C2 is a proper birational
morphism which is an isomorphism except over @,0), and Z = тг~ !(@,0)) is a
curve. Let D and E be the inverse images of the two axes С X {0} and {0} X C.
Then D = D' + D" and E — E' + ?", where D' and E' map isomorphically to
the two axes, while D" and E" are supported on Z. In this case D ¦ [E] is the
point where E' meets Z, and E • [D] is the point where D' meets Z. These two
points may well be different, but one knows they are rationally equivalent,
because Z is a connected curve, all of whose components are rational curves.
D'CE]
Although one may fashion a proof along these lines (cf. [4]), there is now a
simpler proof. Roughly speaking, one blows up X along various subschemes to
reduce to the case where D and E are sums and differences of effective divisors D,
and Ej, such that each intersection of Dt with E} is either proper (in which case the
commutativity is easy) or Dt = E} (when it is evident). See [16, §2.4] for details.
3.5. Applications. Let us apply the preceding results to a situation considered in
the first two sections. If Я,,..., Hd are hypersurfaces (effective Cartier divisors)
on an л-dimensional variety X, we may define, for any fc-cycle a on X, a class
Я, ¦¦¦Hd-aBAk.,(Z),
Z = ПН, П Supp(a). Inductively, this class is defined to be
Hr(H2--Hd-«).
The commutativity law says that this product is independent of the order of the
#,.. If к = d, and Z is complete, this class has a well-defined degree, denoted

/Я, • • • Hd • a. When а = [А^, we omit it from'the notation, and write simply
Hx • • • Hd.
Suppose a nonsingular point P on X, rational over the ground field (R(P) = К ),
is an isolated point of intersection of the intersection of n hypersurfaces Нл,... ,
Я„, n = dim(A'). Shrinking X, assume that Я, meet only at P. Let it: X -> X be
the blow-up of X at P, E = P"~! the exceptional divisor. Then
1г*Я, = m,? + G,,
where m, is the multiplicity of Я, at P; the intersection of G, with ? is the
projective tangent cone Р(СРЯ,). We will show that if these projective tangent
cones do not meet, then
Note first that, since local equations for Я, at P form a regular sequence, the
intersection product Hx • • • Hn is the cycle Z(P)[P], with i(P) defined as in §1.6
or §2.4. Let т] be the projection from E to P. Then since П G, = 0,
O-u^G, ¦•• GJ.
= т,*((тг*Я1-т1?)--. (тг*Яп-т„?)).
Now, by the projection formula, т]+(тг*Я, • • • тг*Ял) = Hx • • • Н„ and
ть^Я,, • • • v*Hlk • ?-*) - Hh • • • Я|4 • тьB<-*).
Since the intersection class E"~k = E • • • ? is in Ak(E), т)*(Е"~к) е ЛЛ(Р) = О
if 0 < к < п. Thus the only terms that survive are
0-Я, •••Яп-(-1)лт1 ...*уи(Д").
Now the restriction of 0^(?) to ? = P" is the dual of the bundle 0P«-.A), so
? • E is represented by minus a hyperplane, and E" is represented by (— 1)и~ !
times a point. Therefore
• • Hn = mx • • • т„[Р].
Let us also reconsider the case where three surfaces Hlt H2, Я3 in a nonsingular
threefold X contain a nonsingular curve С as a (scheme-theoretic) component. Let
it: X -* X be the blow-up of X along C, and let ? be the exceptional divisor,
tj: E -» С the projection. Let
ir*^ = ? + G,.
The hypersurfaces G, do not meet in E, and the intersection of the Я, outside С is
represented by the class tt*(Gi • G2 • G3). Expanding as above, and noting that
again t\*(E) = 0 for dimension reasons, we obtain
The last two terms therefore determine the contribution of С to the intersection
product. The problem is to compute tj *(?')•

28 WILLIAM FULTON
We have seen that E = P{N), where N is the normal bundle to С in X. With
this identification, the restriction of 0?(?) to E (which is the normal bundle to E
in X) is the dual of the bundle 0NA). Referring to §3.1, one expects the formulas
= n,(c,FN(l)J П [P(tf)]) = -c,(iV) П [C],
where c,(iV) is \he first Chern class of the bundle N. In the next section we will
develop the necessary theory of Chern classes for rational equivalence.
Combining these results, it follows that the contribution of С to the intersection
product Я, • #2 • Я3 is
(?",)• c-c,(iv)n[c].
For example, if X = P3, and С is a complete intersection of surfaces of degrees
a and b, N = 0(a) 8 0(ft), deg(C) = ab, so the degree of cx(N) П [C] is
(a + b)(ab), and the total contribution of С to the Bezout number is
as found by Salmon (§1.3). With the machinery of Chern classes, one can also
compute the contribution when С is not a complete intersection.
It is similarly possible to work out the contribution of the Veronese V in the
intersection of five hypersurfaces Я, representing conies tangent to five fixed
general conies (§1.4). Blowing up P5 along V as above one has
where G,,..., G5 are hypersurfaces that do not meet in the exceptional divisor E.
Knowing the Chern classes of the tangent bundles to V = P2 and P5, one knows
the Segre classes of the normal bundle NyP5, and it is a pleasant exercise to verify
that
JG{ ••• G5 = 3264.
This approach to excess intersection problems was developed primarily by B.
Segre [56], with related work by Severi and Todd. All were searching for
constructions which would yield invariants of varieties, generalizing the notion of
genus for curves. For a subvariety К of a variety X, let X be the blow-up of X
along V, E the exceptional divisor, tj the projection from E to V. The classes
т\*(Е') were called the covahants of the imbedding of V in X. We shall see that,
up to sign, they are the inverse Chern classes of the normal bundle of V in X, at
least when V and X are nonsingular. For example, Segre constructed the canonical
classes of a nonsingular variety V by applying this construction to the diagonal
imbedding of V in X = V X V; the formal inverse of Lt\*(E') on V X V projects
to the total Chern class of Tv on V.

4. Chern Classes and Segre Classes
4.1. Chern classes of vector bundles. Eventually one wants a contravariant
"cohomology" theory A*X to go with the covariant theory A*X, and Chern
classes of vector bundles on X should lie in A*X. Although such theories exist, at
this time there is not yet a simple geometric construction of such a cohomology
theory. Indeed, it would be extremely useful to have such a theory, perhaps
analogous to Gore^ky's realization of ordinary cohomology via "geometric
cocycles" [26].
At any rate, any such theory should have "capjproducts" A'X <8> Ak X -»Ak_, Xy
and Chern classes ct(E) <= A'X. In particular, a bundle E on X should determine
homomorphisms
с,(Е)П-
к """* к — i '
by а->с((?)Па. In this section we construct such Chern class operations
directly. They will satisfy properties expected from topolojgyjjlj^
For a line bundle L on a variety (or scheme) X, to define c,(L) П а it suffices
to define c,(L) C\[V], for К a sub variety of X. Choose a Cartier divisor С on V
such that the restriction L\Y of L to К is isomorphic to 0K(C), and set
Note that if L = 0x(D), then сl(J:^J^jL^D-oL:^thQ intersection product of
§3.4. It followsTmrrTtHe^discussion of §14that this operation respects rational
equivalence classes, and satisfies the expected formal properties. For example,
there is ^
for/: Y -* X proper, L a line bundle on X, a acyclec]as$ on Y. Similarly we have
a commutativity property
cx(M) П (Cl(L) Па) = Cl(L) П (с,(М) П а)
for line bundles L, M on X, a e A*X. Thus any polynomial in first Chern classes
of line bundles on X—or on any variety that X maps to—operates on A*X. In
addition there are the elementary formulas:
c,(L® Л/)Па = c,(L) Па + с,(М) Па,
c{(L~l) Па = ~cx(L) Па.
29

Now if E is a vector bundle of rank r on X, define Segre class operators.sJ-E),
st(E) П —:AkX-*Ak_,Xt
as follows. Let p: P(?) -» X be the projective bundle of E, 0?A) the basic line
bundle on P( ?), and for a in A k X set
One shows easily that st(E) - 0 for / < 0, and that so(?) = 1 (i.e. sQ(E) П a = a
for all a). Basic properties such as projection formulas and commutativity of
those classes follow readily from corresponding formulas for first Chern classes of
line bundles.
Now we define Chern class operators
cl(E)n_:AkX-*Ak_lX
by formally inverting the Segre classes (cf. §3.1):
i.e., set co(?)= 1. and c,(?) = -*
plicitly:
), c2(E)
j,(?) 1 О
s2(E) st{E) 1
-s2(E), .... Ex-
Properties such as the projection formula
S](E)
S](E)
and the commutativity property
c,(E) П (cy(F) Па)- c/F) П (c,(?) П a)
follow formally from the corresponding facts for Segre classes. Less obvious but
also true are the vanishing property
c,(?) = 0 for/ > rank?,
and the WhdtBS^L^mJon^^lo.
Ci(E)= L с,.(Е')ск{Е")
for an exact sequence 0 -» E' -* E -» E" -* 0 of vector bundles. There are also
formulas for Chern classes of tensor, exterior or symmetric products.
Although we shall not carry out complete proofs of these statements here, a
basic ingredient is ajpliuing-flnpiy?^ an equation among Chern classes of vector
bundles in a given relation with each other is true if:

(i) the equation is valid when the bundles each* have filti&tions by subbundles
such that the quot^m^im^^rejine_bundles, and
(ii) the given relation is preserved by pu]l-back.
This principle is a simple consequence of the fact that for any bundle E of rank
r on X,
is injective, which follows from the fact that sQ(E) = 1. For on P(?), p*E
contains the universal line bundle LE, with quotient bundle QE: repeating the
process on QE yields a composite /: Y -* X of projective bundles, so f*E is
filtered, and /* injects A * X in A * Y.
If E is filtered, with line bundle quotients L,,..., Lr, the vanishing and
Whitney formulas reduce to showing that ct(E) is the /th elementary symmetric
function of c,(L,),..., cx(Lr). For this, one first verifies directly that Псу(Ь() = О
if E has a nowhere vanishing section; one then may apply this to the bundle
p*E ® LE, which gives
from which the assertion follows easily.
As usual we define total Segre and Chern classes by
c(E)- l+cl(E) + c2(E)+---.
These Chern classes may be used to prove the isomorphisms Ak_rX -* AkE for
a vector bundle E of rank г on a variety or scheme X (§3.2). One proves first the
isomorphisms
which take La, to Ес,(<9?A))' П p*an p the projection from P(?) to X. The
surjectivity of this mapping is proved by a Noetherian induction as in the affine
bundle case; injectivity follows by applying operatorsp*{cx(§E(\))j П ), using
the identities that st{E) = 0 if / < 0, and sQ(E) = 1.
The projective completion P(? Ф 1) contains E as an open subvariety, comple-
complementary to the hyperplane at infinity P(E). From the above isomorphism and the
exact sequence
AkP(E) -+ AkP{E 0 1)-* AkE -* 0
the injectivity of Ak_rX -* AkE follows easily. In addition, one derives a formula
for the inverse isomorphism sE: AkE -* Ak_rX. Given a subvariety К of ?, let V
be its closure in P(? Ф 1). Then,
q is the projection from P(? 8 1) to X, and Q is the universal rank r quotient
bundle of q*{E 8 1) on P(? 8 1). (Note that Q has a canonical section which

32 WILLIAM FULTON
vanishes precisely on the zero section of X in ?, multiplying by the top Chern
class should correspond to intersecting with the zero section.)
4.2. Segre classes of cones and subvarieties. Let Ж be a subvariety of a variety
V. If V and W are nonsingular, or, generally, if the imbedding of Ж in К is a
regular imbedding, one has a normal bundle NWV, and one may construct
invariants of the imbedding by using Chern and Segre classes
c,(NwV)n[W] and J,(iV)n[»1.
In the general case, however, one has only a normal cone С = CWV. We shall see
that, although one does not have a general Chern class formalism for cones, there
is a useful notion of Segre class.
We shall define a total Segre class
s(W,V)eA*W
for any closed subscheme Ж of a variety V. If W = V, set s(W, K) = [V].
Otherwise, let V be the blow-up of V along W, let E = P(C) be the exceptional
divisor, and let i\: E -» W be the projection. The /-fold self-intersections E' =
E • • • E of the divisor E are well defined classes in Ak_,(?)» к = dim(F) =
dim(K), by the construction of §3.4. We set
At least in the nonsingular case, the images of these self-intersection E' were basic
for Segre's construction of invariants [56].
Identifying E with P(C), the restriction of 0p(/T) to E is the dual of the
universal line bundle 0CA) on P(C). It follows that Ei = (- ly-'c^U))' П
[P(C)], and hence
Note that this last expression makes sense for any cone С on a scheme W\ under
the assumption that, for each irreducible component С of C, P(C') is not empty,
we define the Segre class s(C) of the cone С by this formula:
For an arbitrary cone C, the cone С Ф 1 satisfies this assumption, and one may
defines(C)tobes(Ce 1).
Since the restriction of 0p(?) to E is also the normal bundle to E in K, another
definition of the Segre class is
This is a special case of the following important formula.

