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(continued after index)

Robert Friedman
Algebraic Surfaces
and Holomorphic
Vector Bundles
Springer

vi Preface
of \nKx | when X is a minimal surface of general type. The original moti-
motivation of computing Donaldson invariants has however disappeared except
for a brief discussion in Chapter 8 for elliptic surfaces.
It is a pleasure to thank the audience at the lectures which served as
the raw material for this book, as well as David Gomprecht, my course
assistant for the Park City institute, for an excellent job in proofreading
the rough draft of the first part of this book. I would also like to thank
Tomas G6mez and Titus Teodorescu for comments on various manuscript
versions, and Dave Bayer for doing an excellent job with the figures.
New York, New York	Robert FViedman

Contents
Preface	v
Introduction	1
f
: Curves on a Surface	7
Introduction	7
Invariants of a surface	8
Divisors on a surface	9
Adjunction and arithmetic genus	13
The Riemann-Roch formula	15
Algebraic proof of the Hodge index theorem	16
Ample and nef divisors	18
Exercises	22
i.
3 Coherent Sheaves	25
What is a coherent sheaf?	25
A rapid review of Chern classes for protective varieties	27
Rank 2 bundles and sub-line bundles	31
Elementary modifications	41
Singularities of coherent sheaves	42
Torsion free and reflexive sheaves	43
Double covers	46
Appendix: some commutative algebra	51
Exercises	54
S Birational Geometry	59
Blowing up	59
The Castelnuovo criterion and factorization of birational
; morphisms	64
Minimal models	70

viii Contents
More general contractions	72
Exercises	79
4	Stability	85
Definition of Mumford-Takemoto stability	85
Examples for curves	89
Some examples of stable bundles on P2	91
Gieseker stability	96
Unstable and semistable sheaves	97
Change of polarization	99
The differential geometry of stable vector bundles	101
Exercises	108
5	Some Examples of Surfaces	113
Rational ruled surfaces	113
General ruled surfaces	1П
Linear systems of cubics	12J
An introduction to КЗ surfaces	132
Exercises	131
6	Vector Bundles over Ruled Surfaces	141
Suitable ample divisors	14]
Ruled surfaces	14J
A brief introduction to local and global moduli	15?
A Zariski open subset of the moduli space	159
Exercises	163
7	An Introduction to Elliptic Surfaces	167
Singular fibers	167
Singular fibers of elliptic fibrations	172
Invariants and the canonical bundle formula	175
Elliptic surfaces with a section and Weieratrass models	178
More general elliptic surfaces	183
The fundamental group	188
Exercises	190
8	Vector Bundles over Elliptic Surfaces	197
Stable bundles on singular curves	197
Stable bundles of odd fiber degree over elliptic surfaces	200
A Zariski open subset of the moduli space	203
An overview of Donaldson invariants	205
The 2-dimensional invariant	209

Contents be
Moduli spaces via extensions	217
Vector bundles with trivial determinant	225
Even fiber degree and multiple fibers	237
Exercises	240
9 Bogomolov's Inequality and Applications	245
Statement of the theorem	245
The theorems of Bombieri and Reider	248
The proof of Bogomolov's theorem	252
Symmetric powers of vector bundles on curves	257
Restriction theorems	260
Appendix: Galois descent theory	265
Exercises	274
Classification of Algebraic Surfaces and of Stable
Bundles	277
Outline of the classification of surfaces	277
Proof of Castelnuovo's theorem	284
The Albanese map	288
Proofs of the classification theorems for surfaces	290
The Castelnuovo-deFranchis theorem	302
Classification of threefolds	305
Classification of vector bundles	307
Exercises	313
315
Мех	323

Itroduction
|] study of algebraic surfaces is by now over one hundred years old.
ftrfthe fundamental results were established by the Italian school of
с geometry, for example Castelnuovo's criterion for a surface to be
I A895), the theorem of Enriques that a surface is rational or ruled
only if Pa or Pg is zero A905), and in general the role of the canoni-
• in the classification of surfaces. This theory was reworked from
|modern perspective of sheaves, cohomology, and characteristic classes
ieeriee of papers by Kodaira A960-1968) and by the Shafarevich sem-
$A981-1963). In particular, new ideas were developed to attack those
e'in the classification theory which had proved resistant to the
i techniques of the Italian school, for example the classification of
: .surfaces or the structure of the moduli space of КЗ surfaces and
dp with the period map. Another deep result which seems to
! to the classical methods is the Bogomolov-Miyaoka-Yau in-
/? 5- 3cj. Moreover, the new methods could be extended to the
|jf of compact complex surfaces (Kodaira) or algebraic surfaces in pos-
i (Characteristic (Mumford and Bombieri-Mumford). Despite the great
I in understanding algebraic surfaces, many open questions remain.
l>i«xample, the fundamental problem of whether there exists a classifi-
i scheme of some sort for surfaces of general type seems to require a
new insight.
' contrast, the study of holomorphic vector bundles on algebraic sur-
Ikota is much more recent, and effectively dates back to two papers by
8chwaraenberger A961). For the case of algebraic curves, Grothendieck
A966) showed that every holomorphic vector bundle over P1 is a direct
Mm of line bundles (a result known in a different language to Hilbert,
flemety and Birkhoff, and prior to them to Dedekind and Weber). Atiyah
A957) classified all vector bundles over an elliptic curve and made some
jMiminary remarks concerning vector bundles over curves of higher genus.
In 1960, the picture changed radically when Mumford introduced the no-
notion of a stable or semistable vector bundle on an algebraic curve and used
IBOmetric invariant theory to construct moduli spaces for all semistable

2 Introduction
vector bundles over a given curve. Soon thereafter Narasimhan and Se-
shadri A965) related the notion of stability to the existence of a unitary
flat structure (in the case of trivial determinant) or equivalently a flat
connection compatible with an appropriate Hermitian metric. For curves,
much recent work has centered on the enumerative geometry of the mod-
moduli space of curves. Explicit geometric constructions for the moduli space
were given for genus 2 curves by Narasimhan and Ramanan A969) and for
hyperelliptic curves in general by Desale and Ramanan A976).
In this context, Schwarzenberger made the following contributions to
the theory of vector bundles over a surface. In general, for a variety X
of dimension greater than 1, a vector bundle on X is not a direct sum of
line bundles or an extension of line bundles. Schwarzenberger's first paper
studied rank 2 bundles V which are not simple ("almost decomposable"
in his terminology), in other words for which the automorphism group is
larger than C*. He showed, using the existence of a rank 1 endomorphism
on V, that V is an extension by a line bundle of a coherent sheaf of the form
L®Iz, where L is a line bundle and Z is a 2-dimensional local complete
intersection subscheme, and in the case of surfaces X he gave a mechanism
for describing the set of all such extensions with a fixed Z. To do so, he
passed to a blowup X of X in order to be able to replace Iz by a line
bundle of the form Ох{-^2{<ьЕ{), where the Ei are the components of
the exceptional divisor and the a< are nonnegative integers. As part of the
study, he analyzed when a vector bundle on X is the pullback of a bundle
on X.
In Schwarzenberger's second paper, he showed that every rank 2 vector
bundle on a smooth surface X is of the form w.L, where ж: Y —» X is a
smooth double cover of X and L is a line bundle on Y. He then applied
this construction to construct bundles on P2 which were not almost decom-
decomposable; these turn out to be exactly the stable bundles on P2. He showed
further that, if V is a stable rank 2 vector bundle on P2, then the Chern
classes for V satisfy the basic inequality ci(VJ < 4сг(У).
In the years after Schwarzenberger's papers, the study of bundles over
surfaces diverged into two streams. In the first, there were various at-
attempts to generalize Mumford's definition of stability to surfaces and
higher-dimensional varieties and to use this definition to construct mod-
moduli spaces of vector bundles. Takemoto A972, 1973) gave the straightfor-
straightforward generalization to higher-dimensional (polarized) smooth projective
varieties that we have simply called stability here (this definition is also
called Mumford-Takemoto stability, /x-stability, or slope stability). Aside
from proving boundedness results for surfaces, he was unable to prove the
existence of a moduli space with this definition (and in fact it is still an
open question whether the set of all semistable bundles forms a moduli
space in a natural way). Shortly thereafter, Gieseker A977) introduced
the notion of stability now called Gieseker stability or Gieseker-Maruyama
stability. Gieseker showed that the set of all Gieseker semistable torsion

Introduction 3
sheaves on a fixed algebraic surface X (modulo a suitable equivalence
i) formed a projective variety, containing the set of all Mumford
vector bundles as a Zariski open set. This result was generalized by
A978) to the case where X has arbitrary dimension. The dif-
geometric meaning of Mumford stability is the Kobayashi-Hitcbin
ure, that every stable vector bundle has a Hermitian-Einstein con-
unique in an appropriate sense. This result, the higher-dimensional
of the theorem of Narasimhan and Seshadri, was proved by Don-
A985) for surfaces, by Uhlenbeck and Yau A986) for general Kahler
;, and also by Donaldson A987) in the case of a smooth projective
(The easier converse, that an irreducible Hermitian-Einstein con-
defines a holomorphic structure for which the bundle is stable, was
previously by Kobayashi and Lubke.) The geometric meaning of
stability is more mysterious, although Leung A993) has obtained
in this direction. A related general result is Bogomolov's inequality
vector bundles, which follows from the Donaldson-Uhlenbeck-Yau
as well as from various purely algebraic arguments (Bogomolov,
Ье other stream in studying vector bundles consists in analyzing moduli
for specific classes of surfaces (and perhaps specific choices of the
i classes). The case of P2 and more generally P" has received a great
l,',of attention, and moduli spaces of vector bundles on P2 have been
quite explicitly by the method of monads' (Barth, Hulek and
el). Because this subject has been well described elsewhere (see for
i [117]), we do not discuss monads in this book. The case of ruled
has been analyzed by Hoppe and Spindler and also by Brosius.
briefly treated the case of abelian surfaces, but the study of
Г bundles (not necessarily of rank 2) over КЗ and abelian surfaces
r got off the ground with a series of papers by Mukai. This was the state
) art until about 1985, when Donaldson theory gave a powerful impetus
) study of rank 2 vector bundles over surfaces. We shall describe some
в developments arising after 1985 at the end of Chapter 10.
ere is perhaps a third stream which should be mentioned, that of the
rative geometry of the moduli space. By now these questions have
well studied for bundles over curves (Verlinde formula, cohomology
; of the moduli space), and in some sense Donaldson theory is simply a
question about the enumerative geometry of the moduli space of bundles
OW a surface. Deep structure theorems and conjectures in gauge theory,
4ue to Kronheimer and Mrowka and Witten, suggest that there is a very
fimple enumerative structure to this moduli space, but as yet there is no
' to see why this should be true purely within the context of algebraic
pometry.
j goal of this book is to provide a unified introduction to the study of
'•Igebraic surfaces and of holomorphic vector bundles on them. I have tried
to keep the prerequisites to a good working knowledge of Hartshorne's book

4 Introduction
on algebraic geometry [61] as well as standard commutative algebra (see
for example Matsumura's book [87]). Aside from what is contained in [61],
we freely use the exponential sheaf sequence on a complex manifold and
the Leray spectral sequence (typically when it degenerates) as well as basic
properties of Chern classes which are summarized in Chapter 2, and for
which Fulton's book [45] is a standard reference. For the most part, we use
the Riemann-Roch theorem only for vector bundles on a curve or surface,
for which proofe are given in the exercises to Chapter 2. However, we use the
Grothendieck-Riemann-Roch theorem once in Chapter 8 and the Riemann-
Roch theorem for a divisor on a threefold in Chapter 10, without recalling
the general statements. There is also a brief appeal to relative duality in
Chapter 7 and to the existence of a relative Picard scheme for smooth
fibrations of relative dimension 1 in Chapter 9. The appendix to Chapter 9
uses a little Galois theory, and some results which are not used in the rest
of the book use standard facts about group cohomology. The last section
of Chapter 4 assumes some basic familiarity with differential geometry on
a complex manifold, for example as described in the book by Griffiths and
Harris [55], and can be skipped. In Chapter 8, there is a brief discussion of
Donaldson invariants which motivates some of the enumerative calculations
in the rest of the chapter, but which can otherwise be omitted. Of necessity,
I have largely limited myself to the part of the study of vector bundles which
does not involve the heavy machinery of deformation theory or geometric
invariant theory; a few descriptive sections outline the main results.
For the first eight chapters, the plan has been to alternate between the
study of surfaces and the study of bundles on them. This has the pedagog-
pedagogical advantage that, for example, vector bundles over curves are studied in
Chapter 4, then used to describe ruled surfaces in Chapter 5. In Chapter
6, we use the knowledge of ruled surfaces to describe vector bundles over
them, and in Chapter 9 they reappear as part of the proof of Bogomolov's
inequality. Similarly, ruled surfaces are described in Chapter 5 and ellip-
elliptic surfaces in Chapter 7, and the structure of the moduli space of vector
bundles over such surfaces is then described in Chapters 6 and 8. I have
tried to emphasize how the internal geometry of the surface is reflected
in the birational geometry of the moduli space. In the last two chapters,
we drop the strict division of material: Chapter 9 gives a proof of Bogo-
Bogomolov's inequality, which belongs to the theory of vector bundles, as well
as applications to the study of linear systems (in particular pluricanoni-
cal systems) on an algebraic surface. In Chapter 10, we prove the main
theorems on the classification of algebraic surfaces and outline the current
state of knowledge concerning moduli spaces of rank 2 vector bundles over
algebraic surfaces. The proofe of the classification results for surfaces are
old-fashioned, in the sense that they do not appeal to Mori theory. On the
other hand, the old-fashioned proofe may be better adapted to handling
the classification of symplectic 4-manifolds. The point of view of Mori the-
theory and the classification results for threefolds are briefly described toward

Introduction 5
ju of the chapter. Because of the way we alternate between surfaces
vector bundles, it may be a little disorienting to try to read the book
y, and certainly the chapters on surfaces can be, for the most
read independently of the chapters on vector bundles. On the other
jibe later chapters on vector bundles over ruled or elliptic surfaces
heavily on the description of the corresponding surfaces in the chap-
that precede them.
of length and time dictated that many topics had to be
out. For surfaces, I would have liked to devote more time to rational
minimally elliptic singularities and to the classification of surfaces of
egree. For vector bundles, without the main tool of deformation
we are only able to scratch the surface of this rapidly evolving field,
this theory does not seem to be close to a definitive state, it seems
to focus on many concrete examples.
there are many exercises at the end of each chapter, and they
integral part of the book. In particular, many results are left to the
and they are frequently used in later chapters. I hope that the
on examples, both in the text and the exercises, will help to serve
introduction to this rich and beautiful field of mathematics.

Curves on a Surface
Introduction
In this book, unless otherwise specified, by surface we shall always mean
a connected compact complex manifold of complex dimension 2 which is
a holomorphic submanifold of PN for some ЛГ. Thus, "surface" is short
for smooth (connected) complex algebraic surface. By Chow's theorem, a
surface is also described as the zero set in FN of a finite number of homo-
homogeneous polynomials in N +1 variables. The study of surfaces,is concerned
both with the intrinsic geometry of the surface and with the geometry of
the possible embeddings of the surface in FN. Just as with curves, we could
organize this study in order of increasing complexity. In terms of the ex-
extrinsic (synthetic) geometry of a surface in WN, we could for instance try
to study and eventually classify surfaces in Pw of relatively small degree.
Or we could attempt to order surfaces by complexity via some intrinsic
invariants, by analogy with the genus of a curve. This is the aim of the
Kodaira classification, which orders surfaces by their Kodaira dimension.
For this scheme, we have a fairly complete understanding of surfaces except
in the case of Kodaira dimension 2, general type surfaces. We will cover
the broad outlines of the general theory of surfaces. In this chapter, we
will discuss the basic invariants, intersection theory and Riemann-Roch,
and the structure of the set of ample divisors. In Chapter 3, we will dis-
discuss birational geometry. Chapters 5 and 7 will concern some of the main
examples of surfaces: rational and ruled surfaces, КЗ surfaces, as well as
an introduction to elliptic surfaces. Finally, in Chapter 10, we shall give a
general overview of the classification of algebraic surfaces.
We begin with the description of the basic numerical and topological
invariants of a surface.

8 1. Curves on a Surface
Invariants of a surface
A surface X is in particular a complex manifold, and always carries a
canonical orientation from its complex structure. Viewing X as an ori-
oriented 4-manifold, its main topological invariants are its fundamental group
¦к\(Х, *), the Betti numbers bi(X) = bi-i(X), and the intersection pairing
on #2(X;Z). Here by Poincare duality #2(X;Z) ^ #2(X;Z) and inter-
intersection pairing corresponds under this isomorphism to cup product from
#2(X;Z)<g>ff2(X; Z) to #4(X; Z) ^ Z by taking the canonical orientation.
Over R, the intersection pairing is specified by Ьг(Х) and by b?(X), the
number of positive entries along the diagonal when the form is diagonalized
over K. We also let Щ(X) = ^(X) - bJ(X). If X = P2 or if X is one of
an unknown but finite number of surfaces of general type whose universal
cover is the unit ball in C2, then #*(X; R) = R. К X does not belong
to this finite list of examples, then Я2(Х;К) is always indefinite (cf. for
example [40, p. 29, Lemma 2.4]). It then follows from the classification of
quadratic forms over Z [138], [92] that the intersection pairing on #г(Х; Z)
mod torsion is specified by its rank, signature, and type, i.e., whether or
not there exists an element a € #2(X; Z) with a2 = 1 mod 2 or not. (If
there exists such an a the form is odd or of Type I; otherwise it is even
or of Type II.) To decide if a surface is of Type I or Type II, we use the
Wu formula, which says that a2 = a- [Kx] mod 2. Here [Kx] denotes the
homology class associated to the canonical line bundle Kx via c\(K\) and
Poincare duality. Thus, again by Poincare duality, there exists an a with
a2 = 1 mod 2 if and only if the image of [Kx] in #2(X; Z) = #2(X; Z)
modulo torsion is not divisible by two.
There are also the holomorphic invariants of X. The most basic ones are
the irregularity q(X) of X and the geometric genus pg(X) of X, defined
by
q(X) = dimcH0(X;f^) = dimcHl{X;Ox),
P9(X) = dime Я°(Х; П2Х) = dime #2(X; Ox).
Thus, <?(X) is the number of independent holomorphic 1-forms on X and
pg(X) is the number of holomorphic 2-forms on X. We note that the fact
that the two different expressions above for q(X) are equal follows from
Hodge theory, since X is an algebraic surface over C, and do not hold for
an arbitrary compact complex surface or for a surface defined over a field
of positive characteristic; in either case the "correct" definition of q(X)
is 6imHl(X;Ox)- (That the two expressions for ps(X) are equal follows
from Serre duality which holds in general.) Additional invariants are given
by hl*{X) = dimtf^X-.fix) and cj(XJ = [Kx?- The relation of these
invariants to the topological ones is as follows:
bj(X) = 2q(X),

1. Curves on a Surface
b+(X)=2Pg(X) + l.
Here the first two equalities follow by Hodge theory and the last is one
form of the Hodge index theorem for a surface. We also have the Euler
characteristic
X(X) = 1 - h(X) + b(X) - h(X) + 1
= 2 - 2Ъ1{Х)+Ъ2{Х) = 2 -Aq + 2Pg(X) + ftM
and the holomorphic Euler characteristic
X(OX) = h°(Ox) - h\Ox) + ti*{Ox) = 1 - q(X) +pg(X).
There is also Noether's formula (in some sense a special case of the
Riemann-Roch theorem for surfaces) which says that
or in other words that [Kxf + x(X) = 12A - q(X) + pg(X)). An easy
manipulation of the formulas (Exercise 1) shows that Noether's formula is
equivalent to the Hirzebruch signature theorem
6+(X) - 62-(X) = i(c?(X) - 2c2p0).
Beyond this there are the "higher" holomorphic invariants of X, the
p/urigenera Pn(X) = &mH°(X;K^n), defined for n ? 1. Thus, P^X) =
pg(X). It is by now well known [39] that the plurigenera are not in general
homotopy or homeomorphism invariants of X. It has recently been shown
via new invariants introduced by Seiberg and Witten that the plurigenera
are diffeomorphism invariants of X (see for example [16] and [41]). We shall
discuss some of these developments further in Chapter 10.
Divisors on a surface
We recall that a (reduced irreducible) curve С on X is an irreducible holo-
holomorphic subvariety of complex dimension 1. Thus, locally С is described
818 {/(zi> 22) = 0}, where / is a holomorphic function of z\, z%. Of course,
С need not be a (holomorphic) submanifold of X; if it is we say that С is a
smooth curve. A divisor D on X is a finite formal sum $2i п«С« of distinct
irreducible curves Cj, where the щ € Z. The set of all divisors DivX is
thus the free abelian group generated by the irreducible curves on X. The
divisor D is effective if the n, > 0 for all i. An effective divisor D ф 0 will
also be called a curve. We write D > 0 if D is effective and D\ > D2 if
D\ — ?>2 > 0. If fi is a local equation for the curve Cj, then D is locally
described by the meromorphic function П, /Г* > which is in fact holomor-
holomorphic if and only if D is effective. Conversely, a meromorphic function / on
X has an associated divisor (/), which is the curve of zeros of / minus the

10 1. Curves on a Surface
curve of poles of /. All of these constructions also make sense locally on
open subsets U of X. Given a divisor D on X, there is an associated line
bundle Ox(D) whose associated sections on an open subset U of X are
meromorphic functions g on U such that (g) + D\U > 0, where D\U is the
restriction of the divisor D to U in the obvious sense. In particular, if D
is effective, then the constant function 1 defines a global section of D. The
map D *-* Ox{D) is then a homomorphism from the free abelian group
Div X to the group Pic X of line bundles on X under tensor product.
We recall that two divisors D\ and D^ are linearly equivalent (which we
shall write as Dx = D2) if and only if Ox{Di) Э 0лг(?>2) if and only if
D\-D2 = {f) for a globally defined meromorphic function / on X. A linear
equivalence class of divisors will be called a divisor class. Every holomorphic
line bundle L on X is of the form Ox{D) for some divisor D. In fact,
defining a meromorphic section of a line bundle via local trivializations, if
s is a meromorphic section of L, then s has associated to it a well-defined
divisor (s) — D and it is straightforward to show that L = Ox{D). Thus, if
L has a holomorphic section s, then L = Ox(D) for an effective divisor D.
For example, given an effective D, the global section 1 of Ox{D) described
above vanishes exactly along D (viewed as a section of Ox(D), of course).
The group of divisor classes may be naturally identified with Pic X. We
shall call the divisor (or corresponding divisor class) D ample or very ample
if the line bundle Ox{D) is ample or very ample. Divisors are functorial
in the following sense: if ж: X —* Y is a surjective map, then pullback ж*
induces a homomorphism Div У —» DivX. Given a divisor D, it defines a
homology class. One way to see this, for an irreducible effective divisor D, is
to choose a triangulation of X for which the support of D is a subcomplex.
Another way is to use the homomorphism DivX —> H2(X;Z) given by
D >-> c\(Ox{D)), followed by Poincare duality. Here, for a holomorphic
line bundle L, we can define the first Chern class c\{L) directly via the
exponential sheaf sequence
0-»Z-» Ox	> Ox ~* 1,
by taking ci to be the coboundary map Яг(О^) = PicX —»1Р(Х;Ъ). In
any case, we shall denote by [?>] the homology class associated to D.
Our goal now will be to give an algebraically defined intersection pairing
on Div X which agrees in a natural sense with the topological intersection
form under the induced map DivX —> //2(X;Z). We begin by defining a
local intersection number for two curves C\, C% which have no component
in common.
Definition 1. Let C\,Ci be two curves with no component in common
and let i e X. Define C\ -x C2 = dime Ox,x/{fi,h)> where /, is a local
equation for C, at x.

1. Curves on a Surface 11
Note that to say that Ci, C2 have no component in common is exactly to
say that /i and /2 are relatively prime in the ring Ox,x for every x and so
define an ideal such that the quotient ring is a finite-dimensional C-algebra.
It is clear that this quotient is independent of the choice of local equation
and is the part of the scheme-theoretic intersection C\ П C2 supported at
x. Of course this is empty if x ^ C\ П C2, and in this case it is easy to see
that Cx x C2 = 0. Note also:
Lemma 2. The curves C\ and C2 meet transversally at the point x if and
onIyifCi-xC2 = l.
Proof. By definition C\ and C2 meet transversally at x if and only if
(/11/2) = Гох, the maximal ideal of Ox,x at x, and since <?x>/(/i, /2) is a
nonzero C-algebra and thus always has dimension at least 1, this condition
is equivalent to the condition that C\ -x C2 = 1. D
To define an intersection pairing for divisors, we proceed as follows: sup-
suppose as before that C\, C% have no component in common and define
C\ • C2 = 2_^ C\ -x C2.
xex
By hypothesis this is a finite sum.
Lemma 3. IfC\ is a smooth irreducible curve and C\ is not a component
of C2, then C\ • C2 = degC?x(C2)|Ci (here deg is the degree of a complex
line bundle on the curve C\). In particular, in this case С\-Сг only depends
on the iuiear equivalence class of C2.
Proof. Take the exact sequence
0 _» Ox{-C2) ->Ox->OCl->0
and tensor it with Oci- Let /1 and /2 be local equations for C\ and C2,
respectively, at x. An easy calculation using the fact that /1 and /2 are
relatively prime shows that the resulting sequence
16С1ПС3
is still exact (in other words, Torfx'*(C>x,x/(/i),C>x,*/(/2)) = 0). If we
tensor the second exact sequence with 0x(Ca)|Ci» we obtain instead
Thus, Od(C2) has a section vanishing at exactly C\ • C2 points, counted
with multiplicity, and so deg Ocx (C2) =C\-Ci- ?

12 1. Curves on a Surface
Theorem 4. There is a unique symmetric bilinear pairing from Div X to
Z, denoted by (?>i,2?2) which factors through linear equivalence and has
the property that (Ci, C2) =C\-C2 for C\ and C2 distinct smooth curves
meeting transversally.
Proof. Note the following standard lemma:
Lemma 5. Let L be a iine bundle on X. Then there exist two very ample
divisors H' and H" on X such that L = Ox(H' - H"). In particular every
divisor D € DivX is linearly equivalent to a difference of two very ample
divisors. D
To prove Theorem 4, we begin with the uniqueness. Given D\ and ?>2
in DivX write Д = Щ - Щ'. We may assume that all of the Щ, Щ
are distinct and smooth and meet transversally. Thus, necessarily we must
have
A.1)	(Dlt D2) ^ЩЩ-Щ- Щ - #(' • Щ + Щ ¦ Щ.
To see the existence, note that the above formula can be written as
A.2)	(DltD2) = degOx@i)№ - degOxiDJW.
Fix ?>2 and choose smooth curves Щ, НЦ meeting transversally with D2
linearly equivalent to Щ - Щ. For an arbitrary divisor D%, we can define
(?>1,?>2) by formula A.1). Using A.2), we see that {DUD2) only depends
on the linear equivalence class of D\, and in particular does not depend
on the choice of Щ and H". By symmetry the same is true for D%. Thus,
(?>i, D2) is well defined by A.2), and is clearly symmetric. It follows from
A.2) that (Di,D2) is bilinear, and we are done. D
We shall usually denote (D\, ?>2) by D\ • D2. As a corollary of the above
proof, note that, if ?>2 is smooth, then D\ • D% = degCx(jDi)|jD2. In fact,
a similar formula is true for an irreducible curve ?>2, noting that we can
define the degree of a line bundle L on an irreducible curve С in several
equivalent ways:
(i) As the degree of the pullback of L to the normalization С of C;
(ii) By writing L = Oc(Yli ntPt)i where the pi are points in the smooth
part of C, and taking degL = ?V щ;
(iii) Via the exponential sheaf sequence
and the fact that for an irreducible curve we have H2{C\ Z) = Z.
The uniqueness part of the proof of Theorem 4 shows that Dx- D2 =
[?>i] • [D2], where [?)<] is the homology class asociated to Д and we use
intersection product in homology. Finally, two remarks that we shall often

1. Curves on a Surface 13
ее are the following: if Я is ample and D is effective and nonzero, then
I • D > 0. Moreover, if C\ and C2 are distinct irreducible curves, then
7i • Ci > 0, and C\ • C-i = 0 if and only if C\ and Сг are disjoint.
In the rest of this chapter we shall use intersection theory to analyze
urves and linear systems on X.
function and arithmetic genus
hippose that С is a smooth curve on X. Then С ¦ С = degOx{C)\C.
Jy general results on smooth divisors (see for example Hartshorae [61, p.
L82]), Ox(C)\C = Nc/x is the normal bundle to С in X. For an effective
liyisor C, not necessarily smooth or reduced, we shall sometimes define
be normal bundle of С in X to be simply Ox{C)\C. We shall also usually
abbreviate С • С by С2.
a smooth curve C, we also have the adjunction formula
Kc = Kx®Ox{C)\C.
Ibis follows from the normal bundle sequence
0 -> Tc -» TX\C -»JVC/X -» 0,
rhich gives det(Tx|C) = (A"x 1С) = K^1 ® C7X(C)|C. Thus, if д(С) = д
|| the genus of C, then
[
П0К we use the same symbol KX to denote the canonical line bundle and
the canonical divisor (class).
^fbr С any nonzero effective divisor on X, we can still define the dualizing
mmf we by the same formula as A.3)
Щ& significance of u>c is that it is the unique line bundle on the (possibly
¦jngiilftr) scheme С for which Serre duality holds: there is a trace map
Щ}(С;шс) —» С such that, for every line bundle L on C, the induced map
is a perfect pairing (Serre [136] for the case where С is reduced and Barth-
PWere-Van de Ven [7] in general). In particular Н1{С\шс) is dual to
H°(C;Oc). Warning: If С is reduced and connected, H°(C;OC) = С
and the trace map is an isomorphism. In general, however, H°(C; Oc) may
be larger than С (Exercise 3) and thus Нх{С;шс) may have dimension
larger than 1 also.
Far a general nonzero effective divisor С we define the arithmetic genus
of С by the same formula as before
0-6)	2pa(C)-2 = (K

14 1. Curves on a Surface
In case С is reduced and irreducible, 2pa(C)-2 is therefore equal to degwc-
In general an application of the Riemann-Roch theorem on X (see Exercise
8) shows that
A-7)	Pa(C) = l-x(C;Oc).
Thus, if h°(C; Oc) = 1, for example, if С is reduced and connected, then
pa(C) = Ь}\ОС) = h°(C;wc)- As a result we have:
Corollary 6. If С is ал irreducible curve on X, then pa(C) > 0. Thus,
(Kx + C) • С > -2. Moreover, (Kx + C) ¦ С = -2 if and only ifpa (C) = 0,
and otherwise (Kx + C) ¦ С > 0. D
In fact, we can say more than the statement that (Kx + C) • С > 0 if
Pa(C) > 1:
Proposition 7. If С is an irreducible curve on X with pa(C) > 1, then
шс has no base locus. D
For a proof of Proposition 7, see, for example, Catanese [17].
In general it will be useful to have various ways to calculate pa(C). We
begin with the case where С is reduced and irreducible. In this case the
normalization С is a smooth connected curve, and has a well-defined genus
g(C). Let v: С —» С be the normalization map. Now consider the exact
sequence
A.8)	Q-*OC-* vtO6 -> utOd/Oc -> 0.
For x e C, we define the local genus drop at x to be the nonnegative integer
Sx = dime [v*O6IOc\x ¦
Thus, 6X = 0 if and only if a; is a smooth point of C. For example, if x is an
ordinary double point of C, then 6X = 1. Likewise, if x is a cusp point, so
that, locally analytically near x,C is described by the equation y2 = z3, C?
is a smooth curve with coordinate t, and u(t) = (t2, t3) in local coordinates,
we again have 6X = 1. We leave it as an exercise to show that conversely, if
6X = 1, then x is either an ordinary double point of С or a cusp. The genus
drop of the curve С is the nonnegative integer 6 — ]dgc **• N°te that we
can still define the local invariant 6X, and hence 6, if С is only assumed
reduced, but not necessarily irreducible.
Lemma 8. If С is reduced and irreducible, then pa(C) = g(C) + 6. Thus,
Po(C) = 0 if and only if С is a smooth rational curve. More generally,
let С be a reduced but not necessarily irreducible curve on X, and let
С\,...,Сп be the connected components of C, with g(Ci) = <ft. Then
+6 + 1-П.