INTRODUCTION TO INTERSECTION THEORY 33
Proposition. Let тт: V -» V be a proper surjective mor, hism of varieties, of
degree d. Let W be a subscheme of V, W = it" \W\ and let r\: W -* W be the
induced morphism. Then
This is easily proved by blowing up to reduce to the case where W and W axe
Carrier divisors, in which case it follows from the formula f*[W] = d[W].
When d — 1, the proposition expresses the birational invariance of Segre classes.
When the imbedding of W in V is regular, e.g. if V and W are nonsingular, it
gives a formula for s(W, V) in terms of Chem classes of the normal bundle of W
in V. When all four varieties are nonsingular, it gives a remarkable relation
among the Chern classes of the normal bundles; when these Chern classes are
known, it can even be used to complete the degree d.
If Z is an irreducible component of W, the coefficient of [Z] in the class
s(W, V) is the multiplicity of V along W at Z, and denoted {ewV)z. If A is the
local ring of V along Z, and / the ideal of Wy one may show that
length(,4//') = {ewV)z(tn/n\) + lower terms
for t » 0 and n = codim(*F, V). In other words, this multiplicity agrees with
Samuel's multiplicity for the primary ideal / in the local ring A. If Z = W, we
write simply e WV. We shall see that other terms in the Segre classes also appear in
intersection formulas.
It is illuminating to apply these ideas to verify the Riemann-Kempf formula.
Fixing a base point on a nonsingular curve С determines morphisms ud: C(d) -» /
from symmetric products to the Jacobian of C. If Wd is the image of C(d\
1 < d < g, and D e C{d) is a divisor, the Riemann-Kempf formula states that the
multiplicity of Wd at the point ud{D) is (8~dr + r), where g is the genus of C, and r
is the dimension of the linear series \D\ of D. Indeed, one knows that \D\is the
fibre ud](ud(D)), and one may calculate that
where h = c,@(l)) on \D\ = Pr. Since ud maps C(d) birationally onto Wd, the
proposition applies, giving the multiplicity as
f
Jp
g"'*r
pr
Segre classes will appear frequently in these notes. In the study of holonomic
°D-modules, important invariants are constructed by intersecting characteristic
varieties in cotangent bundles with the zero section; as we shall see, all such
intersections can be expressed in terms of Segre classes.
It may be pointed out that, for a vector bundle E, some authors' s^E)
correspond to our $,(? v) = ( — lys^E). The necessity of enlarging their scope to
include general cones dictates our convention.

4.3. Intersection formulas. Recall the situation of the basic construction of
intersection products (§2.7):
W ^> V
hi ig
X «-» У
f
with /: X -> У a regular imbedding of codimension d, V an «-dimensional
variety, g a closed imbedding, W = X П V = g~'(X). Let iV = Л*#*У, С = С„,К
the normal cone, which is a closed, «-dimensional, subscheme of N. We have
defined the intersection product
Another description of X • V may be derived from the last formula in §4.1. Let Q
be the universal rank d quotient bundle on P(N Ф 1), and let q be the projection
from P(N e 1) to W. Then
We use the quotation {ot}A for the ^-dimensional component of a cycle or class
a on a scheme. Using the Whitney and projection formulas, we have
This gives a basic
Intersection formula X- V= {c(N) П j(^, V))n_d.
Note first that if Z is an irreducible component of W of the expected dimension
n - d, then the coefficient of [Z] in X • К is just the multiplicity (ewV)z of К
along W at Z. In particular, in the case of proper intersection, i.e. dim W = n — d.
we recover the formula
the sum over the irreducible components Z of W.
Since A* W = (BA + Wh where the Wl are the connected components of Wy one
has a corresponding decomposition for X ¦ V:
(Notation is abused by writing c(N) n s(Wi9 V) in place of c{N\Wt) n j(H^, K).)
If У° is open in У, and X", K°, and W° are the intersections of Y° with A', F,
and W, then the restriction homomorphism from A*W to A + W° takes X • V to
X" • V°. By means of this localizing principle, which follows immediately from
the construction, it suffices to consider the case when W is connected; similarly

INTRODUCTION TO INTERSECTION THEORY 3 5
one may discard any closed sub varieties of dimension less than n — d> without
loss of information.
Consider the case when the imbedding of W in V is a regular imbedding of
codimension d'. In this case the normal cone С is a subbundle of N. The quotient
bundle E = N/C is called the excess bundle. Since s{Wy V) is given by the inverse
Chern class of C, we deduce from the Whitney formula and the above intersec-
intersection formula the
Excess intersection formula X • V = cd_ d, (E) П [ W ].
In case d' = d, i.e. regular sequences locally defining X in Y remain regular
sequences on K, we recover again the formula X • V = [W].
There is a simple but important refinement of these constructions and for-
formulas. The morphism g: V -* Y can be an arbitrary morphism; it need not be a
closed imbedding. Defining W to be the inverse image scheme g~'(X\ h: W -> X
the induced morphism, one still has the normal cone С = CWV imbedded in
N = h*NxY, and X • V e A + W can be constructed by intersecting [C] with the
zero section in N. The preceding intersection formulas are equally valid in this
generality.
Combined with the birational invariance of Segre classes, this allows an
important reduction procedure. To compute X • V, it suffices to find a proper
birational it: V -+ К for which the class X • V can be computed. For then the
class X • V pushes forward to given X • V. Indeed, if W = n~x{W), and x\:
W -* Wis the morphism induced by it, then
This follows from the formula s(Wy V) = t\*(s(W\ V')) and the intersection
formula. For example, if W is regularly imbedded in V' of codimension d\ with
excess normal bundle E = {h^YNxY/N^Vy then
One may always reduce to this case, with d' = 1, by taking V to be the blow-up
of V along W. Thus many difficult problems can be reduced to the case of
divisors and Chern classes.

5. Gysin Maps and Intersection Rings
5.1. Gysin homomorphisms. If /: X -> У is a regular imbedding of codimension
d, we define Gysin homomorphisms
f*:AkY->Ak_dX
by the formula/*(?л,[^]) = Ln,(X • КД where X • Vt e Л*.,* is the intersec-
intersection product constructed in §§3.3 and 4.3. Verdier's proof that this formula
respects rational equivalence [59] uses the deformation to the normal bundle to
reduce to the known case where d = 1. It goes as follows. Let N = NXY, and let
M° be the deformation space constructed in §2.6. Let / be the imbedding of N in
M° (as a Carder divisor). The complement of N in M° is identified with У X C;
let у be the inclusion of Y X С in M°. Consider the diagram:
Л+itf ^ ^+IM° - Ak + 1{YXC) ^ 0
i'i - Tpr'
AkN —- AkY
a
Here /'* is the Gysin ЬототофЫзт defined for divisors in §3.4. The row is exact
(§3.3C)), and /*° /* = 0 because the normal bundle to N in M° is trivial. Hence
there is a specialization homorm^phism a as indicated, with a(ot) = /*g if /*p =
(pr)*a. For a subvariety К of У, with W = V Г) У, it follows that
where C^Kis the normal cone to Win K. One deduces that/* is the composite
which is evidently well defined on rational equivalence classes.
There is a useful strengthening of these Gysin гютотофги^гш. If /: X -» У is
a regular imbedding of codimension dy and g: У -* У is an arbitrary i
form the fibre square
X' f-* Y'
g'l ig
37

38 WILLIAM FULTON
i.e. X' = XXYY' = g~\X). We define refined Gysin homomorphisms
f':AkY'->Ak_dX'
by the same formula f\V) = X • K. (This intersection product X • V was con-
constructed in Ak_d(V П A") at the end of §4.3; as usual we use the same notation
for its image in Ak_dX\) Similar reasoning shows that /! is well defined on
rational equivalence classes.
The main compatibilities of these Gysin homomorphisms are stated in the
following theorems.
Theorem 1. Consider a fibre square
X' U т
g'i is
X -> Y
f
with fa regular imbedding of codimension d.
(a) Ifg is proper, and a e AkY\ then
/*g*a = g;/!a inAk_dX.
(b) Ifg is flat of relative dimension n, and a e AkY, then
g'7*«=/V« ™Ak+n_dX'.
(c) ///' is also a regular imbedding of codimension d\ set E — g'*NxY/Nx.Y'.
Then, for a e AkY\
f'oL = cd_AE)nf'*a inAk_dX\
(d) Ifg is also a regular imbedding of codimension e, and a e AkY, then
g!/*a=/!g'a inAk_d_eX'.
(e) IfFis a vector bundle on Y', then for all a e AkY', and all i,
fic,(F)na) = c,(f"F) nf'a inAk_d^X'.
For example, if / is proper, then (a) and (c) yield the self-intersection formula:
fora e AkX,
f*f*oi = cd(NxY)n* inAk_dX.
Note that if d' = d in case (c), the assertion is that/!a = /'*a.
The proofs of (a)-(e) follow quite easily from facts we have discussed before:
(a) from the proposition in §4.2; (b) from an analogous formula for pull-backs of
Segre classes by flat morphisms; (c) as in the excess intersection formula (§4.3);
(d) is reduced, as in the discussion at the end of §4.3, to the case of divisors,
which is the main lemma of §3.4; a similar reduction is used in (e).

INTRODUCTION TO INTERSECTION THEORY 39
Theorem 2. Let /: X -» Y and g: Y ^> Z be regular imbeddings of codimensions
d and e. Then the composite gf is a regular imbedding of codimension d + e, and if
a G AkZ, then
(g/)*a=/*(g*a) inAk_d.eX.
The equation (gf)* = /*g* also holds when/is a regular imbedding and either
(i) g and g/"are flat, or (ii) gf is a regular imbedding and g is flat.
For example, if p: E -> X is a vector bundle of rank r, and s: X -+ E is a
section, it follows from (ii) that s*: AkE -* Ak_rX is the inverse isomorphism to
p*\ in particular, s* is independent of the choice of s.
The theorem and the variations stated after it are also valid for the refined
Gysin morphisms. If h: Z' -»- Z is any morphism, and ae^Z', then
with A" = X X z Z'. The theorem is straightforward when Z is a vector bundle
over У, and g is the zero section. The general case is reduced to this by a
deformation to the normal bundle, cf. [59]. We refer to [16, §§6, 17], for the
general statements and complete proofs.
5.2. The intersection ring of a nonsingular variety. If X is an л-dimensional
nonsingular variety (i.e., smooth over the base field), then the diagonal imbedding
5 of X in X X X is a regular imbedding of codimension n. Given абЛДапс!
P e AbX, a product a • p e AmX> m = a + 6 — л, is defined by
a-p = 5*(aX p).
Thus the product on Л * A' is the composite
Note that if a e ЛаК and P e ,4ftW, with K, W closed subschemes of A', then the
product a • p has a natural well-defined refinement in Am(V П W), namely
5!(ot X p), with 8! the refined Gysin homomorphism constructed from the fibre
square:
V П W -» VXW
i i
X -* XXX
8
All the formulas of this section are valid for such refinements, but for simplicity
we write them only in the absolute case.
Define APX to be An_p X. Then the product is a homomorphism
Let 1 e A°X correspond to [X]

40 WILLIAM FULTON
If/: У -> X is a morphism of nonsingular varieties, then the graph morphism
yf:Y-+ YX X
is a regular imbedding of codimension n = &\m(X). Define/*: Л'Л' -> APY by
the formula
Theorem. For X nonsingular, the above product makes A*X into an associative,
commutative ring with unit 1. For a morphism f: Y -* X of nonsingular varieties,
the homomorphism /*: A*X -> A*Y is a ring homomorphism. If also g\ Z -* Y,
with Z nonsingular, then (fg)* — ?*/*•
The theorem follows quite readily from the general properties of intersection
products summarized in §5.1. For example, to prove the associativity of the
product, consider the fibre square:
X i XX X
5| I5x 1
XXX -» XX XX X
1X8
Given cycles a, p, у on X, the equality ot-(p-Y)=(ot-p)-Yis equivalent to the
formula
8*A X 5)*(<x X p X y) = 5*F X l)*(<x X p X y).
This follows either from Theorem l(d), (c), or from Theorem 2. We refer to [16,
§8] for details and refinements.
The formula for /* also makes sense when /: У -» X is any morphism, with X
nonsingular. More generally, one may construct "cap products'*
A'X » AJ-* Aq_fY
by defining/*ot П p, or p -jot, to be ty*(P X a). This makes A + Y into a module
over A*X, and one has the projection formula f *(/*ot О p) = /*(ot) П P, or
in case/is proper.
If Y is nonsingular, and a subvariety X of У is regularly imbedded in У by an
inclusion i, then for any cycle a on У,
a- [X] = /V*(a) in^*y.
More generally, a • [X] = r(ot) e A*(X П Supp(a)). The commutativity prop-
property (Theorem l(d)) is used to prove this.
Since the seminar of Chevalley [10], the intersection ring A*X has been known
as the Chow ring of X. The construction in that seminar was for nonsingular
quasiprojective varieties over algebraically closed fields, and was based on a

INTRODUCTION TO INTERSECTION THEORY 41
" moving lemma". Chow's work in turn was inspired by ideas and constructions
of Seven, many of whose papers were devoted to intersection theory. B. Segje,
Todd, Van der Waerden, Weil, and Samuel were among the others who studied
rings of equivalence classes of cycles. One feature of the present approach,
following [21], is the elimination of any need for a moving lemma. Our approach
is closest to that advocated by B. Segre [56]; related ideas have been proposed by
many others, including Murre, Mumford, Jouanolou, King, Lascu, Scott, and
Gillet.
If V and W are subvarieties of a nonsingular X, the refined intersection class
[V] • [W] = 8![K X W] is in AJV П W\ m = dim V + dim W - dim X. In
particular, any proper m-dimensional component Z of V П W appears in [V] •
[W] with a positive coefficient, the intersection multiplicity i(Z< V • W\ X). Basic
properties of this multiplicity, such as associativity, follows from the refined
versions of the theorems in §5.1. If К and W meet transversally along a nonempty
open subvariety of Z, it follows from our construction that /(Z, V • W\ X) = 1.
The converse is also true. For this criterion of multiplicity one we refer to [34 and
16] for algebraic and geometric proofs.
Although several of the above-mentioned authors indicated that some of their
constructions made sense on singular varieties, the attempt to bring singular
varieties into the general picture was apparently diverted by the notion that it
should be possible to intersect general cycles on a singular variety if rational
coefficients are used. This is possible on normal surfaces and on quotients of
nonsingular varieties by finite groups. For example, the intersection of two
generating lines on a cone over a plane conic is then one-half the vertex. But, as
Zobel [62] points out, this is not possible in general. If X с P4 is the cone over a
quadric surface Q, any two lines in Q are rationally equivalent in X, since they are
rationally equivalent to generators of the cone. But the cone over a line in Q
meets lines in one family of lines in Q transversally, but is disjoint from lines in
the other family. It is interesting that this same cone is used in the example of
Dutta, Hochster, and McLaughlin [14].
5.3. Grassmannians and flag varieties. In general the computation of the ring
A*X, for a nonsingular projective variety Xy is a very difficult problem. One has
A°X = Z, and AlX = Pic(;O,.but for p > 2, there is little general knowledge of
APX. Mumford [45] showed that, for general surfaces, it is impossible to give A2X
any natural, finite dimensional, algebraic geometric structure. Collino [12] has
calculated A*X for X a symmetric product of a curve, and Bloch and Murre [8]
have done the same for certain Fano threefolds. Such calculations use all the
special geometry of the varieties in question; there are very few general algo-
algorithms.
There is an important class of homogeneous varieties, however, for which the
groups APX are finitely generated, and the rings A*X known, at least in principle.

42 WILLIAM FULTON
The group-theoretic approach is probably most satisfactory (cf. [13, 31]), but we
will give more classical descriptions.
The Grassmannian G = Gd(P") of ^-planes in P" is a nonsingular variety of
dimension (d + \)(n - d). Fix a flag
of subspaces, with a, = dim An and set
u(A0,...,Ad) = { L g G|dimL П A, > /,0 < i < d).
Then й(/40,..., Ad) is a subvariety of G, called a Schubert variety. Its dimension
is
E(e,-i)-?«.-<*(«/+0/2.
/=0 i-0
Its class in AtG depends only on the integers 0 ^ a0 < • • • < ad < и, and is
denoted (a0,..., ad). A notation better suited to codimensions, and also used by
Schubert, is to define, for л - d > Xo > ••• >\d> 0,
where at = n - d + / - X,. Then {\0,..., Xrf) is in ^IX|G, where |X| = EjL0\-
A third notation is ctXq Xj.
In the usual Plucker imbedding of Gd(Pn) in P\ N = (JJJ) - 1, the Schubert
varieties are defined by linear equations. If e0,..., en are points spanning P", and
At is spanned by e0,..., ea , the Schubert variety п = U(A0,..., Ad) has an open
subvariety п° consisting of those linear spaces L which can be spanned by
i?0,..., vd with t>, in Л;, but t>, not in the span of e0,..., ea_{. The "reduced
echelon" form of such a basis identifies fl° with the affine space of dimension
Е(д, — /). The complement U — ft° is a union of smaller Schubert varieties.
An inductive argument, using the exact sequence of §3.3C), then shows that the
classes (aOt...tad) generate A4G. We shall see that they form a free basis. As a
first step toward understanding the ring structure on A*G, consider the intersec-
intersection of classes (ao>---> ad) an(* (fy)>---> bd) of complementary dimension. The
dual class to (a0,..., ad) is the class
(n - ad,n -arf_,,...,rt -fl0).
One has the basic duality:
|@,...,<0 if(*b,...,&rf)
(aQt..., ad) • (b0,..., bd) = < is dual to (a0,..., arf),
VO otherwise.
To see this, one may represent (b0,..., bd) by й( J?o,..., Z?rf), where B( is spanned
by the last points en_b,...y en. The Schubert varieties are then seen to meet
transversally in one point when the classes are dual; otherwise they are disjoint.

INTRODUCTION TO INTERSECTION THEORY 43
It follows that the Schubert classes form a free basis for A*G. Moreover, given
any A>cycle a on G, its expression in terms of this basis is described as follows.
For each class (bOi..., bd) of codimension /c, set
^ о
One may use this principle to calculate general products. The reader is invited
to work out /4*G,(P3) this way. In this case Schubert used a special notation:
1 = B,3), g = A,3), gp = @,3), ge = A,2), gs = @,2), and G = @,1) form a
basis, and
g2 = gp + ge> g- gp = g- ge = &»
To see the geometry behind the first equation, note that g2 is represented by the
variety of lines in space meeting two general lines. Moving the lines so that they
meet, this variety degenerates to the union of the variety of lines through the
point of intersection and the variety of lines in the plane of the two lines. Such
arguments, standard in classical enumerative geometry, must be fortified with a
verification of the multiplicities of intersection; for this one may intersect both
sides with a dual basis. Other techniques will be discussed in the next chapter.
With this one may calculate that g4 = 2G: there are two lines meeting four
given lines in general position. If С is an irreducible curve of degree d in P3, and
KC = {/GG,(P3)|/meets C),
then [Vc] = dg. To verify this, one checks that [Vc] • gs = dy since there are d lines
through a general point in a general plane which meet С It follows that there are
2FIdeg(C,) lines meeting four curves C,,..., Q in general position. One may
similarly count the number of common chords to two space curves, and many
other similar problems.
From the fact that the projective linear group acts transitively on Gd(P"), one
may deduce that, after putting varieties in general position via translations by this
group, all intersections will be transversal, so that the naive geometric number
agrees with the intersection-theoretic multiplicity, at least in characteristic zero
[37].
There is a similar description for a general flag manifold, dating from Ehres-
mann [15]. For 0 < d} < d2 < • • • < dr < n, let F = F(dx,..., dr\ n) denote the
flag manifold whose points are flags of subspaces
L. с ••• cLcP"

with dim L, = d(. Fix e0,..., en spanning P" as before. The Schubert varieties in
F are described by an array with r rows
where each row is an increasing sequence of integers between 0 and л, and each
row is a subset of the next. The Schubert variety consists of all flags L, с • • • с Lr
such that L, satisfies the Schubert condition prescribed by the ith row, with
respect to the standard flag. The dimension of this variety is
?(д, - /) + Yj'(bi - 0 + • • • + ?'(ci - 0»
where the primes denote that only trjose terms not counted in the preceding row
are included. The classes of these cycles form a basis for A^(F), and the dual of
such a class is obtained by replacing each row by the dual Schubert condition.
5.4. Enumerating tangents. Let / = F(Q, d\ n) be the incidence variety of points
on d-planes in P". Then A*(I) has a basis of classes of the form (a0,..., ak,...,
ad). Here (a0,..., ad) is a Schubert condition for ^/-planes; if Ao с • • • с Ad is a
fixed flag, with dim Ax = at, then (a0,..., ak,..., ad) is the class of the variety
{ (P, L) e /|dim LC\A^ i, 0 < i < d, and P e Ak),
whose dimension is L(a, - /) + /c. The dual class is
(л - ad,..., n - ak,..., n - i
Let К be a sub variety of Pn of codimension e < d + 1. Let V с / be the
closure of
{ (P, L) e I\P e Kreg, dim(L П TPV) > d - e + l}.
Here Kreg is the nonsingular locus of V, and TPV is the tangent (л — e)-plane to V
at P. Then F is a subvariety of / of codimension d + 1, which measures the
pointed d-planes that touch V. For many enumerative problems involving tan-
tangents, it suffices to compute the class [V] in Ad+'(/).
If M is a linear subspace of codimension d - к + 1, then the class of W is one
of the basic Schubert classes, which we denote \t.k:
= (n - d - 1,л - */,..., n - d+ к - \,n - d + к + 1,..., л - 1, л].
By calculating intersections with dual classes, one verifies that