1. Curves on a Surface 15
Proof. We shall just check the first statement, as the proof of the second
is similar. Since С is connected, H°(Oc) = H°(Oq) and the long exact
cohomology sequence of the sheaf sequence A.8) gives
0 -* H°(v.O6/Oc) -+ H\OC) -+ H\Oe) -+ 0.
A dimension count gives ft^Oc) = ^(Og) +6 = g(C) +6. Q
We will return to the study of 6 in Chapter 3. We now briefly discuss the
nonreduced case. The main tool for studying С in this case is the following
exact sequence: suppose that С and D are two nonzero effective divisors,
not necessarily reduced or irreducible. Here we will allow С and D to have
components in common. Then, by Exercise 9, there is an exact sequence
A.9)	0 -> OD(-C) -> Oc+D ->Oc-+0.
Thus, A.9) allows us to work out x(@c+d) from the knowledge of
X(Od(—C)) and X(®c), and would allow us to work out Л°@с+?>) and
thus h1 (Oc+d) if we could work out the coboundary maps in the associated
long exact sequence (which is usually impossible in general). An important
application of A.9) is to work out Onc0> where Co is a reduced and irre-
irreducible curve. In this case, for n > 2, the exact sequence of A.9) applied
to С = (n - l)Co and D = Co becomes
A.10)	0 -* Oco(-(n ~ 1)CO) -* OnCo -* O(n_1)G0 -* 0.
The Riemann-Roch formula
¦¦*¦/'
Let D be a divisor on X. The Riemann-Roch formula is then:
Theorem 9. The Euler characteristic x(X;Ox(D)) is given by the fol-
bwing formula
Proof. The formula is trivially valid for D = 0. Next, for D = С a smooth
curve, use the exact sequence
0 -* Ox - OX(C) -» OX(C)\C -» 0.
By additivity of the Euler characteristic,
X(OX(C)) = x(Ox) + x(Ox(C)\C).
On the other hand, g(C) = (C2 + Kx ¦ C)/2 + 1 and degOx(C)\C = C2.
Thus, an application of the Riemann-Roch theorem for curves gives
x(Ox(C)\C)=C2-{°2 + KxC Л - C2-KXC

16 1. Curves on a Surface
verifying the formula in this case. In the general case, write D = C\ — C2
where C\ and C% are smooth, which is possible by Lemma 5, and use the
exact sequence
0 -» Ox(d - Ca) -» Ox(Ci) -» Ox(Cx)\C2 -» 0.
Thus, x(Ox(Ci-C2)) = x(Ox(Ci))-x(Ox(Ci)\C2). By the previous case
x(Ox(Ci)) = \ (Cf-CiKx)+x(Ox), and adjunction and Riemann-Roch
on C% give
X(Ox(C,)|Ca) = (d • C2) - \{C% +C2KX)-1 + 1.
Combining these gives the following formula for x(@x(Ci - C2)):
i(C? - d ¦ KX) + x(Ox) - (Cx ¦ C2) + J(Cf + C2 • Jfx)
=i(C? + C22 - 2(d • C2) - d • ^ + C2 • ^x) + x(Ox)
=|((Ci - C2J - (Cx - d) ¦ Kx) + x(Ox),
as desired. П
Note that as a consequence of the Riemann-Roch formula, we recover
the Wu formula for divisors: Kx ¦ D = D2 mod 2.
Closely tied in with the Riemann-Roch formula is Serre duality :
Theorem 10. IfD is a divisor on X, then the vector space H*(X; Ox (D))
is naturally dual to Н*-*(Х; OX(KX - D)). О
Notice that the Riemann-Roch formula is indeed invariant under the
substitution D >-* Kx — D.
Finally, the Riemann-Roch formula is most effective when we have some
criteria for the vanishing of H%(X;Ox(D)). The most famous of these is
the Kodaira vanishing theorem: if D is ample, then Н*(Х;Ох(-О)) = 0
for i = 0,1. Dually H{(X; Ox (Kx + D)) = 0 for i = 1,2. We shall discuss a
generalization of the Kodaira vanishing theorem at the end of this chapter.
Algebraic proof of the Hodge index theorem
We shall give Grothendieck's elegant proof of the algebraic version of the
Hodge index theorem for divisors [57].
Theorem 11. Let H be an ample divisor on X, and let D be a divisor
such that D ¦ H - 0. Then D2 < 0, and if D2 = 0, then DE = 0 for all
divisors E.
Proof. We begin with the following lemma:

1. Curves on a Surface 17
Lemma 12. Let Я be an ample divisor on X, and let D be a divisor such
tiat D2 > 0 and D ¦ H > 0. Then for alln^O, the divisor nD is nonzero
and effective.
Proof. Applying the Riemann-Roch formula and Serre duality to nD gives
,i h°(Ox(nD)) + h°(Ox(Kx - nD)) > x(Ox(nD)) = \D2n2 + O(n).
Thus, for all n » 0 either nD or Kx - nD is effective. However, H ¦ (Kx -
nD) < 0 if n > (Я • KX)/(H ¦ D), so that Kx - nD cannot be effective as
soon as n is sufficiently large. Thus, nD is effective for all n ~3> 0, and it is
nonzero since (nDJ = n2D2 > 0. ?
' Returning to the proof of the Hodge index theorem, let D be a divisor
such that D ¦ H = 0. Suppose first that D2 > 0. By Serre's criterion (see,
for example, [61, II §7]), mH + D is ample for all m » 0. Now D2 > 0
D • (mH + D) = D2 > 0, so applying Lemma 12 to the divisor D
j replaced by mff + D, it follows that nD is effective and nonzero
all n » 0. But then (nD) • H > 0, and so D • Я ^ 0, contradicting our
Bypothesis on D.
Thus, we must have D2 < 0. Suppose now that D2 = 0. If there exists
a divisor E with D • E ф 0, after replacing E by — E we may assume that
Щ-Е> 0. Next replace S by the divisor E' = (Я2)^ - (Я ¦ Я)Я. Then
¦Щ\. H = 0 and D ¦ E' = AР)(О • E) > 0. If we now set D' = mD + E',
Шеа D' ¦ H = 0 and (D1J = 2m(?> • E') + (E1J. For m » 0, ?>' is thus a
divisor satisfying (D'J > 0, D' • H = 0, contradicting the first part of the
firoof. It follows that either D2 < 0 or D2 = 0 and D ¦ E = 0 for all divisors
Щ as claimed. П
Definition 13. A divisor D is numerically equivalent to 0 if D- E = 0 for all
divisors E. Two divisors ?>i and ?>2 are numerically equivalent if ?>i — ?>2 is
numerically equivalent to 0. Note that linear equivalence implies numerical
.•equivalence. We let NumX be the quotient of Div-X" by the equivalence
relation of numerical equivalence.
Let ЩХ; Z) = H2(X; Z) modulo torsion. Then H2(X; Z) is a subgroup
of Я2^; С). According to Hodge theory, H2(X; C) S Я2-°(Л')еЯ1'1(А')е
H°'2(X), where H™(X) ^ НЧ(Х;ПРХ) and HP'"(X) is the complex con-
conjugate of H**(X). Thus, the subspaces H20(X) ® H°>2(X) and Hl-l(X)
are invariant under complex conjugation and are therefore the complexifica-
tions of subspaces of H2(X; R), which we denote by (Я2'0^) ф Н°<2(Х))^
¦nd Я1'1(Х)к, respectively. If D is a divisor, then the image of its associ-
associated cohomology class in H2(X; C) lies in Hl>1(X) as well as in the image
of IP(X;Z), which we can write as Я^Л^ПЯ^рГ). The usual Hodge
index theorem then says that the intersection pairing is positive definite
on (Я2'0^) е Н°-2(Х))Л and negative definite on the complement of an

18 1. Curves on a Surface
ample divisor in Я'^рОн. Combining the algebraic Hodge index theorem
above with the usual Hodge index theorem, we find:
Lemma 14. The natural map from Num-X" to H2(X;Z) П Я
H2(X;C) is an isomorphism.
Proof. There is the natural map from Div X to H2(X; ZJntf1-1 (X), which
is surjective by the Lefschetz theorem on A, l)-classes. Clearly, if D is not
numerically equivalent to zero, then the image of D in iP(X; Z) П Hl<1(X)
is nonzero. Conversely, suppose that the image of D in H2(X; Z)r\H1>1(X)
is nonzero. Let Я be an ample divisor on X. If [D] • H ф 0, then D is not
numerically trivial. If [D]-H = 0, then by the usual Hodge index theorem
[D]2 < 0, and so D is again not numerically trivial. Thus, the kernel of
the map from DivX to tf^jZ) П Н1Л(Х) is exactly the subgroup of
numerically trivial divisors. ?
As a corollary, we have:
Corollary 15. The group Num-X" is a finitely generated torsion free
abelian group, of rank at most Л1'1^). G
We denote the rank of Num-X" by p(X) = p, the Picard number of
X. Over R, the signature of the nondegenerate pairing on Num-X* ® R is
2 — p. The following is then an immediate consequence of the fact that the
signature is well defined:
Corollary 16. Let H be any divisor on X with H2 > 0, not necessarily
ample. Then the intersection pairing on
Hx = {De NumX : D • H = 0}
is negative definite.
Ample and nef divisors
We begin with the statement of the Nakai-Moishezon criterion for ample-
ness ([61, p. 365]):
Theorem 17. A divisor H is ample on X if and only if H2 > 0 and
HC>0forall irreducible curves С on X. D
There is a generalization of Theorem 17 to arbitrary, possibly singular
compact (proper) schemes [98], [69].
Corollary 18. If the divisor H is numerically equivalent to an ampie
divisor, then H is ample. О

1. Curves on a Surface 19
It is easy to see that the corollary fails if we replace ample by very ample.
The next question is to describe in general terms the set of all ample
divisors in Num X. It is easy to see (Exercise 11) that this set is closed under
positive integer linear combinations. It is convenient to work in Num X ®
R. Note that Num-X" igi R = W has an intersection form q, the natural
extension of intersection pairing on Num X. There is a real basis ei,..., ep
of Num X®R such that
Let P С NumXigiR be the subset of vectors ? = ]jTV nei such that q{?) > 0.
Then P consists of two pieces P+,P- where P+ = {[jTV ц^ : x\ > 0} and
Lemma 19. With notation as above:
(i) P± are convex.
(ii) 'Given ? e P+, suppose tbat rj lies in the closure P of P. Then ? ¦ rj = 0
;. • if and only if r) = 0. Otherwise, r) lies in the closure ofP+ if and only
Proof. Part (i) is an easy consequence of the Cauchy-Schwarz inequality
and is left to the reader. For (ii), first suppose that ? • ¦q = 0. Since ¦q € P,
172 > 0. Thus, ? and ¦q span a positive semidefinite subspace of Num-X" ®R,
which can have rank at most 1, and so rj = 0. Now consider the function
P -* {±1} defined by rj *-* sign(? • rj), which is well defined by the first
part of the proof since ? • t] ф 0. This function is constant on the connected
components of P and thus on P+, P— Since ? ¦ ? > 0 and ? • (-?) < 0, we
see that the sign must always be positive on P+ and always negative on
P_. D
Fix an ample divisor H. We can choose our basis above so that H € P+.
Since H ¦ H' > 0 for every ample divisor #', H' € P+ for every ample #'.
We can take the convex hull of the ample divisors in Num X ® R. It forms
a cone in P+ с Num X ® R, the ample cone A(X). If H e A(X) П Num X,
then H is ample by the Nakai-Moishezon criterion, since H2 > 0 and
H ¦ С > 0 for every irreducible curve С
Lemma 20. A(X) is open.
Proof. Let Я be an ample line bundle. For an integral basis d\,...,dp of
NumX, let Di be a divisor corresponding to сЦ. By Serre's criterion for
ampleness, there is a positive integer Ni such that JVjff ± Д is ample for
every i. Thus, A(X) contains the convex hull of the points H ± (I/MR
for every i, and hence contains an open set around H. Now every element

20 1. Curves on a Surface
of A(X) is a finite sum ]jT]fc Afcff*, where A* e R+ and Hk is ample, and
for every к there is an open neighborhood Uk of Afcff* contained in A(X).
It follows that A(X) contains Ylk ^fci which contains an open set around
Lemma 21. Let
A'(X) = {x € P+ : x ¦ С > 0 for all irreducible curves C};
A(X) = {x €~F^ : x ¦ С > 0 for all irreducible curves C},
where P^ is the closure of P+. Then A(X) С А'(Х) С AjX) and AjX) is
the closure ofA(X). Moreover, A(X) is the interior ofA'(X) and of A(X).
Proof. We have already seen that A{X) is an open convex cone. Clearly,
A(X) С А'(Х) С А(Х) and all three sets are convex. Moreover, A(X) is
closed and thus contains the closure of A(X). Conversely, let A € A(X).
Let fi\,..., (лр be a basis for Num X®R consisting of ample divisors, which
exists because A(X) is open. Then, for every n > 0, the set {A + ]?< ^A*» :
0 < U < 1/n} is open in Num X®K, and thus contains a rational point hn.
Clearly, h% > 0 and hn ¦ С > 0 for every irreducible curve C, so that some
integral multiple of hn is ample by the Nakai-Moishezon criterion. Thus,
hn € A(X), and clearly 11%^,,, Л„ = A. It follows that A is in the closure
of A(X). The closure of A(X) therefore contains A(X) and so is equal to
A(X). Finally, it is a general fact that an open convex subset of Rn is the
interior of its closure (this follows easily from, for example, [131, p. 81 ex.
1]), and so A(X) is the interior of A(X) and hence of A'(X). О
Note that, in the definition of A'(X) or A(X), it is enough to consider
only those irreducible curves С with C2 < 0. Indeed, if C2 > 0, then as
С ¦ H > 0 for every ample divisor H,C € P+ and thus A • С > 0 for every
A € P+. Despite the fact that A(X) is described by a countable number
of inequalities defined by integral elements of NumX ® R, its boundary
can be very complicated, in the sense that the boundary can be far from
being a finite polyhedron, even locally. For example, the boundary can be
"round." We also note that there are surfaces X where A' (X) is neither
open nor closed.
Motivated by Lemma 21, we make the following definition:
Definition 22. A divisor D is nef if D ¦ С > 0 for all irreducible curves
C. A divisor D is big if D2 > 0.
In earlier terminology, a nef divisor is also called numerically effective
or pseudo-ample. According to some authors "nef stands for "numerically
eventually free," and we will discuss the reason for this shortly.
Lemma 21 then implies that the divisors in A(X) are exactly the nef
divisors with D2 > 0. In fact this last condition is redundant:

1. Curves on a Surface 21
[jemma 23. Suppose that D is a nefdivisor. Then D2 > 0.
Proof. Fix an ample divisor H. If D2 < 0, there exists a to > 0 such that
D + U)HJ = 0 and (D + tHJ > 0 for t > to. By the Nakai-Moishezon
•riterion, D + tH is then ample for t > t0, t € Q. For such t, some multiple
rf D + tH is then effective, so that D(D + tH) = D2 + t(D ¦ H) > 0.
[.ikewise, D • tf > 0. By continuity, D2 + to(D ¦ H) > 0, and so
0 = (D + t0HJ = D2+ 2to(D ¦ H) + t20H2
= D2 + to(D ¦ H) + to(D ¦ H) + t\H2
> to(D ¦ H) + t20H2 > t%H2 > 0,
i contradiction. Thus, D2 > 0. П
We note, however, that an example due to Mumford [60] shows that there
ndsts a surface X and a divisor D such that D ¦ С > 0 for every irreducible
яре С on X but D2 = 0.
'A standard example of a nef and big divisor on X is obtained by taking a
livisor D such that the morphism denned by the complete linear system \D\
us no base points and is genetically finite onto its image. More generally,
by analogy with ampleness we make the following:
Definition 24. A divisor D is eventually base point tree if for all n » 0,
the linear systen \nD\ has no base points.
, Suppose that D is eventually base point free. Let ipno be the morphism
lefined by \nD\ for n such that \nD\ is base point free, and let Xo be
(he image of X under tpnD- We assume that ipno is generically finite, or
Jquivalently that D is big. From the embedding Xo С Р^, there is an
imple line bundle Lo on Xo> the restriction of OvnA), which pulls back
to Ox(nD). The morphism tpnD has a Stein factorization X —> X —» Xo,
where X is normal and X —» Xo is finite. Let L be the pullback of Lo to X.
Since X -* Xo is finite, L is ample, and it pulls back to Ox (nD). If тг: A" —>
Y is the natural map, then since X is normal (or by the construction
if the Stein factorization) ir*Ox = Ox. Thus, ivtOx(nD) = tt.tt'L =
L®ntOx = L. It follows that F°(A-;Ox(nD)) = Я°(Х;1) and similarly
for H°(X; Ox(mnD)). Hence, for all к » 0 and divisible by n, the image of
X under the morphism (pkD defined by |A;D| is the normal projective surface
X. Moreover, if С is an irreducible curve on X, then (fkD(C) is a single
point if and only if С • D = 0. It follows that X is the uniquely specified
normal surface obtained by contracting all such curves С By the Hodge
Index theorem, the curves С for which С • D = 0 span a negative definite
mblattice of Num-X". Moreover, they must be independent in NumX ® Q:

22 1. Curves on a Surface
Lemma 25. Let C\,...,Cr be distinct curves spanning a negative defi-
definite sublattice of Num X. Then the classes of the C* are independent in
Proof. If not, there exists a relation J2i n»C« = ° ™*п tne "t € Z, not
all 0, and after deleting some of the Cj we can assume that none of the
щ is 0. Since the Q are effective, not all of the щ are positive (otherwise
intersect with an ample divisor). Collecting the negative terms, and possibly
relabeling, there is a relation of the form
j=8+l
where m,j = -щ > 0. Now (?*_, rijCjJ < 0 by assumption. On the other
hand,
) (
i—l	j=e+l
which is only possible if E2'=\ щСхJ = О. Since the lattice spanned by the
Ci is negative definite, it follows that 5ZJ=1 и<С» is numerically equivalent
to zero, which is impossible since the гц are all positive. D
Despite the apparent similarities between being eventually base point free
and being ample, they are very different properties. The major difference
is that there is no numerical criterion for when a nef and big divisor is
eventually base point free (Exercise 7 in Chapter 3). On the other hand,
Mumford has proved the following generalization of the Kodaira vanishing
theorem:
Theorem 26. Let D be a nef and big divisor on the smooth surface X.
Then Н*(Х; Ox(-D)) = 0 for i = 0,1. Dually H\X\ OX(KX + D))=0
for i = 1,2.
We will give a proof of Theorem 26 based on Bogomolov's inequality in
Chapter 9.
Exercises
1.	Show that Noether's formula is equivalent to the Hirzebruch signature
theorem:
b+(X) - b(X) = J(cJ(*) - 2c2(X)).
2.	Let X be a smooth surface in P3 of degree d. Show using standard
facts about the cohomology of P3 that q(X) = 0. (A stronger statement
follows from the Lefschetz theorem on hyperplane sections: X is simply
connected.) Using the fact that Kx = Ox(d — 4) by adjunction and

1. Curves on a Surface 23
the fact that q(X) = 0, determine cf(X) and pg(X). Apply Noether's
formula to find t>2(X) and thus h1'1^). (Of course, you could also,
with somewhat more effort, find ft1'1 (if) directly and then find bi(X).)
When is the intersection form on X of Type I?
3.	Let С be a smooth rational curve on X. For n > 0, find
dimH°(nC;Onc) and Pai'nC) in terms of C2. What can you say if
.i instead g(C) > 0?
4.	Let С be a reduced irreducible curve and x a point of C. Show that
Sx = 1 if and only if x is an ordinary double point or a cusp. (One
direction has already be done.)
5.	Calculate 6X for a singularity of the form y2 = x2fc+1 and also for a
singularity of the form y2 = x2k. What is the local picture of these
singularities?
6.	Let Ai,...,An be distinct complex numbers and consider the local
singularity С given by
Thus, С is a union of n distinct lines meeting at the origin. Compute 6o
directly from the definition in this case. (Note: С is not obtained from
its normalization by identifying the common point on the n branches
if n > 2.) We will see an easier way to compute 6o for this example in
Chapter 3.
7.	Let С be an irreducible curve whose singular locus is a single ordinary
double point x, and let p, q € С be the preimage under the normal-
normalization map v of x. Show that и*шс = Kc <S> Oc (p 4- q)- Which local
sections of this line bundle are the pullbacks of sections of a>c? Analyze
a cusp singularity similarly.
8.	Use the Riemann-Roch theorem applied to x(Px(—C)) to show that,
for every effective nonzero divisor C, pa(C) — 1 - \(C; Oc)- Also show
that, for two such effective divisors С and D,
pa(C + D) = pa(C) + pa(D) + CD-1.
9.	Verily that, in the notation of A.9), the natural map Oc+d —* Oc is
surjective (this is obvious) and that its kernel is Od(-C).
LO. Let Di and D2 be two divisors with D\ > 0. Show that
(D1J(D2J < (D, ¦ D2J
and analyze the case where equality holds. (This is an easy consequence
of the Hodge index theorem which applies to all nondegenerate sym-
symmetric bilinear forms whose intersection matrix has just one positive
eigenvalue.)
Ll. Show directly that, if H\ is ample and H2 is eventually base point free,
then Hi+H2 is ample. Using the Nakai-Moishezon criterion, show that
if Hi is ample and ff2 is nef, then Щ + H2 is ample.

24 1. Curves on a Surface
12.	Let я-: X -» Y be a surjective morphism of surfaces. Given a divisor D
on X, there is a divisor jr.DonK denned as follows: For С irreducible,
7r«C = 0 if ;r(C) is a point, and otherwise ntC = dn(C), where d is
the degree of the map from С to 7r(C). For general D we extend тг« by
linearity. Prove the projection formula n*D ¦ E = D • n*E. Conclude
that 7Г* induces a map Numy —» Num-X", also denoted w*, and that
;r« induces a map NumX —» Numy which we continue to denote by
т».
13.	Let Я be a nef and big divisor on the surface X. Suppose that H =
A + B, where A and В are effective. Show that A • В > 0, with
equality holding if and only if one of A or В is 0. We say that H is
numerically connected. (Let E = aH~ A with о = (А • Wj/H2. Since
0 < A • H < (A + B) ¦ H = H2, 0 < о < 1. By the Hodge index
theorem, E2 < 0. Now estimate A ¦ В = (atf - E) ¦ (A - a)tf + E),
and analyze the case of equality.)

Coherent Sheaves
What is a coherent sheaf?
jLet X be a scheme (or analytic space) with x E X and let R = Ox,x be the
local ring at x. For our purposes X will usually be regular, and we could as
twork in the analytic category, so that the reader can for the moment
R = C{z\,...,zn} to be the ring of convergent power series at the
origin if so desired. There are two paradigms for what a coherent sheaf T
on X should look like:
(i) A locally free sheaf, locally modeled on the free module RN;
(ii) An ideal sheaf, locally modeled on an ideal I C R.
; general torsion free coherent sheaf, roughly speaking, is a blend of these
що' models, and torsion, as we shall see, essentially corresponds to some
torsion free sheaf supported on a proper subvariety of X. Another way to
tnmk of coherent sheaves is the following: begin with a vector bundle V
on X. Its sheaf of regular (or holomorphic) sections is locally isomorphic
to Ox, in any open set where the bundle is trivialized. Moreover, there
is a natural functor from the category of algebraic or holomorphic vector
bundles over X to the category of locally free sheaves of Ox-modules.
Every locally free sheaf arises from a vector bundle, and two vector bundles
are isomorphic if and only if the corresponding locally free sheaves are
isomorphic. However, not every morphism of locally free Ox-modules comes
from a vector bundle morphism. In a local trivialization, a morphism O% —»
0% is just given by a matrix of functions, which need not have constant
rank. However, vector bundle morphisms are required to have constant
rank, so that the cokernel is again a vector bundle. From this point of view,
coherent sheaves are the smallest category of Ox-modules we can obtain
by locally enlarging the category of locally free Ox-modules so that every
morphism has a cokernel, and then gluing these local models together. This
then is the definition that a coherent sheaf of Ox-modules T locally has a
presentation
O? -* O4 - T -»0.

26 2. Coherent Sheaves
The local model of such a sheaf is then a finite Д-module M. The Noether-
ian properties of Д are reflected in the statement (not trivial in the analytic
case) that the kernel of a map of Ox-modules 0% —* Ox k a^BO coherent.
The definition of a coherent sheaf allows us to bring in all the complexities
and beauty of commutative algebra, and it is to be hoped that the chapters
on vector bundles will amply illustrate both of these features. Let us give
some very simple illustrations here of this picture.
Example 1: The meaning of torsion. We suppose for simplicity that
X is smooth. A torsion section s of T corresponds to an element m of the
Д-module M which is annihilated by some /ей. Thus, s is (at least near
x e X) supported on the subvariety {/ = 0}. Conversely, if s is a section
with support on a proper subvariety V с X, let / ф 0 € R be an element
corresponding to the ideal of V, and let V = {/ = 0} Э V. The statement
that s has support on V means exactly that the restriction of s to X — V
is zero, and thus that s restricts to 0 on X - V. Algebraically, this means
that m is in the kernel of the natural map M —> M/ = M ® Л/, where Д/
is the localization of R at /, and then it follows from the definition that
there exists an n > 1 such that fnm = 0. Thus, m is a torsion section,
annihilated by some power of /.
In particular (using finite generation properties associated to coherence)
a torsion coherent sheaf T is precisely one which is supported on a proper
subvariety.
Example 2: Generic rank. Let X again be a smooth (and connected)
scheme, which for simplicity we shall also assume to be affine: X = Spec R.
Let T be a coherent sheaf on X, corresponding to the Д-module N. Then
as Д is an integral domain it has a field of fractions K, and the rank of
N is the dimension of the K-veetoi space N ®д К. Let r be the rank, so
that N ®RK = Kr. Choose a basis vi,..., vT of N ®R K. After clearing
denominators we may assume that Vi is the image of щ е N. Thus, there is
a well-defined map from the free module R1' to N which makes the following
diagram commute:
N
RT 	> RT®RK = KT,
and so RT —> N is injective. If Г is the cokernel, using the fact that
Дг ®л К -> N ®я К -> Г ®д К -> 0
is still exact and that the first map is an isomorphism we see that Т® д К =
0, and thus Г is a torsion sheaf. It follows by Example 1 that there is an
open subset U = Spec Д/ of X such that .F|C/ is free. Its rank is then the
generic rank of T. Pursuing this idea further, it is not hard to show that we

2. Coherent Sheaves 27
ищу stratify X by locally closed subvarieties Sa such that the restriction
of/'toeachSc is free.
Coherent but non-locally free sheaves play two different roles in the study
<rf the moduli of vector bundles:
1. As a way to dismantle a vector bundle V into canonically defined pieces;
j; As the necessary ingredient for compactifying moduli spaces of vector
.'¦ bundles.
V In general, one problem with studying holomorphic vector bundles from
the point of view of algebraic geometry is that a vector bundle is not a very
geometric object in the sense of algebraic geometry, for example in terms
of special configurations of points or divisors on a variety. On the other
band, the existence of many interesting vector bundles implies in some
way that there is a lot of interesting hidden projective geometry on, say,
an algebraic surface: for example, points in special position or systems of
Corvee in special position. One goal of trying to break up a vector bundle
Into more canonical pieces will be to make explicit the link between the
^Kaeeification of bundles and the algebraic geometry of the variety.
In this chapter, we will discuss some of the basics of the theory of vector
bundles and methods for constructing them. After reviewing Chern classes,
we discuss the construction techniques of extensions, elementary modifica-
modifications, and double covers. The chapter ends with some commutative algebra.
In Chapter 4, we will introduce the notion of stability and derive some of
its basic properties. Chapters 6 and 8 describe vector bundles over ruled
and elliptic surfaces. Finally, in Chapter 9, we return to the general theory
prove Bogomolov's inequality for a stable rank 2 vector bundle.
A rapid review of Chern classes for projective varieties
Let V be a vector bundle on a quasiprojective variety X. We can define
the Chern classes of V as follows:
B.1)	Cl(V) = Cl{detV)eH2(X;Z)
it X is defined over C. In general we could use an algebraic substitute for
//^(A'jZ) such as PicX in the algebraic case (in which case we take ci(V)
to be actually equal to det V), or Numl. To define Ci(V) in general, we
first define it for a direct sum of line bundles V = L\ ф • • • ® Ln. In this
case, formally
B.2)
the actual formula is obtained by equating the terms that lie in H2i(X; Z).
A similar formula holds if V, instead of being a direct sum of line bundles,
has a filtration by subbundles Vt such that Vi/V<_i = Ц is a line bundle.

28 2. Coherent Sheaves
To define Ci(V) for a general V, one shows that there exists a variety
Y and a morphism тг: Y -* X for which тг*: H'(X;Z) -* #<(Г;2) is
injective, for all i, and such that tt*V has a filtration as described above.
We have then defined Ci(Tr*V), and one shows that these classes are in the
image of тг* for all i. Then we can set Cj(V) to be the unique class such
that tt*c(V) = a(ir*V). For our purposes, the Chern classes Cj(V) will
lie in ff^.X'jZ). However, for a smooth protective variety X, there exist
algebraic analogues of Н^(Х\Ъ), namely the higher Chow groups A%(X)
consisting of algebraic cycles of codimension г modulo rational equivalence.
For more about the Chow groups, see Fulton's book [45] or Appendix A
in Hartshorne's book [61]. Essentially by definition, Al(X) = Pic A". But
for i > 1, the groups A*(X) are poorly understood and can be quite large.
In any case, one can define Chern classes Ci(V) € A*(X), which are a
much finer invariant of V. For X defined over C, there is is a natural
homomorphism Al(X) —> H2t(X;2) which sends a codimension г cycle to
its fundamental class, and the image of Ci(V) under this homomorphism
is the usual Chern class. Unless otherwise specified, we will always take
The Chern classes so constructed are functorial: <ч(/* V) = /*Cj(V), and
satisfy the Whitney product formula: for an exact sequence
0 -> V -* V -+ V" -* 0
of vector bundles, we have
B.3)	c(V) = c(V')c(V"),
where c(V) is the totai Chern ciass
B.4)	c(V) = l + c1(V) + ---+cn(V).
The idea behind the construction of the Chern classes is called the splitting
principle and has the following useful extension: every "universal" formula
for Chern classes which holds for direct sums of line bundles holds in gen-
general. For example, for the dual bundle Vv of a vector bundle V we have
the formula
B.5)	c(Vv) = c(Vy,
where given a class a = ^ a* € 0 H2i(X; Z), av is by definition the class
?<(-!)'<*• Similar formulas can be given for Sym* V, /\k V, Hom(V, И0,
V ig> W, ... . For example, if L is a line bundle and V is a bundle of rank
r, then
B-6)
On a smooth quasiprojective variety, we can define the Chern classes of
any coherent sheaf. This is because, by a theorem of Serre, every coherent
с2(У® L) = c2(V) + (r - l)d(V) • ci(?) + f Jci(?J.

2. Coherent Sheaves 29
she on a smooth quasiprojective variety admits a finite resolution by
locally free sheaves: there exists an exact sequence of sheaves
0 - En ~*	> ?° -* T -* 0,
where the ?' are locally free. We can then define the total Chern class of
f by the formula
ft is not difficult to show that this definition is independent of the choice of
the resolution and satisfies the Whitney product formula. However, other
properties (for example, the correct definition of pullback or tensor product)
require using the derived functors of tensor product, i.e., the Tor sheaves,
and we refer to Fulton [45] or Borel-Serre [12] for more details. One impor-
important example which we shall use often is the following: let Z be a reduced
irreducible subvariety of X of codimension r, and let j: Z —» X be the
Inclusion map. Then, as a consequence of the Grothendieck-Riemann-Roch
i theorem, Ci(j»Oz) ~ 0 for i < r and
U) = (-1)г-г(г - 1)! [Z],
where [Z] € H2r(X; Z) is the cycle defined by Z. A similar result holds for
a vector bundle V of rank n on Z: Ci(jtV) = 0 for i < r and
From this it is easy to deduce that Ci(j*Oz) = 0 for г < г and Cr(j*Oz) =
(s--l)r~1(r - 1)! [Z] for an arbitrary closed subscheme Z of X of pure codi-
; mension r, where [Z\ is again the associated cycle. Here if the irreducible
components of Z are Z\,..., Za and the length of О г along Ziisrrii, then
[Z] = J2i mi[Zi]- F°r example, suppose that X is a smooth projective sur-
face and that Z is a 0-dimensional subscheme of X supported at p\,..., p*.
Let j: Z —» X be the inclusion. Then Ox,PJIz,Vi is a finite-dimensional
C-vector space. Define ?{ZPi) to be its dimension and let ?(Z) = Yli^Pi)-
Then
c2{j.Oz) = -l(Z) e H*{X;Z) = Z.
Applying the Whitney product formula to the exact sequence
0-*Iz-+Ox-+j.Oz-+0,
we see that c2{Iz) = t(Z). More generally, if X is a smooth surface and V
is a rank 2 vector bundle on X for which there is an exact sequence
0 -* L -» V -» L' ® Iz -> 0,
where Z has dimension 0, then we see that
B9)

30 2. Coherent Sheaves
For an effective divisor D, the formula for c\(j+Od) is easy to verify:
using the exact sequence
0 -» Ox(-D) -+OX-+ j*OD - 0,
and the Whitney product formula, we see that
oo
B.10)	c(j.OD) =
We will need an extension of this formula to the case of a line bundle L
onZ:
Lemma 1. Let L be a fine bundle on the effective divisor D С X, and let
j: D —> X be the inclusion. Then
Proof. We may suppose that L = Op (V2 - Vi) is the line bundle associated
to a difference of two effective Cartier divisors V\, V2 on D. Begin with the
exact sequence
0 -»jtOD(-Vi) -»jtOD -* kltOVl -* 0,
where kit is the inclusion of Vi in X. For simplicity of notation we shall
often drop the j* or k\t and understand that all sheaves are sheaves on
X, and Chern classes are to be taken in this sense. The Whitney product
formula and the calculation B.10) of c(j*Od) gives
A + d(OD(-Vi)) + с2(СЪ(-Уг)) + •••) =
= (l + [D] + [DJ + ---)(l-[V1]+..-)-\
Equating terms gives
С1{ОО(-Ъ)) = [D],
c2(CM-K)) = [Df + (Vi].
A similar calculation with the exact sequence
0 -»j.OD(-Vi) -»j.OD(V2 - Vi) - k2tOV2(V2 - Vi) - 0
finishes the argument. D
We shall also need the Riemann-Roch theorem for vector bundles on
curves and surfaces (see Exercise 17):

2. Coherent Sheaves 31
Theorem 2.
(i) Let V be a vector bundle of rani r on the smooth curve С of genus g,
end let degdet V = d. Then
(И) Let V be a vector bundle of танк г on the smooth surface X. Then
X(X; V) =	C2(V) 4- r\(Ox)-	О
Rank 2 bundles and sub-line bundles
Our goal for the rest of this chapter will be to describe ways to construct
rank 2 vector bundles V over a smooth projective variety X. The simplest
method is to take a direct sum of two line bundles: V = L\ @ Li. Of course,
«6 don't expect to obtain especially interesting bundles in this way. One
#&У to modify this idea is to consider extensions of line bundles, in other
words rank 2 vector bundles V such that there is an exact sequence
If X ie a curve, then all rank 2 bundles can in fact be obtained in this
*f4y. However, for a surface X, most interesting bundles do not have such
i description.
Ц* Tb classify such bundles V, we recall that all such extensions are classified
Jijy the group Exkl{L-2,Lx) - Hl(X\ (L2)~l®Li). (See Exercise 1 for a more
concrete picture of this.) Here an isomorphism between two extensions V
and V (in the strong sense) is an isomorphism of bundles a: V —» V such
that the following diagram commutes:
0 	> Li 	> V 	> L2 	> 0
0 	v Lx 	у V 	у L2 	у 0.
A weak isomorphism of extensions is similary defined, except that we do
not require that the maps Li —y Li be the identity. As we are primarily
interested in V and not in the strong isomorphism class of the extension, we
shall just care about weak isomorphism classes of extensions. Since Lt is a
line bundle, its only automorphisms are C*, and so C* xC* acts on the set of
all isomorphism classes of extensions. Since the sealers are endomorphisms
of V = V, the diagonal subgroup of С* х С* acts trivially, and the weak
equivalence classes of extensions are the quotient of Ext^I^^i) by an
action of C*. It is easy to check that this action is just scalar multiplication,
so that the weak equivalence classes are either the trivial extension L\ ®L2
or are parametrized by a projective space P(Ext1(i-2,?i)).

32 2. Coherent Sheaves
For example, if X = P2, Li is necessarily Opa(oi) and
^	18	- o2)) = 0.
Thus, we cannot obtain any interesting bundles on the very simple surface
P2 in this way.
To make an interesting construction along these lines, consider, instead
of a rank 1 subbundle L\ of V, the following object:
Definition 3. A sub-line bundle of a rank 2 vector bundle on X is a rank
1 subsheaf which is a line bundle.
Note that every rank 2 vector bundle has a sub-line bundle: by Serre's
theorem, V®H has a global section (indeed is generated by global sections)
if if is a sufficiently ample line bundle. Thus, there is an inclusion H~l —¦
V. However, unless V has a canonically defined sub-line bundle, the fact
that it has some such is not very helpful.
Let us give a description of the local picture of a sub-line bundle. Let
R = Ox,x be the local ring of X at x; it is a UFD since X is regular.
Let (p: L —¦ V be a sub-line bundle. After choosing local trivializations <p
corresponds to an inclusion R —» R ф R. Thus, (p is (locally) determined by
v(l) = (f> 9) € Д Ф Д. Now either / and g are relatively prime elements of
R or they are not relatively prime. If they are relatively prime, consider the
map V': R®R-+ R given by ф(а, b) = ag-bf. Clearly, ImV' = (f,g)R = I
is the ideal generated by / and g. The following is an easy special case of
the exactness of the Koszul complex:
Claim 4. If f and g are relatively prime, the sequence
B.11)	0-»Я-^ЯфЯ-^/-0
is exact.
Proof. If (a, b) € KerV', then ад = bf. Since / and g are relatively prime,
Да and g\b. Thus, о = fh and b = gh'. On the other hand, fgh = fgh', so
that h = h'. Thus, (a, b) = (p(h) elnop. О
It is also possible that / and g are not relatively prime. Let t = gcd(/, g),
where t is not a unit in R, and write / = tf, g = tg1. Note that (R Ф
R)/Im(p has torsion, since {f',g') ? Im^j but t{f',gi) € Im^- There is
an induced map <p': R -+ R® R defined by ??'A) = (f',g'), and (p is the
composition of this map with multiplication by t. Alternatively, <p extends
to a map from (l/t)R to R Ф R. Globally, t defines an effective divisor D
on X and we can summarize these calculations in a coordinate free way as
follows:

2. Coherent Sheaves 33
proposition 5.
(i) Let<p: L -* V be a sub-line bundle. Then there exists a unique effective
divisor D on X, possibJy 0, such that the map <p factors through the
inclusion L -> L® Ox{D) and such that V/{L ® Ox{D)) is torsion
bee.
(j|) Jn the above situation, ifV/L is torsion free, i.e., if D = 0, then there
exists a iocal complete intersection codimension two subscheme Z of
X and an exact sequence
0-*L->V->L'®Iz-*0.	О
In particular, if X — С is a curve, then Z = 0 and V is an extension of
line bundles in Case (ii). Thus, since every rank 2 vector bundle has a sub-
line bundle, every rank 2 bundle over a curve can be written as an extension
of line bundles. Let us give a simple application of this, by classifying all
шв rank 2 vector bundles over a curve of genus 0 or 1. (The case of genus
0» a specif case of a theorem of Grothendieck [56], and the case of genus
(??'due to Atiyah [4].)
Theorem e.
Щ Let С = P1 and Jet V be a rank 2 vector bundle over P1. Then V =
* i Ori (а) ф Opi (b) for integers a, b, unique up to order.
(U) Let С be a smooth curve of genus 1 and Jet V be a rank 2 bundle on
С Then exactJy one of the following holds:
(а)	V is a direct sum of line bundJes;
(б)	V is of the form ?® L, where L is a Jine bundle on С and ? is the
(unique) extension of Oc by Oc which does not spJit into the direct
sum Ос Ф Oc;
(c) V is of the form Tv ® L, where L is a Jine bundle onC,p€ C, and
Tv is the unique nonsplit extension of the form
0 -* Oc -* Fv ~* Ocip) -* 0.
Proof, (i) Let V be a rank 2 vector bundle on P1. Since
deg det( V ® OPi (a)) = deg det( V) + 2a,
we may assume that degdet(V) = 0 or -1. Let us first assume that
degdet(V) = 0, i.e., detV = 0P». Then by the Riemann-Roch theorem
applied to vector bundles on P1, we have x(V) = 2. Hence h°(V) > 2.
Choose a 2-dimensional subspace of H°(V) and consider the associated
тлР Opi —* V. If this map is injective, then its determinant det О\х —
Opi —» detV = Opi is nonzero. Hence the determinant map is an iso-
isomorphism. It follows that V = Oplt which is certainly a direct sum of
line bundles. Otherwise, the image of the map O\t -+ V is a line bundle L

34 2. Coherent Sheaves
which has the property that the image of H°(L) contains the 2-dimensional
subspace of H°(V) that we chose. Thus, H°{L) > 2, and so L =* OPi (k)
for к > 1. The map L -+ V factors through a map L ® Or1 @ —» V, where
t > 0 and where the quotient is torsion free and thus a line bundle. So in
this case we can write
0 -¦ Opi (a) -* V -* Opi (-a) -* 0,
with о = к + ? > 1. But the extensions of Opi (-o) by Opi (a) are classified
by H1(OfiBa)), which is zero since о > 0. Thus, the extension splits:
In case degci^V) = — 1, then again by the Riemann-Roch theorem for
vector bundles, x(V) > 1. So there is a nonzero map Ori -* V, which as
above factors through a map Opi (a) —» V with a > 0 and the quotient is a
line bundle. Thus, there is an exact sequence
0 -¦ Opi(o) -» V -* Opi(-o - 1) -» 0.
As Я1(Ор1 Bo - 1)) = 0 for all о > 0, this extension must again split and
V	e* Opi (о) ф Opi (-a - 1). We leave the uniqueness as an exercise.
Next we prove (ii). Let С be a curve of genus 1 and V a rank 2 vector
bundle on C. As before, after twisting V by a line bundle we may assume
that degdet V = 0 or 1. First assume that degdet V = 0. Note the follow-
following:
Claim. Suppose that deg det V = 0 and that there is a nonzero map Lq —*
V	with deg Lo > 0. Then either V splits or V is of the form ?<8>L, where
deg L — 0. In particular, V satisfies (a) or (b) of the statement of the
proposition.
Proof of the Claim. As usual, we can factor the map Lq —> V, so that
there is an exact sequence
0 -+ Li -+ V -* bi -* 0,
with deg L\ > 0 and deg Li = — deg L\. As we have seen, such extensions
are classified by Я1((//2) ® Li), which is Serre dual to H°(L2 <g> (bi))-
If degLi > 0, then degL2 ® (bi) = -2degLi < 0. Thus, H°(L2 ®
(Li)) = 0 and the extension splits, i.e., V = L\ ©L2. If degLi = 0, then
either L2 ® (Li) is nontrivial, in which case H°(L2 ® (Li)) = 0 again,
or L2 ® (bi)~x = Oc, i.e., L\ = L2. In this last case, dimH°(Oc) = 1,
and there is correspondingly a nonsplit extension ? of О с by Ос, unique
up to weak isomorphism. Clearly, in this case, V = ? ® Li- ?
Returning to the proof of (ii), with degdet V = 0, suppose that h°(V) ф
0. Then there is a nonzero map Oc -* V, so that the hypotheses of the
claim are verified. Thus, we see that V satisfies (a) or (b) of the proposition.