where ml is the /th class of V\ namely mt is the degree of the closure of
{ P s Kreg|dim ТРУП A>i- l},
where A is a general (n — e - i — 2)-plane.
For concreteness, consider the case where d = 1, n = 2, and С = К is a plane
curve. Then
[C] = nv + m[ji,
where a? = m0 is the degree of C, m = m, the class of С (§1.2), and
v = fA, = [{(P,/)|/isafixedline}],
И> = ^0= [{ (Л/)|Р is a fixed point}].
With this, one may calculate the number of curves in a given r-parameter
family of curves which are tangent to r given curves in general position. Let
в - {Q,er be an г-dimensional family of plane curves. The characteristics \i'vr~'
of the family are the numbers
jxV~' = #{ t\Ct passes through i general points
and is tangent to r — i general lines}.
Given r curves C,,..., Cr in general position, let n{ = deg(C,), mt = class(Q).
Then the number of curves in the family tangent to С„..., Cr is
This is evaluated by expanding formally, and substituting the characteristics for
each jaV"'
For example, if G is the family of all plane conies, then jx5 = 1, \i4v = 2, and
|x3v2 = 4, as one sees by the fact that the condition to be tangent to a line is a
quadric in P5. For the others one has the Veronese as an excess component (cf.
§1.4), but one may conclude by the duality of conies that \t.'vJ = \t.Jv'\ so
|x2v3 = 4, |xv4 = 2, v5 = 1. Thus if C,,..., C5 are conies, one computes
B[a + 2vM = 3264
conies tangent to five given conies in general position.
It should be pointed out that computation of characteristics can be very
difficult. For the family of all curves of degree > 5, apparently no one has even
guessed what the answers should be.
On the other hand, the above tangency formula is easy to prove, including the
generalization to arbitrary dimensions. Let 6(r) be the closure in / X • • • X / X T
(with r copies of /) of the set
{ (P,,/,) X ••• X (Pr,/r) X /| each P{ is simple
on Ct and /, is the tangent line to C, at ?,-}.

46 WILLIAM FULTON
Consider the projection
/:(?(r) -+ / X ••• X / (rcopies).
Using transversality, one sees that the desired number is the degree of the
intersection class
constructed as in §5.2. Writing out the classes [C/] = m{\i + л,у, the conclusion
follows. Note that, by transversality, any lower dimensional subset of T may be
discarded or added without changing these numbers; in particular, one may take
Г to be a projective variety.
One may realize the equation [C] = m\i + nv geometrically by deforming С to
an л-fold line / via projection from a general point Q:
The condition to be tangent to С deforms to n times the condition to be tangent
to /, plus the sum of the conditions to pass through the points P, where tangents
from Q to С meet /. With this approach the intersection theory can be carried out
on the original parameter space. The essential point is that, for generic such
deformations, the contribution of the "Veronese" of multiple curves remains
constant: no solutions enter or leave this locus of degenerate solutions at either
end of the deformation. For details and other approaches see [16 and 18].
It may be pointed out that the basis for A+(I) used here and in other
enumerative problems, is not the basis one obtains by realizing / as a projective
bundle over Gd(P")y cf. §4.1. The notation for this basis follows Martinelli [43].

6. Degeneracy Loci
6.1. A degeneracy class. Let o: E -» F be a homomorphism of vector bundles of
ranks e and/on an л-dimensional variety X. For к < min(e, /), set
Dk{o) = { x e *| rank(a(x)) < Ac}.
This degeneracy locus has a natural structure as a closed subscheme of X, locally
defined by the vanishing of (k + l)-minors of a matrix representation of a. One
expects Dk(o) to be m-dimensional, where
but in general one can only state that each irreducible component of Dk(a) has
dimension at least m. Our object is to construct a class
Dk(o)<EAm{Dk(o))t
to give a formula for the image of T>k(o) in AmX in terms of Chern classes of E
and F, and to investigate when Dk(a) is determined by the scheme Dk(a).
To construct Dk(o), let d = e - k, and let G = Gd(E) be the Grassmannian
bundle of d-planes in E, with projection it: G -* X On G one has a universal exact
sequence
with rank S = d, rank Q = /c and EG = тг*?. The composite S -* EG^> FG de-
determines a section, denoted .ro, of the bundle S v® 7^. The zero scheme Z(^o) of
this section projects onto Dk(o); let
be the morphism induced by тт. If s0 is the zero section imbedding of G in
S v® i^, one has a fibre square:
Z(so) -* G
4 4 Jo
G ~* S ® jT^
^«
Since .ro is a regular imbedding, we may construct the refined intersection class
•*o[G] G Am(z(so)Y> note thatm = dim(G) - rank(,SfV«> FG). Set
47

Because T>k(o) is constructed by a succession of our intersection operations, it
is compatible with other such operations, e.g. by pull-backs by flat morphisms or
regular imbeddings. In particular, Dk(o) may also be constructed by pull-back
from a universal case. Let
#= Hom(?,F) = Ev® F,
a bundle over X. Inside H there is a subcone Dk consisting of mappings of
rank < k. Locally Dk is a product of X and the variety of e X / matrices of
rank < k\ the latter variety is known [33] to be a reduced, irreducible Cohen-
Macaulay variety of the expected dimension ef - (e - k)(f - k). Giving a
morphism o: E -+ F corresponds to giving a section ta of H. Then Dk(a) —
Note that the assertions about the dimension of Dk(o) follow from this statement.
In addition, it follows that
Dk(o) = [Dk(o)]
precisely when depth(Z\(a), X) = codim(Dk(o), X) = (e - k)(f - k)\ this
means that for all x e Dk{o\ the ideal of Dk(a) in &x x contains a regular
sequence of length (e - k)(f - k). If X is Cohen-Macaulay, e.g. nonsingular, this
is equivalent to Dk(o) having the expected codimension. Without this depth
condition, even if Dk(a) has the right codimension, Dk(a) will be a cycle whose
support is Dk(o) but whose coefficients are smaller than those in [Dk(o)].
It remains to compute the image of Dk(a) in AmX. By the theory of §5, one has
in AmG. The required class is then the image of this class in AmX. The answer, to
be verified in the next section, is the Giambelli-Thom-Porteous formula:
Here l^^(c) denotes the determinant of thep by p matrix
Cq Cq+\ "' Cq+p~\
Cq-\ Cq Cq+p-2
Cq-p+\ Cq
and c(F- E) = c{F)/c(E) = c(F) ¦ s(E).
This formula yields a geometric construction for the Chern classes c,(F) of a
bundle F of rank /. Let e = f - i + 1, and let E be the trivial bundle of rank e.
Then o: E -» F is given by e sections slt..., se of F, and Df_t{o) by the locus
where these sections become dependent. Then Df_,(a) is a class in An_l(Df_i(a))
which represents c,(F) П [X], since ^(ciF - E)) = c,(F). If F is generated by
its sections, and $,,..., se are chosen generically, then

INTRODUCTION TO INTERSECTION THEORY
49
6.2. Schur polynomials. For any
any commutative ring, and any
define Дх(с)t0 be det(cx§+,_,), i.e.
.2. Schur polynomials. For any formal series c
any commutative ring, and any finite sequence
ne Дх(с)t0 be det(cx+,_,), i.e.
formal series c=l+c,+c2 + --- with ct in
finite sequence X = (Xh...,X?/) of integers,
\, x
det
c\d-d+\
Note that adding a string of zeros to X does not change Дх(с). Usually we will
assume X is a partition, i.e. X, > X2 >
0, so X partitions |X| = EX .
Then Дх(с) is the Schur polynomial corresponding to X. If one represents X by a
Young diagram
with X, boxes in the /th row, the conjugate partition is obtained by interchanging
rows and columns. A basic formal identity is
(О ДДс) = <
if |jl is the conjugate partition to X. Another is
where the sum is over all |л = (\l,,..., \id+,) with
and |[jl| = |X| + m. More generally there is a Littlewood-Richardson rule for the
coefficients NX[ipo( an arbitrary product [41]:
Some useful formulas for top Chern classes can be expressed in terms of Schur
polynomials:
where/ = rank F, and the subscript is repeated e = rank E times.
(") ctop{S2E)^
(iii> ctop(A2?) =
Here S2E and Л 2E are symmetric and exterior powers, and e = rank E [39].
We may use (i) to complete the proof of the Giambelli-Thom-Porteous formula.
It suffices to verify that, with the notation of §6.1, and any sequence X =

ir*(Ax(c(Fc - S)) П [G]) = k^ciF- E)) П [Л'],
where\л - (X, - /с,..., X^ - k).x
Note that Ax(c(Fc - 5)) = ДхМ^с ~ fg) * C(Q))- Expanding this determi-
determinant, one is reduced to showing that, for a e A * X,
if/, = ... =id = k,
otherwise.
See [36,3, or 16] for details.
Schur polynomials have been used by Navarro Aznar [47] to define local
invariants of a coherent sheaf fona variety A" at a point x. Let r be the generic
rank of $\ and X a partition with r^\,^\2^---. Shrinking X if necessary,
one can find a birational proper map it: X -* X such that the quotient of i
its torsion subsheaf is the sheaf of sections of a vector bundle E. Define
It follows from the birational invariance of Segre classes (§4.2) that this is
independent of choices. When Ф is the sheaf of differentials, these classes were
studies by Le and Teissier. MacPherson's local Euler obstruction is an alternating
sum of some of these invariants.
6.3. The determinantal formula. There is a similar formula for a more general
determinantal locus. Let o: E -* F be a vector bundle homomorphism as before,
and let_Kbe a flag of subbundles of E:
О с К, с • • • с Vr с Е.
Let v, = rank Vn X, = / - v, + /, m = dim(X) - EX,. Set
Й(К; a) = { x e X\ dim(Ker(a(x)) П V,(x)) >/,Ui< r).
A similar construction to that in §6.1 constructs a class U(V) a) in Anl(U{V: a)),
with analogous properties. The determinantal formula states that the image of
V\ a) in Am(X) is the cap product of
det
-V2) cX2(F-V2)
- K) cK(F-Vr)
with the fundamental class [X]. As in §6.1, the proof of Kempf and Laksov [36]
carries over to arbitrary varieties, cf. [16, §14],
This applies to the Grassmannian X= G = Gd(P") = Gd+{(E)y with E a
vector space of dimension n + 1, and a the canonical projection from EG to the
universal quotient bundle Q. A flag of subspaces Aoa ••• сЛ^сР" corre-
corresponds to a flag Vo с • • • <z Vd<z E of subspaces, with dim V, = dim A, + \. The
Schubert variety is the corresponding degeneracy locus

Set a, = dim(^j), \, = n - d + i - ar The determinantal formula then yields
Giambelli 's formula:
For example, the m th special Schubert class
°m ~ (m) ~ [{ L G ^1^ meets a given (n - d - m)-plane}]
is equal to cm(Q), for m = 1,..., n — d.
The formula B) of §6.2 becomes Fieri's formula
the sum over n - d > \i0 > Xo ^ • • • > \t-d> \d> 0 with L[a, = EX, + m. Note
that <|jl0, ..., \ir) = 0 if \i0 > n - d or r > d, corresponding to the facts that
c,-(G) e 0 for i > л - d and ^BV) = <^,('Sf) = 0 for / > d. Similarly the Lit-
tlewood-Richardson rule specializes to a general formula for multiplying Schubert
classes.
If X is a nonsingular subvariety of P" = P(?), there is a canonical vector
bundle homomorphism
a: E ® 0^A) -* NxPn
which is the composite of the quotient maps E <8> 0A) -» TP» on P", and
Tp»\x ~* Nx^n-If flags .4, if are chosen as above,
fl(K; a) = { x e Л1 dim TXX П Л, > i, 0 < i < i/}.
The degree of this locus is theprojective character X(a0,..., ad). The determinan-
tal formula gives a formula for these extrinsic invariants in terms of the intrinsic
Chern classes of Tx> and a hyperplane section. The classes and ranks of X (cf. §1)
are special cases, corresponding to partitions X = A,.".., 1,0,..., 0) and their
conjugates X = (/, 0,..., 0).
6.4. Symmetric and skew-symmetric loci. There are similar formulas for bundle
maps a: Ev-> E which are symmetric (ov= a) or skew-symmetric (ov= — a).
Such correspond to sections ta of S2E or Л 2Е. The locus Dk(a) is defined as in
§6.1, but now its expected dimension m is, for к < e — rank( E),
m = dim(X) - (e ~ ^ + *) (symmetric),
m = dim( A") -r" ) (skew-symmetric, к even).
There are classes denoted DJ(a) or D"(o) in Лт(/)Л(а)) in each of these cases.
The analogous formulas are
These formulas also date from Giambelli; modern versions have been given by
Barth, Tjurin, Jozefiak-Lascoux-Pragacz, Harris-Tu, and Damon. A particularly

52
WILLIAM FULTON
simple treatment has recently been given by Pragacz. The calculations depend on
calculating Gysin push-forwards for «rr: Gd(E) -> X as in §6.1. All such push-for-
push-forwards are known "in theory", but it requires ingenuity to find useful general
formulas. One such [35] is, for any X = (X,,..., ХД v = (v,,..., vk), а€ЛД,
where[jl = (X, - /с, Х2 - fc,..., \d - /c, v,,..., v*).
For applications, cf. Harris-Tu [29], one needs generalizations to symmetric or
skew-symmetric bundle maps a: Ey -* E ® L, for La line bundle on X. Lascoux
and Pragacz also makes these formulas explicit, as follows. Given partitions
X = (X,,..., \e) and |jl = (fA,,..., \ie)y say that |jl < X if \it ^ X, for 1 ^ / < e,
and define
X, + e - i
Set t = (d,d
e = rank(?),
- 1,..., 1) and & = (</- 1,</-2,..., 1). Then for d = e - k,
D?(o) -
Here X is the conjugate partition of X. When L = M®2, they follow from the
preceding cases and the formal identity [39]
^(c(e * м)) = E ^Л
The case for general L can be deduced from this case.

7. Refinements
7.1. Dynamic intersections. Consider our basic intersection theory setup
W <-> V
i i
X ^ Y
f
with / a regular imbedding of codimension d, V an л-dimensional variety,
W= X П V. The normal cone С = С^К is imbedded in the normal bundle
N = NXY to * in Y. Let
[c]-E*,[c,]
be the cycle of C. Each irreducible component C, of С is a subcone of N\ let
Z, = ^ \C,) be the support of C,. Let ty be the restriction of NtoZ,. Then C, is a
subvariety of Nn and we may set
By construction, the class Lm,a, represents the intersection product X • V in
An_d{W). Whenever some Z,*= W there is more information in the classes a,,
with their multiplicities m,, than in the class Em,a, on W. For any closed subset
Z of W, set
Z,cZ
and call (A'- V)z the part of X • V supported on Z.
One way to refine this class further is to have a section s of the bundle N other
than the zero section. Then s'[C,] is a well-defined class onr'fQcZ,, which
re/wes a, (i.e., s'[C,] maps to a, by the inclusion of s~ '(C,) in Z,). Suppose N is
generated by a finite dimensional space Г of sections. One can show that for any
closed Z с W there is a nonempty open F(Z) с Г such that for all j e TjZ),
dim s~\C) = n - d, so s'[C] is a well-defined (n - d)-cycle, and the part of
sl[C] contained in Z is precisely (A' • V)z.
Suppose the imbedding X -* Y is deformed to a family %-* Y X Г of imbed-
dings; we assume Г is a nonsingular curve, % is flat over Г, the imbedding of %
in У X Г is regular, and the imbedding Xo -+ Y X {0} over 0 e T is the given
imbedding. This deformation determines in a well-known way a Kodaira-Spencer
section of the normal bundle N, which we denote by s%.
53

If the generic intersection of X( with V is proper, one may define a limit
intersection cycle lim,_0 Xt ¦ К as follows. Consider the fibre square:
% -+ VXT
i i
% -* YXT
The components of % that project dominantly to T have relative dimension n - d
over T. The intersection cycle % • (V X T) in An_d+l{%) therefore specializes
to a well-defined cycle on the fibre over 0 (cf. §3.4); this cycle is denoted
lim,^o*r* V-
One can show that lim,_0 .Y, • К is supported on s7<[(C\ and that this limit
class refines s][C]. It follows that, if Z is given, then for any deformation % for
which s.x belongs to F(Z),
limX- V = slx[C].
In particular, the part of the cycle \imt^0Xt ¦ V supported on Z represents
(Л' • K)z, for sufficiently general deformations, cf. [40,16].
For example, if X = #, X • • • X 'Hd and Y = P" X • • • X P" (d copies), with
Ht hypersurfaces in P", one may construct such deformations by varying equa-
equations for the #,, as in §1. In case d = л, V = P", it follows that the degree of
(Л' • V)z is the number j(Z) constructed by the Severi-Lazarsfeld method. Thus
the (refined) static construction of intersection products (using normal cones)
yields the same information as the dynamic construction (using deformations).
7.2. Rationality of solutions. In much of our geometric discussion, we have been
tacitly assuming that the ground field is the complex numbers, or at least
algebraically closed. No such assumptions are needed for the basic constructions,
however. If one begins with cycles Lfl,[J^] defined over a given ground field Ky all
our operations can be carried out with such cycles.
The degree of a zero-cycle T,n,[P,] on X is 1>ДД(Р,): K]t where R(P) denotes
the residue field of the local ring of X at P. For a zero-cycle or class a on a
complete variety X over Ky we let /a denote its degree.
Suppose K,,..., Vr are subvarieties of a complete smooth variety X over K,
with E codim( Vx, X) = dim X. Then our construction produces a cycle class
Vx ••• KGA0(nVt)
whose degree is/[K,] •••[Kr].
For example, if К = R, and f[Vx] • • • [Vr] is odd, it follows that П V, must
contain real points. Indeed, a zero-cycle ЕиДР,] on П Vt which represents
K, • • • Vr cannot have all R(Pt) = C. Note that this argument can be used on
each component of П Vr For example, if certain points of proper intersections
are known, their contributions can be subtracted; if an odd number remains,
there are additional real points in П Vr Similarly one may subtract contributions
from components of excess intersection.

INTRODUCTION TO INTERSECTION THEORY 5 5
A pleasant application of these ideas is to a simple -algebraic treatment of the
Borsuk-Ulam problem [2]. Let S" be the sphere X$ + • • • + Xl = 1 in R"+1. If
g,,..., gn are odd polynomials in R[^o> • • •» ^J>tnen there is a point jcgS" such
that all g,(x) = 0. To prove this, for any odd g of degree d, set
where glJ) is the homogeneous part of g of (odd) degree j. By the previous
paragraph, g*,..., g* have a common nontrivial solution (xQ,..., xn). Multiply-
Multiplying by a positive scalar, one may assume this point is in 5", in which case it is the
required solution. It follows that for any n polynomials (or continuous functions,
by approximation), there is a point jcg5" such that each takes the same value at
antipodal points; one applies the preceding to the odd parts of the functions.
For X = P" one may prove such results by using deformations, and the
compactness of P"(R). The approach with refined intersections is simpler; it
works for any X, and gives analogous results for any field all of whose finite
extensions are a power of a fixed prime.
The question of how many solutions of real equations can be real is still very
much open, particularly for enumerative problems. For example, how many o>f
the 3264 conies tangent to five general (real) conies can be real?
7.3. Residual intersections. In our basic situation for constructing intersection
products (§7.1), there may be a distinguished subscheme D of the intersection
scheme W. A natural candidate for the contribution of D to the intersection
product X - V is the classs
m = dim(K) - d, d = codim^, Y). Our object is to construct a residual scheme
R and a class, denoted R, in Am(R) so that one has a
Residual intersection formula
in AmW.
We have a diagram
D Д W ^ V
ig if
X - Y
with / a regular imbedding, W = f~l(X). Assume first that the composite ja
imbeds D as a Cartier divisor on V. The residual scheme R to D in W is trie
subscheme of V whose local equations are obtained by dividing local equations
for Win К by a local equation for D in V\ then W = D U R, with ideal sheaves
on V related by

Set E = g*NxY ®j*®y(-D) = g*NxY <8> (NDV)V. One verifies that the normal
cone CRV is a subcone of the restriction ER of E to R. Then one may define the
residual class R to be the intersection class of the /i-cycle [CRV] by the zero section
of the bundle ER\
With these definitions, the residual intersection formula is valid. To prove it, one
blows up V along Z to reduce to the case where W — D + R as a divisor, in
which case the excess intersection formula applies; see [16, §9] for details.
For example, if the imbedding of R in К is regular of codimenion d\ then
R = cd_d,(E/NRV)n[R].
In particular, if d' = d, then R = [R].
For arbitrary ?>, one can blow up V along D to achieve the situation just
studied. Let it: V-* К be this blow-up, <тт*?> = D and тг~ \W) = W. Let R be the
residual scheme to D in W and R the residual class in AmR just constructed. If
one sets R = ъ(к) and R = it*(R), the desired residual intersection formula
results.
7.4. Multiple point formulas. Laksov developed a version of this residual
intersection formula to prove a double point formula for a morphism/: X -* Y of
nonsingular varieties of dimensions n and m. In this case (/Х/)~'(ДУ) contains
kx\ the residual scheme will be the locus ?>'(/) of double point pairs. If X is
complete either projection from X X X to X maps D'(f) onto the double point
locus D(f) in X The projection of the residual intersection class R is the double
point class, denoted D(/), in Aln_m(D(f)). One deduces from the residual
intersection formula the
Double point formula
Щ/)=Г/*[Х]-(с(ГТг)с(Тх))т_„П[Х].
For example if X is a curve of genus g, Y = P2, and /maps X birationally onto a
curve of degree n, one has the classical formula
degD(/) = ("-0("-2)-2g.
By construction D(/) is constructed from the residual scheme R to the
exceptional divisor P(T^) in the blow-up of X X X along the diagonal Д x. When
this scheme R has the expected dimension In - m, then D(/) is the projection of
the cycle [R]. Note that R may have components inside PG>), as happens e.g. for
plane curves with cusps. When m = n + 1 and / maps X birationally and finitely
onto its image in У, one expects D(/) to be the cycle determined by the conductor
ideal, but this has only been proved for n = 1.
For triple point and higher multiple point formulas the situation is more
complicated; however, when / is a proper immersion which is completely regular
(i.e., for any distinct points xt e X with the same image у ?¦ Y, the images of the
tangent spaces Tx X are in general position in TVY), the answer is quite simple.
Let Yk be the set of points in Y which are the images of к or more distinct points

INTRODUCTION TO INTERSECTION THEORY 57
of X, and let Xk = f~\Yk), both with reduced scheme structure. Then one has an
inductive formula of Herbert:
[хк]=Г[Ук-Л-сЛГТу/тх)(л[хк.х1
d = dim Y - dim X. Indeed, following Ronga, if X% denotes the set of unordered
fc-tuples of distinct points of X with the same image on У, one has a fibre square
к к ~ I ~* к
i i
X -> Y
with Xj mapping birationally onto Xy An application of the excess intersection
formula (§4.3) then yields Herbert's formula.
For more general mappings significant progress has been made, primarily by
Kleiman, Le Barz, and Ran. Their results are most satisfactory when the multiple
points occur only in a "curvilinear" way; they can be used to deduce enumerative
formulas for secant lines to varieties in projective space.
The excess intersection formulas can also be used to study fixed points of
correspondences on a nonsingular complete variety X. If Г is a variety, cycle, or
equivalence class of cycles on X, with dim Г = dim X, then the virtual number of
fixed points is the intersection number /Г • Д of Г with the diagonal Д in X X X.
When X = P", one recovers formulas of Pieri [49]. Conversely, Pieri's work may
be seen as an important precursor of modern intersection theory.