2. Coherent Sheaves 35
So we may assume that h°(V) = 0. Choose a point peC, and consider the
vector bundle V®Ocip)- By the Riemann-Roch theorem, x{V®Oc{j>)) = 2
and thus h°(V ® Ocip)) > 2. Choose a 2-dimensional subspace of H°(V ®
Ocip)) an<^ consider the map Oc -* V ® Ocip)- К the image of this map
is a line bundle, then as in the proof of A) there is a line subbundle Lo of
V ® Ocip) with deg Lo > 2. Thus, V has a subbundle L\ = Lo ® Oc(-p)
with degLi > 1. So we can apply the claim (and in fact V splits). In
the remaining case the induced map <p: Oc —¦ V ® Ocip) is an inclusion.
By comparing determinants, since we have detO^ = Oc and degdet[V <8>
&c(p)] = 2, the determinant det <p must vanish at some point x € C. Thus,
there is a nonzero v in the fiber СфС of Oc at x with <px(v) = 0. But since
the induced map from H°(OC) to the fiber of Oc over x is an isomorphism,
there exists a section s € H°{OC) whose restriction to the fiber over x is
v. Hence the induced map <p\Oc • s vanishes at x. Thus, the induced map
Oc -* V ® Ocip) vanishes at x to order at least 1, and perhaps elsewhere.
So there is a subbundle of V ® Ocip) oi the form Oc(d), where d is an
effective divisor on С containing x in its support. Thus, V contains the
subbundle Odd) ® Oc(—p), which has degree > 0. Again by the claim, V
satisfies (a) or (b).
Finally, we must consider the case where deg det V = 1. By the Riemann-
Roch theorem, x(V) = 1. Thus, h°(V) > 1 and there is a nonzero map
Qc ~* V- К this map vanishes at some point, then we have an exact
sequence
0 -» Li -> V -* L2 -* 0,
withdegLi = d > 1 and degL2 = 1-d. So Л1^)®^) = ^{(Li)'1®
L%) = 0 since
deg(Li)-1 ® L2) = 1 - Id < 0.
Thus, V = L\ ф L-i,. The remaining case is where О с is a subbundle of V.
In this case we have an exact sequence
0 -¦ Oc -» V -» Oc{q) -* 0,
where q is a point of C. This extension either splits, in which case V satisfies
(a), or it does not split, in which case V satisfies (c). ?
Remark. Suppose that V satisfies (ii)(c) above. Then it is a straightfor-
straightforward exercise (Exercise 2) to show that V = V <8> F for every line bundle
F on С with F®2 = Oc- More generally, for every p and q in C, there is
a line bundle L, unique up to multiplying by a 2-torsion line bundle, such
that Tv ® L = Tq. Otherwise, the descriptions of V in the three cases of
(ii) above are unique, up to permuting the factors of a direct sum of line
bundles.

36 2. Coherent Sheaves
We return to the general problem of understanding rank 2 vector bun-
bundles. The above discussion suggests that we should reverse the analysis of
Proposition 5 and try to construct vector bundles as extensions
0 -¦ L -» V -* V ® Iz -» 0,
where L and V are line bundles on X and Z is a local complete intersection
codimension 2 subscheme. Thus, we must analyze Ext1 (I/ ® Iz,L). (It
follows from (ii) of Lemma 7 below that the nontrivial weak isomorphism
classes of extensions will be parametrized by PExtx(L' ® Iz,L).) Now in
general there is a local to global spectral sequence for Ext groups, which
gives in our case a spectral sequence with E2 term
ep,q = hp(X. Ext4<jj ®iZjl) =* Extp+»(L' 9 Iz,L).
This spectral sequence is really just a long exact sequence, because of the
following:
Lemma 7. Let Л be a regular local ting and let I = (f,g)R be an ideal
of Я generated by two relatively prime elements. Then:
(i) Нотд(/, R) Si R and the isomorphism is induced by the natural re-
sttiction map Нотд(Д, R) —» Нотд(/, Я);
(ii) Нотд(/, /) = R and again the isomorphism is induced by the natural
restriction map Нотд(R, R) —» Нотд(J, Д);
(iii) ExtUl, R) = Ех4(ЯД,Я) Si R/I;
(iv) Extn(/, Я) = 0 for к > 2.
Proof. Apply the functor Нотд(-,Я) to the resolution B.11) of /. We
obtain the short complex
0 «- ExtH(/, Я)<-Я*^-ЯФЯ<- Нотд(/, Я).
Here the transpose of ip is the map (a, b) >-* af + bg. By the arguments
used to prove the exactness of B.11), it follows that Нотя(/, Я) = Я and
that Ext}j(/, Я) = R/I. A check left to the reader shows that the restriction
map Нотд(Я, Я) -+ НотдA, Я) in fact is an isomorphism. Prom the exact
sequence
0 -¦ / -+ R -* I/R -» 0,
we see that there is an inclusion
HomH(/, /) С НотдG, Я) ^ Нотя(Я, Я) = Я.
On the other hand, Я С Нотд(/,/) by multiplication, and thus
Нотд(/,/) = Я acting by multiplication. The higher Ext group» are zero
since / has a short free resolution. Finally, the isomorphism ExtH(/, Я) =

2. Coherent Sheaves 37
Ext2a(R/I> R) is an immediate consequence of the above by applying the
Jong exact Ext sequence to
»0.	?
Note that, just as in Claim 4, the lemma is again a special case of the
calculation of ExtR{I, R) for an ideal / generated by a regular sequence.
Fbi the global case, Ham(L' ® IZ,L) = (I/) ® L and Extx(ll ® /г, L)
IfM bundle supported on Z. Tracing through the above construction,
ft is not to hard to identify this line bundle with det(/z/lf)v ® {L')'1 ®L.
Here, since Z is & local complete intersection, Iz/^z m a 1°саиУ ^гее rank 2
sheaf on Z and its dual is by definition the normal bundle of Z in X. This
definition agrees with the usual definition in case Z is smooth, and is the
generalization of the definition of the normal bundle of a divisor D on X.
We can now replace the Ext spectral sequence by a long exact sequence
0 -» H\(L')-1 ®L)-* Extx(L' 9 IZ,L) -»
L).
In case X is a surface, Extl(L' <8> /z,L) = Oz-
Next we need to decide when an extension V, which о priori is just a
coherent sheaf, is in fact locally free. If V corresponds to an extension class
J 6 Ext1 B/ ® Iz, L), we have the image of f in H°(Ext\ (I/ ® Iz, L)), and
there is the following theorem of Serre:
Xbeorem 8. The extension corresponding to ? is locsdly free if and only
ц the section ? generates tie anea?Ext*{L' ® Iz,L), i.e., the natural map
defined by ? is onto.
Proof. This is a local question: Let M be the Д-module Vx, where as usual
R = Ox,x- Thus, there is an exact sequence
0->R->M-*I->0.
Applying Нотд(-, R) to this sequence, there is a long exact sequence
Нотя(Д, R) Л Ext^il, R) -» Ext^M, R) ~* Ext^R, R) = 0,
and the image of Id € Нотд(Д, R) in Extn(/, R) corresponds to the value
of the extension class in the stalk over x. Moreover, for к > 2 this sequence
shows that ExtkR(M, R) 3* Ext?(/, R) = 0. Thus, ExtkR(M, R) = 0 for all
к > 1 if and only if ExtR(M, R) = 0 if and only if ? generates Ext}j(/, Д).
To conclude, we use the following theorem of Serre whose proof is deferred
to Theorem 17 below:

38 2. Coherent Sheaves
Theorem 9. Let Rbea regular Noetberian local ring and let M be a unite
R-module. Then M is bee if and only if Ext'Д(М, R) = 0 for all i > 1.
Corollary 10. Suppose in the above situation that X is a surface and that
Я2((Ь')~' ig> L) = 0. Then there exist locally free extensions V of L' ® Iz
byL. D
Example. On P2, take L = II - Or». Then Я2@1«) = О, and so there
exist locally free extensions V on P2 of the form
We leave it as an exercise to show that, if Z ф 0, then V is not a direct
sum of line bundles.
In many situations, the corollary is not sufficient, and we will need to
analyze the Ext exact sequence further. We shall only consider the case
where X is a smooth surface.
Claim 11. There is a commutative diagram
L'®Iz,L) 	> H°{Extl{L'®IZ,L))
I-	I	i"
I/igi/z,!) 	> Ext2(Oz,L),
where the bottom row is the exact sequence obtained by applying the long
exact Ext sequence to the exact sequence
0 -+ L' ® Iz -* L' -* Oz -> 0.
Proof. The spectral sequence for Ext gives an isomorphism
Ext2(Oz,L) S H°(Ext2(Oz,L)) й
From this, the commutativity of the above diagram is a straightforward
consequence of the compatibility of the Ext spectral sequences with the
long exact sequences associated to the Ext sheaves. D
Thus, it will suffice to analyze the image of Extx(L' ® Iz,L) in
Ext2@z>?), noting that the isomorphism
Н°{Ех?{О2,Ц) и H°{Ext\L' 9 Iz,L))
has the property that a section of Ехг1{Ь' ® Iz,L) is a generating section
if and only if the corresponding section of Ext2(Oz, L) is generating. Now
we can apply Serre duality on the surface X to the Ext groups above, in

2. Coherent Sheaves 39
the form which says that, for a coherent sheaf T on X, Ext*(.F, Kx) is dual
tajf2"'^!^)- Thus, the sequence
'X' Extl(L'®Iz,L) -» Ext2(Oz,L) -> Ext2(L',L) -» Ext2(L' ® /Z,L)
Щ dual to the sequence
¦_ h°(L' ® L ® jK"x) ¦- Я°A' ® L ® Kx
now suppose that Z = {pi,..., pn} consists of distinct (reduced) points.
Oz — ©jCpi *nd the duality pairing between Ext2@z,?) and
¦ B^(Oz) is local, in the sense that it is induced by a direct sum of local
nondegenerate pairings Ext2(OPi, L) ® Я0(С„<) —» С. Choosing a trivial-
iiation of L at each pit we may identify Ext (Oz,L) = Cn and similarly
|ЯрЯ°(Ог)- Thus, the pairing between Ext2{Oz,L) and H°{OZ) is of the
farm (x,y) = SiAt^tJ/i) where Aj is a nonzero complex number. We then
the following result:
theorem 12. A locally free extension of I/ ® Iz by L exists if and only
Uftyery section of L' ® L~x ® J^x wiich vanisies at atf but оле о/ tie p,
at the remaining point as well.
*roof. Let a be a section of Ext2 (Oz, L). Then s is the image of a section
i^Ext^L' ® IzyL) if and only if d{s) = 0, where 9 is the connecting
(Dmomorphism in the Ext long exact sequence. Now d(s) = 0 if and only
$Щв), e> = 0 for all e e H°(L' ® L~l ® Kx) if and only if (в, в(е)) = О
да all e e H°(L' ® L~l ® Kx), where 9(e) is the image of e in
Я°@2) = H°((L' ® L ® Kx) ® Oz).
(Here we denote by {,) any of the Serre duality pairings involved.) Choosing
local coordinates for X and local trivializations of the bundles L and V
at the p^ we may identify d(e) with a vector (eb... ,en) € Cn, and we
may similarly identify s with a vector (si,..., sn). Finally, as we have seen
above, the pairing {,) may be identified with the diagonal form (x,y) =
S"=i AjXjj/i, where the А* Ф 0, since it is a sum of local nondegenerate
pairings. Now a locally free extension exists if and only if there exists an
в as above with s< / 0 for all г which is liftable to Ext^-t' <g> h,L), or
equivalently if and only if (s,d(e)) = 0 for all e 6 #°(I/ ® L~l ® ifx)-
The proposition is now an immediate consequence of the following linear
algebra lemma:
Lemma 13. Let (,) be the bilinear form on Cn given by

40 2. Coherent Sheaves
where At Ф 0. Let W be a vector subspace of Cn. Then there exists an
a = (si,..., sn) € Wx with Si Ф 0 for ail г if and only if W does not
contain 6i = @,..., 1,..., 0) for any i.
Proof. If W contains 6i for some i, then (s,6i) = AjSi. Thus, if s € Wx,
then Si = 0. Conversely, if every s € Wx satisfies st = 0 for some i, then
Wx С иГ=1 Hu where Щ is the hyperplane { s € Cn : Sj = 0 }. As С is
infinite, there exists an г such that W± С Щ. But then W Э #tx = C&,
and so 5j € W for some г. D
Definition 14. We say that Z = {pi,... ,pn} has the CayJey-Bacharach
property relative to the linear system L' ® L~l ® Kx if every section of
L' ® L~l ® Kx which vanishes at all but one of the Pi vanishes at the
remaining point as well.
For example, it is well known that every cubic passing through eight of
the nine points of intersection of two cubics meeting transversally passes
through the ninth intersection point as well [55] (this is also a conse-
consequence of Lemma 17 in Chapter 5). Thus, nine such points have the
Cayley-Bacharach property relative to Орз(З). Taking L = 0pa(-3) and
V = Оря(З), so that V ig> L~l <g> Kx = Орз(З), the above arguments show
that, for Z the set of nine points of intersection of two transverse cubics,
there is a unique locally free extension V up to isomorphism which sits in
an exact sequence
0 -» Oi«(-3) -¦ V -» Op»C) ® lz -* 0.
We leave it as exercise to show that in fact V is the trivial bundle.
For a less trivial example, if we take sufficiently many points {p\,..., pn}
in general position on X, then there will be no section of L' ® L <8> Kx
which vanishes at all but one of the pi. Thus, vacuously Z has the Cayley-
Bacharach property relative to L'®L~l®Kx- In this way we can construct
rank 2 vector bundles VonX with det V — L® L' such that c2(V) is
arbitrarily large.
More generally, a set of n points {pi,.¦¦tPn} in general position on X
will impose independent conditions on sections of L' ® L~x <8> Kx ¦ For such
points, we can count how many vector bundles we can construct by Theo-
Theorem 12. However, this number is usually much smaller than the dimension
of the moduli space of vector bundles. In this way, the existence of many
vector bundles implies that, for infinitely many linear series \D\ there must
be many configurations of points in special position with respect to |D|.

2. Coherent Sheaves 41
Elementary modifications
Definition 15. Suppose that X is regular and that D is an effective divisor
gn X. Denote by j: D -+ X the inclusion. Let V be a rank 2 bundle on
jf and L a line bundle on D, and suppose that we are given a surjection
yj*-* j*L. Define W as the kernel of the given surjection V -> j,L; thus
is an exact sequence
yfe say that W is a elementary modification of V.
Lemma 16. An elementary modification is locally free. Its Cbern classes
are given by
= ca(V) - d(V) • [D]
.4 .
af. Locally on X, with Л = Ox,*, D is defined by an equation t and
eurjection V —» j»L is given by a surjection ^: ЙФЛ-» Я/tA, Since
is free, ф Ufts to a map ip: RФ Д -+ R which reduces mod t to ^.
Since y>: Я ф R ~* R is surjective mod t, it is surjective by Nakayama's
\emm&. Since R is free, the surjection <p is split. The kernel of <p is a rank 1
eummand of Д Ф R, and since Л is local it is a free rank 1 Я-module and so
tsomorphic to R. This gives a possibly different direct sum decomposition
Ы R@ R for which <p is of the form \x,y) i-» у (mod ?). Thus, locally W
j? isomorphic to R ф tR = Я ф Я and so W is locally free. The statement
?jtf»ut the Chern classes of W follows from the Whitney product formula
ftnd Lemma 1:
= A + c,(V) + etiV) + ¦ ¦ ¦ )A - [D] - [Z7]2 + j.Cl(L) + [Z?]2 + • • ¦)
= 1 + (Cl(V) - [D]) + (c2(V) - a(V) ¦ [D] + j.d(L)) + ¦¦¦.
Equating terms gives the statement of Lemma 16. D
Given an elementary modification 0 -+ W -* V -* j,L -* 0, the surjec-
surjection V —* jtL gives an exact sequence
0 -» L' ~* V\D — L -» 0,
where V is a line bundle on D, defined as the kernel of the map V\D —» L.
Thus, there is an induced surjection W —» j*L'. Since detW^ = detV ®
Ox(-?>), it follows that
det{W\D) =

42 2. Coherent Sheaves
Thus, Ker(W\D -» L') is the line bundle L ® 0X(-.D). We leave it as an
exercise to show that if U is given by the elementary modification
0 -*U-* W -* j.L' -» 0,
then U = V ® Ox(-D)- In other words, repeating an elementary modifi-
modification in the obvious way essentially brings us back to where we started.
Elementary modifications are the first case in understanding the struc-
structure of an inclusion of a rank 2 bundle W in another rank 2 bundle V, by
analogy with the case of sub-line bundles. Needless to say, the general case
is much more subtle (see Exercise 10).
We can also reverse the construction of elementary modifications as fol-
follows: given a rank 2 bundle W on X and a line bundle L on the divisor
D С X, we can try to construct extensions V of j»L by W. Such extensions
are classified by the Ext group Ext^.L, W), and an easy exercise gives
Ext^j.L, W) Si H°(D; (W ® Ox {D))\D®L~l).
We could also see this by noting that if W is an elementary modification
of V, then V, or equivalently Vv, is also obtained as an elementary modi-
modification
0 -¦ Vv -¦ W -* j.{L-1) ® OX(D) -* 0,
and these are classified by
'J.lL-1) 9 OX(D)) = H°(D; (W ® OX{D))\D ® L~l).
Singularities of coherent sheaves
In this section, we will try to determine when a coherent sheaf on a regular
scheme X is locally free, and, in general, try to attach some invariants to a
coherent sheaf in order to determine how bad its singularities are. As these
questions are local, it suffices to consider the corresponding problem for a
regular local Noetherian ring R and a finite Д-module M. We shall denote
the maximal ideal of R by m. We begin with the following theorem due to
Serre (already stated as Theorem 9):
Theorem 17. Let R be a reguiar local Noetberian ring and Jet M be a
finite R-module. Then M is free if and only if Ext jj(M, R) = 0 for all i > 1
if and onJy if Ext^M, N) = 0 for all i > 1 and all R-modules N.
Proof. Clearly, if M is free, then Ext*fl(M,jV) = 0 for all i > 1 and
all Д-modules JV and, in particular, Ех^д(М, R) = 0 for all г > 1. Con-
Conversely, suppose that Extn(M, R) = 0 for all i > 1. First we claim that
Extjj (M,N) = 0 for all finite Д-modules N. Indeed choose a surjection
Д —+ N —¦ 0 with kernel Ni. Thus, there is an exact sequence
0 _» Ni -* ДП1 -» N -* 0.

2. Coherent Sheaves 43
1Ъе long exact Ext sequence and the hypothesis that ExtlR(M, R) — 0 for
all t > 1 giv6 &a exact sequence
Extfl(M, Rni) -» ExtR(Af, N) - ExtR(M, JVi) -» Ext?(M, Д) = 0.
, Ext}j(Af,iV) —» ExtH(Af, JVi) is an isomorphism. Continuing in this
we choose a surjection Я -* Ni with kernel ЛГ2 and see that
(,Ni) a Ex4(M,7V2). Eventually we obtain ExtdR(M,Nd-i) ?
(-1(M,iV(i), where d = dimfl. By the Hilbert syzygy theorem [87],
Щ$ЧМ, Nd) = 0. Thus, ExtR(M, N) = 0.
:'• 1b finish the proof of Theorem 17, choose a surjection Я" —» Af and thus
дп exact sequence
We claim that this exact sequence splits, i.e., that there is a map M —> Й"
ШЛе identity map Af —* M. If so, then M is a summand of a free
iule, and thus it is projective. Since R is local, this will imply that
тве. '
lee 'that the above exact sequence splits consider the associated exact
Ext sequence
"^	Ношд(M, Rn) ~* Нотя(М, М) — Ext^M, N).
Jlnce ExtJi(M, N) = 0, we can indeed lift Id € Нотд(М, M) to an element
^	as desired. ?
'¦! Given an Я-module M, recall that the support Supp M is defined to be
jtt'6 Specie : Mp ф 0} = У(Аппл(АГ)), where, for a an ideal of R, V(a)
'» the closed subset of Spec R of prime ideals containing o, corresponding
to the support of the subscheme R/a. In particular, SuppAf is a closed
eubecheme of Spec R, and is the smallest closed subscheme 5 such that the
ebeaf M on Spec R corresponding to M is zero on Spec R — S.
Definition 18. With M as above, the singular support of M is the set
S(M) = {pe Spec R : Mp is not locally free overflp}
Thus, S(M) is a closed subset of codimension at least 1 in SpecR, the
coherent sheaf M corresponding to M is locally free on Spec R - S(M),
and S(M) is the smallest closed subset with this property.
Torsion free and reflexive sheaves
Our goal in this section will be to describe two classes of sheaves with
rather mild singularities. As in the previous section, we shall only consider

44 2. Coherent Sheaves
the local case where R = Ox,x, or more generally R is a regular local
Noetherian ring, and M is a finite Д-module.
Definition 19. Let Mtors be the torsion submodule of M and define the
rank of M to be the dimension of the if-vector space M <S>r K, where К is
the fraction field of R. M is torsion free if Мюш = 0. A coherent sheaf T
on a regular scheme X is torsion free if and only if the Д-modules Fx are
torsion free for every x € X.
Proposition 20. The following are equivalent:
(i) Mu,n = 0.
(ii) There exists an inclusion M С Rn for some n.
(iii) There exists an inclusion МСД» where n = rank M.
Proof. Clearly, (iii) implies (ii) implies (i). To see that (i) implies (iii),
note that Мюга = 0 is equivalent to the natural map M —» M <8>д К = Kn
being injective, where n = rank M. Let e\,..., en be a K-haais for M ®д А"
and let mi,...,mjt generate M over R. Write пц = X)"=i r»jej> where the
t\j € K. If r € Я is a common denominator for the r^, i.e., rrij € R for
all i,j, then
proving (iii). ?
Thus, all torsion free modules are given as submodules of Д" for some n.
The standard example is the case of an ideal /, i.e., a submodule of R. In
general, ideals / will not be free. Indeed, since rank/ = 1, / is free if and
only if it is projective (recall that R is local) if and only if it is principal.
Corollary 21. Mtore = Ker(M -* Mvv).
Proof. Clearly, Mum Q Ker(M i-» Mvv). To see the opposite contain-
containment, we may as well replace M by M/MtOre and show that in this case
(M torsion free) M injects into Mvv. Using Lemma 20, choose an inclusion
МСД" for some n, and consider the following diagram:
M 	> Яп
I
From this, the injectivity of M —+ Mvv is clear. ?

2. Coherent Sheaves 45
! The proof of the following, which is somewhat technical, is deferred to
the last section of this chapter (see also [117]).
**»
Proposition 22. IfM is torsion free, then codim S(Af) > 2. /n particular,
has dimension I, tien every torsioa free Д-moduJe is free.
& Ffcr example, if M = (f,g)R is an ideal generated by two relatively prime
elements of Д, neither one of which is a unit, then it is easy to see that
SIM) is the scheme Spec R/(f, g)R which has codimension exactly 2.
'There is an obvious restatement of Proposition 22 for coherent sheaves:
jet X be a regular scheme and let T be a torsion free sheaf on X. Then
there exists a closed subscheme Y of X of codimension at least 2 such that
- Y is locally free.
[JDwflnition 23. A finite Д-module M is reflexive if the natural map M —»
is an isomorphism. Hence a reflexive module is torsion free. Reflexive
saves on a regular scheme are similarly defined.
ilVe have the following results concerning reflexive modules, whose proofs
ven at the end of the chapter (see also [117]).
Proposition 24. The following are equivalent:
(i) M is reflexive.
There exists a finite R-module N such that M = iVv.
M is torsion free, and, for all ideals a of R such that codim V(a) > 2,
the natural map #°(SpecR,M) -+ #°(Specfl - V(a),M) is an iso-
isomorphism, where M is the coherent sheaf on Spec R naturally associ-
associated to M.
V/
Ш
Proposition 25. If M is reflexive, then codim S(M) > 3. Hence if
dim Л < 2, every reflexive R-module is free.
We leave to the reader the formulation of the above results for a reflexive
sheaf on a regular scheme.
Using Corollary 21 for a regular local ring of dimension 2, we see that
every torsion free Д-module M canonically sits inside a free rank 2 Д-
module, namely Mvv. Conversely, we can obtain every torsion free R-
module as follows: starting with Дп, choose a submodule M such that R/M
has finite length. Then M is a torsion free Д-module with Mvv = Дп. The
simplest example of this construction is ф"=1 Izt С Д", where Izt is the
ideal of a O-dimensional subscheme of Spec Д. Of course, not every M is of
this form.
Our final result about reflexive modules is the following.

46 2. Coherent Sheaves
Lemma 26. A rank 1 reflexive Я-module M is free.
Proof. Let M be a rank 1 reflexive Я-module and set X = Spec Я and
У = S(M) С X. If M is the sheaf on Л" induced by M, then M|X - У
is a locally free Ox-y module of rank 1 and is thus a line bundle. Since
codimy > 2 by Proposition 25, Pic(A" - У) = Pic Я = 0, for example,
by [61, II.6.5 and 6.16]. Thus, M\X - Y =* OX-y- Since M is reflexive,
it follows from (iii) of Proposition 24 that M = H°(X - Y, M\X - Y) =*
H°(X-Y,OX-y) = R. ?
Thus, to find examples of reflexive modules which are not free, we must
consider Я-modules of rank at least 2, where dim Я > 3. One way to
produce such examples is the following: start with R a regular local ring of
dimension at least 3, and choose a regular sequence (t,f,g). Consider the
Я-module M defined by the short exact sequence
where the map R -* R3 sends 1 to (t,f,g). We leave it as an exercise to
show that M is reflexive. If S = R/tR and /, g are the induced elements of
S, we may think of M as a typical 1-parameter deformation of the torsion
free S-module S Ф J, where J = {f,g)S.
Double covers
In this section, we give a method for constructing rank 2 vector bundles
via double covers. All of the results described here are essentially due to
Schwarzenberger [134].
Let X and У be two smooth varieties and let /: X -+ У be a finite
morphism of degree 2. If D is a divisor on X and Ox{D) is the associated
line bundle, then f*Ox (D) is a rank 2 vector bundle on У. This procedure
is, apart from taking direct sums of line bundles, perhaps the simplest
method for constructing rank 2 bundles. To analyze the direct image, we
begin by recalling some standard facts about double covers.
Let У be a smooth variety and let В be a reduced effective divisor on
У. Suppose that Oy(B) is divisible by 2 in Pic У and let L be a choice of
a square root, i.e., L®2 = Oy{B). Then we construct the double cover X
of У branched along В associated to L as a subvariety of the total space
of the bundle L:
X = {x:x®2= s},
where з is any section of L®2 corresponding to B. It is easy to see that X
is smooth if and only if В is smooth. In this case f*L = Ox(B), where
we have identified В with its inverse image on X. Conversely, given / :
X -+ У, we can recover В and L as follows. Let ь: X -* X be the sheet

2. Coherent Sheaves 47
Interchange involution. Then В is the image under / of the fixed set of i
<фА fjOx = ®y ® L~x, where the direct sum decomposition corresponds
to taking the +1 and -1 eigenspaces of t.
V Xet Div X be the group of divisors on X and similarly for Div Y. There
¦ the natural map /»: DivA" —+ Div У which is defined on a reduced
Icible effective divisor D as follows: /»(!>) = rf(D), where г is the
of f\D: D —» f(D). Hence r is either 1 or 2. In general we extend
Ijp all divisors by linearity. (Of course, we can define /» for any finite
morphism by the same procedure; see also Exercise 12 in Chapter 1.) It is
g to see that this /, is compatible with the map /»: lP(X) -* IP{Y)
by Poincare duality. In addition, /»/* is multiplication by 2 in
and f*f.{D) = D + l{D) for all D e DivA". We can similarly define
ty cycles of any degree. It is easy to check directly that the projection
ormula f,(D-f*E) - f,DE holds if, for example, D and fE are effective
ttVfaore meeting transversally; the general case is done in [45].
i that /»(?>) is a divisor whereas ftO\{D) is a rank 2 bundle. The
hip between the two is given by the following (cf. also [61, IV, p.
'Ex. 2.6]):
Proposition 27. We have the foJJowing equality in Pic Y:
ci(/,0x(D)) = [det/.Ox(D)] = [ДВ] - [L].
Ш particular the linear equivaience class off.D depends only on the linear
tquivalence class of D, and so f, induces a homomorphism from Pic X to
,r af. For D = 0, we have cy(ftOx) = [det(CV ei)] = [L~% which
Verifies the formula in this case since /»0 = 0. Next consider a general
divisor D = Y^i niDi, where the D, are distinct reduced irreducible divisors
OB X. The proof will proceed by induction on n = JV |n»|- К n = 0, D = 0
add we have verified the proposition in this case. Otherwise, choose some
Щ Ф 0. If rij < 0, then we set D' = D + Dj. Thus, D' = ?V rajД, where
rt{ = n< if г ф j, and nj = rij + 1, so that \n'j\ = \rij\ — 1. Consider the
exact sequence
0 - Ox (D) - Ox (D') - OPi (D') -» 0.
Now /»is an exact functor since / is finite and therefore affine. Thus, there
is an induced exact sequence
0 -» f.Ox(D) - f.Ox(D') -» /,0Oi(D') - 0.
The restriction of / to ?),• is a morphism of degree r = 1 or 2, and
1*Оо5 is a vector bundle of rank r on f(Dj). By the comments after
B-8), ci(f,ODj) = r[f(Dj)] = [f,Dj]. By the Whitney product for-
formula we thus have ci{f,Ox{D)) = cx{f*Ox{D')) - [f.Dj]. By induction,

48 2. Coherent Sheaves
ci(/.C?x(Z?')) = [/.?']-№ Thus,
ci(/.Ox(D)) = \f,&\ - [Ц - \f.Di\ = [f.D] - [L].
This concludes the argument in case some rij < 0, and a similar argument
handles the case where all щ are > 0. ?
A straightforward argument shows that the homomorphism from Pic X
to Pic У in Proposition 27 is just the norm homomorphism
H\X;O*X) ^ H\Y;O*Y).
To illustrate Proposition 27, let /: P1 -¦ P1 be the degree 2 map given
by taking the double cover of P1 branched at two points. Then in the above
notation В consists of two points and L = Opi(l). Thus, ci(/»CV(d)) =
d — 1. We leave it as an exercise to show that, for d = 2k even, /»Opi (d) =
Ori(k) Ш CV(fc - 1), whereas for d = 2k + 1 odd, /»Opi(d) = CV(fc) e
CV(fc).
Our next result is a calculation of C2(f»Ox(D)), originally proved by
Schwarzenberger via the Grothendieck-Riemann-Roch theorem.
Proposition 28. With /: X -» У as above and D a divisor on X, the
following equality holds in H*(Y; Z[i]):
b(f.Ox(D)) = m.DJ - f.(D2) - f.D ¦ L).
Proof. To begin with, we note the following:
Lemma 29. There is a natural exact sequence
0 - Ox(l(D)) 9 f'L-1 - rhOx(D) - OX(D) - 0.
Proof. The natural map f*f,Ox(D) ~* Ox(D) is surjective because /
is an affine map. The kernel is thus the line bundle detf*f*Ox(D) ®
Ox(-D). Using Proposition 27, det f*f*Ox(D) = f*(OY(f*D) ® L'1) =
Ox(f*f*D) ® /'L-1 = OX(D + l(D)) <8> f*L~l. Combining gives Lemma
29. ?
Returning to the proof of Proposition 28, the Whitney product for-
formula applied to the exact sequence in Lemma 29 gives f*C2(f*Ox(D)) =
C2{f*f*Ox(D)) = D ¦ i(D) - D ¦ f*L. On the other hand, we have
= D2 + t{Df + 2D ¦ t(D)

2. Coherent Sheaves 49
|nd thus, in H*(X; ®Z[±]), D ¦ l{D) = \f\hDf - \D* - \t{Df. Thus,
(with all coefficients in Z[i])
= f.rc2(ftOx(D)) = f
-{/.D)-L,
where at the last stage we have used the projection formula and the fact
»t(DJ = /»(O2). Dividing by 2 then gives the formula in Proposition
D
>le. Let Q ^ P1 x P1 be a smooth quadric in P3 and let n: Q -» P2
ie projection of Q onto a plane. Then Q is a double cover of P2 via тг;
ich locus is a smooth conic. Hence, in the notation of this section,
?A). A basis for PicQ is given by {/ь/2}, where the U are lines
; to the two distinct rulings. Thus, 7r»(/j) is a line in P2. Let
Ъе the rank 2 bundle 7r»O<j(o/i + bf2) on P2. Using Propositions 27
28, **V..» = O,»(a + b - 1) and c2{Va,b) = |((o2 - a) + (б2 - 6)).
,We leave it as an exercise to determine when Va,b is a direct sum of line
||l The, following result of Schwarzenberger says that every rank 2 bundle
ФИ* a curve or surface arises as the push forward of a line bundle on a
double cover. However, the nonuniqueness of the construction makes this
Jesuit mainly of theoretical interest.
Proposition 30. Let У be a smooth curve or surface and let V be a rani
t%vector bundle on Y. Then there exists a smooth double cover f: X —» Y
tad a fine bundle M on X such that V = /»M.
Proof. Let P = P(VV) -^+ У be the projectivized bundle associated to V
(here our sign conventions are the opposite ones to those in, for example,
[61] or EGA). Then for the tautological bundle OP{\) on P, we have [61,
Ш, p. 253, Ex. 8] 7T.Op(l) = V. Let L be a sufficiently ample line bundle
on Y and let V = OP{2) ® tt'L. Then V is very ample on P [61, II, 7.10].
If X is the linear system corresponding to L', then for X 6 X and € a
fiber of тг, either ? С X or X meets ? in exactly two points, counted with
multiplicity.
Claim 31. There exists a smooth X e X such that the induced map
/: X —* У is finite and so is a doubie cover.

50 2. Coherent Sheaves
Proof of the Claim. Consider the incidence correspondence I c X xY
given by
/ = {(Л',р):7Г-1(р)сЛ'}
and let tti, 7Г2 be the projections of / to X, Y, respectively. Then tti(/) is
the set of X which contain some fiber of тг, and we will be done by Bertini's
theorem if we show that tti(J) is a proper subset of X. Let dim^ = N.
Then we claim that dim / < N - 1 (it is here that we use the assumption
dimY < 2). To see this, let p € Y and let I - тг^). Then dimTT^^) =
dim{A" eX:?cX} = h°{P; L' ® It) - 1, where h is the ideal sheaf of L
Consider the exact sequence of sheaves
0 -* L' ® It -* U -» Oi{2) -* 0.
For L sufficiently ample on Y, for example, L = Lf2 where Op(l) <S>7r'L0
is very ample on P, we claim that the map H°(P;L') -» H°{i\Ot{2)) is
surjective. Indeed, since Op(l) igi tt*Lo is very ample on P and restricts to
Ot(l) on ?, the map H°(P;OPA) ®n*L0) -» H°(e;Oi(l)) is surjective.
But then L' = (Op(l) ® 7r*L0)®2, so that H°(L') contains the image of
Sym2H°(OP(l) ® tt'Lo), and thus the image of H°(L') in H°(e;OtB))
contains the image of Sym2 H°(?; Ot(l)) = Я°D OtB)).
As <timH°(e;Oi{2)) = 3, it foUows that dim^'Cp) = h°(P; L' ® It) =
{h°(L') - 1) - 3 = N - 3. Hence dim/ = dimTr^1^) + dimY < N - 1.
Thus, 7Ti(J) must be a proper subset of X, proving the claim. D
Returning to the proof of Proposition 30, choose X as in the claim. It
suffices to prove that /»(ОрA)|Л") = V. Apply 7г» to the exact sequence
0 -* OP{\) ® Op(-X) -* Op{\) -* Op{l)\X -* 0.
We thus obtain a map V = тг,ОРA) -»тг,(ОРA)|Х) = /,(OPA)|X). To
see that this map is an isomorphism, note that for all у 6 Y, Op{l) <8>
Op{-X)\*-\y) = 0p.(-l) and that #°@pl(-l)) = Hl{Ovi{-l)) = 0.
Thus,
Д°тг.@рA) ® Op(-X)) = R}n,{OP{l) ® Op(-X)) = 0,
proving that the map V —» f*OP{l)\X is an isomorphism. D
Consider the example У = P2. Every smooth double cover тг: X -» P2 is
branched along a smooth plane curve of degree 2d, and if d > 2, then the
generic such double cover X has Pic X ^ Z, with a generator pulled back
from Op» A). But тг.тг'Оря (n) = Oja (n) Ф Op» (n - d), and in particular it
is a direct sum of line bundles. Conversely, this construction realizes the
direct sum of two line bundles as the direct image of a line bundle on an
appropriate double cover. Thus, the existence of nontrivial rank 2 vector
bundles on P2 implies that there are many configurations of plane curves

2. Coherent Sheaves 51
Ct and Сг in special position, in the sense that C\ is smooth and C2 is
everywhere tangent to C\.
Appendix: some commutative algebra
Our В0*! k *° Prove some of the more technical results used above. We
by recalling the following definitions (cf. [87] or [139]:
Definition 32. If Л is a regular local ring and M is a finite Я-module,
define the projective dimension proj. dim M, by any of the following equiv-
equivalent ways:
1. the minimal length of a free resolution of M;
? sup{/fc : there exists a finitely generated Я-module N such that
^Ё4
i the equivalence of B) and C) is an argument analogous to the proof
f^Theorem 17.
vKPor M Ф 0, let depth M be the maximal length of an M-sequeace
|j^,.,.,Xfc e Я, i.e., Xi € m and for all i the map M/(xi,...,Xi_i)M -+
M/(xi,-.¦ ,Xi-i)M induced by multiplication by Xi is injective, where
 means that there is no such Xi. More generally, for о an ideal of
! set depth0 M to be the maximal length of an M-sequence xb..., x*
. that Xi € a for all i. Let us collect some salient facts about depth:
Lemma 33. Let Я be a reguiar local ring of dimension d.
:(i) depth0M = inf{depthMp : p € V(a)}.
\ (li) proj. dim M + depth M = d.
(Ш) The local cohomoiogy group Hla{M) is 0 for aJJ г < к — 1 if and only
'''¦ if depth, M > k.
(iv) If a is an ideal of Я with codim a > k, then Я* (Я) = 0 for all г < к -1.
Proof, (i) This is [87, p.105].
(ii) This is the dimension formula of Auslander-Buchsbaum [87].
(ffi) This is [61, p. 217, Ex. 3.4].
(iv) It suffices by (i) and (iii) to show that depth Rv > к for all primes
p € V(d). Since Я is regular, Яр is regular, and thus
depth Яр = dim Яр = d - dim V(p) = codim V(p).
Since a has codimension at least k, codim V(p) > к and thus depth Яр >
к. ?

52 2. Coherent Sheaves
Lemma 34. For every R-module M,
dim Supp ExtlH(M,R) <d-i,
i.e., codim Supp Ext д(М, R) > i.
Proof. If p € SuppExt^(M, R), then the localization ExtlR(M, R)p is not
0. As there is a natural isomorphism
Ext'H(M, R)p » ExtlHp(Mp, Rp),
Ext^^Mp, Rp) Ф 0, and hence proj. dimMp > z, where the proj. dim is as
an Яр-module. As Rp is again a regular local ring of dimension d-dim V(p),
from
proj. dim Mp + depth Mp = d — dim V(p),
we obtain dimV(p) < d - i for all p € SuppExt'fi(M, R). Thus,
dim Supp Ext*H(M, R) < d - i.	D
Definition 35. M is a fcth syzygy module if there exists an exact sequence
0 -> M -* R -* Я -¦	> RT1-1 -» Rnk.
Lemma 36. Let M be a fcth syzygy module. Then:
(i) proj. dimM <d — k.
(ii) codim S(M) >fc + l.
(iii) For all ideals a of R, deptho M > k.
Proof. We claim that, for all i > 1, dim SuppExt'H(M, R) < d - к - i and
that, in particular, Supp ExtXR{M, R) = 0 if i > dI - fc. Thus, by definition
dim S(M) < d-k-1. Moreover, we must have Ext'H(M, R) = 0 for г > d-k,
so that proj. dimM < d - к as well. We will prove the claim by induction
on k. For к = 0, i.e., no condition on M, this is just the syzygy theorem. If
now M is a (k + l)Bt syzygy module, then there exists an exact sequence
0 -¦ M -* Я™ -¦ M' -> 0,
where M' is a kth syzygy module. An argument with the Ext exact se-
sequence shows that Ext'H(M, R) =* Extj^M', R) for all г > 1. Thus, by the
inductive hypothesis dim Supp ExtR(M, R) = dim Supp Ext'^^M', Д) <
d-fc-(i + l) = d-(fc + l)-i. This completes the inductive step and thus
the proof of (i) and (ii).
To see the last statement, it suffices by (i) of Lemma 33 to prove it for
prime ideals p. Now since Rp is a flat Д-module, Mp is a fcth syzygy module
as well (for Др). Thus,
proj. dim Mp > dim Др — к

2. Coherent Sheaves 53
да! 80, since
pro j. dim Mp + depth Mp = dim R p,
<«e have depth M9 > k. D
Proposition 37. The following are equivalent:
ft'Af is torsion free.
iV'Af is a first syzygy module,
•fbr all proper ideals a, deptho M > 1.
Proof. The equivalence of (i) and (ii) follows from Proposition 20. Next
note that, if f-m = 0 for / e RJ Ф 0 and m e M,m ^ 0, then for a = (/)
м have deptho M = 0. Conversely, if M is torsion free, then it is a first
" ¦ module and so deptha M > 1 by (iii) of Lemma 36. D
38. If M is torsion free, then codim S(M) > 2. In particular,
dimension 1, then every torsion free Д-moduie is free.
This is immediate from Lemma 36. D
ition 39. The following are equivalent:
Af is reflexive.
fttbere exists a finite Я-module N such that M ^ ATV.
Af is a second syzygy module.
for all ideals aofR such that codim K(a) > 1, deptho M > 1, and for
Pj аи ideals a of R such that codim V(a) > 2, deptho M > 2.
W is torsion free, and, for all ideals aofR such that codim V(a) > 2,
the natural map /7°(Speci?,M) -¦ H°(SpecR - V(a),M) is an iso-
morphism, where M is the coherent sheaf on Spec R naturally associ-
to M.
: **roof, (i) =ф- (ii) is trivial (take iV = M). (ii) =*¦ (iii): Choose a presen-
tation
Dualizing gives an exact sequence
0-»Nv =M-> Rni -+Д,
le., Af is a second syzygy module, (iii) =>• (iv): Since a second syzygy
module is torsion free, H°(M) = 0 for all proper ideals a. Thus, if M is а
second syzygy module we have an exact sequence
0 -> M -* Дп> -> M' -> 0

54 2. Coherent Sheaves
with M' С Д™2 and hence torsion free. Applying the long exact sequence
of local cohomology, we obtain
0 = HZ(M') - H\{M) -» H\{Rr>).
From (iv) of Lemma 33, if codim V(a) > 2, then Я](ДП1) = О and hence
H\(M) = 0. The statement (iv) then follows from (iii) of Lemma 33.
To see that (iv) ==>• (v), note that deptho M > 1 for all a is equivalent
to M being torsion free. Let a be an ideal with codim V(a) > 2. Then by
hypothesis deptho M > 2, and so H\{M) = 0 by (iii) of Lemma 33 again.
Setting M = the sheaf on Spec Д naturally associated to M, we have a
long exact sequence
0 -» НЦМ) -+ #°(Spec Д, M) -> tf°(Spec R - V(a), M) -> НЦМ).
Thus, H°(SpecR,M) ^ H°(SpecR- V{a), M).
Finally, to see that (v) =* (i), since M is torsion free we have M«-» Mw
and codim5(M) > 2. Let M be the sheaf on SpecR naturally induced by
M and Mvv that induced by Mvv. We have a commutative diagram
H°(Spec R-S(M),M) 	 H°(Spec R-^
H°(SpecR,M) —?-» H°(Speci?,Mvv).
By hypothesis / is an isomorphism. Since Mv v is the dual of the Д-module
Mv, by using the implication (ii) ==> (iii) g is an isomorphism. Thus, h is
an isomorphism, and so M = Mvv, i.e., M is reflexive.
Corollary 40. If M is reflexive, then codim S(M) > 3. Hence if dim R <
2, every reflexive R-module is free.
Proof. Since a reflexive Д-module is a second syzygy module, the corollary
follows immediately from Lemma 36.
Remark. A) The proof that (vi) implies (i) also showsthat for a general
torsion free Д-module M, Mvv = H°(Spec Д - S(M), M). For example, if
/ is an ideal in Д such that codim Spec R/I > 2, then /vv = Д.
B) One can more generally show the following. An Д-module M is a fcth
syzygy module if and only if, for all j < к and for all ideals a of Д with
codim V(a) > j, depthoM > j. There is also an equivalent statement in
terms of local cohomology.
Exercises
1. Let Li and L2 be line bundles on a scheme X. Give an alternate
description of Ext1^»^!) = H1((La)~1 ®^i) as follows. If V sits in

2. Coherent Sheaves 55
the exact sequence
0 -> Li -» V -> L2 -> 0,
then the transition functions for V can be taken to be upper triangular,
with the diagonal entries transition functions for L\ and Li- How does
the nonzero off-diagonal entry transform? Define in this way a section
of Hx{(L-?)~^ ® L\), independent of choices, and then show that this
procedure can be reversed. Generalize this to vector bundles V\, Vi of
arbitrary rank.
Prove the uniqueness assertions of the remark after the proof of The-
Theorem 6: if Tv is the nonsplit extension of Oc(p) by Oc> where С is
an elliptic curve, then Tv ® L = Tv if and only if L is a line bundle
of order 2. Conclude that Нот{Тр,Тр) = Ос ®LX®L2® L3, where
the Li are the nontrivial 2-torsion line bundles on С Moreover, for all
qeC, there exists an L such that Tv®L = Tq (here L is unique up
to a 2-torsion line bundle).
Let V be a vector bundle on P2 which sits in an exact sequence
0 -* O^ -* V -> Iz -* 0.
If Z ^ 0, show that V is not a direct sum of line bundles.
Let Z be a set of nine points in P2 which is the transverse intersection
of two smooth cubics. Show that there is a unique locally free extension
V up to isomorphism which sits in an exact sequence
0 -> 0i«(-3) -»V-» Op2C) ® Iz -* 0.
Show that in fact V is the trivial bundle.
Show, for a rank 2 vector bundle V, that V* ^ V ® det V~x. Now
suppose that
0 -¦ W -> V -* j,L -* 0
is an elementary modification of V, where j: D -* X is the inclu-
inclusion of a smooth divisor. Show that V* is obtained as an elementary
modification of Wv', and thus that V is an elementary modification of
6.	The line bundle Opi(l) is generated by two global sections, and thus
there is a surjection Or* ф Opa —» j»Opi(l), where j: P1 -¦ P2 is the
inclusion of a line in P2. Show that the elementary modification
0 -> V -> Ov Ф OP2 -> j»Opl A) -> 0
denned by this map satisfies ci(V) - —[?\, where ? is a line in P2,
c2(V) = 1, and H°(V) — 0. Is V a direct sum of line bundles?
7.	Let 0-+W-+V-+j*L—»0bean elementary modification defined
by the line bundle L on the divisor D, and let L' be the kernel of the
surjection V\D -* L. Thus, there is an induced surjection W -* L'.

56 2. Coherent Sheaves
Show that if U is the induced elementary modification
then U is just V <g> Ox(-D).
8.	Let X be regular, let D be a divisor in X, and let V be a rank 2 vector
bundle on X. Suppose that there is an exact sequence
0->H-*V->H'->0,
where Я and #' are line bundles, and let L — H'\D. For the elemen-
elementary modification W corresponding to the surjection V —¦ j,L, show
that there is an exact sequence
0 -¦ H'(-D) ->W->H->0.
9.	Using the above exercise, show that for a smooth curve C, if V is
an arbitrary rank 2 bundle, then there is a sequence of elementary
modifications beginning with V and ending with a bundle of the form
(Ос Ф Ос) ® L, i.e., up to a twist by a line bundle, V is obtained
via elementary modifications from the trivial bundle. (Starting with
an arbitrary exact sequence 0 —¦! Я —> V —¦ Я' —» 0 and applying the
previous exercise, we can assume that deg H' is sufficiently negative
and thus that the exact sequence splits. Now work with both factors
and use the fact that, on a smooth curve of genus g, every divisor of
degree at least g is effective.)
10. Let R be a regular local ring and F: R2 —¦ R? a homomorphism of
Я-modules, corresponding to a 2 x 2 matrix. Suppose that the matrix
is of the form (* M with gcd(/,p) = 1. Show that R2/F(R?) s
/	)	h h
where Iz = {f,g)R and Ie = tR, where t = fh2 — ghi is the
determinant of the matrix.
(Working a little harder, one can show that for every 2x2 matrix
whose entries are relatively prime, there exists a choice of bases for
the domain and range R?'a for which the first column has relatively
prime entries. Note that if ? is singular, for example nonreduced, then
the modules Iz/Ie can be very complicated.)
11. Let Я be a regular local ring of dimension at least 3, and choose a
regular sequence (t,f,g). Consider the Д-module M defined by the
short exact sequence
0-»Я-»Д3->М->0,
where the map R -* R3 sends 1 to (t,f,g). Show that M is reflexive.
(Show that there is an exact sequence
0 _> Mv -> Я3 -+ / -> 0,
where / = (tj,g), and that Ext}j(/,fl) = 0.) What is S(M) in this
case?

2. Coherent Sheaves 57
i.2. Let M be a complex manifold of dimension at least 2 and let Y с М
be a closed analytic subspace of codimension at least 2. Suppose that
L is a line bundle on M—Y. Is it true that L extends uniquely to a line
bundle on M? (Compare [55, Example 4, p. 49] for the case M = C2,
, using the exponential sheaf sequence.) If L extends to a coherent sheaf
on M, then L extends to a line bundle on M. Conclude that a reflexive
;. rank one coherent sheaf on a complex manifold is locally free. (Locally
around a point of Y, show that L has a holomorphic section in M-Y
and is thus associated to a divisor, which extends across Y.)
18. Let /: P1 —¦ P1 be the degree 2 map given by taking the double cover
" , of P1 branched at two points. Show that, for d = 2fc even, /»Opi (d) =
Op» (fc) ©OPi (fc -1), whereas for d = 2fc +1 odd, /»Opi (d) = Opi (fc) ©
(Ppi(fc). (It suffices by the projection formula to do the cases d =
0,1. The case d = 0 is done. For d = 1, write /»0pi(l) = dpi (a) ©
Opi (b) and calculate H°{Ofl A)) = H°(/,Opi A)), and Я°(/,ОР1 A)®
°)	)))
())
|Ц> Let Q ^ P1 x P1 be a smooth quadric in P3 and let 7r: Q -> P2 be the
double cover obtained by projecting Q onto a plane. Let Va,b be the
rank 2 bundle 7r»0Q(a/i -f bf2) on P2, where }\ and /2 are the two
rulings on Q, as in the example at the end of the last section. Show
that, if a = b, then Vatu = 7r,@Q®7r*0i«(a)) = Op*{a) ® Op*(a - 1).
If |a-&| = 1, say b = a + 1, then show that Va,a+i = Орз(а)фОрз(а).
In all other cases, show that Va^ is not a direct sum of line bundles.
16.	Let X be a regular scheme. Given a rank 2 bundle W on X and a line
bundle L on the divisor D с X, show that
Extl(j,L, W) й Я°(Д (W ® 0х(Я))|Я ® Ь-1).
Jto. Let X be a regular scheme and Z a codimension 2 local complete inter-
intersection subscheme of X. Verify that there is a canonical isomorphism
Ext1Az, Ox) — det Nz/X, where Nz/X is the dual of the locally free
rank 2 sheaf Iz/Iz-
17.	Prove the Riemann-Roch theorem for vector bundles V of rank r on a
smooth curve or surface (Theorem 2). (For a curve C, show first that
there is some exact sequence
where L is a line bundle and V' is a vector bundle of rank r — 1, and
apply induction. For surfaces S, show that there is an exact sequence
0 -> L -> V -> 0,
where L is a line bundle and V is a torsion free sheaf of rank r - 1.
Thus, there is another exact sequence

58 2. Coherent Sheaves
where W is a vector bundle and Q has finite support. Apply induction
and use B.8).)
18. Let /: X -> У be the double cover constructed by choosing a square
root L of the line bundle Oy (B), where У and В and therefore X are
smooth. Show that k^X; Ox) = ti(Y; Oy)®hi(Y;L-1) for all i. Also,
by using the natural inclusion f*Ky —> Kx and local coordinates,
show that Kx = f*KY ® OX(B) = f*(Ky ® L), viewing В as a
smooth divisor on X. Conclude that f,Kx - Ky ® (Ky <g> L) and
thus that pg(X) = pg(Y) + h°{Y;KY ® L). Finally, if X and У are
surfaces, then #? = 2K\ + 4(tfy • L) + 2L2. For example, if У = P2
and В is a smooth curve of degree 2d, show that pg (X) = 0 if and only
if d = 1,2, that otherwisepg{Y) = (d ~ l\ and that ^ = 2(d-3J.

this chapter, we describe the basic properties of surface birational ge-
gentry. After reviewing the operation of blowing up a point on a surface
and the relationship between the invariants of the blown up surface X
$,fbe invariants of X, we prove the Castelnuovo criterion for blowing
irn a curve. Using the Castelnuovo criterion, we show that every bira-
aal morphism between two smooth surfaces is a composition of blowups,
i discuss various notions of minimal models. At the end of the chapter,
discuss more general contractions to normal surfaces, with particular
ention to rational singularities and rational double points.
Birational Geometry
powing up
operation of blowing up goes by many names in the literature: a-
s, monoidal transformation, standard quadratic transformation,....
We begin with a review of standard facts about blowing up. Let X be a
Surface and let p be a point of X. Let p: X —* X be the blowup of X at
the point p. Let E = p~l(p) be the exceptional divisor. Thus, E = P1 and
AetgNE/x = -1, so that E2 = -1 and ? • Kx = — 1 by the adjunction
formula. We may describe X locally as follows: let U be a coordinate neigh-
neighborhood of p, with coordinates x,y centered at p. Let U = p~l(U). Then
У is covered by two coordinate charts Ui and U2. Here U\ has coordinates
x*,^ and the map p is given by x = x', у = х'у'. The chart U2 has coordi-
coordinates x", y" and in these coordinates p is given by a; = x"y", у = у". We
glue t/i - {y' = 0} to U2 - {x" = 0} via
У" = x'y'.
The exceptional divisor E is defined in Ui by ж' and in U2 by j/". From this
it is easy to check that E = P1 and that NE/X is identified with OPi(-l).

60 3. Birational Geometry
Thus, degNE/x = —!• Also recall the universal property of X [61, p. 164]:
if tp: Y —»X is a morphism such that (р~1(тх)Оу is the ideal of a Cartier
divisor on У, then tp factors through X: there exists a unique morphism
tp: Y —» X such that <p — po (p.
We will frequently use the following lemma:
Lemma 1. Let щ be the maxima/ ideal of p in X. Then
/n / v\ / m?' tfe = -n<0,
*	I Ox, Ife>0.
Proof. For a neighborhood U of p, let / € T(p~l(U),Ox(aE)). Thus, /
defines a section of Ox over p-1(C/) — E = U — {p} or in other words
a holomorphic function on U - {p}. By Hartogs' theorem / extends to
a holomorphic function on U. It follows that there is a natural inclusion
j: ptOx(aE) •—> Ox for all a. If a > 0, we can reverse this inclusion
too: if д € T(U,Ox), then p*g is a section of Ox over p~l{U) and thus a
section of Ox(aE) over p~x(U), i.e., a section of p*Ox{aE) over U. Clearly,
J(P*9) = p and so p,Ox{aE) = Ox.
For а = —n < 0, we must determine the image of j. Equivalently, given a
function g holomorphic in U, it suffices to determine when p*g e Ox{aE).
Now we can write g(x, y) = J2T=mТ&Л1* !/)> where gv is homogeneous of de-
degree v and gm Ф 0. Thus, m is the multiplicity multpp and g € m™; indeed
g 6 mJJ1 — m™+1. Working, for example, in U\ and using the coordinates
described above, we have
P'9(x',y') = <>(*', *V) = (x')m(9m(hy')+x'g'),
where x'g1 vanishes along E and gm(l, у") does not vanish identically on
E. Thus, we see that
p'g e T{p~\U), Ox(-mE)) - T^-^U), Ox(-(m + l)E)).
It follows that p*g € T{p-l{U),Ox{-mE)) if and only if g e m™, which
was what we needed to complete the proof of Lemma 1. ?
Definition 2. Let С be a nonzero effective divisor on X. We define the
multiplicity multp С to be multp g, where g is any local defining equation
for С at p. It is easy to check that this definition is independent of the
choice of g. In this case, the proof of Lemma 1 shows that we can write (as
effective divisors) p*C = mE + C, where C" is an effective nonzero divisor
on X and С does not contain E as a component (C — E is not effective).
We call C" the proper transform of C.
We leave as an exercise the statement that C" • E — m and that the
scheme-theoretic intersection С" П E can be identified with the projective
tangent cone to С at p.

3. Birational Geometry 61
We turn now to the structure of Pic X and Num X.
^position 3.
(i) P* '• fc -^ "~* ^*c an^ P*: Num X —* Num X are iojective.
jji) PiX 'PicX®Z[E]
jjii)	[
'Щ) NumX = p* Num X © Z[?], and this direct sum is orthogonai with
Щ] respect to intersection pairing on X and X.
Pr The proof of (i) in the case of p*: Pic X -» Pic X follows from the
projection formula p,p*L = L <g> p*O^ = L. The proof for p*: NumX —»
NfumA' also follows from the projection formula and the definition of p».
[UTb see (ii), let Pic' X be the set of L e PicX such that Aeg{L\E) = 0.
rhus, Pic' X is the kernel of the natural restriction map Pic X —* Pic E =
JSJJ)Note that degO^(a?)|? = -a, so that, identifying a[?] € Z[E] with
' (?), we have a splitting Pic X = Pic' X®Z[E]. To prove (ii) it therefore
to identify Pic' X with p* PicX. Clearly, p* Pic Jf is contained in
. Conversely, suppose that L € Pic' X. Consider L\X - E = X - {p}.
Pic(X- {p}) = PicX. the line bundle tonX- {p} extends uniquely
> a line bundle on X (compare the proof of Lemma 26 in Chapter 2). Let
U be the extension of L\X -E to a line bundle on X; in fact M = {p,L)y v.
^ien p*M|X - E ^ L|X - E. Thus, p*M ® L has a xegular nowhere
irtmishing section over X - E = X - {p}. Such a section extends to give
|lmeromorphic section of p*M ® L~l which can only have zeros or poles
igE. Thus, p*M = L <g> Ox(aE) for some integer a. As both p*M and
liave trivial restriction to E, a = 0. This concludes the proof of (ii).
То see the direct sum part of (iii), it again suffices to show that every
6 NumX with D ¦ E = 0 is of the form p*D'. Representing D as the
numerical equivalence class of a line bundle L with deg L\E = 0, this follows
fcom (ii). The fact that the direct sum is orthogonal is an easy consequence
jrfthe projection formula, which implies that p*D ¦ p*D' = D • ptp*D' =
#'¦• D' and that p*DE = D- p,E = 0. D
t The proof also shows that the projection of NumX onto the factor
p'NumX = NumX is given by p». Likewise, the projection PicX —*
p*PicX = PicX is given by L н (p«L)vv. In general we see that
p»(p*M®Ox(aE)) is Mif a> 0andisM®m^ if a- -n < 0.
Here are some easy consequences of Proposition 3 (which could also be
checked directly, cf. Exercise 1).
Proposition 4.
(i) If С is a curve on X and multp С = m, then the proper transform C"
of С satisfies С" Е =m.
(ii) With notation as in (i), (CJ =C2-m2.

62 3. Birational Geometry
(iii) Kx=p'Kx+E.
Proof. Write p*C = С + mE, where C" is the proper transform of C.
Since E and p*C are orthogonal,
C'E = (p*C - mE) ¦ E = -mE2 = m.
Next we note that (CJ = CP-m2. Finally, we can write Kx = p*Kx+aE
for some integer a. By adjunction Kx • E + E2 = -2, and thus a — 1. ?
Corollary 5. с?(ЛГ) = с?(A") - 1 and pg(X) = pg(X). More generally
Pn(X) = Pn(X) for all n > 1.
Proof. The equality for <%(Х) follows immediately from (iii) above. To
see the other statement, note that
H0{X;K%n) = H°{X;p*K^n ® Ox(nE))
= H°(X;pt[p'K%n
Now by the projection formula, if n > 0,
P^Kf1 ® Ox{nE)\ = Kf1 <8>p.Ox("
by Lemma 1, since n > 1. Thus, Н°(Х;^|") =* Я°(А"; ^|"), and so
a
Thus, the invariants Pn are unchanged after blowing up. Such a state-
statement would fail if we had considered the equally "natural" bundle Kxl —
detTx, for example.
Finally, we note that q(X) is also unchanged under blowing up:
Proposition 6. q(X) = q(X).
Proof. One proof uses the topological fact that Hl(X;Z) e* Hl{X;Z);
indeed an easy Mayer-Vietoris argument shows that iri(X;*) ^ iri(X;*)
via рф. A second proof uses the fact that Pic0 A' = Pic0 X, where Pic°X
is the component containing the identity of the complex Lie group Pic X,
by Proposition 3, together with the fact that q(X) = dim Pic0 X. A third
proof uses the Leray spectral sequence for the map pt (or the easy special
case that is in [61]) as follows. We shall show that Rlp*Ox = 0. Then by
the Leray spectral sequence
H\X;OX) = H1(X;p.Ox) = H\X;OX) = q(X).
To see that R}p*Ox = 0, we apply the formal functions theorem [61, p.
277] to p and Ox:
Rlp*Ox = lim Hl(nE;OnE).

3. Birational Geometry 63
1b evaluate Hl{nE\ Опе), use the exact sequence
0 -¦ OB(-(n - 1)E) - OnB — O(n_i)B -> 0
of A-10) of Chapter 1. Since Ojs(-(n - l)E) = Opi(n - 1), it follows that
В1(Ое(~(п ~ 1)-®)) = 0 for all n > 0. Thus, by induction, starting with
the case n = 1 where we know that Я1 (?; Од) = p(.E) = 0, we see that
Н1(Опе) = 0 f°r every n > 1. This concludes the proof that Rxp,Ox = 0
thus of Proposition 6. ?
To conclude this section, we will say more about the arithmetic genus
of an irreducible curve. Let С be a (reduced) irreducible curve on X and
suppose that peC. Let m be the multiplicity of С at p, let X be the blowup
of X at p, and let C" be the proper transform of C. We shall compare pa{C)
tfith pa{C).
-Proposition 7. a,(C) =pe(C) - m(m2~1).
|»roof. By our general formulas
= {p*Kx + E) ¦ {p*C - mE) +C2-m2
itad this is a restatement of Proposition 7. ?
Note that pa(C) < pa(C) unless m = 1, i.e., p is a smooth point of
C. Since pa(C) > 0, we cannot continue blowing up singular points indef-
indefinitely: if we successively blow up the singular points of C, then all of the
singular points on the proper transform, and so on, we eventually arrive at
a surface Y and a proper transform of С which is smooth. This is embed-
embedded resolution for curves on a surface. Keeping track of the total change in
po(C), we arrive at the classical formula for 8P:
yr-* mq(mq — 1)
where the q are the "infinitely near" points to p and are defined as follows:
they include p, all the points on С mapping to p via p, all the points
on blowups of X lying on C" and mapping to p via the composite map,
and so on. If С consists (locally analytically) of the union of m distinct
lines meeting at a point p (the case of Exercise 6 of Chapter 1), then
6P = m(m -

64 3. Birational Geometry
The Castelnuovo criterion and factorization of birational
morphisms
In this section we review the main facts about birational morphisms. We
begin by recalling the following theorem, due to Van der Waerden, which
is the "easy" special case of Zariski's Main Theorem.
Theorem 8. Let it: Y -+ X be a birational morphism between two
smooth varieties. Let у 6 Y and let x = ir(y)- Then either there is a
Z&riski open subset U of X containing' x and a morphism U —* Y which is
an inverse to 7r, or there exists a hypersurface V on Y containing у such
that the Z&riski closure of 7r( V) has codimension at least 2 on X.
Proof. There is an inclusion of local rings Ox,x ? Oy,y and the two rings
have the same field of fractions. An easy argument shows that Ox,x = Oy,v
if and only if ж is invertible on some Zariski open subset of X containing x.
Writing Oy,y as the localization of C[h,..., tn)/I for some ideal /, we can
write U = fi/9i, where fitgi € Ox,x- By standard commutative algebra
[87], Ox,x is a UFD. Thus, we may assume that ft and gi are relatively
prime in Ox,x- If л is a unit for all г (i.e., gi(y) ф 0), then Ox,x — Oy,v
and the first conclusion of the theorem holds. Otherwise, gi is not a unit
for some г, so that {ir*gi = 0} defines a hypersurface V on У containing y.
Moreover, in Oy,y T*fi = n'gi • U, and so both ж* fi and 7r*p< vanish on V.
Thus, fi and gi vanish on ir(V), and so the closure of ir(V) is contained in
{/« = 9i — 0}- Since fi and p,- were assumed relatively prime, {fi — gi =
0} has codimension 2 in X, and so the second alternative of Theorem 8
holds. D
We can now prove the following fundamental result, known as the Castel-
Castelnuovo criterion:
Theorem 9. Let У be a smooth surface, and iet E be & curve on У such
that E = P1 and E2 — — 1. Then there exists a smooth surface X, a point
p € X, and an isomorphism from the blowup X of X to Y such that E is
the image of the exceptional curve on X.
Proof. The are two steps to the proof. In the first and easier step, we
construct the surface X. The second step shows that X is smooth and
identifies У as the blowup of X.
Step I. To find X, choose a very ample divisor H on Y. We may also assume
after replacing Я by a multiple that #*(У; Oy(H)) = 0. Let а = H E > 0.
Consider the linear system corresponding to the divisor Я + аЕ. Note that
(Я + аЕ) ¦ E = 0. We claim that the linear system \H + aE\ has no base
points and that the image of the corresponding morphism tp: Y —* PN is

3. Birational Geometry 65
a surface X such that ip(E) is a single point p 6 X and ip\Y - E is an
isomorphism from Y - E to X - {p}. To see this, note that since a > 0,
ny+aE| contains the subseries \H\ which, since Я is very ample, separates
points and tangent directions on Y—E. Next we claim that |Я+а.Е| has no
base points along E. Since OY(H+aE)\E = Ob, it will suffice to show that
the map H°(OY(H+aE)) -> H°(Oy(H+aE)\E) = Я°@?) is surjective.
?he cokernel of this map is Hl(OY(H + (a - l)E)).
Claim. For 0 < к < a + 1, Я^СМЯ + kE)) = 0.
Proof of the Claim. Consider the exact sequence
0 -¦ OY{H + (k- 1)E) -> OY(H + kE) -> OE(H + kE) -+ 0.
By assumption H\OY{H)) = 0. Moreover, Hl(E;OE(H + kE)) =
B1(Opi(a - к)) and this group vanishes for к < a + 1. Thus, by induc-
induction on A we see that Hl{OY{H + kE)) = 0 for 0 < к < a + 1. D
SjtfThus, \H+aE\ has no base locus and so defines a morphism <p: Y —»P^
for some N. Since (Я + aE) ¦ E = 0, v(?^) is a point p. Moreover, given
д< point q € У — E, since |Я| has no base locus, there exist curves in
J# + aE\ vanishing along E but not along q, by using the subseries \H\.
Ttiue, v? separates y(?) = p from the points of Y — E as claimed. The
morphism defined by ip maps У onto a projective surface Xq. Taking X
s the normalization of Xq gives the candidate for the blowdown of У.
i concludes the proof of Step I.
j|tep II. Let p = tp(E) as above. We must identify У with the blowup
Ш X —* X of X at p. First we claim that X is smooth at p. It suffices
fcO show that the completion Ox,P of the local ring of X at p is a formal
power series ring. Since X is normal and ip is birational, Ox = 4>+OY. By
the formal functions theorem,
But the sheaf OY/m?OY is supported on E and is thus annihilated by
aome power of Ie = OY(-E). In particular there is a surjective map
Oy/OY(-NE) -* Oy/m?CV and it will in fact suffice to show that
Шп#°(У; OY/OY(-nE)) is a formal power series ring. Consider the exact
sequence
0 -+ OY(-nE)/OY(-{n + l)E) -> OY/OY(-(n
-+ OY/OY(-nE) -* 0.
Note that OY(~nE)/OY(~(n + l)E) is identified with OPi(n) and thus
H\OY{-nE)/OY{-{n + 1)E)) = 0.

66 3. Birational Geometry
It follows that the map
H°(OY/OY(~(n + 1Щ) -» H°(OY/OY(-nE))
is surjective for every n > 0. For n = 1, we have the inclusion
H°{OY{-E)/OY{-1E)) с H°(Oy/Oy{-2E)).
As OY(-E)/OY(-2E) S Opi(l), dimH°(OY{-E)/OY(-2E)) = 2, and
we can choose zf\ z{21' a basis for H°(OY(-E)/OY{-2E)). For all n > 1,
we can choose z[n\z^n) € H°(OY{-E)/OY{-{n + 1)E)) mapping onto
can be identified with the isomorphism Symn#°(CV(l)) -+ H°(OPi(n)).
From this, an easy induction shows that the C-algebra map
C[z[n\ z<n)] - H°(OY/OY(-(n +
is surjective for every n. Taking Zi = limzj1**, it also follows that the
induced map
C[[zb г,]] - Urn Я0(У; OY/OY(-nE))
is surjective and hence an isomorphism (since Ox must have Krull dimen-
dimension 2). Thus, X is smooth at p.
Let trip be the maximal ideal of p. Clearly, (<^~1mp)Oy С OY(—E) — Ie-
Moreover, the images of z\ and &i generate Ie mod J\, since Opi(l) is
generated by its global sections, and thus z\ and z2 generate Ie- It follows
that {ip~lx4p)OY = Ie- Thus, by the universal property of blowing up the
morphism ip factors through the blowup p: X —* X: there is a morphism
ф: Y —> X such that ip = p о ф. Clearly, ф is birational. Let E be the
exceptional curve on X. Since Y is protective, ф is surjective, and so we
must have ф{Е) = Ё. In particular, ф(Е) is a curve on X. But by Theorem
8 above, if ф is not an isomorphism, then there must exist a curve С on
Y such that ф(С) is a point. As ф is an isomorphism onY — E, the only
possibility for such a curve С is С = E, but we have seen that ф{Е) is
again a curve. Thus, ф is an isomorphism, identifying У with the blowup
of X at p. D
Definition 10. A curve ? on a smooth surface У such that E S P1 and
E2 = —1 is called an exceptional curve. The smooth surface X obtained
from У via Castelnuovo's criterion is called the contraction of У along E,
or the contraction of E. We also say that X is obtained from У by blowing
down E.
Here are two other characterizations of exceptional curves:

3. Birational Geometry 67
па 11- E is an exceptional curve if and only if E2 = E • Ky = — 1 if
ind oniy if Я2 < 0 and E ¦ Ky < 0.
The proof is left as an exercise.
Next we turn to the factorization of birational morphisms.
ГЬеогет 12. Let 7r: Y —» X be a birationaJ morpiism. Then тг is a
xmpoeition of bJowups.
Proof. Given x ? X, suppose that ir~l is not defined at x. Then by Theo-
«n 8, я~1{%) = С = Uj Ct is a curve on Y, and by Zariski's connectedness
ieorem [61, p. 279], С is connected. We will find a component Cj of С such
ihat Ci-Ky < 0 and Cf < 0; by Lemma 11, Cj is therefore exceptional and
те can contract it by Castelnuovo's criterion. The result is a new surface Y.
Chere is an induced continuous map #: Y —* X. It is easy to see that ж is
i оюгрЫвт: supposing that U is an affine neighborhood of x, contained in
in for some n, the coordinate functions on An define functions on ir~1(U)
md thus functions on jr^)-^ = 'r~I(^)-{p}> where p = 7r(Ci). These
ien extend to regular functions on *"*(?/) by Hartogs' theorem, and so
I'isa morphism. We may reapply the argument to the morphism Y —* X,
rating that since С has only finitely many components this procedure will
itop.
To find Cj, we shall first prove the following two claims:
Claim 1. There exist positive integers r, with Ky = ir*Kx + Ylj rjCj
phere D is an effective curve disjoint from C.
Claim 2. There exists an i such that ?\ r,(Cj • Cj) < 0 and Cf < 0.
Proof that the claims imply Theorem 12. We have Cf < 0 and
Ci-KY= ?\ r,(Cj • Cj) < 0. Thus, d is exceptional.
Proof of Claim 1. Choose local coordinates Zi,Z2 at x € X. Given j,
choose a smooth point у € С,- and let W\, v>2 be local coordinates for Y
centered at y. Then zt = гДадь^г) and given the local generating section
\o = dzi A dz2 of Kx, we have
d(«>U>)
Since Cj is mapped to a point, the coefficient of dw\ Л du>2 vanishes along
Cj. Thus, the natural map ir*Kx —» #V defined by pullback vanishes to
positive order along the Cj (and perhaps elsewhere, but not otherwise in a
neighborhood of C), and this is the statement of Claim 1.
Proof of Claim 2. Choose a very ample divisor Я on X. Thus, Я2 > О
and ir*H ¦ Cj =0 for all j. There exists a curve in |H| passing through x,

68 3. Birational Geometry
so that 7Г*Я = H' + J2j SjCj with Sj > 0 for all j, where Я' is a curve in У
which does not contain any of the Cj in its support. Moreover, by Zariski's
connectedness theorem ir*H is connected, and so H' • С, Ф 0 for some i.
For every j,
0 = 7г*Я • Cj ^H'Cj+J^ siiPi ¦ Cj) + Sj{CjJ.
Moreover, H' • C, > 0 and (C* • Cj) > 0 if i Ф j, and by the connectedness
theorem at least one of these is > 0. Thus, SjCj < 0, and therefore we
must have C? < 0 for all j.
Suppose now that 53,-tj(C< • Cj) > 0 for all i. By the Hodge index
theorem, since Y^jrjCj ^ orthogonal to ir*H, which has positive square,
(Ej rjCjJ < 0, and (?V r^C,J = 0 if and only if ?\ г;-С7,- is numerically
trivial. But
(EJ = Er- Er>(Ci • c>) > о
so that (J2j rjCjJ = 0. It follows that JV '"j^j is numerically trivial. But
H' ¦ Ci > 0 for all i and Я' • Cj > 0 some г. Since the г< are positive, this
implies that H' ¦ Q]V r,Cj) > 0, and so ?V г;С, is not numerically trivial,
a contradiction. Thus, there exists an i such that 53jrj(C« • Cj) < 0, as
claimed. ?
We shall analyze the above arguments for more general contractions at
the end of the chapter.
Next, we have the following result on the elimination of indeterminacy
of rational maps:
Theorem 13. Let f: X —» У be a rational map from the smooth surface
X to a projective variety Y. Then there exists a sequence of blowups Xn —»
¦¦¦ -* Xo = X and a morphism f:Xn-*Y such that / and / agree on a
Zariski open subset of Xn.
Proof. Clearly, we may assume that У = P" and that f(X) is nondegen-
erate, so that / corresponds to a linear system С on X without fixed curves.
We may further assume that N > 1 and that С actually has base points. If
p is a base point for C, let p: Xt -* X be the blowup of X at p and let E be
the exceptional curve. If D e С and D' is the proper transform of D, then
p*D = D' + kE. Let fco be the minimum possible value for к as D ranges
over the elements of С Thus, p*D — koE is effective for all D ? C, and there
exists a Do € С such that p*Dq — кцЕ = D\ is the proper transform of Do.
Since p is a base point, fc0 > 1. Thus, (p*D0 - koEJ = Dg - fc§ < D§. Set
С = {p*D - kQE : D e ?}. Thus, С consists of effective divisors and the
base points of С are the base points of С other than p together with some
possible base points along E. The only possible fixed curve of С would be