8. Positivity
8.1. Positivity of intersection products. When cycles meet properly their inter-
intersection product will be an effective cycle, i.e. a sum ЕлД^-] with nt > 0. If two
cycles are equivalent to cycles which meet properly, their product is represented
by (equivalent to) a positive (or zero) cycle. For general excess intersections,
however, this is not possible: if E is the exceptional divisor of the blow up of a
nonsingular surface at a point, then jE • E = — 1.
From our construction of intersection products via cones in normal bundles, it
is natural to expect that suitable positivity of the normal bundle will guarantee
the positivity of intersection products.
Recall that a line bundle L on a variety X is ample if some positive power L® n
is the pull-back of 0A) for a projective imbedding of X in projective space. If a is
a &-cycle on X, define
degL(a)=/c1(L)/cna.
For a sub variety V of X, let degL(K) = degL[K]. Since hyperplanes can always
be moved to meet subvarieties properly, degL(a) > 0 whenever a is equivalent to
a positive cycle, i.e. nonzero effective cycle.
A vector bundle E on X is ample if the canonical line bundle 0?v(l) on P(? v)
is an ample line bundle; note that 0?v(l) is a quotient bundle of the pull-back of
E to P(? v). In general ampleness is preserved by direct sums, by tensor exterior
and symmetric products, by passing to quotient bundles, and by pull-backs by
finite morphisms.
To investigate the positivity of intersection products it suffices to consider the
intersection class of an irreducible cone С with the zero section in a vector bundle
E on a variety X. Let a = s?[C] be this intersection class. We assume that
dim(C) ^ rank(is). Fix an ample line bundle L on X.
Theorem, (a) // E is generated by its sections, then a is represented by an
effective cycle.
(b) // E ® Lv is generated by its sections, and Supp(C) = X, then degL(a) >
(c) // E is ample and generated by its sections, then a is represented by a positive
cycle.
(d) If E is ample, then degL(a) > 0.
59

Of these statements, (a) and (b) are quite easy to prove; they follow from the
case where E is trivial, and one may induct on the rank. We refer to [16, §12] for
the proofs of (a)-(c).
The most difficult is (d). For example, when С is the zero section, and
dim X = rank(?) = n, then a = cn(E) П [X]. The assertion that cn(E) > 0 is a
theorem of Bloch. and Gieseker [l]r Their proof, valid-in characteristic zero, used
resolution of singularities and the hard Lefschetz theorem. By using intersection
homology [27] one may avoid resolution of singularities and extend the result to
arbitrary characteristic, but we do not know a more elementary proof of the key
assertion that cn(E) * 0. We do not know, in (d), if some positive multiple of a
can be represented by a positive cycle. For the proof of (d), see [20].
The theorem applies to the intersection products X • V e Am{W) constructed
from our basic construction, provided the pull-back N of NxY to W has the
required positivity. For example, if NXY is ample, and К is a subvariety of Y with
dim К ^ codim^, У), such that [V] is equivalent to a positive cycle whose
support meets X, then V must itself meet X. Indeed, degL(Ar • V) - degL(X • a)
> 0.
If X is nonsingular and its tangent bundle Tx is generated by its sections, it
follows from (a) that all intersections of effective cycles have effective representa-
representatives. Indeed, the normal bundle to the diagonal imbedding of X in X X •-• • X X
(r copies) is the direct sum Tx Ф • • • 0 Tx (r - 1 copies).
If X = P" and L = 0A), then Tx ® Lv is generated by its sections, so (b) holds
for all intersections on P". Let K,,..., Vr be subvarieties of P", and let Vx • • • Vr
be the intersection product constructed from the diagram
Xx
i
X
X
V,
X
i
X
by the prescription of §7.1. If [С] = ЕтДС,] is the cycle of the normal cone to
П Vj in K, X • • • X Vn then we have a decomposition
with a, a class on Z, = Supp(C/). By the theorem, each a, is represented by а
positive cycle, and degta,) ^ deg(Z,) > 0. In particular,
П deg(K) = E*Meg(af.) > E*Meg(Zf).
Note that each irreducible component of П^ appears as some Z,, and each
m, ^ 1, so this refines the Bezout theorem of §2.3.
8.2. Positive polynomials and degeneracy loci. Let a: E -* F be a homomor-
phism of vector bundles of ranks e and / on an я-dimensional variety X. For
к < min(e, /), the expected dimension of the degeneracy locus Dk(o) was seen in
§6.1 to be m = n - (e - k)(f - k).

INTRODUCTION TO INTERSECTION THEORY 61
Proposition. Assume E v<8» F is ample, and m > 0. Then Z)A(o) =*= 0, and [юг
any ample line bundle L on X,
To prove this, let H = Hom(?, F) = ? v<8> F, and let Dk с Н be the cone of
maps of rank < A: (§6.1). Then a corresponds to a section ta of Я, and the
degeneracy class D^(a) is the intersection class t],[Dk]. Since the normal bundle
to ta is Я, which is assumed to be ample, the theorem of §8.1 yields degL(DA(a))
> 0. The Giambelli-Thom-Porteous formula for DA.(o) completes the proof.
A similar construction shows that if E is any ample vector of rank e on an
л-dimensional variety X, then for any partition \ of n with e^X1^X2>">
0,
/дх(с(?))>0.
To prove this one takes a vector space V of dimension n 4- e, and a flag of
subspaces К, с V2 с • • • с К with dim(F;) = e + /-\, Let Я = Нот(К^, Е\
and set
Gx = {ф е Я|ШтКег(ф) П J/ ^ /, / = 1,...}.
Then Я is ample, and a corresponds to a section ta of H. By the determinantal
formula, Дх(с(?)) П [Л'] = r*[fixl' whose degree is positive by the theorem of
§8.1.
Any polynomial P(с,,..., ce) e Q[c,,..., ce] of weight n can be written
uniquely in the form
the sum over partitions X with e^X, ^ ••• ^Х„>0, and EX, = n.
Theorem [20]. Ifax ^ Ofor all X, and some aK > 0, /Лед?
/or a// flw/?/e vector bundles E of rank e on all n-dimensional varieties X. Con-
Conversely, if some ax < 0, there is an ample E for which jxP(c(E)) < 0.
To see the last statement, let К be a Schubert variety representing the dual class
to {X,,..., X,,} in G = G,,(P"+?>). Then
jP(c(Q)) = ax
for Q the universal quotient bundle on G. Let L be a very ample line bundle on
G. For any к > 0 there is a finite surjective morphism /: X -* V such that
f*L = M®A for a line bundle M on X. Then E = f*Q <8> M is ample on *, and if
F{ t) is the polynomial

then fxP{c{E))= deg(X/V)F(\/k)\ this is negative for sufficiently large k4
since F@) = ax < 0.
A similar analysis shows the positivity of products of Schur polynomials in
Chern classes in two or more ample bundles, and the existence of degeneracy loci
D'l(o) or D'ls{o) for symmetric or skew-symmetric bundle maps o: E v -* E <8> L,
when the expected dimension is nonnegative, and Sym2(?) <8> L or Л 2Е <8> L is
ample.
Although we have been using rational equivalence, the natural equivalence for
most of these questions is numerical equivalence. Two cycles a, a' on a complete
variety X may be said to be numerically equivalent if / P П a = / P П a' for all
polynomials P in Chern classes of vector bundles on X. When X is nonsingular, it
follows from Riemann-Roch that this is equivalent to requiring /p • a = /p • a'
for all cycles p on X.
If the expected dimension m of Dk(o) is at least 1, and E v<8> F is ample, then
Dk{o) must be connected [19]. The analogous assertion is open for symmetric and
skew-symmetric bundle maps. These assertions would follow from the general
conjecture that for any ^-dimensional subvariety К of an ample bundle of rank e,
and any section s of K, s~\V) is connected, provided к > e. Note that the
nonemptiness of s~ \V) follows from the theorem of §8.1, for any к > е.
8.3. Intersection multiplicities. Let K,,..., Vr be subvarieties of an «-dimen-
«-dimensional nonsingular variety X meeting properly at a point P (assumed to be
rational over the ground field). Let i(P) = /(?, K, • • • Vr\ X) be the intersection
multiplicity. By shrinking X, we may assume the Vx intersect only at P.
Let tt: X -* X be the blow-up of X at P% E = P"~l the exceptional divisor. Let
Vt с X be the blow-up of Vl at P, i.e., the proper transform. Note that Vt П E is
the protective tangent cone P(CPVt), whose degree in E = P"~' is the multiplicity
ePVt of V{ at ? (§4.2). Note also that the intersection product Vx • • • Kr on ? is a
well-defined class in A0(E). Then
fx ••• Vf.
For two curves on a surface the curves Vx and V2 must intersect properly, and one
may continue blowing up; this results in Noether's formula for i(P) as the sum of
the products of the multiplicities at all infinitely near points. (One can prove (*)
in general using either deformation to the normal bundle to P in X, or the
residual intersection formula.)
In general, the V, need not intersect properly. Since intersections on X can be
negative, it is not so obvious that Vx • • • Vr must be nonnegative. Using the
theorem of §8.1, however, one can show that there is a decomposition
with m, > 0, a, a cycle on a subvariety Z, of П V, = П P(CPV,)t
deg(a,)> degZ,- > 0.

INTRODUCTION TO INTERSECTION THEORY 6 3
The union of the Z, is ПК,. In particular, the "error term" / K, ••• Vr is
bounded below by the sum of the degrees of the irreducible components of
ПР(С^)с E = P"-1.
When the Vt are hypersurfaces the proof is an easy application of the theorem,
since the normal bundle to K, in X is ample on Vt n E. For general V{ the proof is
more complicated, since the normal bundle to the diagonal imbedding of X in
X X • • • X X is not ample near E. See [16, §12.4] for details.