3. Birational Geometry 69
E, but, by the choice of fco, E is not a fixed curve of C. Hence ?' has no
fixed curves. It follows that (DiJ > 0 for Dx e ?'. Clearly, С ^ ?', and
the rational map from Xx to PN defined by С agrees with / away from E
and the points of X\ corresponding to points of indeterminacy of / other
than p. If С has no base points, we are done. Otherwise, continue this
procedure. If Dk denotes a typical element of the linear system at stage к,
then 0 < D\ < D\_x. Thus, this procedure cannot continue indefinitely,
and eventually we reach a base point free linear series. D
Corollary 14. Let /: X —*Y be a birationai map between two smooth
surfaces. Then there exists a smooth surface Z and morphisms tti : Z —* X,
7г2: Z —* Y, such that tti and тгг are sequences of blowups and f о щ = Л2
in the sense of rational maps (i.e., where defined).
Proof. Blow up X until / becomes a morphism, and let the resulting
surface and morphism to X be denoted 714: Z —¦ X. Then the induced
morphism 7Г2: Z —* Y is a birational morphism, and thus by Theorem 12
it is a sequence of blowups. D
Example. Consider the rational map /: P2 —¦» P2 defined by
(zo, zi, z2) •-> A/г0, l/zi,
in homogeneous coordinates. This map (called a Cremona transformation)
is defined as long as (zo,zi,z2) ? {A,0,0), @,1,0),@,0,1)}. We leave it
as an exercise to show that / becomes a morphism on the blowup of P2 at
these three points and that the birational morphism from the blowup to
the target P2 consists exactly in contracting the proper transforms of the
three lines joining pairs of points in {A,0,0), @,1,0), @,0,1)}.
A base point p of a linear series С is called a simpJe base point if the
following holds: if p: X —¦ X is the blowup of X at p, the linear series
С = {p*D -E-.DeC}
has no base points along E. It is easy to see that p is a simple base point
if and only if the general element of ? is smooth at p and two general
elements have different tangent directions at p. If p is not a simple base
point, then we say that С has an infinitely near base point at p. In general,
given a complete linear system \D\ and a point p ? X, we can consider
the linear system С = \D - p\ of all curves in D passing through p. We
say that p is an assigned base point of L. Any other base points of С
(including infinitely near base points at p) are unassigned base points. We
could likewise consider the linear system of all curves in \D\ containing
p which either have a given tangent direction at p ox are singular at p
(this corresponds to looking at \p*D — E — q\, where q e E). Likewise,
the linear system of all curves in |D| containing p which are singular at p,

70 3. Birational Geometry
corresponding to \p*D — 2E\, is denoted by \D — 2p\. These are examples of
assigned infinitely near base points. Note that \D — p\ defines a morphism
on X if and only if \D — p\ has no unassigned base points.
We note the following consequence of Corollary 14:
Corollary 15. Let /: X —¦» У be a birational map between two smooth
surfaces. Then Pn(X) = Pn(Y) and q(X) = q{Y).
Proof. By Corollary 14, there exists a surface Z and morphisms Z —> X,
Z -»У which are repeated blowups. Using Corollary 5 repeatedly, Pn{X) =
Pn{Z) — Pn(Y). A similar argument handles q. Q
Minimal models
Definition 16. An algebraic surface X is minima] if it contains no excep-
exceptional curves. An algebraic surface X is a minima/ mode] of a surface Y if
there exists a birational morphism Y —+ X (necessarily a blowup of X, by
Theorem 12), such that X is minimal.
Lemma 17. For every surface Y, there exists a minima] model X.
Proof. If У is minimal, we are done. Otherwise, there exists an exceptional
curve E on Y. Let Y\ be the smooth surface obtained by contracting E. If
Y\ is minimal, we are done. Otherwise, continue. At each stage we obtain
a surface Yn such that rankNumyn = rankNumyn_i - 1, by Proposition
3. Thus, this procedure must terminate. ?
On the other hand, the minimal model of a surface У need not be unique.
The simplest example where this fails is the case where X is a surface con-
containing a curve С = P1 with C2 = 0. If У is the blowup of a point p € C,
then the proper transform C" of С is = P1 and satisfies (CJ = -1- Thus,
it can be contracted, and the resulting surface X' is a candidate for a new
minimal model for У. More generally we could construct examples of this
type if X contains a smooth rational curve С with C2 > 0, by blowing
up several points on С until the proper transform has self-intersection —1,
or even more generally if С is any curve on X such that the embedded
resolution has genus 0 and nonnegative self-intersection. As this last ex-
example suggests, the set of all minimal models can be very complicated.
For instance, an example of Kodaira which we shall discuss in Chapter 5
shows that there exist blowups of P2 which have infinitely many exceptional
curves; of course, they cannot be pairwise disjoint. So it is important to
find cases where the minimal model of У is unique, in a strong sense:

3. Birational Geometry 71
Definition 18. A surface X is a strong minimal model of У if X is mini-
minimal, there exists a birational morphism /: Y -* X, and if У is a blowup of
У and g: Y —* X' is a, birational morphism to a smooth surface X', then
there exists a morphism h: X' —* X such that f = hog, where / is the
composition У —*Y —* X.
It is easy to see that the morphism h above is unique, and that a strong
minimal model of Y is unique up to isomorphism. More generally, suppose
that Y\ —+ У2 is a birational map. Then if /1: Y\ —» X is a strong minimal
model of Y\, there exists a birational morphism /2: У2 —¦ X commuting (in
the sense of rational maps) with the given birational map Y\ —¦» Y%. We
see this by blowing up Ух until the rational map to У2 becomes a morphism
and applying the definition with X' = Уг-
We then have:
^heorem 19. Suppose that Pn(Y) ф 0 for some n > 1. Then every
Щ1 dl of У is a strong minimal model.
Proof. Let X, У, and X' be as in the statement of Definition 18. Since the
hypothesis Pn(Y) ф 0 is unchanged under blowups, we may (in the notation
of Definition 18) assume that Y = Y. Factor the morphism /: У —¦ X into
a sequence У = Xn —> Xn-i -*¦¦¦-* Xq = X, where Ei is an exceptional
curve on Xi and Xj_i is the contraction of Ei. The morphism У —> X'
may also be factored. Suppose that we have factored it by first blowing
down the curve F С У, and let g': Y -» У be the result of contracting F.
If, we can show that there is a morphism h': Y' —* X such that / = h' о д',
then by repeating the argument with У replacing У (and noting that
Pn(Y') = Pn(Y)) we can continue until we reach a morphism from X' to
X.
Suppose that En is disjoint from F. Then we can contract En on У = Xn
and the image of En on X'. Continue in this way until one of the Ei is not
disjoint from (the image of) F. Note that this must happen at some stage
since X is minimal and so F cannot map isomorphically onto a curve in
X of square — 1. If Ei = F we are done. Otherwise, Ei ¦ F > 1. Thus,
Ei -{Ei + F)>0 and F-(Ei + F)> 0.
Now suppose that H e \пКхг \ is effective; we know that such an H exists
since Pn(Y) ф 0. Since Ei and F are exceptional H ¦ Et = H ¦ F = -n.
Thus, Ei and F are components of H. We may write H = rE, + sF + H',
where r, s > 1 and H' is effective and does not contain either Ei or F in
its support. Suppose for example that r > s and write H = (r — s)Ei +
s(Ei + F) + H'. Then
-n = HF = (r- s)(Ei ¦ F) + s(F ¦ (Ei + F)) + F ¦ H' > 0,
which is a contradiction. D

74 3. Birational Geometry
of the line bundle Ox{D) to p~x{U) is trivial. Here the necessity is ob-
obl
vious, and the sufficiency follows since, if Qx(D)^p~l{]J) =	),
then p,(Ox(D)\p-l(U)) = p,{Ox\p~l(U)) = Ov, since, as X' is normal,
p.Ox = Ox'¦ Let Ci,...,Cr be the irreducible components of p~x(x). By
the exponential sheaf sequence, we have an exact sequence
0 -» Hl{jT\U);Z) -» Я1^^);©) - Hl(p-\U);O*)
- H2{p-\U);Z) - H2{p-\U);O),
where Яг(р~1(С/');О*) = Pic(p~'(^)) is the group of holomorphic line
bundles on p (?/¦). If U is a Stein contractible neighborhood of x, then
an application of the Leray spectral sequence shows that H%(p~l(U); O) =
H°(U;R*p*O). (Here the Stein condition means that the higher cohomol-
ogy of a coherent analytic sheaf on U is zero.) Moreover, H2(p~l(U);Z) ?
Zr, and the map Hl{p-l(U)\O*) -> H2(p-l{U);Z) is the obvious map
which sends a holomorphic line bundle L to the vector whose zth compo-
component is deg(L|Ci). Finally, as p has relative dimension 1, R2pmO = 0. Thus,
for U as above, there is an exact sequence
H°(Rlp.Ox) - Pic (p'HU)) - Zr.
We see that we have proved the following statement:
Lemma 22. Suppose in the above situation that Rlp*Ox — 0. Then there
exist arbitrarily small analytic neighborhoods U' of \Jt Ci such that every
line bundle L in U' is specified by the integers L • C"<. In particular, if
L ¦ Ci = 0 for all i, then L is trivial in a neighborhood of |Ji C*. ?
Here the neighborhoods U' in Lemma 22 are the sets of the form p~1 (U),
where U is a Stein contractible neighborhood of x.
Definition 23. An isolated surface singularity x € X' is rational if
Е}ж»Ох = 0 for every resolution тт: X —> X'. It is easy to see in fact by
the factorization of birational maps that Rln,Ox = 0 for some resolution
тг: X —> X' if and only if R}ir*Ox = 0 for every resolution тг: X -* X'.
To check if a singularity is rational, suppose that тг: X -* X' is a resolu-
resolution of the singularity x with тг-1(р) = \Jt Ci. Then we have the following
consequence of the formal functions theorem:
Lemma 24. The singularity x is rational if and only if, for every effective
nonzero cycle Z = ?V щСи Hl{Z\ Oz) = 0.
Proof. By the formal functions theorem RlirtOx — 0 if and only if
HmH1{Z;Oz)=0,

3. Birational Geometry 75
феге the limit is over the inverse system of effective cycles Z supported
%\)iCi- Thus> if Hl(Z>°z) = 0 for all such Z, then Rlir,Ox = 0.
fjonversely, we claim that the map lim Hl(Z\0z) -* Hl(Z;Oz) is sur-
(ective for every Z. It suffices to show that, if Z' > Z, then the map
iPiPz1) -* HX{OZ) is surjective. But writing Z' - Z = Z" > 0, there is
§П exact sequence
0 -* OZ"(-Z) -> Oz> -» Oz -> 0.
corresponding long exact sequence is surjective on fl^'s because, as
Z is a curve, H2(Z"; Oz,,(-Z)) = 0. It follows that HX{OZ-) -* Hl{Oz)
jbeurjective for all Z' > Z, and so lim Hl(Z;Oz) -* HX{Z;OZ) is sur-
jjective. Since lim HX{Z;Oz) = 0, Hl(Z\Oz) = 0 for every Z supported
mple. If С is a smooth rational curve with C2 < 0, then by Exercise
Chapter 1 the result of contracting С gives a rational singularity.
Q = ?j Ci is the exceptional set of a rational singularity, then since
>(Oc) — 0) every component Ci of С is smooth rational, two components
"meet at at most one point, and there transversally, and the dual com-
: associated to jj< Ct has no cycles, i.e., is a tree. Here given a collection
¦'cunres d on a surface X, we define a complex by letting the vertices
the d and connecting two different vertices corresponding to C4, Cj by
ly Ci ¦ Cj edges. This complex is called the duaJ complex or dual graph
Ci. We can also weight the complex by assigning the positive integer
to the vertex corresponding to d. However, the above necessary con-
confer a rational singularity are by no means sufficient (see Exercise
Ф. Artin [3] has shown the following criterion for a singularity to be rational:
Theorem 25. The singularity x is rational if and only if, for every effective
nonzero cycle Z — ?\ riiCi, pa{Z) < 0.
Proof. If the singularity is rational, then hl{Oz) = 0 for all Z supported
on Ц Ci. Thus,
Pa(Z) = 1 - h°(Oz) + h\Oz) < 1 - h°(Oz) < 0.
Conversely, suppose that pa{Z) < 0 for all Z supported on \JtCi. We
must show that in this case hl{Oz) = 0 for all such Z.ll Z = d, then
Paid) = hl(OCi) = 0 and, by Corollary 5 of Chapter 1, pa(d) = 0
implies that d is smooth rational. Choose an arbitrary effective nonzero
Z = J^riiCi. We shall show by induction on n = ^TV щ that hl(Oz) = 0,
the case n = 1 having been dealt with above. We claim that there is an г

76 3. Birational Geometry
such that -Cf + Z ¦ d < 1. For, if -Cf + Z ¦ d > 2 for all i, then
Kx ¦ Ci + Z ¦ Ci = -2 - Cf + Z ¦ d > 0,
and thus Kx • Z + Z2 > 0. It would then follow that
contradicting the assumption that pa(Z) < 0.
For a choice of г such that -Cf + Z-d<l, consider Z' = Z - d and
take the exact sequence (using A.9))
0 -» OCi {-Z') -+OZ-+ Oz. - 0.
By the inductive hypothesis H1(Ozi) = 0. Moreover, Od(-Z') is a line
bundle on the curve d = P1 of degree ~d • Z' = Cf - d ¦ Z > -1, and
so Hl{OCi(-Z1)) = 0. It foUows that HX{OZ) = 0 as well. П
There is the following analogue of the Castelnuovo criterion for a rational
singularity [2]:
Theorem 26. Let X be a smooth projective surface and let {C\ ,...,Cn]
be a collection of curves spanning a negative definite sublattice of NumX
such that H1(Z;Oz) = 0 for every effective nonzero cycle Z = Х^тцС,.
Then there exists a normal projective surface X and a birational morphism
tt: X —» X such that the irreducible components of the 1-dimensional fibers
of it are exactly the d. Moreover, the singularities of X are rational.
Proof. We shall imitate as far as possible the easy part of the proof
of Castelnuovo's theorem. Choose a very ample divisor H on X with
Hx(Ox{H)) = 0. Let A = spanz{Ci,...,Cn} with its induced intersec-
intersection form. The linear form defined byCGA>-+/f-Cis strictly positive on
the generators d- Since the intersection form on A is negative definite and
in particular nondegenerate, the homomorphism A —¦ Av induced by the
intersection pairing has finite cokernel. Thus, after replacing H by nH we
may assume that there exists Z = Y^i fhd such that H ¦ С = —Z-Clox
all С 6 A. Next we claim that all of the coefficients n< are strictly positive.
Indeed we can write Z = Z++ Z_, where Z+ is the sum of the terms with
positive coefficients, Z_ the sum of the terms with negative coefficients. If
d € SuppZ-, then Z+-d > 0 and Z-d = -H-d < 0. Thus, Z- d < 0.
It follows that (—Z_J > 0, contradicting the negative definiteness. Like-
Likewise, there cannot exist a Ci with щ = 0. Thus, all of the coefficients of Z
are strictly positive, and Z is effective.
Now by construction (H+Z)-d = 0 for all г. Consider the exact sequence
0 - OX(H) -» OX{H + Z)-* OZ(H + Z) -» 0.
Applying the exponential sheaf sequence to the (possibly nonreduced) divi-
divisor Z, we see that the hypothesis HX(Z; Oz) = 0 implies that PicZ = Zn

3. Birational Geometry 77
Ш the map L •-» LCi. Thus, Oz(H + Z) =Oz- Consider the long exact
Kpbomology sequence associated to
?,	0 -» OX(H) -» OX(H + Z)-> OZ(H + Z) -» 0.
follows from the assumption that Hl(Ox{H)) = 0 that the section
ik>#°@z) = H°(OZ(H + Z)) lifts to a section of H°(OX(H + Z)).
x]H + Z\ has no base locus along Z. Since |# + Z\ contains the
ies \H\, it has no base locus elsewhere and so defines a morphism
trf X' from X' to a projective variety X', necessarily a surface since
i (я + Zf = (H + Z) ¦ H + (H + Z) ¦ Z = H2 + H ¦ Z > Я2 > 0.
Spring the Stein factorization of the morphism X —* X' as in Chapter 1,
discussion after Definition 25, gives the normal surface X. Clearly, all of
pej^ingularities of X are rational. ?
most important examples of rational singularities are the rational
feubie points, whose dual graphs are depicted in Figure 1 below (the num-
i above the vertices are the coefficients of the corresponding curves in
^fundamental cycle, which is defined in Exercise 8).
111111
An •¦••¦•
E6
1	4	A	?	L
• • m	*	9
42
2	3 4	3	2 1
2 4 6 5 4 3 2
Es •	•	«	•	•	•	•
1з
Figure 1

78 3. Birational Geometry
To define the rational double points, suppose that {Ci,..., Cn} is a col.
lection of n distinct curves such that d = P1 and Cf = —2 for all i, and
such that the span of the Cj is negative definite. In this case, the dual com-
complex of the set {Ci ,...,Cn] has connected components which form a graph
of type An,n > 1, Dn,n > 4, or En,n = 6,7,8; this is just a statement
about when lattices spanned by vectors of square —2 are negative definite
(Exercise 16). Assuming for simplicity that the dual complex is connected,
we can then contract the curves Ct and the resulting singularities are by
definition the rational double points. In fact the analytic type of the sin-
singularity is determined by the dual graph, and all of these singularities are
locally hypersurface singularities in C3. Note that Kx • Ci — 0 for all i.
Indeed this is one of the real reasons for the importance of the rational
double points: the components of a resolution of a rational double point
are among the curves which are orthogonal to Kx- We will discuss the
properties of the rational double points in greater detail in the exercises.
At the moment, we note:
Theorem 27. The rational double points are rational singularities.
Proof. Since Kx ¦ Z = 0 for every effective nonzero Z whose support is
contained in Ц C,, 2pa(Z) — 2 = Z2 for every such Z. As the intersection
pairing is negative definite for a rational double point, Z2 < 0 if Z ф 0.
Since Z2 < 0 and is even, 2pa(Z) = 2 + Z2 < 0 as well. Thus, by Artin's
criterion (Theorem 25) rational double points are rational singularities. ?
As a consequence we can prove the following theorem of Mumford:
Theorem 28. Suppose that Kx is nef and big. Then it is eventually base
point free, and the normal surface corresponding to the image of the mor-
pbism defined by \nKx\, n ;» 0, is the surface with rational double points
obtained by contracting all the (Bnitely many) smooth rational curves С
on X with C2 = -2.
Proof. Let С be an irreducible curve such that С ¦ Kx = 0. Then since
K\ > 0, C2 < 0 by the Hodge index theorem and thus 2po(C) - 2 < 0.
It follows that pa(C) = 0 and that С is a smooth rational curve. If we
consider the set of all such C, then they are independent in Num X, by
Lemma 25 of Chapter 1, and so there are only finitely many. Again by the
Hodge index theorem, the lattice they span is negative definite, and thus
there exists, by Theorems 26 and 27, a normal surface X with just rational
double point singularities which is exactly the contraction of the C's. By
Lemma 22 and the discussion preceding it, Kx descends to a line bundle
on X, which we shall denote by шх (in fact it is the dualizing sheaf of X).
Also ш2х = K\ > 0, and if D is any irreducible 1-dimensional subvariety
on X, and D' is the unique irreducible curve on X mapping onto D, then

3. Birational Geometry 79
'Ш-,?) = Kx • ГУ > 0, since Kx is nef and D' is by construction not one
шю curves С with Kx ¦ С = 0. By the general version of the Nakai-
/jfehezon criterion for a singular scheme, шх is ample. In particular, шх
eventually base point free and the same must be true for Kx. Moreover,
Hjniorphism defined by \nKx | factors through X and has the same image
he morphism defined by w®n. Since шх is ample, this image is exactly
D
i' theorem of Bombieri shows that it suffices to take n = 5 in the above
lit. We will give a proof of this theorem in Chapter 9.
fees
Jet С be a curve on X and let X —+ X be the blowup of X at p. Let
the proper transform of С Show that С ¦ E = m and that the
icheme-theoretic intersection С" П E is the projective tangent cone to
I. .etC
7 be
method of proof of Theorem 13 to the linear system | H—p\,
srhere Я is a line in P2 and p is a point. Show that if H' is the proper
W ;r/&nsform of H on the blowup of P2 at p, then \H'\ has no base points
ind defines a morphism from the blowup to P1. What are the fibers
it this morphism?
%', For the examples y2 = i2fc+1 and y2 = x2k of Exercise 5 of Chapter 1,
;alculate the infinitely near points and their multiplicities and obtain
mother calculation of 6o-
Ш Prove Lemma 11, that ? is an exceptional curve if and only if E2 =
E • KY = -1 if and only if E2 < 0 and E ¦ KY < 0.
Verify the statements about the Cremona transformation, and show
that the rational map / corresponds to the linear system of quadrics
passing simply through the points A,0,0), @,1,0), and @,0,1).
Щ Show that the germ of an isolated normal surface singularity has a
strong minimal model in the sense of Definition 18. (Follow the proof of
Theorem 19 until you reach a contradiction to the negative definiteness
theorem.)
% Let С be a smooth cubic in P2 and let D be a smooth plane quar-
tic meeting С transversally in 12 points {pi,...,Pi2}- Let X be
the blowup of P2 at {pi,-¦ ¦ ,Pu} and let D' be the proper trans-
transform of D. Show that |D'| is base point free and that OX{D') =
тг*ОраD) ® Ох{—"^цЕг), where the Ei are the exceptional curves
on X. On the other hand, show that, for a general choice of points
(Pi. • • • ,Pi2}, the line bundle L = 7r*0p2D) ® Ox(- Z)i E<) restricts
to a line bundle of degree 0 but infinite order on the proper transform
C" of C, and so the linear system corresponding to Z,®" has C" as a
fixed component for every n > 0. In fact, for a general choice of the
Pi, there does not exist any line bundle on X for which the restriction
to C" is the trivial line bundle.

80 3. Birational Geometry
8. (The fundamental cycle.) Suppose that Ci,...,Cn are irreducible
curves spanning a negative definite lattice and that \Jt Ci is con-
connected. Show that there is a unique effective nonzero cycle Zo such
that ZQ ¦ Ci < 0 for all i, Zo ¦ d < 0 for at least one г, and ZQ щ
minimal with respect to the above property. (If Z\ = J^ щСг and
Z2 = ?j mid both have the above property, so does
Z =
Thus, there is a minimal such cycle if any exist. To show that it exists,
we can use the argument of the first part of the proof of Theorem 21.)
This ZQ is called the fundamental cycle. .
9. Show that we can find the fundamental cycle as follows: choose Z\ = Ci
for any г. Now suppose that we have inductively found Z\,...,Zi. If
ZiCa < 0 for all a, then stop. Otherwise, there is a Ca with Zi-Ca > 0.
Set Zi+i = Zi +Ca. Show by induction that Zo - Zi is effective for
every i, and thus that this procedure must terminate with Zq. Such a
sequence of curves Z\,..., Zn = Zq is called a computation sequence.
Conversely, if Ci,...,Cn are a collection of curves such that Cf < 0
for all г and such that this sequence terminates, then the Ci span a
negative definite lattice. (Use the second part of the proof of Theorem
21.)
10.	If Zo is the fundamental cycle of |Ji d as above, where \Jt Ct is con-
connected, show that H°(Zo;Oz0) — C. (Induct on a computation se-
sequence, using A.9).) Thus, pa(Z0) > 0.
11.	(Artin.) With notation as in Exercise 8, the configuration {C\,..., Cn}
contracts to a rational singularity if and only if H1(ZQ;Oz0) = 0,
where Zq is the fundamental cycle, if and only if pa(Zo) = 0. (The
equivalence of the two versions follows from the last exercise. To see
that H1{Zq\Oz0) = ° implies that Hl{Z;Oz) = 0 for every Z sup-
supported in \Jid, it suffices to show that Hl(nZ0;Onz0) = 0 for all
n > 0. Use the condition Hl{ZQ; Oz0) = 0 to show that all the C{ are
smooth rational, and that for a computation sequence we must have
Zi- Ca < 1. Now consider the sequence
0 -» 0c0(-nzo ~ Zi) -» OnZo+zi+1 ~* OnZo+Zi -» 0,
using induction on i and n.)
12.	(A dual form of Artin's criterion [43].) With notation as in Exercise 8,
suppose that \Jt Ci contracts to a nonrational singularity. Then there
exist nonnegative integers 1Ц with щ > 0 for at least one t such that
(Kx + ?)i щСг) Cj >0 for all j. (Consider a minimal element under
the partial ordering > in the set of all curves С = ?4 етцС, such that
hx{Oc) Ф 0. If С = Ci for some г, then Ci is not a smooth rational
curve, and we can apply adjunction to Kx +C Otherwise, for Ci < C,

3. Birational Geometry 81
let С = С - Ci- Then hl{Oc) = 0. From the exact sequence
0 -» OCi{-C) -+Oc-+ Oc -»0,
we see that h}{OCi{-C')) ф 0. By duality
h\OCi{-C')) = h°
and so degcji + С ¦ d > 0. Now apply adjunction.)
Let Ci,.--,Cn be a chain of curves, so that Ci- Cj ф 0 if and only
if» = j ± 1. Suppose that C? < -2 for all г. Show that the C, span
a negative definite lattice, by finding the fundamental cycle of С and
using Exercise 9. Show also that, in this case, if the Ci are all smooth
rational curves, then the singularity is rational. Next consider the case
of curves Ci, C2, C3, D, with d ¦ C}¦ = 0 for г ф j and Ci ¦ D = 1 for
all». Suppose that Cf,D2< -2, and show that the d and D span
a negative definite lattice. Moreover, if the Ci and D are all smooth
rational curves, then the singularity is a rational singularity. Finally,
consider the case of curves C\, C2, <?з, Ci, D, with d Cj = 0 for i Ф j
and Ci ¦ D = 1 for all i. Suppose that C?, D2 < -2, and show that
\ the Ci and D span a negative definite lattice if and only if Cf < — 3
for some г or D2 < -3. Moreover, if the Ci and D are all smooth
rational curves, then the singularity is a rational singularity if and
onlyifD2< -3.
thet Zq be the fundamental cycle of a rational double point. . Show
i that Zq = -2 (the terminology from the theory of root systems is
'that Zq is the highest root.) Conversely, if Zq is the fundamental cycle
of a rational singularity, then the singularity is a rational double point
if and only if Z% = -2.
Compute the fundamental cycles for the rational double points as fol-
follows. The rational double point of type An corresponds to a chain
of curves Cu ¦ ¦ ¦, Cn with d ¦ Ci+i = 1, d ¦ Cji = 0 if j'• ф i ± 1
and Cf = —2 for all г, and the fundamental cycle for such configu-
configurations has been computed in Exercise 13. Otherwise, consider curves
Ai,..., Лр_1, Bi,..., Bq-i,C\,..., Cr-i,D such that each of the con-
configurations Ai,...,Ap-U Bi,...,Bq-i, Ci,...,Cr_i is a chain (i.e.,
Ai ¦ Ai+i = 1, Ai ¦ Aj = 0 if j ф г ± 1, and similarly for the В
and С curves), Ai • Bj = 0 for all г and j, and similarly for Ai ¦ С к
and Bj ¦ Ck, and Лр_! • D = B,_! • D = Cr~i ¦ D = 1, whereas D
meets no other curves. Lastly we assume that each curve in the list is
smooth rational and of square —2. Here Dn corresponds to the case
(p, q,r) = B,2, n - 2), and En to the case (p,q, r) = B,3, n - 3). The
lattice spanned by the curves is called а Тр,ч>г lattice and the dual
graph is called a TPi9iI. graph. (See Figure 2 on the next page.) Show

82 3. Birational Geometry
that the fundamental cycle Zq is given by
C\ + 2Ci Л	+ 2С„_з +2D + Ai + Bi,	in case ?>„,
Bi + 2B2 + 3D + 2C2 +Ci+ 2Ai,	in case Ee,
Ci + 2C2 + 3C3 + AD + 3B2 + 2Bi + 2Аг,	in case E7,
2Ci + 3C2 + 4C3 -I- 5C4 + 6?> + 4B2 + 2Вг + ЗАг,	in case Es.
In particular, using the last part of Exercise 9, conclude that the lat-
lattices in question really are negative definite.
m •	m • •	• •
lp-1
Figure 2
16.	Every connected configuration of curves of square —2 spanning a neg-
negative definite lattice spans a lattice of type An, Dn, Еь,Ет,Е%. (Every
sublattice of such a lattice is negative definite. Thus, if we are given
a chain Ai,... Ak which is a subset of the given set of curves, then
(Ai -I	1- AkJ = —2 and the lattice spanned by A = A\ Л	\- Ak
and the remaining curves is still negative definite. More generally, if X
and С are elements of square —2, then X ¦ С = 0 or ±1. Show that the
dual graph of the set of curves is a tree and (by considering the configu-
configuration Ci+Ci+Cs+Ci+D, where С{С, = 0 if i Ф j and d-D = 1 for
1 = 1,..., 4 and showing that it is not negative definite) that in fact the
dual graph is either a chain (case An) or that of a TP,,,P graph. Show
that such a graph must contain T2i2n_2 or Т2д„_з, and in fact must
contain T2i3itl_3 except in case Dn. Now argue that if a Tp,q,r graph
properly contains a T2]3in_3 graph and the corresponding Гр,г lattice
is negative definite, then there has to exist a curve C, meeting either
A\,B\, or C\, with CZ0 = 1. Using the previous problem, show that
we must be in case Ee, E7, E&. In particular, the TPi,iT. lattice is nega-
negative definite only for (p,g, r) = B,2,n - 2), B,3,3), B,3,4), B,3,5).)
17.	Let {Ci,... ,Cn} be the components of the resolution of a rational
singularity with fundamental cycle Zq such that Z\ = — 1 (and such

3. Birational Geometry 83
that Ui С» k connected). Show that one of the Ci is an exceptional
curve, and that the blown down configuration also has fundamental
cycle of square — 1. Conclude that the Ci can be successively contracted
in some order to a smooth point.
More generally, Artin has shown [3] that, with Ci and Zq as above, if
the singularity is rational, then its multiplicity is given by — Z\ and
its embedding dimension by —Zq +1. In particular, by using Exercise
14, we see that a rational singularity is a rational double point if and
only if it is a hypersurface singularity, if and only if its multiplicity is
2 (hence the name rational double point). From this, one can work out
the analytic type of the equations for the rational double points: they
are
(n > 4),
Ап:
Dn:
Ев:
E7:
Es:
x2+yU
*2 + yV
x2 + y3 +
x2 + y3 +
x2 + y3 +
• zn+1
+ zn~
z4,
yz3,
-z5.
Finally, there is yet another approach to rational double points: they
are locally analytically isomorphic to the quotient of the germ of C2 at
the origin by a finite subgroup of SLB,C), and as such were studied
Ijy Klein. For a very readable discussion of these and other properties,
we refer the reader to [28] as well as [20].

4
Stability
In this chapter, we shall define stability and investigate some of its elemen-
elementary properties. After giving simple examples of stable bundles over curves
and P2, we describe unstable and strictly semistable bundles carefully and
look at what happens when we change polarizations. The final section,
which is not necessary for the rest of this book, describes the differential
geometry of stable bundles.
Definition of Mumford-Takemoto stability
Let X be a smooth projective variety of dimension d and let H be an ample
line bundle on X.
Definition 1. For V a torsion free coherent sheaf on X, the normalized
degree Hh{V) of V with respect to H is the rational number
We shall omit the subscript when the line bundle H is clear from the con-
context. Note that if we replace Я by аН, о € Z+, then (iaH{V) = ad~ 1(m(V).
We can also define the normalized degree with respect to a nef and big line
bundle H, as well as with respect to a Kahler metric, by replacing H by
the Kahler form cj.
The idea is that the normalized degree is a measure of the "degree"
of V with respect to H. For example, if V = OX{D) is the line bundle
associated to an effective divisor D, then fi(Ox{D)) is indeed just degZ?.
The normalized degree has the following convexity property with respect
to exact sequences.
Lemma 2. Suppose that
0 -* V -» V -* V" -» 0

86 4. Stability
is an exact sequence of nonzero torsion free sheaves on X and H is an
ampie line bundle on X. Let \i = \1ц. Then
and equality holds at either end if and only if fi(V) = fi(V")
Proof. Let n' = rankf and n" = rank^". By the Whitney product
formula, cx{V) = cx{V') + ci(F"). Thus,
So fi{V) = Ap(V') + A - X)n(V"), where 0 < A < 1. Thus, if say n{V) <
H(V"), then (i{V) < (i(V) < fi{V"), with equality if and only if fi(V) =
We also have the following property of normalized degree for subsheaves
of equal rank.
Lemma 3. Suppose that W is a subsheaf of the torsion free sheaf V, with
rankVF = rankV. Then (i(W) < n(V). Moreover, ifV and W are vector
bundles, then either fi{W) < fx(V) or W = V.
Proof. We shall just consider the case where W and V are vector bundles
of rank r, and leave the general case to the exercises. In this case there is
the associated map det W —* det V, which is a nonzero map of line bundles.
Thus, det^ = det W®OX{D), where D is effective, and D = 0 if and only
if det V = det W, in other words if and only if the inclusion map W —» V is
an isomorphism. In this case W = V. Otherwise, D is an effective nonzero
divisor, and so
/i(V) = -rHd~l ¦ Cl(V) = ±[Hd-1 ¦ (Cl(W) + D)]
> -Hd-1 ¦ ci(W) = fi(W).
r
Thus, n(W) < fi(V). D
We now give the definition of Mumford-Takemoto stability which for the
purposes of this book will simply be denoted stability.
Definition 4. For V as above, V is H-stable (resp., Я-semistabJe) if, for
all coherent subsheaves W of V with 0 < rankVF < rankF, we have
(i{W) < (i(V) (resp., n{W) < fi{V)). We call V unstable if it is not
semistable and strictly semistabie if it is semistable but not stable. (The
convention that unstable is the opposite of semistable is unfortunately by
now well established.) Finally, a subsheaf W of a torsion free sheaf V with
0 < rank W < rankF is destabilizing if fi{W) > fi{V).

4. Stability 87
Thus, for example, line bundles are (vacuously) stable. Moreover, V is
ff-stable if and only if it is аЯ-stable, for а ? Z+, and similarly for semista-
bility Furthermore stability and semistability only depend on the numerical
equivalence class of H. If X is a curve, then \in (V) = deg ci (V) is indepen-
independent of the choice of H. However, for dimX > 2, if rankNumX > 2, then
the definition of stability depends on the choice of the numerical equiva-
equivalence class of H. We shall return to this point later.
Next let us work out the meaning of stability in the case where V is a rank
2 vector bundle on a surface X. In this case, we need only check stability
with respect to rank 1 torsion free sheaves W. But since W is torsion free,
Wvv is a reflexive rank 1 sheaf on X and is thus a line bundle L (either
since dimX = 2 or since rankVF = 1). Hence W itself is of the form
L® Iz for some ideal sheaf Iz, where dimZ = 0, and ci(VF) = ci(Wvv).
Moreover, since V = Vvv, there is an inclusion of VFVV = L in V. Thus,
we see that it suffices to check the degree Ah for sub-line bundles of V.
Now suppose that the quotient V/L is not torsion free. Then as we have
seen in Proposition 5 of Chapter 2, the map L —* V factors through the
inclusion L —+ L® Ox(E), where E is an effective, nonzero divisor, and
V/(L®OX(E)) is torsion free. As H-{L®OX{E)) = HL + HE > HL,
the normalized degree of L can only increase. Thus, it suffices to check the
degree of all sub-line bundles L such that V/L is torsion free.
Lemma 5. Let V be a torsion free sheaf on a smooth projective variety
X.
(i) V is stable if and only if, for all coherent subsbeaves W of V with
0 < rank W < rank^ such that V/W is torsion free, (i{W) < (i(V).
(ii) V is stable if and only if there exists a line bundle F such that V ® F
is stable if and only if for all line bundles F,V(g)F is stable.
(iii) V is stable if and only ifVyy is stable.
(iv) V is stable if and only if for all torsion free quotients Q of V with
0 < rankQ < rank^, we have fi(Q) > fi(V).
(v) V is stable if and only ifVv is stable.
Moreover, all of the above statements hold if we replace stable by
semistable.
Proof. We shall only check these statements in the case where V is a rank
2 bundle on a surface X. In this case, we have already seen (i), and (ii) is
a consequence of the fact that /z(L ® F) = (i(L) + H ¦ F and /j,(V ® F) =
(i(V) + H ¦ F by B.6) of Chapter 2. Part (iii) is obvious in our case since
V = Vvv, and (iv) is a consequence of Lemma 2. Finally, (v) holds in our
case as a result of (ii), since Vv = V ® det V. D
Lemma 6. Let
о -»v -* v -* v" -* о

88 4. Stability
be an exact sequence of nonzero torsion free sheaves, with fi(V) = fi(V) =
fi(V"). Then V is semist&ble if and only ifV and V" are semistable, and
V is never stable. In particular, if V and V" both have rank 1, then V is
semistable.
Proof. Suppose that V is semistable. Let W is a subsheaf of V. Then W
is a subsheaf of V and so fx(W) < n(V) = fi(V). Thus, V is semistable.
Likewise, if W is a subsheaf of V", then the inverse image W of W in V
satisfies fi(W) < n(V) = /i(V')- Using the exact sequence
it follows from Lemma 2 that n(W) < fi(V) = fi(V") as well. Thus, V" is
also semistable.
Conversely, suppose that V and V" are semistable. Let W be a nonzero
subsheaf of V. Let p: W —> V" be the projection and let W = Kerp,
W" = Imp. Thus, there is an exact sequence 0 -+ W -* W -* W" -* 0. We
may assume that both W and W" are nonzero, since otherwise W Si W\
say, and thus (i(W) = /i(PF') < д(У) = fi(V). But if Ж" ^ 0, then
either rankW" = rankV", in which case n{W") < n{V") by Lemma 3,
or rankW" < rankF". By hypothesis V" is semistable, and so fi(W") <
n(V") = fi(V). A similar argument handles fi(V). Hence, by Lemma 2,
H(W) < max(u.(W'),/i(W")) < n(V) as well. D
Proposition 7. If if: Vi —+ V% is a nonzero homomorphism between two
stable torsion bee sheaves V\ and Vz with fi(Vi) = n(Vz), then tp is injective
and is an isomorphism ifVi and V-i are vector bundles or ifVi = Vi-
Proof. Suppose that ip is not injective. Then 1пкр = W is a proper torsion
free quotient of Vi and thus n(W) > д(И) = n(Vi). Thus, W must have the
same rank г as V2, since V2 is stable. By Lemma 3, however, n{W) < /^(Vb),
a contradiction. Hence ip is injective. Thus, W is a subsheaf of V2 with
fi(W) = ц(У2), and so by stability rankW = rankVlj. Again using Lemma
3, if Vi = W and V2 are vector bundles and (i(Vi) = n(V2), then W = V2
and ip is an isomorphism. The case where V\ = V2 is not necessarily locally
free follows from the general fact that an injective map from a coherent
sheaf (on a projective scheme) to itself is necessarily an isomorphism, which
we leave as Exercise 1. ?
Corollary 8. If V is a stable torsion free sheaf, then V is simple, i.e.,
EndF={A-ld:AeC}.
Proof. If tp is a nonzero endomorphism of V, then it is an isomorphism, by
Proposition 7. The proof of Corollary 8 is then a standard Schur's lemma
argument: every nonzero element of End V is invertible, so that End V is a
finite-dimensional division algebra over C. Thus, End V = C. Alternatively

4. Stability 89
p e X such that V is locally free at p and consider the induced map
the stalk ipp: Vp -* Vp. If A is an eigenvalue of ipp, then the map <p - A • Id
be an isomorphism and so must be 0. Hence ip = A • Id. D
•k. An argument similar to the proof of Proposition 7 shows that,
> V4, is a nonzero homomorphism between two semistable torsion
sheaves V\ and V2 with fi(Vi) = д(Уг) and such that at least one of
is stable, then either tp is injective or the rank of its image is equal
rank of V2.
EJjtamples for curves
Щ us give some examples of stable and semistable bundles for the case of
9.
;(/Руег Р1, there are no stable rank 2 bundles. The only semistable rank
\,2 bundles are the bundles O?i(a) Ф Opi(o).
'tLet С be a curve of genus 1. The only stable rank 2 bundles over С
sre the bundles of the form Tp ® L, where L is a line bundle on C,
C, and Tp is the unique nonsplit extension ofOc(p) by Oc, as in
f ^Theorem 6 of Chapter 2. The only strictly semistable rank 2 bundles
|k('f>ver С are either of the form L\®Ij2, where deg L\ = deg L^, or of the
f'''-fyrm E®L, where Lisa line bundle on С and ? is the unique nonsplit
extension of О с by Ос-
These statements all follow easily from Theorem 6 of Chapter 2
J'from Lemma 6, with the exception of the statement that Tp ® L is
It suffices to prove that Tp is stable. Note that ц(Тр) = 1/2. If L is
ieub-line bundle of Tv such that the induced map L —> Oc{p) is nonzero,
ben degL < degOc[p) = 1, and degL = 1 if and only if L = Oc(p)- But
a this last case the extension is split, contradicting the definition of Tp.
[bus, degL < 0 < \i.{Tp) = 1/2. If the induced map L -* Ocip) is zero,
hen L С Ос- In this case degL < 0 < n{Fp) again. Thus, Tp is stable. ?
For curves of genus at least 2, we have the following existence result:
Ibeorem 10. Let С be a, curve of genus at Jeast 2.
(i) If L is a line bundle on С with degL = — 1, then there exist stable
bundles V over С of the form
0 -» L -» V -* L~x -» 0.

t. otauuuy
(ii) If M is a line bundle on С with degM = 1, then there exist stable
bundles V over С of the form
in fact, every such extension which is nonsplit is stable.
Proof, (i) For a curve С of genus g and a line bundle L on С of degree
—1, the set of extensions V of the form
is classified by Hl{C; L®2). Since deg L®2 = -2 < 0, we have h\C\ L®2) =
g +1 > 3 by Riemann-Roch. In particular nonsplit extensions always exist.
Let V be such a nonsplit extension. When is V stable? Since (i(V) = 0,
we would like to show that, for all sub-line bundles F of V, deg F < 0. If
the composite map F —» L is zero, then the image of F is contained in
L, so that F = L ® OC{D), where D is effective. In this case deg F < -1.
Otherwise, the map F —* L-1 is nonzero. Thus, F~1®L~1 has a section and
so degF < 1. If degF = 1, then F = L and the sequence is split, contrary
to hypothesis. If deg F < 0, then F is not destabilizing. So the only problem
is when deg F = 0. In this case F = L~l ® Oc(—p) and the map F —¦ L^1
is, up to a nonzero scalar, the natural map L~x ®Oc{— p) —* L~x. Thus, by
construction, we are in the following situation: V is an extension such that
the natural map L~l ®Oc{-p) —» L~l lifts to a map L ®Oc{—p) -* V.
To see when this is possible, consider the exact sequence
® Oc(~p),V) -
-» H\C; (L-1 ® Oc(-P))-1 ® L).
Since g{C) > 2, dim Я°(С; Oc(p)) = 1, and the obstruction to lifting the
essentially unique element of HomCL ® Ос{-р),Ь~1) = Н°(С;Ос(р))
to an element oiUom{L~l®Oc{-p), V) lies in Я^С;L®2®Oc(p))- Now
we have a commutative diagram
H°(C;OC) —?—+ Hl{C;L®2)
I-
where д is the coboundary map in the appropriate long exact cohomol-
ogy sequence. Moreover, the image of д{1) is just the extension class. So
a nonzero element of Hom(L^1 ® Oc{— p),L~l) can be lifted to a homo-
morphism L ® Oc(—p) —¦ V exactly when the extension class is in the
kernel of the natural map
H\C; L®2) -> Hl{C\

4. Stability 91
Рог a fixed p 6 C, we have an exact sequence
0 -> Lm -» L®2 ® Oc(p) -» Cp - 0,
where Cp is a skyscraper sheaf at p with stalk C. Since deg L®2 ® 0c (p) =
-1, there is an inclusion С = Я°(СР) С H\C\ L®2). In fact, Я1 (С; L®2) is
the dual of H°(C; KC®L®~2), and the linear system denned by Kc®L®~2
is base point free (since deg Kc ® L®~2 — 2g). Via Serre duality, the line
tf°(Cp) С ЯЧС;^®2) defines a point in Р(Я°(С; Kc ® L®~2)v) which
is the hyperplane of all sections vanishing at p, and this point is then the
image of p under the morphism defined by Kc ® L®~2. We see that V is
stable if and only if the extension class corresponding to V, viewed as an
element in Ш\С\ L®2), does not lie on the image of C. Since РЯ^С; L®2)
is a projective space of dimension at least 2, the general such extension class
Will indeed correspond to a stable bundle.
(ii) Let us first show that there exist nonsplit extensions of M by Oc- The
set of all such extensions is equal to Hl(C\M~x). Since degM~x = —1,
the Riemann-Roch theorem gives dimHl{C\ M~x) = g > 2. In particular,
there exist nontrivial extensions V. If V is a nonsplit extension, then V is
stable by an argument identical to that given in the proof of Theorem 9
for^p. ?
ftlMore generally, along these lines we shall show (Exercise 3) that, for
iVery line bundle L on С of negative degree, there exist extensions of L~l
Щ L which are stable. Moreover, if deg L > -{g-1)/2, then the "generic"
extension of L by L is a stable bundle V such that Hom(L', V) = 0 for
all V such that degL' > degZ- and L ф L'. A theorem stated classically
fey Corrado Segre [135] (and rediscovered by Nagata [110]) asserts that,
for every rank 2 vector bundle V on С with det V = 0, there exists a line
bundle L of degree at least -{g - l)/2 such that Hom(L,^) ^ 0. Thus,
there is an exact sequence
0 -> L' -* V -+ (L'y1 -> 0,
where degL' > -(g - l)/2. Moreover, for the "generic" stable bundle V,
if g is odd, then there are exactly Iя line bundles L of degree -(g - l)/2
Such that Hom(L, V) Ф 0, and for every other line bundle M such that
Hom(M, V) ф 0, we have deg M < -(g - l)/2.
Some examples of stable bundles on P2
In this section, we shall give some of the elementary properties of stable
rank 2 bundles on P2. In particular we shall show that a rank 2 bundle
on P2 is stable if and only if it is simple. Many of the following arguments
work equally well for any surface X with Pic X = Z.

92 4. Stability
Throughout this section, V denotes a rank 2 bundle on P2. By stability
we mean stability with respect to 0ргA). In fact as PicP2 = Z, there is a
unique notion of stability for P2.
Lemma 11. For a rank 2 vector bundle V on P2, there exists an integer
ky and a sub-line bundle Орг(ку) ofV such that, for every integer k, if
there is a nonzero map Opi(k) —* V, then к < ky. Moreover, the quotient
of V by a sub-line bundle of V isomorphic to Ojn(ky) is torsion free.
Proof. For ? > 0, V <g> Op2 {?) is generated by its global sections, and thus
for ? < 0 there exists a nonzero map OP2(?) -» V. Using Proposition 5
of Chapter 2, there exists an effective divisor D on P2 such that the map
Op*(?) -» V factors through the inclusion OP2(?) -> OP2(?) ® OP2(D) and
the quotient is of the form L' ®IZ- Now OP2{D) — OP2(m) for some integer
то and L' = Cpz(A;') for some integer k'. Thus, there exists an integer к
and an exact sequence
0 -» Ota(k) -> V -> Opa(*') ® /z -» 0.
It follows that, if n > max{A;, Jk'}, then
Hom(OP2(n), V) = Я°(^ ® Орт(-п)) = 0.
Thus, there is a largest integer kv such that Hom(Oi«(A;v), V) ф 0; indeed
fcv < max{A;,Jfc'} in the above notation. Clearly, for a sub-line bundle of V
isomorphic to O^ky), the quotient must be torsion free, for otherwise as
before the map would factor through Op2{kv) ® Of2(D) = Ор2(ку + то)
for some positive integer то, contradicting the maximality of ky- ?
Note that we do not claim in the above lemma that there is a unique
sub-line bundle ОР2(ку).
Lemma 12. V is staWe if and only if2kv < d, where detF = Op2(d),
and V is strictly semistahle if and only if 2ky = d.
Proof. V is stable if and only if for all nonzero maps Op2 (к) —> V, we
have к < \d if and only if ky < \d. The proof of the second statement is
similar. D
Corollary 13. A rank 2 bundle V over P2 is stable if and only if it is
simple.
Proof. If V is stable, then it is simple by Corollary 8. Conversely, suppose
that V is not stable, and let detF = Op2(d). Then there exists an exact
sequence
0 - OV2(k) -> V

4. Stability 93
with к + к' — d and 2k > d. Thus, k' < k, and there is an inclusion
Of(k') ® Iz Q Oja{k') С Орз(к). Hence there is a nonzero map V -»
Or'ik') ® Iz —* Op2(k) —* V which is not multiplication by a scalar.
Therefore V is not simple. ?
Remark. If V is not stable and not of the form O^{k) © О^а{к), then
ty is easy to check that Е.от{Орг{ку),У) has dimension 1, i.e., that the
fjctension
Hh;	0-> OP2(kv)-> V-> Орг(к') ® Iz-> 0
is canonically determined by V. Moreover, since ky — k' > 0, we have
JP{Ov2{—к')®Ор2(ку)) = 0, so that we may in fact construct all such ex-
extensions via Theorem 12 of Chapter 2. This result goes back to Schwarzen-
berger, who analyzed nonsimple bundles ("almost decomposable bundles"
in his terminology) and used the method of double covers to show the
existence of simple bundles.
• Next we give some examples of stable bundles on P2. For example, con-
consider the bundle V defined in Exercise 6 of Chapter 2. This bundle has
Cx(V) = Op2(-1) and kv < -1 since H°(V) = 0. Thus, V is stable by
Lemma 12. Another example of a stable bundle is the tangent bundle Gp2
(Exercise 6). In fact, these bundles agree up to a twist by a line bundle. As
an exercise, we ask the reader to show that the bundles Va,b constructed
in the example after the proof of Proposition 28 in Chapter 2 are stable as
long as о Ф b or 6 ± 1.
We conclude this section by studying stable rank 2 bundles on P2 with
Ci(V) = 0 and сг(У) small. Let us begin with the following:
Proposition 14. Let V be a rank 2 bundle on P2 with ci(F) = 0, and let
(i) V is stable if and only ifkv<0 if and only ifh°(V) = 0.
(ii) If V is stable and с < 5, then there exists an extension
(iii) Conversely, ifV is given by an extension (*) as above, then с = ?(Z) — 1
and V is stable if Bind only if Z is not contained in a line.
(iv) If V is stable, then с > 2.
(v) Let V be given by an extension (*). Tien h°(V <g> OP2(l)) is equal
to dim/io(CP2B) <g) Iz) + 1, i.e., to 2+ the dimension of the space of
conies containing Z (or 1 if there are no conies containing Z).
Proof, (i) Using Lemma 12, since det V = 0, V is stable if and only if
every sub-line bundle of V has strictly negative degree, which is clearly the
case if and only if V has no sections.

94 4. Stability
(ii) Applying the Riemann-Roch formula to V ® CVA), we obtain
x(V®O?2(l))=6-c.
Hence, if с < 5, then either h°(V ® CVA)) > 1 or h?(V ® Opa(l)) =
°
(-1) ® JKpa) > 1. Since a section of V ® CV(-l) ® #i* is
equivalent to a nonzero map (Dps D) —» V, it follows from (ii) that since
ky < 0 this case cannot occur for stable bundles V. Thus, we must have
h°(V® CV»A)) Ф 0- Hence there exists a nonzero map Opa(-l) -> V. The
cokernel of this map must be torsion free, else there would exist a nonzero
map from Орз(к) to V with к > 0, contradicting (i). Thus, we obtain the
exact sequence (*), proving (ii).
(iii) Suppose that V is given by (*). By B.9) of Chapter 2 we have
с = ?{Z) - 1. By (ii) above, V is stable if and only if ft°(V) = 0. Since
Я1(Р2,Ор2(-1)) = 0, h°(V) = h°(Op2(l) ® Iz), and this is nonzero if and
only if there exists a section of Op2(l) vanishing on Z, i.e., if and only if Z
is contained in a line.
(iv) We may clearly assume that с < 5, so that V is given by an exten-
extension (*). Since every 0-dimensional subscheme of P2 of length at most 2 is
contained in a line, if V is stable, then necessarily ?(Z) > 3 and so с > 2.
(v) This follows from the exact sequence
0-*Of2 -*V®Opi(l) -+CVB) ®/г-+0.	D
We shall next describe the bundles V corresponding to the first few
choices of с
Proposition 15. Suppose that V is a stable rank 2 bundle on P2 with
ci(F) = 0 and c2(F) = 2. Tien there is an exact sequence
0 -> ОМ-2) © Op»(-2) -> (Срз(-1)L -» V -» 0.
Moreover, two subbuncUes Wi and W2 of (Cpz(—I)L, both isomorphic to
Орз(-2) ©0p2(-2), give isomorphic quotients V if and oniy if there exists
a bundle automorphism of (Орг(-1)L taking WTi to W?.
Proof. Using (ii) and (iii) of Proposition 14 we may write V as an ex-
extension (*) with ?(Z) = 3 and Z is not contained in a line. It follows,
for example, by [61, Prop. 4.1, p. 396] that Z imposes independent con-
conditions on conies, so that h°((Op2B) <g> Iz) = 3. Next we claim that
ОрзB) ® Iz is generated by its global sections, so that the natural map
(Opz(-l)K —> ОрзA)®1г is surjective. One way to see this is to note that,
again by [61, Prop. 4.1, p. 396], there exists a smooth conic С containing
Z corresponding to a section of CpzB) <g> Iz- The quotient of CpzB) ® Iz
by Орг is then (Op2(C)/Op2) ® Iz, which is a line bundle of degree 1 on С
and so is generated by its global sections. Thus, 0i«B) ® Iz, too, must be
generated by its global sections.

4. Stability 95
putting this together, there is a natural surjective map (Cp2(—I)L —» V,
#hich is more invariantly given by the natural map H°(V ® Oi«(l)) ®
^a(-l) -¦ V. Let W be the kernel. We must show that W ? Op*(-2) ©
jJ3^i(-2), or equivalently that W ® CVB) ?! Орг © Орт• By the Whitney
product formula, ci(W ® CVB)) = c2(W ® 0p»B)) = 0. FVom (iv) of
ition 14, W ® OpzB) cannot be stable. Let Jfc > 0 be the largest
such that there exists a nonzero map Op* (k) —> W<g>Op2 B). If к = 0,
СргB) is an extension of Opz by Op2 which must necessarily split,
we are done in this case. To rule out the possibility that к > 0, note
VT®Oi«B) isasubbundleof (Op2(-l)L®Op2B) = (Op2(l)L. Hence
y nonzero map Op (A) -» W ® Opz B) induces a nonzero map Opz (k) ~*
1fPia(l)L. Thus, if A; > 0, then к - 1 and the map Op (A) -¦ W ® Op2B)
no zeros. But then W ® Ci«B) is an extension of Op2(-1) by
sjsjWe have proved all of Proposition 15 except the last statement, which
i from the more general observation that any isomorphism Vi —> V2
lV1> * two stable bundles as in the theorem induces an isomorphism
® ОргA))® Opi(—1) —¦ H°(V2 ® Cpz(l)) ® Opi(—1) which is com-
! with the natural maps to V\, V2. ?
$In the exercises we shall also show that every stable rank 2 bundle V on
kwith ci(V) = 0 and C2(V) = 2 is of the form V2,-i for. a unique double
• /: Q —»P2, in the notation of Exercise 14 of Chapter 2. Each double
: /: Q —* P2 of the plane by a quadric is specified completely by its
locus, which is a smooth conic in P2. In this way, we can identify
щв moduli space of stable rank 2 bundles V on P2 with cx (V) = 0 and
ШУ) = 2 with the space of conies in P2, which is an open subset of the
Objective space |0i«B)| = P5.
Next we turn to the case с = 3.
Proposition 16. Suppose that V is a stable rank 2 bundle on P2 with
Sbi(V) = 0 and Cb{V) = 3, and write V as an extension (*). If no three of
tile points in Z are collinear, then there is a natural exact sequence
0 -» Ор*(-3) -» (O,«(-l)K -» V -> 0.
¦The set of all such V is then identified with an open subset of the space of
nets of conies in P2 which is isomoiphic to the Grassmannian GC,6).
Proof. Using the exact sequence
0 -¦ Of2 -» V ® ОрзA) —> OpaB)/z -¦ 0,
and the fact that there is a smooth conic through four points Z (pos-
(possibly infinitely near) as long as no three are collinear, we easily check as
above that with assumptions on Z as above, the natural map (Oi«(-1)K =
H°(V® CVA)) ® Opa(-l) -» V is surjective. The kernel, which is a line

96 4. Stability
bundle, is necessarily Op2(-S), and all rank 2 bundles V so obtained are
stable. Any bundle map Орз(—3) —¦ (Op2(—I)K is determined by a generic
element in Я0((Ор2B)K), or more invariantly, by a generic net of con-
conies. D
The proof of the following for the cases с = 4,5 is similar to the above
proofs and so will be omitted.
Proposition 17.
(i) Suppose that V is a stabie rank 2 bundle on P2 with ci(V) = 0 and
= 4, and write V as an extension (*). Then V may be written
as an extension
0 - 0p*(-l) e CV(-l) -» V -» (Iz/Ic) ® Of(l) -* 0,
where С is a conic containing Z. This extension is canonical if there
is a unique conic containing Z, or equivalent]/ by Proposition 14 if
h°(V®OP2(l)) = 2.
(ii) Suppose th&t V is a st&ble rank 2 bundle on P2 with ci (V) = 0 and
C2(V) = 5. Then f may be uniquely written as an extension (*) if and
only if Z does not lie on a conic. ?
We shall deal with the case ci(V) = ОрзA) for small values of c2(V) in
the exercises.
Gieseker stability
For constructing compact moduli spaces, another notion of stability due to
Gieseker has proved to be extremely important.
Definition 18. Let p\{n) and рг(") be two real valued functions with
domain the natural numbers. Then p\ x рг (resp., <) if, for all n » 0 we
have pi(n) < рг(и) (resp., <). Let V be a torsion free sheaf on X of rank
г and H an ample line bundle. Define the normalized Hilbert polynomial
PH,v{n) = (l/r)x(V® H®n). (It would amount to the same thing if we had
used the function (l/r)h°(V ® H®n) instead.) Then V is Gieseker stable
(resp., semistable) if for all coherent subsheaves W of V with 0 < rank W <
rank V, we have Ph,w -< Pl,v (resp., x).
If С is a curve, then for V a vector bundle of rank г and degree d on C,
it follows from the Riemann-Roch theorem that
pH,v(n) = hn + - + l

4. Stability 97
where h = deg H. Thus, V is Gieseker stable if and only if it is Mumford
stable, and similarly for Gieseker semistability. For a surface X and a vector
bundle V of rank г on X, we have by a slightly tedious calculation using
B.6) and Theorem 2 (the Riemann-Roch theorem) of Chapter 2,
The constant term is just {x{V))/r. It follows that V is Gieseker stable
if and only if, for all rank s subsheaves W of V with 0 < s < r, either
fi{W) < n{V) or n(W) = n(V) and x(W)/s < x(V)/r. Thus, we have
proved the following lemma in the surface case for vector bundles (although
it is true in general):
Lemma 19. IfV is Mumford stable, it is Gieseker stable. IfV is Gieseker
eemistable, it is Mumford semistahle. D
:i>
The normalized Hilbert polynomial has the same convexity properties
as the normalized degree (Exercise 9). Thus, Gieseker stable bundles have
some properties which are similar to those enjoyed by stable bundles. For
example, a Gieseker stable bundle is simple. On the other hand, Gieseker
stability differs in many ways from ordinary stability. For example, it is
possible for V to be Gieseker stable but for V ® F to be Gieseker unstable,
where F is a line bundle on X.
Unstable and semistable sheaves
Why is stability a good definition? I don't really know the answer to this
question. But for many questions either Mumford or Gieseker stability is
exactly what is needed. One partial answer to the question above is that un-
unstable and strictly semistable bundles are much simpler to understand than
stable ones. In fact, we can canonically construct an unstable bundle out of
semistable torsion free sheaves of lower rank by successive extensions (this
is called the Harder-Narasimhan nitration). Likewise, a strictly semistable
bundle V with /x(V) = ц is a successive extension of stable torsion free
sheaves of smaller rank, all with normalized degree equal to /x, although
in this case the corresponding filtration on V is not necessarily canonical.
We shall just make this explicit in the case of a rank 2 bundle on a sur-
surface. In any case, the above says that the stable bundles are the interesting
ones, since they are the ones we don't know how to describe canonically!
From this point of view, stability is a nondegeneracy condition. Another
partial answer to the above question is provided by the discussion in the
next section of the complex differential geometry of stable bundles.

98 4. Stability
Let us now prove the above statements in the rank 2 case, leaving the
general cases as a series of exercises.
Proposition 20. Suppose that V is an unstable rank 2 bundle. Then there
exists a unique sub-line bundle F of V with torsion free quotient such that
H(F) > n(V). Indeed, ifL is a sub-line bundle of V such that ц(Ь) > n(V),
then L is a subsbeaf of F and n(L) < n(F), with equality if and only if
L = F.
Proof. If V is unstable, then there exists some sub-line bundle F of V
with torsion free quotient such that n(F) > /x(V) (Lemma 5). Thus, there
is an exact sequence
0 -¦ F -* V -* F' ® Iz -* 0,
where fi(F') < n(V). Now let L be a sub-line bundle of V such that ц(Ь) >
fi(V). We claim that the composite map L —* F'®Iz is zero. For otherwise
there is a nonzero map L —> F', so that, by Lemma 3, ц(Ь) < fi(F') <
H(V), a contradiction. Thus, the map L —> V factors through F, and so, if
the quotient is torsion free, then L = F. D
Proposition 21. Let V be & semist&ble but not stable rank 2 bundle.
Then exactly one of the following holds:
(i) There is a unique sub-line bundle FofV with /x(F) = fi(V). The
quotient V/F is necessarily torsion free, and V is given canonically as
an extension
0 -» F -> V -* F' ® Iz -* 0.
(ii) There are exactly two distinct sub-line bundles F and GofV with
H(F) = n(G) = /j,(V). In this case V = F@G.
(iii) V = F © F, and there are infinitely many sub-line bundles with nor-
normalized degree n{V), exactly corresponding to tie choice of a line in
Afore precisely, the following holds: Suppose that V is an arbitrary rank 2
vector bundle which is given given as an extension
0->F-*V->F'®Iz->0,
and such that n(F) = fi(V). Then V is H-semistable and either F is the
unique destabilizing sub-line bundle with torsion free quotient or Z = 0
and V - F © F', i.e., the extension splits.
Proof. We shall just prove the last statement, leaving the remaining ones
as an exercise. By Lemma 2, as /x(F) = fi(V), we also have fi(F') = HF' =
ti(V). Thus, by Lemma 6, V is Я-semistable. Let M be a sub-line bundle
of V such that H ¦ M > fi(V). If the map M -> V factors through F, then

4. Stability 99
Ц#= F®Ox{~D), where D is effective. Thus, HM = HF-HD < HF,
¦ith equality holding if and only if D = 0 and Af = F. So in this case M is
JMtabilizing only when M = F. Otherwise, the induced map M -* F'®IZ
inonzero. Thus, F' = M®OX(D) for an effective divisor D, where D = 0
"' when Z = 0 and M = F'. In this last case the extension
rly splits. Assuming that the extension does not split, we have H- M =
_#•?><#• F'-l and so fi(M) = HM < fi(V) - 1 < /j,(V).
в, М is not destabilizing. ?
¦№>¦¦
Rhange of polarization
v
that H\ and Я2 are two ample divisors (although the proofs go
in case the Hi are just assumed to be nef and big). When does
Й exist a vector bundle which is Hi -stable but is not ^-stable? We shall
at this question for rank 2 vector bundles on a surface X (it can
for higher rank also, but the analysis requires Bogomolov's
and will be discussed in the exercises to Chapter 9). Most of
results and further developments can be found in [126]. Fix a rank
or bundle V with detF = Ox (A) for some divisor Д. Instead of
; the classes ct(V) = A and c2(V), we will use the classes w2(V) = Д
I 2 € NumX/2NumX andpi(adF) = d(VJ - 4ca(F). (Here sAV is
I'kernel of the trace map Hom(V,V) -* Ox, and C2(adF) = ic2(V) -
J = —pi(ad V).) The advantage of this choice is that the classes wi(V)
l(iPi(adF) are unchanged if we replace V by V ® F, where F is a line
e. For V fixed, we set w = w2{V) and p = pi(ad V).
sition 22. In the above notation, an H\-stabh bundle V is not
if and onJy if there exists a sub-line bundle Ox(D) of V with
free quotient such that
Hx ¦ BD - Д) < 0 < H2 ¦ BD - Д),
and such that
p < BD - ДJ < 0.
Moreover, Ox (D) is the unique sub-liae bundle of V with torsion free
quotient with the above properties. Finally, V is strictly semist&ble with
respect to an ampie divisor which is a convex combination of Hi and H2.
Proof. V is not Яг-stable if and only if there exists a sub-line bundle
Ox(D) of V with torsion free quotient such that
Я2 • ?> = nH2{Ox{D)) > hh2{V) = H2 ¦ Д/2.

100 4. Stability
Thus, Я2 • B?> - Д) > 0. Since V is Hrstable, Hy • BD - Д) < 0. Now
2D - Д is orthogonal to a convex combination of Hi and Щ, so by the
Hodge index theorem BD - ДJ < 0, with equality only if 2?> - Д is
numerically trivial. However, B?> - Д) • Hi < 0, so that this last case does
not occur. There is an exact sequence
0 - OX(D) -* V -» Ох(Д - D) ® /z -> 0
for some 0-dimensional subscheme Z. Thus,
c2(V) = -D2+D-A+?(Z) > -D2 + DA.
So
B?> - ДJ = 4P2 - 4D • Д + Д2 > -4c2(V) + d (FJ = p.
The uniqueness of Ox(D) follows from Propositions 20 and 21: either V is
#2-unstable and Ox{D) is the unique destabilizing sub-line bundle, or V
is H2-semistable. However, as V is 7/i-stable, it cannot be a direct sum of
line bundles, so that only the first case of Proposition 21 can occur. Finally,
if we choose H to be a convex linear combination of H-y and Щ which is
orthogonal to 2D-Д, then Я is ample and fiH(Ox(D)) = nH(Ox(A.-D)).
Thus, by Lemma 6, V is strictly semistable. D
The meaning of the conditions of Proposition 22 is as follows. Given
? € Num X, С is a ciass of type (w, p) if the mod 2 reduction of ? is w and
p < <2 < 0. For such С we define the wall W< = <-"- с А(Х), where A(X) is
the ample cone of X, provided that W^ ф 0 and W^ ф А(Х), i.e., provided
that there exists an ample divisor orthogonal to ? and provided that С is
not numerically trivial. In fact, ? determines an oriented wall, meaning
that A(X) — W> has two connected components and on one of them the
linear from (•?) is positive. Notice that С is not uniquely determined by the
(oriented) wall, but that any two classes defining the same oriented wall are
positive rational multiples of each other. In particular, since B > p, only
finitely many С define the same oriented wall. There is also the following
standard fact, for whose proof we refer to [38]:
Proposition 23. For fixed w and p, the set
{W^ : (, is a. class of type (w,p)}
is locally finite in A{X). D
We shall refer to the set of walls described in Proposition 23 as the walls
of type (ti),p). The connected components of the complement of the set of
walls of type (w,p) are called the chambers of type (w,p). It follows that
the definition of an L-stable rank 2 vector bundle V with «^(V) = w and
Pi(ad V) = p only depends on the chamber containing L, provided that L
lies in a chamber, in other words, does not lie on a wall of type (w,p). The

4. Stability 101
chambers in A(X) for the walls of type (w,p) are the algebro-geometric
analogue of the chambers needed to define the Donaldson invariants for a
4-manifold with b% = 1. We shall return to this point in Chapter 8. Finally,
let us note that we can in some sense reverse the analysis of Proposition
22 to find bundles which are Hi -stable but Яг-unstable:
Proposition 24. Suppose that Hi and Я2 are two ampie divisors, and
that W4- is tie unique wall of type (w,p) separating Hi and Я2, and assume
further that С • Яг < 0 < ? • Я2. Let V be given by a nonsplit exact sequence
0 - OX{D) -» V -» Ox(A - D) ® Iz -* 0,
where С = 2D - Д and w2 (V) = w, pi (ad V) = p. Tien V is Hx -stable and
Яг-unstabie.
Proof. Clearly, if <Я2 > 0, then V is #2-unstable. We claim that V is Яг
stable. Suppose not, i.e., suppose that there exists a sub-line bundle Ox(F)
of V with torsion free quotient and such that hhi(Ox(F)) > /j,Hi(V). Since
X?HX < 0, OX(F) ф OX(D) and in fact OX(F) is not asubsheaf of OX(D).
Since Ox(F) is not a subsheaf of Ox(D), it follows from Proposition 20
applied to the Яг-unstable bundle V that hh3(Ox(F)) < fiH2(V)- Hence
T) • Hi > 0 > T) • Яг, where rj is the class 2F — Д. Clearly, rj is a class of type
(w,p) separating Hi and Я2, so that т? is proportional to ?¦ Now choose an
ample Я € W- = Wv (for example, a suitable convex' combination of Hi
and Я2). By Lemma 6 V is Я-semistable. By applying Proposition 21 to V,
since OX(F) Ф OX(D), we must have V = OX{D)®OX(F), contradicting
the assumption that V was not split. ?
Note that, if the above extension is split, then V = Ox (D) © Ox (A - Щ
is unstable for every Hx not on the wall W^. Finally, for ?(Z) » 0, we can
use the discussion of Chapter 2 to find bundles V corresponding to nonsplit
extensions for which V is ffi-stable and Яг-unstable.
The differential geometry of stable vector bundles
In this section, we assume that the reader knows a little complex differential
geometry (which can be found, for example, in [55]) and describe some of
the special properties of stable vector bundles on Kahler manifolds. We can
interpret these results as giving another answer to the question raised at
the beginning of the last section: Why is stability a good definition for a
holomorphic vector bundle?
Let M be a manifold and let ? be a C°° complex vector bundle on M
of rank r. Recall that a connection on ? is a C-linear map D from C°°
sections of E to sections of AX(E) = E® A1(M), where here Ay(M) is the

102 4. Stability
bundle of C°° 1-forms on M, satisfying the Leibniz rule: for all sections s
of E and C°° functions / on M,
D(fs) = fDs + s®df.
It follows that the difference of two connections is a C°° 1-form with coeffi-
coefficients in End E, and in fact the space of all connections is an affine space for
A1(EadE). There is a natural extension of D to an operator from AP(E)
to AP+1(E), where AP(E) is the vector bundle of p-forms with coefficients
in E, by requiring the graded Leibniz rule
s)=d<fi®s + (-1)рф® Ds.
The curvature D2 of the connection D is a C°° section of A2(EadE) =
A2(M) ® End E. Choosing a local basis s-y,..., sr of C°° sections, we can
identify a section s with a vector of functions and we can write Ds =
ds + As, where A is a matrix of 1-forms, called the connection matrix.
We frequently use the letter A to denote the connection D as well. In this
case the curvature D2 is locally given by the matrix Fa = dA + A Л A,
which transforms as a section of A2(End.E). The vector bundle E (or more
precisely the pair (E,D)) is flat if D2 = 0, in which case we say that D
is integrabie. As a corollary of the Frobenius theorem, if E is flat and M
is simply connected, then E is trivialized by global sections s\,... ,sT such
that Dsi = 0 for all i. More generally, for an arbitrary manifold M, flat
vector bundles E correspond to representations of жу(М,*) into GL(r,C).
Given bundles Ei and E2 and connections Di on Ei, there is a naturally
induced connection D = D\ © D2 on E\ © E2, and D2 — D2 © D\ in the
obvious sense. Likewise, there is an induced connection D\ <g) Id -f Id ®D2
on Ei®E2, and its curvature is equal to D\ ®Id + Id ®D%. A connection D
is reducible if E is a direct sum E = Ei © E2, where both E\ and Ец have
positive rank, and D = ?>! © D2 for some connections Dv and D2 on E\
and E2, respectively. The connection E is irreducible if it is not reducible.
In the cases of interest, E will have a Hermitian metric (•, •), and D will
be compatible with the metric in the sense that
(Dsi,s2) + {si,Ds2) = d(si,s2).
Thus, if Si is an orthonormal basis with respect to the inner product, then
the connection matrix A is skew-Hermitian, or in other words it lies in the
Lie algebra u(r) of the unitary group U(r). We say that the connection A
is unitary or Hermitian. In this case, the curvature, computed in a local
orthonormal frame, is a skew-Hermitian matrix of 2-forms. The flat vector
bundles E whose connections are compatible with a Hermitian metric es-
essentially correspond to representations of ni(M,*) into U(r). For unitary
connections, we will take reducible to mean that E = Ey ® E2 is an or-
orthogonal direct sum of (nonzero) vector bundles Ei, and Di is a unitary
connection on Eit such that D = Di + D2.