9. Riemann-Roch
9.1. The Grothendieck-Riemann-Roch theorem. If ? is a vector bundle on a
complete variety X, let \{E) denote its Euler characteristic:
Motivated by some ingenious calculations of Todd, Hirzebruch discovered the
Hirzebruch-Riemann-Roch formula for expressing x(?) terms of Chern classes of
? and of the tangent bundle of a nonsingular variety X:
x(?)=/ch(?)-td(rj.
Here ch and td denote the Chern character and Todd class respectively. The
Chern character ch(?) of a vector bundle ? of rank e on a variety X is the sum
ch(?) = e + c, + i(c? - 2c2) + i(cf - 3c,c2 + 3c3) + • • •
= E Pk/kU
where c, = c;(?), and pk is the sum xf +
explicitly.
Pk = det
1 0
2c,
kcL
+ jc* in Chern roots xt of E\
Then ch(?) = ch(?') + ch(?") if 0 -* ?' -?-?"- 0, and ch(? 9 F)
ch(?) • ch(F). Similarly the Todd class td(?) is defined by
td(?) = 1 -h 4c,
2)
They are related by the formal identity
A) L(-l)'ch(A'?v) = c,(?
Note that td(?) = td(?') • td(?") if 0 -> ?'-*?-?"
65

Both x and ch are additive on exact sequences of vector bundles, the former by
the long exact cohomology sequence. Let К °Х denote the Grothendieck group of
(algebraic) vector bundles on X\ it is the free abelian group on isomorphism
classes [E] of vector bundles on X, modulo relations
[?] = [?'] + [?"]
for any exact sequence 0 -* ?' -> E -> E" -* 0 on X. If X is complete, then x
determines a homomorphism from K°X to Z. For a nonsingular complete variety
X4 set K(X) = K°X, A(X) = A*X, and let
r:K(X)-* A{X)q = A(X)®Q
be the homomorphism given by t(?) = ch(E) • td(Tx). So HRR reads: x(?) =
jt(E).
If/: X -> У is a closed imbedding of nonsingular varieties, there is an induced
homomorphism/* from K(X) to А"(У), determined by
if 0 -+ Fm -+ Fm_ ,-*.••-* Fo -*/„?-» 0 is a resolution of the sheaf /„? by
vector bundles Fr Assume X and У are complete, and consider the diagram:
K(X) ^ K(Y) i Z
Л(У) - Q
One sees that to prove HRR on X it suffices to know HRR on У (so the right
square commutes), and the commutativity of the left square. When X is projec-
tive, one may take У = P". Then /C(P") is generated by [<?(/)] for / = 0 n,
and the verification of HRR for these line bundles amounts to a formal identity.
So the essential part of the proof of HRR, for X projective and nonsingular, is to
verify the commutativity of the left square of the diagram. That is, for all
a<= K(XL
(•) ch(/,a) • tdG» = /,(ch(a) • tdG»).
Let N be the normal bundle to X in У. Since td takes sums to products.
By the projection formula, (*) is equivalent to
(••) ch(/,a)=/*(td(N)-l-ch(a)).
Let us first verify (*¦) on a simple example, v.here everything can be
calculated explicitly: A4s an arbitrary nonsingular variety, У = P(N Ф 1), where
N is an arbitrary vector bundle on X, and /: X -+ У is the zero-section imbedding
of X in N, followed by the open imbedding of N in P(N e 1). Let p: Y -* X be

INTRODUCTION TO INTERSECTION THEORY 67
the bundle projection, let Q be the universal quotient bundle on P(N ® 1), and
let d = rank(N) = rank(Q). Let s be the section of Q determined by the projec-
projection of the trivial factor in p*(N Ф 1) onto Q. The zero scheme of s is precisely X.
It follows that for any P e A{Y),
B) /•(/*&)-Э'/•[*]-c,(fi)-P.
The section s determines a Koszul complex
0- Л'(?*-> Arf-'ev-* ••• - Л'?)*-¦ 0y-/,0^-0
which is a resolution oi f+§x. It follows that for any locally free sheaf E on A',
/*? has a resolution
0- AdQv®p*E -* ••• -* Л lQs/®p*E-+p*E-+f,tE->0.
Therefore,
From A), the right side is cd{Q) ¦ td(Q)~l • ch( p*E). Using B), and noting that
f*Q = N and f*p*E = ?, one has
C) ch(/,?W,(td(A0~'-cfa(?))
which proves (* *) in this case. This model also shows, via identity A), where the
Todd classes come from.
To prove (* *) in general, consider the deformation to the normal bundle
(§2.6). In order to have a projective parameter space, we deform from the given
imbedding at 0 e P1 to the normal bundle imbedding at oo e P1; i.e., M is the
blow-up of Y X P1 along X X {oo}:
X
XP1
'oT
X
-*
—>
F
p(n e l) -f
л \
it Sjx
M
у w
- {00}
л р<
T
- {0}
Неге/: Я* -* P(N e 1) is the preceding model. Now let E be any vector bundle
on X. We must show that equation (* *) holds. Let q: M -* Y be the composite
of the blow-down map from M to УХР1 and the projection to Y. Since
qj0 = idr, it will suffice to compute the image of сп(/*.Е) in A(M)Q.
Let Ё = (pr)*? be the pull-back of E to X X P1, and let
be a resolution of F*(E) on M. Since M is flat over P1, it follows thaty^G. is a
resolution of f+E on Y and j*G. resolves f*E on M^. In particular, since У is

68 WILLIAM FULTON
Write ch(/\) in place of L(- l)'ch(F,), for any complex F. of vector bundles.
Using the projection formula, we have
7o.(ch(/,?)) =yo.(chU*G.)) = ch(G.) -jo.[X].
Note that jQ.[X] = kJP(N Ф 1)] + /*[У], since 0 = [div(p)] = [MQ] - [A/J.
Therefore
ch(</.) -jo.[X] = *,(ch( *•</.)) + /*(ch(/*G.))
= *,(<*(/*?))+ 0.
But ch(/*?) was calculated for the model. So we have
7o.(ch(/,?)) - k*(u(td(Ny] • ch(?)))
in /4(M)Q. Applying q4 to both sides yields the required formula (* *), since
On an arbitrary variety Xy let A'oA' denote the Grothendieck group of coherent
sheaves on X. Tensor product makes K°X into a commutative ring, and КоX into
a module over K°X. If/: Y -» X is a morphism, the pull-back of vector bundles
defines a ring homomorphism/*: К °X -* K°Y. If /is proper, there is a push-for-
push-forward
Неге Я7*^аге Grothendieck's Л/g/ier Л/'есг images: R'f + У is the sheaf associated
to the presheaf G -* H'(f~l(U),^). One has the projection formula /*(/*^ <8> P)
= a <8> /„, p for a e A:°X, P e K*Y.
For any A' there is a homomorphism from K"X to К„Х taking a bundle ? to its
sheaf of sections, or a -> a <8> [0x]. If X is nonsingular, it follows from the fact
that every coherent sheaf $ on X has a finite resolution by locally free sheaves
that this homomorphism is an isomorphism. Set K(X) = K°X = KoX, and
A(X) = A*X = A*X\ both К and A then become covariant for proper mor-
phisms, as well as contravariant. Grothendieck realized that the Riemann-Roch
problem could be formulated as the comparison of these two push-forwards, via
the Chern character.
Theorem (GRR). For any proper morphism /: X -* Y of nonsingular varieties,
and any a E K(X),
ch(/»(a)) • tdGY) -/,(ch(a) • td(T,)).
In other words the homomorphism т is covariant for arbitrary proper mor-
phisms. To prove that т commutes with/*, it suffices to factor/into a composite
gh, such that the commutativity with g* and Л* is known. For X quasiprojective,
one can find such a factorization

with h a closed imbedding and g the projection. We have proved GRR for closed
imbeddings; since K(Y X P") is generated over K(Y) by the classes [(?(/)], the
calculations that proved HRR for P" also prove GRR for g. For extensions to
varieties which may not be quasiprojective, see [17].
For a closed imbedding /: X -* Y, the same reasoning yields a Riemann-Roch
formula without denominators, i.e. a formula
c,(f*E)=f*{P,_d(c(E),c(N)))
for certain polynomials P} of weight j in the Chern classes of E and TV,
d = codim(;f, У). In particular, c,(/J^]) = 0 for 0 < / < d4 and
From this, or from a similar deformation argument, one proves a formula for the
Chern classes c,G» of the blow-up У of У along X.
9.2. The singular case. The transformation т from К to AQ defined for
nonsingular varieties in the previous section extends uniquely to the category of
arbitrary varieties, in the following sense. For every algebraic scheme X over a
given field К there is a homomorphism
satisfying the properties:
A) If/: X -» У is proper and a e KoX, then
/•т*(а) = ту/,(а).
B) If a E /Co* and p e K°X, then
C) If К is a subvariety of X, then
Txi®v) = lv] + lower terms.
D) If /: X -* У is smooth with relative tangent bundle Tf, then for а
These properties uniquely determine т. Indeed, one only needs (I), B), C) for
V = P", and D) for open imbeddings, to characterize т; note that none of these
conditions refer to Todd classes. When X is nonsingular, т^(а) is given by the
formula ch(a) • tdGV) of §9.1.
For arbitrary X, and а е KoXy one may often calculate т^(а) by finding a
proper morphism it: X' -* X with X' nonsingular and а' е K°X' with it*a' — a.
Then by A),
Tr(a)-ir»(ch(o')-td(rr)).
At least in characteristic zero such X\ a' always exist, although X' may not be
connected. Similarly using Chow's lemma, one may determine rx for arbitrary X
from the construction of i> for X' quasiprojective. What must be proved is that
such constructions are independent of choices.

If X is a variety which admits a closed imbedding in a nonsingular variety M,
then tv may be constructed as follows. For a coherent sheaf vF on X, let E. be a
resolution of °J on M. Note that ch(?.) = E(- l)'ch(?.) E A{M)Q restricts to
zero in A(M - X)Q. Thus ch(?.) must be the image of some class in A*{X)Q.
MacPherson's graph construction [4] produces such a localized Chern character
chMx(E.)mA*(X)Q.Then
The graph construction is an important generalization of the deformation to the
normal cone, useful for constructing characteristic classes on their natural loci.
For any scheme X% define the Todd class Td( X) of X by
If X is nonsingular, Td(X)= td(Tv) n [*]. Then GRR extends to arbitrary
varieties in the following form. If/: X -+ У is a proper morphism, $ e K°X, and
there is an element /*(P) e KY such that
then by A) and B),
/,(ch(P) n Td( *)) - ch(/,P) n Td(y).
This includes the generalization of Grothendieck's theorem to nonprojective
smooth varieties. In addition one has the HRR formula for a vector bundle E on
an arbitrary complete variety X,
E(-l)'dimH'(X?)= [ ch(E)DTd{X).
Jx
Another corollary of the general RR theorem is that the induced homomorphism
is an isomorphism, for any algebraic scheme X.
When/: X -* У is a regular imbedding there are push-forwards/*: K°X -* K°Y
and pull-backs/*: .КоУ -* K*X. Assume that У can be imbedded in a nonsingular
variety. Then any locally free sheaf E on X can be resolved on У, and one sets
if /\ is a resolution of/*?, and
where G. is a resolution of/*0^, and %i{G-®y($) are tne homology sheaves of
the complex G.<8>t^ У. Then one has the RR formulas
for a vector bundle ? on A' or a coherent sheaf ^ on У; here N is the normal
bundle to X in У. Such formulas were first proved in Grothendieck's seminar

INTRODUCTION TO INTERSECTION THEORY J \
SGA 6 and by Verdier [59]. The second gives an adjunction formula relating Todd
classes:
= td(AO~! n/*TdG).
There are similar compatibilities with exterior products. In particular,
Note that for any complete Xy
?(-!)'dim Н'{Х,вх) = f T6(X).
Jx
The above properties of Todd classes generalize classical facts about the arith-
arithmetic genus. For example, the constancy of arithmetic genus in flat families
generalizes to the fact that Todd classes are compatible with specialization.
For a recent interesting application of the singular Riemann-Roch theorem in
local algebra, see Morales [44].