4. Stability 103
If E is a Hermitian vector bundle and D is a connection which is com-
compatible with the metric on E, then we can consider the characteristic poly-
polynomial
tr-k
'k\^)
fc=O
Here the coefficients сь(Е) turn out to be closed forms of degree 2k rep-
representing the Chern classes of the vector bundle E. Thus, for example,
cy{E) = (i/2n)traceD2. Note that, if D is flat, then d(E) = 0 for all
i>0.
Now suppose that M is a complex manifold, so that d = д + д. Let
пр'4 (М) be the vector bundle of forms of type (p, q), and, for a complex vec-
vector bundle E, define QP'q(E) similarly. If E is holomorphic, then В is well de-
defined on C°° sections of E, and we say that the connection jD is compatible
with the complex structure if тг^ф) = В, where тг0'1: А'(?) -¦ fi0'1^)
is the projection induced from the projection of the l-forms on M to the
@, l)-forms. In this case n°'2(D2) = 0, in other words, the curvature has
no component of type @,2). Conversely, if ? is a C°° vector bundle and
D is a connection on E such that ir°'2(D2) = 0, then there exists a unique
holomorphic structure on E for which D is a compatible connection (this
is an easier special case of the Newlander-Nirenberg theorem on integrable
almost complex structures). For example, it is easy to see that a flat com-
complex vector bundle on M has a natural holomorphic structure. Every holo-
holomorphic vector bundle E with a Hermitian metric has a unique unitary
connection D which is compatible with the complex structure [55, p. 73].
We shall refer to D as the compatible unitary connection associated to the
metric. In this case, since B2 = 0, D2 has no component of type @,2),
and since it is skew-Hermitian, it has no B,0)-component either. Thus,
the curvature D2 lives in ?lx<1(E) (and is skew-Hermitian). It follows that
the Chern classes Ck{E) are represented by real forms of type (k,k). If
D = Di + ?>2 is a reducible connection on E, corresponding to a direct
sum decomposition E = Ev © E2, then it is easy to check that the bundles
Ei are again holomorphic and the Dt are compatible with the complex
structures.
Suppose in addition that M is a Kahler manifold with Kahler metric ш.
If E is a holomorphic vector bundle on M with a Hermitian metric and D is
a unitary connection on E which is compatible with the complex structure,
then D2 is a (l,l)-form with coefficients in End?, and so D2 Л ш"
is a form of type (n,n) with coefficients in End.E. Thus, we can write
D2 л w" = F ¦ ш", where F is a C°° section of End?. Note that, up to
a positive scalar, F is the same as the contraction of D2 with ш (since u>n
is n! times the volume form on M).

104 4. Stability
Definition 25. Let M be a compact Kahler manifold with Kahler met-
metric ш and let E be a holomorphic vector bundle on M with a Hermitian
metric. If D is a unitary connection on E which is compatible with the
complex structure, then D is a Hermitian-Einstein connection if, in the
above notation, F = A Id for some constant A.
If F = Aid for some constant A, then D2 A w" = Aldw™ and so
trace(D2) A w" = rAw". Thus, integrating over M, we find that
w".
-2тгг f ci(E)Mj"-1 =r\ f
Jm	Jb
Here fM w" = n\ vol(M) and the formula says that, up to universal pos-
positive constants, A = -гцш(Е), where we define цш by analogy with the
normalized degree to be
Thus, if E = Ei © Ei and each Ei is a vector bundle of positive rank with
a Hermitian-Einstein connection Dt, then D = Di + D2 is a Hermitian-
Einstein connection on E if and only if fiu(Ei) ~ ^(Ег). On the other
hand, if Ei has a Hermitian-Einstein connection D* for i = 1,2, then the
connection ?>i ® Id + Id ®Di is always a Hermitian-Einstein connection on
Ey ® ?2-
We turn now to the study of Hermitian-Einstein connections in special
cases. If ? is a holomorphic line bundle, a connection D is a Hermitian-
Einstein connection on E if and only if D2 Auin~1 is a constant multiple of
wn. Starting with a fixed metric | • |2 on E, there is an associated compatible
unitary connection Do. If we replace | • |2 by eft| • |2, then Do is replaced by
D = Do +dh and ?>g by Dl + ddh (compare [55, p. 73]). By the Эа-lemma
([55, p. 149]), there exists a choice of h so that D^+ddh is a harmonic A,1)-
form. Every such form can be written as Aw + t/>, where A is constant and ф
is a harmonic A, l)-form satisfying JM ip Л ш"'1 = 0, since ш is harmonic
and ь)п is nonzero in cohomology. Now since M is a Kahler manifold and
ф is harmonic, the form ф Л шп~1 is a harmonic (n,n)-form and is thus a
constant multiple of the volume form, so it must be identically zero, since
its integral over M is 0. Hence ф is pointwise orthogonal to шп~1 and so
(Aw + ф) А шп~1 = Aw™. Thus, we see that, if D2 is harmonic, then D
is a Hermitian-Einstein connection. In fact, there is a unique choice of a
metric on E, up to a constant factor, so that the corresponding compatible
connection D is Hermitian-Einstein. It suffices to show that, for a real-
valued C°° function h, if ddh is pointwise orthogonal to w", then h is
constant. If we set f = ddh, then f is an imaginary A, l)-form pointwise
orthogonal to w", and /Mf A f A w"~2 = 0 by Stokes' theorem, since
f = ddh. On the other hand, a slight generalization of the proof of the
Hodge index theorem in [55, p. 125], shows that ?A?Aw"~2 is a nonpositive

4. Stability 105
multiple of u", which is 0 if and only if ? = 0. Thus, Bdh — 0. It is easy to
see that the maximum principle holds for such functions h. In fact, using
the Kahler identities, it is easy to check that in our case Bdh = 0 implies
that h is harmonic. Since M is compact, h must be constant. We conclude
that there is a C°° real valued function h, unique up to a constant, such
that the compatible connection with respect to ел|-|2 is Hermitian-Einstein.
There is also an equivalent formulation in terms of bundle automorphisms
of a fixed unitary bundle which we shall not describe here.
Next we consider the meaning of Hermitian-Einstein connections on a
vector bundle E with су (Е) = 0 for compact complex manifolds of dimen-
dimension 1 or 2. If M is a compact complex curve and Cy(E) = 0 as a real
cohomology class, then D is a Hermitian-Einstein connection on M if and
only if D2 = 0, in other words if and only if D is a flat unitary connection.
Suppose that M is a compact Kahler surface with Kahler form w. Then
the Hodge *-operator acts on 2-forms, and *u> = w. In fact, for a general
4-manifold M, the bundle of C°° 2-forms A2(M) splits into the (+1) and
(-1) eigenspaces for *: A2(M) = U2+(M)®tiL(M), and likewise for A2(E).
Thus, given a connection D on E, we can split its curvature into two parts,
D\ and D2_. We say that D is self-dual if D2__ = 0 and anti-self-dual
if D\ = 0. In case M is a Kahler surface, it is easy to check that the
complexification of п\ (M) is just П2-°(М) ©fi°'2(M) © A°(M)C ¦ ш, where
A°(M)c is the bundle of complex-valued C°° functions on M, and that the
complexification of U2_(M) is the orthogonal complement-to w in Q}'l(M).
Thus, a connection D on E is anti-self-dual if and only if the curvature
D2 is of type A,1) and orthogonal to ш. Hence, if ? is a holomorphic
Hermitian vector bundle with cy (E) = 0 and D is the unitary connection
on E compatible with the complex structure, then D is anti-self-dual if and
only if D is a Hermitian-Einstein connection.
Note that the definition of the normalized degree рш enables us to define
w-stability for a general Kahler metric on a compact Kahler manifold M,
by copying Definition 4. In case ш is a Hodge metric associated to an ample
divisor H, then w-stability is the same as Я-stability. The following is the
main result concerning Hermitian-Einstein connections:
Theorem 26. Let M be a compact Kahler manifold, with Kahler form
ш, and let E be a holomorphic vector bundle on M. If there exists а Яег-
mitian metric on E whose associated compatible unitary connection is an
irreducible Hermiti&n-Einstein connection, tien E is w-stable. Conversely,
if E is ui-stable, then there is a Hermitian metric on E whose associated
compatible unitary connection is an irreducible Hermitian-Einstein connec-
connection on E, and this connection is unique up to C°° bundle automorphisms
ofE.
The case dimM = 1 was proved by Narasimhan and Seshadri [112]. In
this case, if Cy(E) = 0, the theorem essentially asserts that a stable bundle

106 4. Stability
V on M such that degdetF = 0 is equivalent to an irreducible unitary
representation of ni(M, *). The uniqueness part of the general statement
was proved by Kobayashi [70] and Liibke [84]. Donaldson [23] showed the
existence for an algebraic surface, and then Uhlenbeck and Yau [146] proved
the result for a general Kahler manifold. (See also [25] for a proof for smooth
protective varieties.)
We shall prove one very special case of a small part of Theorem 26. Sup-
Suppose that E is a rank 2 holomorphic vector bundle on M with a Hermitian-
Einstein connection D. We claim that either E is w-stable, in other words
that, for every holomorphic sub-line bundle L of E, цш(Ь) < цш(Е), or
that there exists a holomorphic sub-line bundle L of E with Hw(L) =
цш{Е) and E =* L© L' for some holomorphic line bundle L' such that
fiu(L') = fiu(L) = цш(Е) (in which case E is strictly w-semistable). To
see this, suppose that L is a holomorphic sub-line bundle of E. Note that
L has a Hermitian-Einstein connection since it is a line bundle, and so we
can consider the induced Hermitian-Einstein connection on E® L~l. Now
рш{Е ® L~x) = цш(Е) — fiu,(L). Moreover, E ® L~v has a holomorphic
section and цш(Ь) > цш(Е) if and only цш(Е <S> L~l) < 0. Thus, after
replacing E by E ® L~y, it suffices to show: if E is a holomorphic rank 2
vector bundle with цш(Е) < О and a Hermitian-Einstein connection, then
E has a nonzero holomorphic section if and only if цш{Е) = О and in this
case E — Ом © L' for some holomorphic line bundle L' with цш{Ь') = О.
To see this last statement, we begin by proving the identity, for an arbi-
arbitrary compatible unitary connection D,
2д*д = D*D -
where Cn is a positive real number and * represents the formal adjoint of the
appropriate differential operator. Indeed, if Л denotes the operator which
is given by contraction with ui, then we have the Kahler identities [55, p.
Ill]:
d*=i[d,\);
1,0)
)- o
(D1-0)* = i\d,
and similar identities hold for the A,0) and @,1) parts of D, which we can
write as Dlfl and 8. Thus, for @,1)- or A,0)-forms, we have
and so
D*D = ((Dl'°)* + a*)(?>ll() + d)
= -iA(Dlfid - dD1'0),
since the B,0) part of D2 is 0. Now D2 is equal to its A,1) component,
namely Dl>°d + dD1'0, and so iAD2 = i\(D1'°B + dD1'0). Thus,
D*D = i\D2 - 2i\D1'°8

4. Stability 107
= i\D2 + 2д*д,
where we have used the Kahler identities again. Since AJD2 = F up to a
positive factor, we are done.
t In the Hermitian-Einstein case, up to a positive scalar, F = — щш{Е).
Thus, after replacing с„ by another suitable positive constant, we find, for
i/hplomorphic section s of E, that
D*Ds - СпЦш(Е)8 = 0.
tyring the inner product of this expression with s itself and integrating
pjyerAf gives
\\Ds\\2 - CnibiE) f \s\2 = 0.
Jm
в Ф 0 and Ци(Е) < 0, the only way that this expression can be
|;fa if Цш(Е) = 0 and Ds = 0, in other words * is a covariant constant
m of E. In this case, it follows that * does not vanish at any point of
id so defines an inclusion of Ом as a holomorphic subbundle. Let L'
le orthogonal complement to the corresponding C°° subbundle of E.
i L' is a C°° complex line bundle, and since Ds = 0 it is easy to see
$ D induces a connection on V. Thus, E = Ом © L' as C°° bundles
^ such a way that the connection D = ?>i Ф ?>2, where Dx is the trivial
tion. But then ?>2 defines a holomorphic structure on V. Finally,
Ы fiu(E) = цш(Ь'), as desired. ?
|liPur final result is the analytic proof of Bogomolov's inequality, which
ftmly state in the rank 2 case:
§#¦' •
эгеш 27. Let E be a. rank 2 holomorphic vector bundle on M and let
the & Hermitian-Einstein connection on E. Then jTwDc2(^) - с?(Е)) Л
> 0, or in other words JMpi(adE) Л ы""^< 0. Moreover, if
Awn~2 > 0 and сi(E) = 0, then D is a flat connection.
. In a local orthonormal frame, the curvature matrix D2 looks like
/ Xu) + ia ц> \
\ -<p Хш + г/З)'
where a,/?,<p are pointwise orthogonal to шп~1 and a,/? are real. Thus,
Dc2(?) - cf (E)) Л wn is represented by
[4A24/?4BAi( + /?)J] Aw",
where we have used a A wn~x = /? A wn-I = 0. Expanding this out, we are
left with

108 4. Stability
Again by looking at the analytic proof of the Hodge index theorem, we
see that -A/4тг2) [(a - /?J + 4tp А ф] A wn~2 is a pointwise nonnegative
multiple of the volume form on M, and thus JM(\с^{Е)-с{{Е))Ашп~2 > о.
If equality holds, we must have a = /? and <p — 0. If in addition Cy (E) = 0,
then Л = 0 and a + /? = 2a is a real A, l)-form pointwise orthogonal to
w" which is exact, since it represents cy(E). Thus, JMaAaAun~2 = 0.
It follows as in the discussion of Hermitian-Einstein metrics on line bundles
that a = 0. Thus, D2 = 0 and so D is fiat. ?
If /мDс2(.Е)-с?(E))Aun~2 = 0 but ci(E) ф 0, then E need not be flat.
For example, if ? is of the form E' ® L, where E' is flat and L is a holo-
morphic line bundle with Cy{L) ф 0, then fMDc2(E) - c((E)) A wn~2 = 0.
However, not all w-stable rank 2 bundles E satisfying JMDc2(E) — с{(Е))л
шп~2 = 0 are of the form ЕУ ® L, where E' is flat and L is a holomorphic
line bundle (see Exercise 3 in Chapter 6).
Exercises
1.	Let X be a scheme, proper over a field k, and let F be a coherent sheaf
on X. Then an injective map <p from T to itself is an isomorphism.
(Since Hom(.F, T) Is finite-dimensional, tp satsifies a polynomial equa-
equation, and since <p is injective we can assume that this polynomial has a
nonzero constant term. Clearly, tp satisfies such a polynomial equation
on each fiber TlmxT, which is a finite-dimensional vector space. Thus,
on each fiber tp is injective and therefore surjective, so it is surjective
by Nakayama's lemma.)
2.	Let V be a semistable rank 2 bundle of degree d on a curve С of genus g.
Show that, for every line bundle L on C, if degL >2g— [d/2] -1, then
H^VtoL) = 0, and if degL > 2g- [d/2], then V®L is generated by
its global sections. (If HX(V®L) ф 0, then H°(Vy ®L~l®Kc) фО,ао
that there is a nonzero map from V to a line bundle of degree [d/2] — 1.
Likewise, if degL>2g- [d/2], then Hl(V®L®Oc(p)) = 0 for every
peC.)
3.	Let С be a smooth curve of genus at least 2 and let e be a positive
integer. Let L be a line bundle of degree —e > —(g — l)/2. Show that,
for the generic extension
V is stable and there does not exist a nonzero map from a line bundle
L' to V unless L = V or degL' < degL. (Imitate the proof of Theorem
10 as follows: suppose that V is an extension corresponding to 4 €
H^L2). By Riemann-Roch, dimtf^L2) = 2e + g - 1. Thus, the set of
all extensions is parametrized by p2e+s-2 Suppose that L' is another
line bundle of degree -d, where d is allowed to be negative. If deg L' >
degL and L' ф L, then Hom(L',L) = 0, and so there must exist an
element of Hom(L', L~x) which lifts to an element of Hom(L', V). Now

4. Stability 109
I/ = L~l <8 Oc(—-D), where ?) is ал effective divisor of degree d + e.
The method of proof of Theorem 10 shows that ? is in the kernel of
;he map Hl{L2) ~* Hl{L2®Oc(D)). Show that this kernel is a linear
jpace of dimension d + e, and so we need ? not in the union, over all
jUvisors D of degree d + e, of a linear space of dimension d + e — 1 in
pae+s-2 Compare 2e + 2d-l with dim P2e+s-2 = 2e + g - 2 and use
i<e<(9~ l)/2- What happens for e = (g - 1)/2?)
If V is a rank 2 bundle on a surface X, use the splitting principle and
the Riemann-Roch theorem for vector bundles to show that
X(Hom(V, V)) = c?(V) - Ac2{V) + ix(Ox).
If V is a rank 2 stable bundle on P2, then cf (V) < 4c2(V). (As V
is simple, /i°(Hom(F,F)) = 1. By Serre duality, h2(Hom(V,V)) =
h°(Hom(V, V) ® /ifpa) = 0 since ATj» = CV(-3) С Орз- Now apply
the previous exercise.) Show that if V is a rank 2 strictly semistable
bundle on F, then cf (F) < 4c2(F).
Suppose that 0 < о < 6 < с are three integers. Show that there exist
Vector bundles V which fit into an exact sequence
о -¦ Op2 -»Opi(a) e Орз (ь) e cv (c) -»v -»o.
What is A;y in this case and when is V stable or semistable? Use this
to show that the tangent bundle of P2 is stable, via the Euler exact
sequence
2
о-»Opa -»0cv(i)
0
Let V be a rank 2 bundle on P2 with det V =
(a) Show that if V is given by an extension
0 -» OPi -» V -» Opi A) ® /z -+ 0,
hen V is stable if and only if Z ф 0.
b) Conversely, show that, if c<z(V) < 4 and V is stable, then V is
jiven by an exact sequence as in (a), with с > 1.
|c) When does a locally free extension as in (a) exist, and how many
u-e there?
[d) Show that there is a unique stable bundle W, up to isomorphism,
mch that det W = Opa(l) and c2(W) = 1. In particular, Gpa(-l)
and V(l), where V is the bundle of Exercise 6 of Chapter 2, are both
isomorphic to W.
(One approach is as follows: Writing W as an extension as in (a), W
is specified up to isomorphism by Z = {p}. So it suffices to show that
there exists one bundle W on P2, say W = вРа(-1), such that for
every p € P2, there exists a section of W vanishing at p. Now use the
description W = ф^/

110 4. Stability
For another approach, suppose that W and W are two isomorphic sta-
stable bundles with the correct Chern classes. Use the Riemann-Roch the-
theorem and the splitting principle to conclude that H°(Hom(W, W')) ^
0, and finish by Proposition 7.)
8.	Deduce (l)-C) of Proposition 21 from the last statement.
9.	Prove the analogue of Lemmas 2 and 3 for Ph,v, restricting yourself
to the case of vector bundles on a surface. Deduce that Lemma 6,
Proposition 7, and Corollary 8 hold for Gieseker stable bundles.
10.	In the notation of the example after the proof of Proposition 28 of
Chapter 2, show that Va,b is stable if о Ф 6,6 ± 1. (If /: Q -> P»
is the double cover map and Va,b = f*OQ(afi + bf2), let Op2(k) be
a sub-line bundle of Vab- Then there is a.nonzero map f*Op2(k) -»
OQ(afx+bf2).)
11.	Let V be a stable rank 2 bundle on P2 with a(V) = 0 and c2(V) = 2.
Then there is a unique conic С in P2 such that, if /: Q —> P2 is the
associated double cover, then V = f*C>QBf\ - f2). To see this, argue
as follows:
(a) For the existence of such a double cover, first note by the previous
exercise that for a fixed conic С and associated double cover f-Q—*
P2, the bundle Vc = f,OQBfi - f2) is stable. Next show that, if
g € AutP2 = PGLC,C), then g*Vc = Уд-цС)- Thus, it suffices to
show that PGLC, C) acts transitively on the set of stable bundles V
with Ci(V) = 0 and c2(V) = 2. To see this, note by B) of Proposition
14 that V is given by an exact sequence
0 - CW-1) -» V -* CVA) <g> Iz -* 0,
where ?(Z) = 3. First, we can assume that Z consists of three distinct
noncollinear points; the proof of this is deferred to (b). Now PGLC, C)
acts transitively on the set of such points. Finally, the connected com-
component of the stabilizer of a set of three noncollinear points in P2 is
C* x C*. Show that this group acts transitively on the set of locally
free extensions inside
here you will need to identify the term Oz more intrinsically as
det^Vz/P2 ® Cp2(~2), where Nz/v is the normal bundle to Z in P2.
(b) Show that we can find a map Орг(-1) —> V such that the cokernel
is Qpz{l) ® Iz, where Z consists of three distinct points (necessarily
not collinear), as follows: we have the exact sequence
0 -> Op2 -> V ® Op2(l) -> Op2B) ® Iz -> 0,
where СТрз B) ® Iz is generated by its global sections. Show more gen-
generally that, if W is a rank 2 bundle on a smooth surface X which fits
into an exact sequence
0 -> Ox -» W -* L ® Iz -> 0,

4. Stability 111
X® Iz is generated by its global sections, and the map H°(X; W) —*
H°(X;L® Iz) is surjective, then there is a nonzero Ox -* W such
that the cofcemel is of the form L®Iz> with Z' smooth. (This is a local
question: If locally the map is of the form 0 -* R —> R(BR -* Iz -* 0,
with 1 >-¦ (/, 9), (a, b) t-+ -ag+bf, and given tv, t2 € C, the hypotheses
say that we can find a section of W which has the local form (F, G),
where
F = h (fhi + l)+mi+f,G = t2(gh2 + 1) + m2 + S,
where hi, h2 are arbitrary and my, m2 e m-Iz- For generic small values
of ti and t2, F = G = 0 will consist of smooth points.)
(c) To see that С is uniquely associated to the bundle Vc, show that,
for a line l С P2, Vc\t = Ot © Ot if t is not tangent to C, and Vc\t =
Oe(l) © Ot(-l) ii ? is tangent to C. (If I is not tangent to C, then
Vc\? = (fe)*OpiA), where ft is the restriction of / to f~v{t) which is
a hyperplane section of Q с P3. Thus, Ci(Vfc|*) = 0, h°(Vc\t) = 2, and
h%Vc\l®Oe(-l)) = 0. If I is tangent to C, then F = /~l(?) = /iU/2,
Vhere /i and /2 are two fibers of different rulings of Q. Tensoring the
exact sequence
0 -» f*Op2(-?)-* OQ -» OF -» 0
by OqB/x — /2), there is an exact sequence
0 - 0qB/, - ЫГОМ-*) - OeB/i - /2) -» OfB/i - /2) - 0.
Applying /», it follows that Vc\t = /»CfiB/i -/2)- Now use the exact
sequence
У'	0 -» OFBfl - f2) -» Oh (-1) © ОЛB) -» Cp -» 0
to conclude that c^Vfcl^) = 0, /i°(Vb|^ ® O/(-l)) = 1 and
()) )
Slt'l Let V be a torsion free sheaf of rank r on the smooth protective variety
X. We define det V to be the sheaf (Дг V)v- Thus, det V is a reflexive
rank 1 sheaf on X and hence is a line bundle. Show that C\{V) = det V
and that the proof of Lemma 3 carries over to the case where V and
W are torsion free.
IS. Give a proof of Lemma 5 in general.
14. Let V be a torsion free sheaf of rank r on the smooth projective variety
X, and let Я be a fixed ample divisor. We shall always abbreviate ци
by /x. Show that the set
MW) : W is a subsheaf of V, 0 < rank W < r}
is bounded above, in the following steps. First show that there is a
filtration of V by subsheaves {0} = F° С F1 С • • • С FT = V such
that the successive quotients F'/F* are torsion free of rank 1, and
thus of the form Li®Iz, where L is a line bundle and Z is a subscheme

112 4. Stability
(possibly empty) of codimension at least 2. Now, if rank W = 1, show
that ci(W) = Li - [E] for some effective divisor E, and thus that
(i(W) < fi(Li) for some i. In the general case where rank W = k, find
a similar filtration for Д* V and compare ^(Д* W) with n(W) by the
splitting principle.
15.	Continuing Exercise 14, let Цо be the maximum value of (i(W), for IV
a subeheaf of V with 0 < rank W < r. Show that, if Wi and W2 are two
subeheaves of V with n{W{) = n{W2) = A«o, then n(Wi + W2) = no-
Conclude that there is a maximal W with n(W) — /io, that V/W is
torsion free, that every subeheaf W of W has n{W) < (J.{W), and
that every subsheaf W" of V/W has fi{W") < цо- Finally, argue by
induction that there is a canonical filtration (the Harder-Naiasimban
Sltr&tion) {0} = F° С F1 С • • • С Fk = V' such that F'/F' is
torsion free and semistable for every i and such that ^(jF*+1/F*) <
^(jpi/jri-i) for every j.
16.	Suppose that V is a semistable torsion free sheaf with p.(V) = /j.
Show that there is a filtration {0} = F° с F1 С • • • С Fk = V such
that F'/jF*-1 is torsion free and stable for every i and ^(F*/F*~1) =
p for all i. Such a filtration (which is not in general canonical) is
called a Jordan-Holder filtration of V. One can also show that the
associated graded sheaf ©<(F*/F*~1) is independent of the choice of
the filtration, i.e., is canonically associated to V.

>me Examples of Surfaces
Ettional ruled surfaces
chapter, we give a leisurely tour of some of the important classes of
certain blowups of P2, ruled surfaces, and КЗ surfaces. Our plan
[фё to begin with certain linear systems with assigned base points on P2
then to see where this study leads. Linear systems of plane curves of
degree with assigned base points have been extensively studied and
to a rich source of surfaces with interesting projective geometry. We
у to touch upon some of these examples where appropriate,
simplest example is that of lines. Let Я be a line in P2. The complete
system |#| gives the "embedding" P2 ^ P2. If we choose a base point
;JPai, then \H -p\ defines a base point free linear series on the blowup of
; p (Exercise 2 of Chapter 3). The fibers of the induced morpbism to P1
the proper transforms of the lines passing through p, and so all of the
i are = P1. The exceptional curve E meets each fiber H' transversally
i one point, and is thus a section with self-intersection — 1. Thus, we have
«fhibited the blowup of P2 at one point as the rational ruled surface Fi,
pul we will return to this example shortly. If we consider the linear system
МГ — p — q\, then on the two point blowup of P2 the linear system consists
of a single curve, necessarily a fixed component, and for general choices of
three points p, q, r the linear system \H — p — q — r\ is empty since there
are no lines passing through three general points. Thus, we shall have to
move up a degree, to consider linear systems of conies.
If we take the complete linear system of conies \2H\, we get an embedding
B + 2\
of P2 in a projective space of dimension I	1—1 = 5. The image of P2
in P5 is a surface of degree 4, the Veronese surface. Imposing one assigned
base point p gives a very ample linear system on the blowup Fi of P2 at p.
Here the base point free linear system on Fi corresponding to \2H — p\ is
\2ir*H - E\. The image of Fi in P4 has degree Bir*H - Ef = 3. Note that
Bn*H - E) ¦ (ж*Н — E) = 1, so the images of the proper transforms of
the lines through p are embedded in P3 as (disjoint) lines. For this reason

114 5. Some Examples of Surfaces
the image of Fx in P4 is called a (cubic) scroll. As Bтг*Я -E)-E = 1, the
image of E is also a line in P4.
Now suppose that we consider \2H - pi - рг|, where pi and рг are two
distinct points on P2. It is easy to verify directly that the corresponding
linear system on the blowup X of P2 at the points pi and P2 has no base
points and defines a morphism from X to P3, such that the images of the
exceptional curves Ei corresponding to Pi are (disjoint) lines. But if ? is the
line joining p\ and рг, then its proper transform on X is it*? -Ei-E2==?')
and Bтг*Я — Ei — E2) ¦ ?' = 0. Thus, ^' is contracted under the morphism
X-tP3. The image of X has degree 2 and thus is a smooth quadric
Q in P3. As is well known, a smooth quadric in P3 is isomorphic to the
surface P1 x P1 = Fo. The quadric Q has two distinct families of lines fa
and /2, with f2 = 0 and /i • /2 = 1, and there are two different morphisms
Q —> P1 corresponding to the two projections. In fact ft = ir*H — Ei, where
/1 + /2 — 2ж*Н - Ei- E2- Thus, X is the blowup of the quadric Q at a
point q, with exceptional divisor ?', and the proper transforms of the two
lines /1 and /2 passing through 9, namely ¦k*H—Ei-(-k*H-Ei—E2) = Щ
and n*H — E? — (k*H — Ei — Ei) = Ei, are exceptional curves. Blowing
these down gives back P2. Finally, the blowup of P2 at two points, or in
other words the blowup of Fi at a point not lying on the exceptional curve,
is isomorphic to the blowup of Fo at a point.
We may now define the rational ruled surfaces Fn, for n > 0, inductively:
we suppose by induction that we have constructed the surface Fn, with the
following properties:
1.	There exists a morphism тг: Fn —> P1 such that all fibers of тг are iso-
isomorphic to P1;
2.	There exists a curve а С Fn, the negative section, such that а ¦ f = 1
for every fiber / of тг and a2 = -n.
Then we construct Fn+1 as follows: blow up a point p G a. Let Fn be the
blowup_and let E the exceptional curve. There is an induced morphism
from Fn to P1. The proper transform of a is a new curve a' with (a'J =
—n — 1. Since f-a = 1, there is a unique fiber (which we shall again denote
by /) passing through p, which necessarily meets a transversally, and its
proper transform /' on the blowup is a curve disjoint from a' andjmch
that (fJ = -1. Clearly, E + f is a fiber of the morphism from Fn to
P1. Thus, if we contract /' via the Castelnuovo criterion we obtain a new
surface Fn+b together with a morphism Fn+i —> P1, all of whose fibers are
isomorphic to P1. Moreover, the image of a' on Fn+i is a curve satisfying
(a'J = —n - 1 and a' ¦ f = 1 for every fiber of the new morphism to
P1. This completes the inductive construction of the Fn. The procedure of
passing from Fn to Fn+1 is called an elementary transformation. Note that
if instead we blow up a point of Fn not on a and then contract the fiber
through this point, the result is Fn_i and we have performed the inverse
of the elementary modification which starts with Fn_x and yields Fn.

5. Some Examples of Surfaces 115
There are at least three reasons why the Fn are important:
They are exactly the (geometrically) ruled surfaces over a rational base
!'curve;
[They provide almost all of the examples of the nondegenerate surfaces
'iia'VN of smallest degree;
(Along with P2 itself, and excluding Fx which blows down to P2, they
'все a complete set of isomorphism classes of minimal models for P2.
#ore we discuss these properties more fully, let us describe the numerical
variants of PicFn and the nef and ample cones of Fn.
1. g(Fn) = Pm(Fn) = 0 for ali n > 0 and aiJ m > 0. furthermore
b
roof. These statements follow from Corollary 5 and Proposition 6 of
|M>ter 3, using the standard results for P2. D
Unma 2. Let f be a fiber of the map Fn -» P1 and Jet a be the negative
. Then PicFn й NumFn ? F2(Fn;Z) S Z • [/] © Z • a. Thus,
iroof. Clearly, PicFi = Z • [/] ® Z • a, and the general case follows by
mparing PicFn with PicFn+i. Since the intersection pairing on Z • [/] ф
fjij is nondegenerate, PicFn = NumFn. The statement about Я2, and
Й the calculation of C2(Fn), follows from the exponential sheaf sequence,
^is a standard comparison of Я2 of a blowup with Я2 of the original
(rface, which follows easily from the Mayer-Vietoris sequence and is left
the reader. D
Proposition 3. Let / and a denote the fiber and negative section of Fn.
4e following are equivalent:
ffi) The divisor аа + bf is nef;
Щ а > 0 and b > an;
fiii) \aa + bf\ is base point free;
(iy) \а<т + bf\ has no fixed components.
Proof, (i) implies (ii): If off + 6/ is nef, then (off + bf) ¦ f = а > 0 and
Ьаа + ь/)а- -an + b > 0.
v (ii) implies (iii): Write aa + bf = o(ff + nf) + (b~ an)f = o(ff + nf) + cf.
It suffices to show that a + nf is base point free and that c/ is base point
free provided that с > 0. The second statement is clear. To see the first,
suppose that к > п and consider the exact sequence
0 -> Ор„(kf) -> Ор„ (ff + kf) -* Oa(k - n) -> 0.

116 5. Some Examples of Surfaces
Claim 4. The induced map H°(Ofn (a + kf)) -> Н°(О„(к - n)) is onto.
Proof. The cokernel of this map lives in H1(Ofn(kf)). We claim that,
for all к > 0, we have ^(Ch^kf)) = 0. For к = 0 this is the statement
that g(Fn) = 0. The general statement follows by induction on k, using the
exact sequence
0 -» OfAkf) -» 0rn((* + 1)/) -» O/ -» 0,
and the associated long exact cohomology sequence. ?
Returning to the proof of Proposition 3, we see that, for the case к = n,
we can lift the nonzero section of Og to a section of H°(Ofn (a + nf)), and
thus \a + nf\ has no base points along a. Since it also contains the base
point free series \nf\, it has no base points elsewhere either.
(iii) implies (iv): Trivial,
(iv) implies (i): Trivial. ?
Lemma 5.
(i) aa + bf is effective if and only if a,b>0.
(ii) aa + bf is ample if and only if а > 0 and b > an.
Proof. Clearly, if a, b > 0, then aa + bf is effective. Conversely, if aa + bf
is effective, then using the fact that |/| and \a + nf\ are base point free,
we have a — (aa + bf) ¦ f > 0 and b = (aa + bf) ¦ (a + nf) > 0. If aa + bf
is ample, then a > 0 and (aa + bf) ¦ a = —an + b > 0. Conversely, if a > 0
and b > an, then by applying (i) we see that (aa + bf) • С > 0 for every
effective divisor, and moreover (aa + bfJ = 2ab — a2n = aBb — an) > 0.
So aa + bf is ample by the Nakai-Moishezon criterion. D
Of course, it is easy to give a direct argument that aa + bf is ample if
a > 0 and b > an, along the lines of the proof of Proposition 3.
The next result describes the morphism denned by a + kf for к > п.
We recall that a rational normal curve of degree n is the image of P1 in P™
under the embedding denned by the complete linear system associated to
Opi (n).
Proposition 6. If к > n, then the linear system \a + kf\ is very ample. In
this case, dim \a + kf\ = 2k — n + l, and iftp is the corresponding morphism
?n -» p2*-i+i; fjien ip(Fn) is a surface of degree 2k - n. Moreover:
(i) cp(f) is a line for every fiber f;
(ii) tp(a) = Ci is a rational normaJ curve of degree к -n inside a Pk~n с
2fc

5. Some Examples of Surfaces 117
l(ffi) For every choice of a smooth section a' 6 \a + nf\, necessarily disjoint
from a, tp{a') = Сг is a rational normal curve of degree к contained
, in a P* С P2fc-"+1, disjoint from the Pfc~n.
Moreover, tp(?n) is obtained as follows: start with Ci and Ci- For each
point p e Ci, there is a unique point q e Ci such that тг(р) = n(q), where
jr: Fn —* P1 iS the natural шар. Then ip(Fn) is the union over all p € C\ of
Ifce iines pq.
:ч For k = n, the morphism tp corresponding to \o + nf\ contracts a and
is the cone over a rational normal curve in P".
Proof. We leave these statements for the reader to work out, using Claim
4. ?
It is a standard fact that, if X С Р^ is an irreducible nondegenerate
surface (i.e., is not contained in a hyperplane), then degX > N — 1. In
feet, if degX = N — 1, then either X = (p(Vn) for some n and some tp
corresponding to \a + kf\, к > n, or X is the Veronese surface in P5.
General ruled surfaces
Definition 7. A smooth surface X is a (geometrically) ruled surface if
there exists a morphism тг: X —> С, where С is a smooth curve, such that
all fibers of ж are isomorphic to P1. The surface X is ruled if there exists a
morphism тг: X —* С such that at least one fiber of тг is isomorphic to P1.
Next we shall show that every ruled surface is obtained by blowing up a
geometrically ruled surface:
Lemma 8. If тг: X —> С is a ruJed surface, then the morphism тг factors
through a smooth biovedown X —> X, where X is a geometrically ruled
surface.
Proof. If / is an irreducible fiber of тг, then /2 = 0 and / = P1. Thus,
Kx ¦ f = -2. Now suppose that / = Si n«C« *s a reducible fiber n*t of тг
(here t is a point of С and as a reduced curve 7r~1(t) = [Ji d). Since /2
ind / • Kx are independent of the choice of /, we still have /2 = 0 and
Kx ¦ f = —2. By the connectedness theorem, |J^ C, is connected. Thus,
"or every j there exists а к such that Cj ¦ Ck > 0. It then follows from
) = Cj ¦ (Ei nid)that C? < ° for еуегУ 3- Moreover, Kx ¦ (Ei П*С) = ~2>
ю that there exists a j such that C? < 0 and Cj • Kx < 0. Thus, Cj is
sxceptional and we can contract it. Continuing in this way, we eventually
•each a stage where this is no longer possible, i.e., all fibers are irreducible.
\t this point we claim that the resulting surface X is geometrically ruled.

118 5. Some Examples of Surfaces
Indeed all fibers are irreducible, but there might exist a multiple fiber of n,
i.e., a point t eC and a positive integer m such that, as divisors, ir*t = rnC
with m > 1. However, (roCJ = /2 = 0 and Kx ¦ (mC) = -2. Thus,
C2 = 0 and Kx • С is even and divides -2. It follows that Kx ¦ С - -2
and ro = 1. ?
Note that if X is not minimal, the blowdown X -» X is never unique. In
fact, at the second to last stage we must have started with a geometrically
ruled surface and blown up a point p. The proper transform of the fiber
through p is then an exceptional curve and we may contract it. The resulting
birational map from X to a new geometrically ruled surface is called an
elementary transformation, just as for the rational ruled surfaces. If the
genus g(C) of the base curve is at least 1, then it is not too difficult to
show that every birational map from X to X', where X and X' are two
geometrically ruled surfaces over C, is obtained as a sequence of elementary
transformations. (See also Exercise 2 of this chapter and Exercise 9 of
Chapter 2.) In particular a ruled surface over С is birational to С х Р1,
which we can also see directly from Theorem 9 below. Finally, again if
g(C) > 1 and X is a (not necessarily geometrically) ruled surface over C,
then the morphism ¦n: X —> С exhibiting X as a ruled surface is unique.
Indeed, if тг': X —* С is another morphism to a smooth curve C" such
that a fiber /' of тг' is isomorpbic to P1, then the restriction of тг to /' is a
morphism from P1 to C. Since every morphism from P1 to С is constant, it
follows that the fibers of тг' are also fibers of тг and thus that тг = тг'. More
intrinsically n is given by the Albanese map of X (which we will define in
Chapter 10).
From now on, unless otherwise specified, we shall take ruled to mean
geometrically ruled. There are natural examples of geometrically ruled sur-
surfaces: take a rank 2 vector bundle V over the curve С and consider the
surface P(V) together with the obvious morphism тг: P(V) —¦ C. (Note
that our conventions are opposite to those of EGA [59] or [61], so that the
fiber of 7Г over x G С is the projective space associated to the fiber of V
over x. Hence, if 0P(V)A) is the tautological line bundle on P(V), then
7r»Op(v)(l) = Vv.) The morphism ж gives P(V) the structure of a geomet-
geometrically ruled surface over C, and the next result says that all geometrically
ruled surfaces arise in this way:
Theorem 9. Let i:X-»Cbea (geometrically) ruled surface.
(i) There exists a section a on X, i.e., an irreducibJe curve a such that
а ¦ f —I for every fiber / of тг.
(ii) X is isomorphic to P(V), where V is the dual of the rank 2 vector
bundle РРж*Ох(а)-
(iii) V(V) = V(V) if and only if there is a holomorphic line bundle L such
that V =* V ® L.