10. Miscellany
10.1. Topology. For a space X, H'X denotes the ordinary (singular) cohomology
of X, with integer coefficients. For a closed subspace Z of X, H'(X, X - Z)
denotes relative singular cohomology. A useful homology theory for the study of
possibly noncompact spaces is the homology with locally finite supports, or
Borel-Moore homology, which we denote by Ht X. If X is imbedded in an oriented
real л-manifold M, then
A)
Taking M — R", this isomorphism may be used to define H(X. If Z is closed in X,
and U = X - Z there is a long exact sequence
B) ••• - #I + I{/ - H(Z -* H,X -* H,U - ...
and there are cap products
n
Any complex /:-dimensional variety V has a fundamental class cl(K) which
generates H2kV s Z. For a complex variety X there is an induced homomorphism
D) c\x:AkX-*H2kX
which takes ЕлДК,] to ?л,с1(Р;). That cl^ respects rational (or algebraic) equiva-
equivalence is a special case of the proposition which follows.
Any regular imbedding /: X -* Y of codimension d determines an orientation
class uXY e H2d(Y, Y - X). If X and Y are nonsingular, uXY is determined by
the equality ux Y П clG) = c\(X). If У is a vector bundle over X, and / is the
zero section, ux Y is the Thorn class of the bundle. For the general case see [4].
Proposition. Let i: X -+ У be a regular imbedding of codimension d, V a
k-dimensional variety, f: V -* Y a morphism and W = f~\X). Then
dw(X- V)
The proof can be achieved by reducing via familiar methods to the case of
divisors, where it is straightforward [16, §19]. It follows from this proposition that
all our intersection constructions are compatible with those in topology; in
particular, the (refined) intersection product on nonsingular varieties maps to the
73

topological product described in §3.1. For varieties meeting properly at a point in
a nonsingular variety, it follows that the intersection multiplicity is given by the
linking number of the intersections of the varieties with a small sphere about the
point.
One may define two cycles a, a' on a complex variety X to be algebraically
equivalent if there are subvarieties Vt of X X С, С a complete nonsingular curve,
and r,, t2 e С with
(cf. §3.3). If X is nonsingular, one has a filtration of the codimension p cycles on
X:
Rat' X с Alg' X с Horn' X с Num' X с ZPX
consisting of cycles rationally, algebraically, homologically and numerically
equivalent to zero. For p = 1 these groups and the factor groups are quite well
understood: Alg1 AyRat1 X is the Picard variety, Alg1 X = Horn1 X,
Num1 X/Alg1 X = H2(X)tors is finite, and Z'A'/Num1 X'\s finitely generated and
free.
For p > 1, ZpX/Homp X is always finitely generated, so Z'AyNum' X is free
abelian. We have mentioned that, for surfaces, Alg2 AyRat2 X can be "infinite
dimensional". Griffiths showed that Alg2 X can differ from Horn2 X when X is a
threefold, and Clemens [11] has improved this to show that Horn2 Ay Alg2 *need
not be finitely generated. The principal tool for studying this problem is an
Abel-Jacobi map
Нот'AyRat''*-»./''*
to the /?th intermediate Jacobian of X.
For p = 2, J. P. Murre has recently showed that the image of Alg' AyRat' X is
an abelian variety, universal for "regular" homomorphisms of Alg' AyRat' X to
abelian varieties.
10.2. Local complete intersection morphisms. Consider for simplicity the cate-
category of varieties which admit closed imbeddings into nonsingular varieties. Then
any morphism/: X -* Y admits a factorization/ = pi
with / a closed imbedding and p smooth; if X с М, М nonsingular, one may take
P = Y X M, p the projection. We call / a l.c.i. morphism if for some (and hence,
in fact, for any) such factorization, / is a regular imbedding. If p has relative
dimension n and / has codimension e, then d = e - n is independent of the
factorization, and is the codimension of /. In addition, one has the virtual tangent
bundle

INTRODUCTION TO INTERSECTION THEORY 7 5
One verifies that such notions are independent of factorization by comparing a
factorization through P and through P' with the diagonal factorization:
If /: X -* У is a Lei. morphism of codimension d, f determines Gysin
homomorphisms
f*:AJ-*Ak_dX
by/*a = /*(/?*a), where p* is flat pull-back (§3.3) and /'* is the Gysin homomor-
phism for regular imbeddings (§5.1). Similarly there are refined Gysin homomor-
homomorphisms
f:AkY'^Ak_dX>
for any Г -> У, with X' = X X Y Г, by/*a = r(p'*( a)), where pf is the induced
(flat) morphism from P X y Г to У.
The Riemann-Roch formulas of §9.2 generalize to Lei. morphisms/: X -* Y:
If i: X -+ У is a regular imbedding of codimension d, and У is the blow-up of У
along X, then the morphism /: У -* У is a Lei. morphism of codimension zero
(by the lemma in §2.4). Consider the fibre square:
gi if
X -> У
The exceptional divisor X is P(N)y N = NxYy and the excess bundle E is the
quotient bundle on P(iV);
One can show that there are split exact sequences
with fl(a) = (cd_ ,(?) П g*a, -**a) and 6(P, y) =7*P + /*y. Moreover there is
the following general formula for /*, involving Segre classes. For any /c-dimen-
sional subvariety V of У, let V be the blow-up of V along V П X. Then
/•[K] = [F] +jME)ng*s(VnX,V))k
in ^^У. See [16, §6.7] for details.

10.3. Contravariant and Invariant theories. We have mentioned the problem of
giving a geometric construction of a suitable contravariant ring-valued
("cohomology") theory A* to pair with the covariant ("homology") theory A * we
have been studying. At present there are several definitions of such rings A*X,
each with its uses as well as defects:
A) For quasiprojective varieties X, one may define [4, Appendix]
the limit over all morphisms /: X -» Y from X to nonsingular varieties Y, with
A*Y as in §5.2. There are pull-backs /*: A*X -> A*X' for any morphism /:
X' -» X, cap products
with the usual projection formula, and vector bundles have Chern classes in A*X.
For complex varieties there are homomorphisms
c\:A*X-+H2*X
to cohomology. With this theory one also has the desirable properties
Pic( X) s AXX and ch: K°XQ ^ A*XQ.
However there are few other functorial properties known. For example, one
would like Gysin homomorphisms
for a proper Lei. morphism /: X -» Y of codimension d\ it is nbt clear how to
construct such/* for this theory, even for smooth projections.
Note that if X is nonsingular, A*X is the same as that constructed in §5.2. It
follows that this theory A* is the finest possible contravariant theory agreeing with
the given theory on nonsingular varieties.
B) For any X one may construct an operational theory A*X as follows [23, 16]:
an element с of Л''A' is a collection of homomorphisms
cx.:AqX'-*Aq_pX'
for all X' -* X, and all q, compatible with all our other intersection operations.
Precisely, one requires that if /: X" -» X' is proper (resp. flat, resp. a regular
imbedding) with X' -* X given, then for a e A * X" (resp. p е/1Д')
<*(/•«)-/•<>(«) (resp.cr,(/*p)=/Vr(P)).
The ring structure on this A*X is constructed by composing homomorphisms.
This theory has pull-backs, cap products, Chern classes, and also Gysin homo-
homomorphisms /* for l.c.i. morphisms/. However, the map from Pic^) to^'A'need
not be an isomorphism, and we do not know a homomorphism from these groups
A*X to cohomology H*X, for complex varieties X.
These operational groups are useful because of their formal properties. For any
series of operations that finally end up with a class in a group A+X—e.g. for any

enumerative problem—there is no loss at all in using them. Even less is known
about computations of these A*X than in the classical case, however. One can at
least show that APX = 0 for p < 0 or p > dim X; our proof that A*X is
commutative uses resolution of singularities, so is known only in characteristic
zero.
When X is nonsingular, this A*X also agrees with that in §5.1. Note that this Л*
is the coarsest theory with this property and with a theory of cap products
compatible with intersection products.
C) Mumford [46] has used the image of the first of these theories in the second.
D) One may define APX to be Hp(X,\Kp), where \Kp is Quillen's sheaf of
higher ^-groups [52]. When X is regular, Quillen proved Bloch's formula:
n = dimCA"). Gillet [25] has constructed Chern classes in these groups, cap
products, and some Gysin homomorphisms.
E) Another possibility has been proposed by M. Levine, in order to extend
results about vector bundles by Murthy and Swan, and Collino, to general
singular varieties.
It is most useful if the covariant and contravariant theories are part of a general
bivariant theory [23]. This should assign to any morphism /: X -» У a graded
abelian group
a*(xIy)
with products, for/: X -> У, g: У -> Z,
Ap( X^
push-forwards, for/: X -¦ У proper, g: Y -» Z,
and pull-backs, for/: X -* У, h: Y -> У,
with/': X X Y Y -* Y the induced тофЫзт. These three options should satisfy
various compatibility axioms. Then one sets
id id id
whre pt. = Spec( К). The products for the composite X -* X ~* X and X-+ X -*
pt. give "cup" and "cap" products.
One point of such a theory is that orientations for morphisms /: X -» У should
determine classes in A*(X -> У). For example, a flat morphism or a l.c.i.

morphism / of codimension d should determine a class [/] in All{ X -> Y). And
such a class determines Gysin homomorphisms
/*: AJ-Ak.dX, /*« = [/]'«,
When a class in A*X lives naturally on a locus Z с A'—as has been a frequent
theme in these lectures—, the class should really be a class in A*{Z -* X). For
closed imbeddings Z ^ X% A*{Z -* X) should function as local cohomology
groups A *7 X.
I
In topology there is a natural bivariant theory H*(X -» Y). If one imbeds X in
R", one may define
1 ~Л Я' + "(У X R\ Y X R" - X).
At present we have only an operational bivariant theory for rational equiva-
lence: a class in Ap( X -* Y) is defined to be a collection of homomorphisms from
AqY' to Aq_p(X X у Y') for all Y' -> У, all g, compatible as in B) above. One can
show that A~4(X -* pt.) is isomoфhic to AqX4 that there are orientation classes
[/] for flat and l.c.i. morphisms, and that our constructions of degeneracy classes,
residual intersection classes, local Chern classes, etc., all belong to appropriate
bivariant groups.
When X -> У is a closed imbedding the local cohomology groups #?(У, "\р)
used by Gillet look promising for a sharper bivariant theory. D. Grayson has
pointed out, however, that for general /: X -* У, if one imbeds A' in a nonsingular
Ms the groups H*(Y X Af, У\ *) are not independent of the imbedding.
There is a satisfactory bivariant theory specializing to Ко and K\ which should
agree with the ideal rational equivalence theory ® Q. The objects of Ko( X -+ Y)
are complexes on X of finite Tor dimension over У.
On singular varieties the intersection homology theory of Goresky and Mac-
Pherson [27] have led to many new insights. One does not know an analogous
theory lying between A*X and A + X. The place of algebraic cycles in their theory
is not very well understood.
10.4. Serre's intersection multiplicity. If two subvarieties K, W of a nonsingular
variety X meet properly at a point P, Serre [57] showed that the intersection
multiplicity /(?, V • W; X) is given by the formula
where A is the local ring of X at Л and / and J are the ideals of V and W. A
unique feature of this formulation is that, at least in its statement, it requires no
reduction to the diagonal. This definition makes sense, in fact, for any regular
local ring A, whether it contains a field or not. In this generality the positivity of
this multiplicity remains an open question. Recently, Dutta, Hochster, and

INTRODUCTION TO INTERSECTION THEORY 79
MacLaughlin [14] have shown that the natural generalization of this conjecture to
modules of finite projective dimension is false, even in the geometric case. In the
process they produce some interesting resolutions of modules, which cannot be
pull-backs of complexes of vector bundles from any nonsingular variety.
For arbitrary varieties V, W on a nonsingular X, the virtual sheaf
is an element of Ka(VП W). One can show that, if т is the Riemann-Roch
homomorphism (§9.2), then
т(Тог*(К, W))=V-W+ lower terms
in A *( V П W)Q, even in the case of excess intersection.
With Faltings' recent solution of the Mordell conjecture via solutions of
conjectures of Tate and Shafarevich, one may anticipate a renewed interest in
intersection theory on arithmetic varieties. For such applications it is important to
bring in the infinite primes, as in [1].

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