5. Some Examples of Surfaces 119
proof» All fibers of тг are smooth curves, in the sense of divisors. Thus,
г is a smooth morphism, and locally on X in the analytic topology тг is a
iroduct. In particular there exist local analytic sections of тг around every
>pint of С
.Next, let Г be a smooth complex curve, not necessarily compact, and
0t jr: Y -* T be a smooth and proper holomorphic map from the com-
ilex surface Y to T such that all fibers of тг are isomorphic to P1. Sup-
uee that a is a section of Y, in other words a curve in Y such that
\г<Г= 1 for all fibers / of тг, and consider the coherent sheaf Д°тг„ CV(<r).
Ince H°{f;OY(<T)\f) = Я^Р^Орф)) has dimension 2 for every fiber
J Л°7г»Оу(<т) is a holomorphic vector bundle of rank 2. If moreover
$*»Oy (&) is holomorphically trivial, choose two generating sections s0, Si
if ПРп*Оу(<г)- The pulled back sections of тг*Д°тг»С)у(<т) define an iso-
isomorphism Y -* Р(Д°тг.Оу(<т))у = T x P1. It is easy to check that the
SOmorphism Y —> Р(Д°тг,Оу(<т))у is canonical, i.e., does not depend on
|e choice of the sections. Thus, for arbitrary Y such that there exists a
ion a as above, there is a global isomorphism Y -* Р(й°7г»Оу(<т)у).
articular (ii) of the theorem follows from (i). Conversely, suppose that
(^), and let a be a meromorphic section of V. Then s defines a
a of X, so that (i) follows from (ii). We will show that every X is
(j;he form P(V), thereby proving (i) and (ii).
an open cover of C, say {Ua}, for which we can find holomorphic
fcal sections aa of тг for which Д°тг«Ог-1([/а)(<та) is trivial. Fix an isomor-
i/>a: flo7r*Cir-i(Ua)(<Ta)v —» Ofja. There is an induced isomorphism
|jj ^Ua) -*Uax P1. On the overlaps Ua П Щ, let Aa0 = фа о V^1 be
p2i><; 2 matrix with coefficients in Oc{UaC\Up) and let Д,^ be the induced
of PGLB, Oc(Ua ПUp)). Now it is easy to check directly from the
instruction that, over Ua П Up П Щ, АарАр^ = Aal, in other words, that
щ is a 1-cocycle for the cover {Ua} with values in the nonabelian sheaf
|(drLB, Oc)- On the other hand, it need not be the case that the lifted
Spchain Aap is a 1-cocycle for the sheaf GLB, Oc)- Indeed, all we can say
ithat, over UanUpr\U^, there exists an element /а/з7 € Oc(Ua
juqh that
Zllaim. The 2-cochain fap^ is a 2-cocycJe for {Ua} with values in 0c.
Proof of the Claim. We must show that, over Ua 0 Up П t/7 П Us,
»r equivalently that /р-уб1а^61ар\1арб = 1. By definition, it suffices to show
hat
Z^A^pKAapApsA-l) = Id.

120 5. Some Examples of Surfaces
First rewrite this expression as
and then use the fact that (Ap-,A-,sApg) lies >n tne center to further rewrite
it as
AaSA~gA~^(A^A^A^)A0SA~l = Id,
as claimed. D
Claim. H2{C;Oc) = 0.
Proof of the Claim. This follows from the exponential sheaf sequence
on С and the fact that H2 (C; Oc) = Я3 (C; Z) = 0. D
Thus, possibly after passing to a refinement of the cover {Ua}, fap^ is a
Cech coboundary of vap, say. In other words, fap^ = vapvp~,v~}j. Consider
the isomorphisms A'a/3: ОЪаПЩ -* О2, nUfj defined by A'a0 = v~lAap.
We leave to the reader the check that Aa0A'^ = A'ay. Thus, the Aap are
the transition functions of a rank 2 vector bundle V on C. Moreover, it is
easy to check from the construction that X ^P(V), compatibly with the
projection to C. For instance, this is clear over Ua since both ruled surfaces
are products over Ua, and over the intersections Ua П Up the transition
functions Aap and A'ap agree as elements of PGLB,Oc(Ua П Up)).
Finally, suppose that P(V) = P(V')- Let ? = OP{v)(i) be the tauto-
tautological line bundle over P(V) and let С = 0р(у/)A) the tautological line
bundle over P(V") = P(V). Thus, canonically тг.? = Vy and тг,?' = (V")v.
On the other hand, ?® {C)~l has degree 0 on the fibers, and so is trivial
since the fibers are all P1. By base change тг»(? ® (?')-1) = L ш a. line
bundle on C. Moreover, the natural map 7г*тг.(? ® (С')~х) -> С ® (C')~l
is an isomorphism, and hence ?® {C)~l = tt'L. Thus,
It follows that L ® (^')у ^ Ку, and thus V ^ V' ® L. D
The calculations of the above proof can be summarized as follows: there
is an exact sequence of sheaves
O-O^- GLB,Oc) -» PGLB,Oc) -» 0,
with Oc in the center of GLB, Oc)- Thus, there is an exact sequence (of
pointed sets)
HX{OC) -» Hl(GLB,Oc)) -* H1(PGLB,OC)) -* H2(OC) = 0.
Every Px-bundle over С defines a unique element in H1^; PGLB, Oc))-
By the exactness of the above sequence, this element can be lifted to an
element of #X(C; GLB, Oc)), corresponding to a rank 2 vector bundle over

5. Some Examples of Surfaces 121
C, and two such lifts differ by an element in H1(Oq), in other words the
two possible vector bundles differ by twisting by a line bundle on C-
In general, as is customary, we will call any curve a on a geometrically
ruled surface X such that a ¦ f = 1 a section. Thus, ж defines a bijection
between a and C, which is an isomorphism since С is smooth. We now
analyze the geometry of ruled surfaces in more detail.
•'Lemma 10. Let X be a geometrically ruled surface. There is a one-to-
#jne correspondence between sections a of X and rank 2 vector bundles
V such that X = P(V), together with a choice of a nowhere vanishing
section ofVv. More precisely, given a section a, the bundle V is defined by
Kv = lPn,Ox(cr). Moreover, with Vv = If^»Ox{a), we have Op(v)(l) =
Ox(ff)- There is an exact sequence
E.1)	0 -> Oc -» V -» L - 0,
щЬеге L is a line bundle on С with degL = a2, and hence c\{V) = — cr2.
Finally, ifr is another section of X, then there exists a unique line bundle
$qnC such that Ох(т) = Ox(a) ® тг*Л.
Proof. The correspondence between sections of X and quotients of Vv
is a general fact [61, II, 7.12]. In our case, we can see this follows: fix a
Section a and let Fv = Д°тг.С»х(^)- By the proof of Theorem 9, X —
P(V). Moreover, the surjection ir*ir»Ox{<r) = тг* Vv —> Ox{o) gives a dual
ilusion Ox{-<t) —> n*V which identifies Ox{-a) with OP(v)(-l) and
i Ox{a) with <9P(v)(l). Consider the exact sequence
;¦•¦	O-*0X -* Ox(<t)-> 0<,((т)-* 0.
Apply Д°тг» to this sequence. The direct image Rlv%0x = 0 since
№(f;Of) = 0, and Д°1г,Ох = Op. Thus, there is an exact sequence
j,	0 -> 0C -» V^V -> Oc(a) -» 0,
where we identify a with С via ж and thus О„(а) with a line bundle on C,
denoted by Oc(v) = L. Moreover, deg Oc{a) = a2-
To go the other way, suppose that V is a rank 2 bundle and that there
exists an exact sequence D.1). Then, for every p e C, we let ap be the point
in P(V^) defined by the vanishing of the linear form on Vp corresponding
to the image of a nonzero element of H°(Oc) in (VV)P. It is easy to check
that these are inverse constructions.
Now suppose that т is another section. As Ох{т — <т) is a line bun-
bundle whose restriction to each fiber / has degree 0, Д1тг»Ох(т - a) =
0, and ЕРж„Ох(т — о") is a line bundle Л on С such that the nat-
natural map тг*Д°7г»<!7х(т - <r) —* Ох(т - a) is an isomorphism. Thus,
Ox{r) = Ox{o) ® тг*Л. Moreover, A is determined by this property since
Л = Д°тг»7г*Л, by the projection formula, and so A = Д°тг»(!7х(т - ^)- D

122 5. Some Examples of Surfaces
Proposition 11. PicX = PicC® Z. More precisely, there is an exact
sequence
0 -> PicC -^ PicX -.Z-.0,
where the second map is given by L i-» deg(L|/), and the choice of a section
splits this sequence.
Thus, NumX =* Z • / Ф Z • a.
Proof. Clearly, the map L i-» deg(L|/) is a surjection from PicX to Z
which is split by the map 1 >-* Ox{a). We must identify the kernel with
PicC via 7Г*. If L e PicX and deg(L|/) = 0, by base change tt.L = Л is a
line bundle on С and the map ir*7r»L —* L is an isomorphism. Thus, every
L such that deg(L|/) = 0 is in тг'РкС, and the arguments of Lemma
10 show that ¦к*: Pic С -¦ Pic X is an isomorphism onto its image, with
inverse тг». ?
We next want to find a generalization of the negative section of Fn. For
a rank 2 bundle V over C, we could try to consider
max{deg L : there exists a nonzero map L —* V}.
However, this number changes when we replace V by К® Л for a line bundle
Л on С of nonzero degree, and instead we consider
e(V) = max{2 deg L — deg V : there exists a nonzero map L —* V}.
It is easy to see that e(V) < oo. For example, if there is an exact sequence
0 -» Li -» V -» L2 -* 0, then e(V) < |degZ/! - degL2|-
Proposition 12.
(i) e(V) = e(V ® Л) for all line bundles X on C.
(ii)e(V0 = e(Vv).
(iii) Suppose that V = L\ ® L2, with degLi = d\ > degZ/2 = <fe- Then
е(^) = di - d2 > 0.
(iv) — e(V) = min{<72 : a is a section of V}.
(v) V is stabie if and only if e < 0 and semistabJe if and only ife < 0.
Proof. Part (i) is clear, and (ii) follows from (i) and the fact that Vv =
V®det V. For (iii), in that notation it is clear that the maximum possible
degree of a line bundle L for which there is a nonzero map L —» V is di.
Thus, e(V) = 2dx - (dx + d^) = di - d^. For (iv), first suppose that there
is an exact sequence
Then 2 deg L - deg V = deg L. Thus, if L is chosen so that 2 deg L - deg V is
maximal, after replacing V by V ® Л for some line bundle Л we can assume

5. Some Examples of Surfaces 123
that V/L = Oc (note that for such L, V/L is necessarily torsion free). In
this case there is the dual exact sequence
a by Lemma 10 there is a section a of X with a2 — -degL = -e(V).
jjonversely, if a is a section, then by Lemma 10 there is a line bundle Л
end an exact sequence
L	0 -» Oc -» Vv <8> Л -» M -» 0,
^ritb deg M = <r2. Thus, e(V) = e(Vv ® A) > 0 - degM = -a2. It follows
that a2 > -e(V) for every possible choice of a. Thus, we obtain (iv).
Finally, (v) follows from the definitions. ?
ffflt follows that e(V) only depends on the ruled surface X = P(V), and
^ will also use the notation e(X). Note that e(V) > 0 if and only if V is
table, and in this case there is a unique maximal destabilizing sub-line
lie. There is thus a unique section of X of negative self-intersection.
$ also Exercise 1 for a generalization.
iple. Over P1, every rank 2 vector bundle is a direct sum of line
After twisting, we see that every ruled surface over P1 is of the
i P(Opi ® 0pi (n)). Clearly, this ruled surface is just Fn, and e(Fn) = n.
||iOver an elliptic curve C, every ruled surface is either V(Oc®L) for a line
He L with degL = d > 0, or P(?), where E is the nontrivial extension
fOc by Oc (Theorem 6 of Chapter 2) or ?(TP). The possibilities for e
in the first case, 0 in the second, and -1 in the last case.
J
if)'By the Segre-Nagata theorem, we always have the inequality e(V) > -g,
where g is the genus g{C). (See the remarks after Theorem 10 in Chapter
Щ For example, in case ci(V) = 0, there always exists a line subbundle L
|f V with degL > -(g - l)/2 and thus e(V) > -{g - 1).
.•'¦' Next let us determine the numerical invariants and the canonical bundle
Lemma 13. If X is a ruJed surface, then q(X) = g(C) and pg(X) = 0.
Proof. By base change R°ir*Ox = Oc and Я*тг,С>х = 0 for i > 0. Thus,
t;he Leray spectral sequence gives H1(X;Ox) = Hl{C;Oc), which has
dimension g(C), and H2{X\ Ox) = H2(C; Oc) = 0. П
Lemma 14. Let a be a section of X and write Оа{о) = -к*Ос{А)\а for
юте divisor d оя С. Then

124 5. Some Examples of Surfaces
Thus, if degd = d, then the numerical equivalence class of Kx is —2a +
Bg - 2 + d)f and Kx = -8(g - 1).
Proof. We may write Kx = aa + тг*Ь for an integer a and a divisor class
b on C, by Proposition 11. By adjunction Kx • / = о = -2. Next, applying
adjunction to a = C, we see that
giving (as divisor classes) Kc = -2d + b + d = b-d. Thus, b = Kc +d.
The remaining statements are then clear. ?
Another way to see Lemma 14 is as follows: Since ¦к is smooth, the
relative canonical line bundle Kx/c is an invariantly defined line bundle
on X satisfying Kx = Кх/с®к*Кс. On the other hand, there is the Euler
exact sequence
0 -» Ox -» **V ® Op(v)(l) - Tx/C - 0,
where TX/c = ^xjc ^ *^e relative tangent bundle. Thus, KX/c =
ir*(det V)~* ® Cp(v)(—2). In case a is a section of V, so that det V corre-
corresponds to the divisor Oa(—a) and 0p(v)(-2) = Ox(-^a), we recover the
statement of Lemma 14. (See also [61, III, ex. 8.4.])
Finally, let us determine the ample cone of a ruled surface X. Let /
be the numerical equivalence class of a fiber and let a be a section with
a2 = -e.
Proposition 15. If e > 0, then cur + bf is ample if and only if а > 0 and
b > ae. If e < 0, then aa + bf is ample if and only if а > 0 and b > \ae.
Proof. We shall just prove the statement when e > 0, referring to [61, V
2.21] for the case e < 0. If e > 0 and acr+bf is ample, then (aa + bf)f = a
and (aa+bf)a = b-ae. Thus, о > 0 and b > ae. Conversely, suppose that
a > Oandb > ae. Then (aa + bf)f = a> 0 and (aa + bf)-a = b-ae > 0.
Moreover, (aa + bfJ = — a2e + 2ab = oB6 — ae) > ab > 0. Now suppose
that С is an irreducible curve on X with С Ф f,a. Write С = па + mf.
Then / • С = n > 0 since / moves in a base point free system and С is not
contained in a fiber and a ¦ С = —ne + m > 0 since С фа. Thus,
(aa + bf) ¦ С — —one + am + bn> —ane + a(ne) + (ae)n = one > 0.
The Nakai-Moishezon criterion then implies that aa + bf is ample. D

5. Some Examples of Surfaces 125
Linear systems of cubics
yfe turn now to the study of linear systems of cubics in P2. While the study
jtf; linear systems of conies led us to general ruled surfaces, here we will be
Jed to elliptic surfaces and u КЗ-like" surfaces.
|y Let pi, • • • ,Pn € P2 be distinct points and let X be the blowup of P2 at
|he points Pi, with Et the exceptional divisors. By analogy with the case of
i we want to determine when |3ff - J2iPi\ induces an ample divisor
f
iy-
Theorem 16. Let X be the blowup of P2 at the distinct pointspi,... ,pn,
With n < 8, and Jet D = Зтг'Я - f^ Ei7 where п: X -* P2 is the blowup
дор. Then D is ampie if and only if no three of the points pi are coJiinear,
ПО six lie on a conic, and, ifn = 8, the pi are not ail contained in a plane
цЫс such that one of the pi is a singular point. In this case, D is very
!ifn<6.
We begin with the following lemma on distinct points pi,... ,pn e
Lemma 17. Suppose that the points pi,... ,pn are contained in an irre-
irreducible cubic Do, and that none of the points is a singular point of Do-
f Ь) Ifn < 8, the proper transform D of Do on the bJowup X of P2 at the
f> pi is nef and big, and D = -Kx-
|ii) If С is an irreducible curve on X with DC = 0, then C2 = -2 and
;i С is a smooth rational curve.
The linear system \D\ is base point free if n < 7, whereas there is a
unique base point of \D\ ifn = 8, and in this case \2D\ is base point
Г tee-
(iv) If n < 6, the morphism defined by \D\ is birational to its image and
J exactiy contracts the smooth rational curves С on X with C2 = -2.
Ifn= 7, the morphism defined by \D\ has degree 2 onto P2.
Proof. Note that, since none of the points pi is a singular point of Do,
D - Зтг'Я - Y,i Ei, where ж: X -> P2 is the blowup map. Thus, D2 =
9 - n > 1 and D is big. As K& = -ЗЯ, we see by Proposition 4 of
Chapter 3 that Kx = -Зтг'Я + YiEi = ~D- Since D is irreducible, it
follows that D ¦ С > 0 for every irreducible curve С Ф D and D ¦ D > 0
by the above, so that D is nef. If С is a curve such that С ¦ D — 0, then
by the Hodge index theorem C2 < 0. Moreover, as D = —Kx, we have
2po(C) - 2 = Kx ¦ С + С2 = С2 < 0. Thus, ра{С) = 0, С is rational, and
С2 = -2.
To determine the base points of |D|, since there is a section of Ox(D)
vanishing exactly along D, the base points of \D\, if any, are contained in

126 5. Some Examples of Surfaces
D. Consider the exact sequence
0 -» Ox - 0x{D) - 0х(Л)|Л -» 0.
Since X is regular, Яг(С>х) = 0 and so every section of H°(OX{D)\D)
lifts to a section of H°{OX{D)). The line bundle OX(D)\D is a line bundle
of degree D2 = 9 - n on the curve D. Moreover, pa(D) = 1, which follows
from the corresponding fact for Do or by adjunction for D: 2pa(D) — 2 =
Kx ¦ D + D2 = -D2 + D2 = 0. Now it is well known that for a smooth
elliptic curve D, a line bundle of degree at least 3 is very ample, a line
bundle of degree 2 has no base points and the corresponding morphism
maps D to P1 with degree 2, and that every line bundle of degree 1 is
of the form 0o(p) for a unique point рб Д which is thus a base point.
Similar results hold for an irreducible plane curve of arithmetic genus 1,
with essentially the same proofs. Thus, OX(D)\D has no base points if
D2 > 2, i.e., n < 7, and is birational to its image as long as D2 > 3, i.e.,
n < 6, and the same must hold for OX(D). Clearly, an irreducible curve С
contracted by \D\ satisfies С ¦ D = 0 and thus С is a smooth rational curve
with C2 = — 2. We see that we have proved all of the statements of the
lemma, except the one about ID in case n = 8. In this case, use instead
the sequence
0 -» OxiD) - OXBD) -» OXBD)\D -» 0.
The above sequence for OX(D) and the fact that H1{OX{D)\D) = 0 as
long as D2 > 1 shows that Hl{Ox{D)) = 0, and we can argue as before,
using the fact that OXBD)\D has no base points. ?
Note that the above lemma shows that eight points pi,•• • ,Ps lying on
an irreducible plane cubic, and not singular points of the cubic, determine
a ninth point pg lying on the cubic, and every other plane cubic passing
through pi,...,ps also passes through pg. In other words, Pi,• •.,Рэ have
the Cayley-Bacharach property with respect to Op»C) (Definition 14 in
Chapter 2).
Our next concern will be to find out when a set of at most eight points
{pi> • • • iPn} in P2 is contained in an irreducible cubic, and when there exist
smooth rational curves С on the blowup X with C2 = —2.
Lemma 18. Suppose that n < 9, and that no three of the pi are collinear
and no six lie on a conic. Then there exists an irreducible cubic containing
Pit • • • iPn- Ifn<7, we can assume in addition that the cubic is smooth.
Proof. The dimension of the linear system of cubics in P2 is 9. Thus, any
set of no more than nine points is contained, in a cubic. Given pi,... ,pn
with n < 9 with no three of the p* collinear and no six on a conic, we can
always find a pn+i which does not lie on any line through two of the p, or
any conic through five of the pj. Thus, p\,... ,pn+i also satisy: no three

5. Some Examples of Surfaces 127
%t the jh are collinear and no six lie on a conic. So we may assume that
'n = 9. Suppose that pi,. ¦ ¦, pg is contained in a reducible cubic, necessarily
the union of a line and a (possibly reducible) conic. At most two of the
points can lie on a given line. Thus, the remaining seven lie on a (possibly
jugular) conic, contradicting the hypothesis.
SfeNow the above argument implies that, for n < 8, the linear system
^mtaining pi,... ,pn has no fixed curve. The only possible singularities of
? general member of the linear system, by Bertini's theorem, are at the
base points. To prove the last statement, it suffices to find an element of
the linear system |3# — $^iP«l which is smooth at the p*. We may assume
that n = 7. Consider a line ? containing pi and рг and a conic с containing
the remaining five points. If с were reducible, then it would be a union of
two lines, and thus three of the p; would have to be collinear. Thus, с is
Irreducible and therefore smooth. The singular locus of ?+с is then exactly
jf П c. If Pi lies on a singular point of (. + c, then p,- 6 l Л с and so either I
contains three of the pi or с contains six, contrary to hypothesis. Thus, the
general cubic containing pi,..., pn is smooth at pi,..., pn. ?
!','!¦ '¦
• iNext, we must determine when there exist smooth rational curves С on
X with C2 = —2. Such a curve must be the proper transform of a plane
curve of degree d. Thus, С = dir*H + ][\ <цЕи with о, > О and
(dir*H + ]T aiEkf = d2 - 53 a2 = -2;
Solving for d — (%2i ut)/3, this becomes
By the Cauchy-Schwarz inequality, (]T\ o,J < r ?V <*?> where r is the num-
number of the o, which are nonzero. Thus, we must estimate |(r - 9)(?V a2).
First suppose that a, = 0 or 1 for all i. Then 3d = r and d1 = r — 2. Thus,
d2 - 3d + 2 = 0, so that (d - l)(d - 2) = 0 and d = 1 or 2. So if С is
effective, there is either a line or a conic containing all the points p; for
which о, = 1. The only way to have C2 — —2 in this case is for three of
the points to lie on a line or six on a conic.
In the remaining case, at least one of the сц is > 2. Thus,
A(r-9)(X>2)<i(r-9)(r
Direct inspection shows that this is always < —2 unless r = 8. So there is
no solution unless r = 8 and at least one of the o, is greater than 1. An
easy if somewhat tedious calculation rules out the possibility that r = 8

128 5. Some Examples of Surfaces
and two of the a, are greater than 1 or one of them is greater than 2. We
are left with 3d = 7 + 2 = 9, so that d = 3, and С is the proper transform
of a cubic passing through seven points with multiplicity 1 and a remaining
point with multiplicity 2. Thus, if the points Pi satisfy the hypotheses of
Theorem 16, then D is ample. Conversely, if the points fail to satisfy all
of the conditions of Theorem 16, then we can use the discussion above to
locate a curve С on X with D ¦ С < 0, so that D cannot be ample.
To finish the proof of Theorem 16, we note that, under the hypotheses
of Theorem 16, D2 > 0 and D ¦ С > 0 for every irreducible curve C, so that
D is ample by the Nakai-Moishezon criterion. To see that D is very ample
when n < 6, note that we have shown that \D\ defines a morphism from X
to P9"" which is birational to its image. One argument that |D| separates
points and tangent directions on X proceeds along the lines of Lemma
17; see [61, V.4.6] for details. Another argument, which can be generalized
to the case where D is not necessarily ample, goes as follows: since D is
ample, there exists а к > 0 such that kD is very ample. For this k, we claim
that the natural morphism Н°(Ор*(к)) -» H°(Ox(kD)) is surjective (here
d = 9 - n). It follows that the morphism X -> VN defined by kD factors
as follows: consider the morphism X -* P** defined by D, followed by the
Veronese embedding P* -» Vм defined by \Opa (fc)l- Then the image of X is
contained in a linear subspace PN С PM and this is exactly the embedding
of X in P^ via kD. Thus, the original morphism X —> P"* must have been
an embedding as well.
So we must show that H°(Opa(k)) -> H°(Ox(kD)) is surjective for all
к > 0. We argue by induction, the case к = 0 being clear. For к > 0, choose
a smooth curve in \D\, which we will also denote by D. Then the image
of D in P* is an elliptic normal curve in Pd~1, in other words the image
of the smooth elliptic curve D in P"* under a complete linear system of
degree d > 3. By [61, IV, Ex. 4.2], D is projectively normal, i.e., the map
H°(Opd-i(k)) -» Н°(Оо(к)) is surjective for every k. Now consider the
diagram with exact rows
°	°		>0
I	i	I
H»(Ox((k-l)D)) 	> H°(Ox(kD)) 	> H°(OD(k)).
The right-hand vertical arrow is surjective for all fc, and the left-hand ver-
vertical arrow is surjective by induction. Thus, the middle arrow is surjective
as well, and so D is very ample. D
Arguments similar to those above show that, in case n < 6 and D is
nef but contracts some curves С with C2 = -2, the image of X under the
linear system D is normal and has just rational double point singularities
(see also the remarks after Definition 24 of Chapter 1). The surfaces X
above with —Kx ample are called del Pezzo surfaces. We should also add

5. Some Examples of Surfaces 129
x P1 to this list, since -Kx = 2Д + 2/2, where the ft are the fibers of
! projections of P1 x P1 to the two factors. The del Pezzo surfaces have
! following property: if D = -Kx is very ample and embeds X in FN,
1 the lines on X с WN are exactly the exceptional curves on X. Indeed,
JJ.JJ is an exceptional curve, then E ¦ D = -E ¦ Kx = 1, so that E has
Cgree 1 as a subvariety of T?N and is thus a line. Conversely, if ? С X С ?N
1 a line, then ? • D = 1, so that ? ¦ Kx = —1 and ? is a smooth rational
в. Thus, ? is exceptional. In case n = 7 and D is ample, the morphism
by D is a finite morphism from X to P2 and the branch curve is a
'smooth quartic plane curve. In this case, we could also analyze when D is
nef but not ample in terms of the branch curve.
' i The most famous example of a del Pezzo surface is of course the cubic
! in P3. It is a fact which we shall not prove here (see [55], [61], etc.)
; every smooth cubic surface in P3 is in fact the image of the blowup of
1 at six points in general position as above. The lines on the cubic are: the
/6\
с disjoint exceptional curves, the proper transforms of the I I = 15 lines
two of the Pi, and the proper transforms of the ( ) = 6 conies
sing through five of the pt, for a grand total of 27. Similarly, we could
: out the total number of exceptional curves on any of the X with —Kx
ample. The configuration of the lines and the projective geometry of the
Jeurfaces X in ?N are a source of a vast amount of really intricate geometry.
^Chere is also a deep connection with sub-root systems of the root systems
sJBe, E7, Es- One reason is the following: for a del Pezzo surface X of degree
¦id, the lattice [Kx]x С H2(X; Z) is, up to sign, a root lattice of type Eg-d,
Щ least for d — 3,2,1. In fact, viewing X as the blowup of P2 at r = 9 - d
<points, let H be the pullback of the hyperplane class and let Eu..., ET be
Jasses of the exceptional curves. Then [Kx]x is spanned by the classes
- 2?2, • • •, Er-i - Er, H - Ei - E2 - Ez, which are elements of square
-2 whose associated Dynkin diagram is that of type Er. There are also
connections with the possible rational double point singularities which can
lie on a generalized del Pezzo surface, for which we refer to [20].
1 One reason for the importance of the del Pezzo surfaces is that they
are surfaces of "almost minimal" degree. If —Kx is very ample on X,
then it embeds X as a surface of degree 9 - n in p9-". Thus, the del
Pezzo surfaces are examples of surfaces of degree d in F1, for 3 < d < 9.
Conversely, it is a beautiful theorem of classical algebraic geometry that a
nondegenerate smooth surface of degree d in P^ is either a del Pezzo surface
or the embedding of P1 x P1 via |2Д + 2/2|. In particular it follows that
d < 9. The general idea of the proof is as follows. One first shows that if X
is a smooth surface in P* of degree d and d > 4, then X is an intersection
of quadrics in P1*—this follows by looking at the hyperplane sections, which
must be elliptic curves embedded by a complete linear system of degree d,
and using classical facts about them. From this it follows easily that there

130 5. Some Examples of Surfaces
can be only finitely many lines lying on X. Projecting X to Pd~1 from a
point of X not lying on any line, we obtain a new surface X' in Pd~1 of
degree d — 1. We can repeat this process until we reach a cubic surface in
P3. But each time we do so, we introduce a new line, disjoint from the ones
previously introduced by projection. Moreover, the cubic surface in P3 has
27 lines, but a maximal set of mutually disjoint lines can have at most six
elements. Thus, d < 3 + 6 = 9, and X is obtained from a cubic surface
by blowing down exceptional curves, from this it is easy to show that our
examples of such surfaces exhaust the possibilities.
Next let us determine the ample cone of a del Pezzo surface. More gen-
generally, we make the following definition [38]:
Definition 19. A rational surface X is a good generic surface if Kx = -D
for some irreducible curve D and there are no smooth rational curves С on
X with C2 = -2.
In fact, it follows from classification theory that a good generic surface
is automatically rational. (See also Exercise 3.) As we have seen, if D2 > 0,
a good generic surface is a del Pezzo surface. There exist good generic
surfaces for all possible choices of D2 < 0 as well; in fact, we leave it as an
exercise to show that the blowup of n generic points on Do, where Do is a
smooth cubic in P2, is a good generic surface for every n > 0. However, for
n > 9, we do not have an explicit description of what exactly the genericity
condition is.
Proposition 20. If X is a good generic surface and D is an irreducible
curve on X such that D = —Kx, then the irreducible curves of negative
self-intersection on X are exactly the exceptional curves and possibly D, if
D2<0.
Proof. If С is an irreducible curve with C2 < 0 and С Ф D, then CD > 0.
If С ¦ D > 0, then CKX = -C ¦ D < 0 and С is exceptional by Lemma 11
of Chapter 3. Otherwise, С ¦ Kx = 0 and C2 < 0, so that C2 = -2 and С
is a smooth rational curve. But we have assumed that no such curves exist
on A". ?
In particular, we see that if D2 > 0, the walls of the ample cone A(X)
are exactly the classes of the exceptional curves, whereas if D2 < 0, the
class D = —Kx defines an additional wall. Thus, for example, for the cubic
surface X, a divisor D is ample if and only if D ¦ ? > 0 for each of the 27
lines ? on X.
Next we shall give a characterization of the exceptional classes on X, in
case D2 > 0.

5. Some Examples of Surfaces 131
Proposition 21. Let X be a good generic surface with Kx > 0 and let F
be a divisor on X with F2 = F-Kx = — 1. Then there exists an exceptional
curve E with E = F as divisor classes.
Proof. Of course, the main point here is that F is just some divisor, not
necessarily the divisor associated to an irreducible curve, and we need to
show that F is effective. As before, let I? be an irreducible curve with
D = -Kx as divisor classes. First note that, since D2 > 0 by hypothesis,
D • С > 0 for every effective divisor C. Furthermore, if D ¦ С = 0, then C2 <
0, by the Hodge index theorem, and the case С Ф D, С2 < О is impossible
since then С would be a smooth rational curve of self-intersection -2. If
C2 = 0, then С and D must span a 1-dimensional lattice in Num X, by the
Hodge index theorem. Thus, С is a positive rational multiple of D.
By the Riemann-Roch formula,
h°(Ox(F)) + h2(Ox(F)) > \(F2 - F ¦ Kx) + 1 = 1.
jThus, either h°(Ox(F)) ф 0, so that F is effective, or h2(Ox(F)) Ф 0, so
that by Serre duality Kx - F = —D - F is effective. Suppose that —D - F
is effective. Now D ¦ (~D - F) = -D2 - D • F < -1, which is impossible.
Thus, F is effective. Moreover, D ¦ F = 1. If F = J^t riid, where the C» are
irreducible and rij > 0, then D • C< = 1 for exactly one C<, with гц = 1, and
D ¦ Cj = 0 for i Ф- j. By the remarks at the beginning of the proof, d = E
is irreducible and Cj is a positive rational multiple of D for all г Ф j. Thus,
there exists an irreducible curve E with Kx • E = — D ¦ E = — 1 such that
F = E if D2 > 0, and F = E + rD with r > 0 if D2 = 0. However, in this
last case we have F2 = -1 = E2 + 2r. Thus, E2 = -1 - 2r < 0, so that ?7
is an exceptional curve, E2 = -1 and r = 0. ?
Let us consider the case n = 9 above, i.e., D2 = 0. Then we have the
following example of Kodaira:
Proposition 22. If X is a good generic surface with Kx = 0, then choos-
choosing a fixed exceptional curve Eq, there is a bijection from the set of all
exceptional curves E on X to the lattice (Kx)x/Kx = Z8. In particular
there are infinitely many exception^ curves on X.
Proof. Given an exceptional curve E on X, map it to the image of E-Eo e
(Kx^/Kx, noting that Kx • E = Kx ¦ Eo and thus E - Eo 1 Kx.
Conversely, given a class e e (Кх)х/Кх, choose a lift of e to a class
F € Kj( and consider Eo + F. We are free to modify this by a multiple of
Kx, say mKx- Now, if a = Eo ¦ F, then
(Eo + F + mKxJ = -1 + 2a + 2m + F2,
where F2 is even since F ± Kx- Thus, for a unique choice of m we have
(Eo + F + mKxJ = -1- Since (Eo + F + mKx) • Kx = -1 as well,

132 5. Some Examples of Surfaces
Eo + F + mKx = E is the class of an exceptional curve, by Proposition
21, and we have constructed the desired bijection. ?
To find an example satisfying the hypotheses of Proposition 22, it follows
from Exercise 8 that the blowup of nine points in P2 in general position is
a good generic surface. In this case, there is a unique (smooth) plane cubic
passing through all nine. However, if the points are in somewhat special
position, a whole new phenomenon occurs, and we see our first examples
of elliptic surfaces, which we shall study more systematically in Chapter
7. Let ш note here, however, that we can indeed find a morphism to P1
from certain blowups of P2 at nine points. Let px,... ,pg be contained in
a smooth cubic and let D be the proper transform of the cubic. Consider
the exact sequence
0 _* Ox -» OX(D) -» OD(D) -» 0.
Since D2 = 0, Od(D) is a line bundle of degree 0 on D. If it is in fact trivial,
then since Hl(Ox) = 0 we can lift a nonvanisbing section to a section of
Ox{D). It follows that h°(Ox(D)) = 2 and that \D\ defines a morphism to
P1, one of whose fibers is D. Note that every exceptional curve E satisfies
E • D — 1 and so the exceptional curves E are sections of the map X —+ P1.
Conversely, every section E is isomorphic to Pl and E-Kx = — E-D = —1,
so that E is exceptional.
A similar construction works if Od(D) is a line bundle of finite order on
D. In this case, if m is its order, then \mD\ defines a morphism to P1 whose
general fiber is a smooth elliptic curve, and such that one fiber is mD. We
leave this as an exercise.
An introduction to КЗ surfaces
By definition а КЪ surface X is a surface such that Kx = 0 and q{X) =
0.	Thus, КЪ surfaces are one natural generalization of an elliptic curve.
The other possible generalization is an abelian surface or complex torus of
dimension 2. Here a complex torus of dimension 2 is a compact complex
surface of the form С2/Л, where Л = Z4 is a discrete subgroup of C2. The
torus is an abelian surface if in addition it is a smooth projective surface, or
equivalently carries a Hodge metric. An abelian surface X also has Kx = 0,
but q(X) = 2, and these are the only two classes of surfaces with Kx = 0,
or in other words with a nowhere vanishing holomorphic 2-form. Despite
the apparent similarities between complex tori and КЗ surfaces, it is the КЪ
surfaces which have received by far the lion's share of attention in surface
theory, partly because many statements which are elementary to prove for
abelian surfaces have very difficult proofs in the case of КЪ surfaces.
Here are some of the basic examples of КЪ surfaces:
1.	A smooth surface of degree 4 in P3;

5. Some Examples of Surfaces 133
I; A smooth complete intersection of a quadric and a cubic surface in P4;
I, A smooth complete intersection of three quadrics in P5;
[, A double cover of P2 branched along a smooth sextic curve;
i. (Kummer surfaces.) Let A be an abelian surface. Then there is an in-
involution t: A —> A defined by i(a) = —a. This involution has 16 fixed
points, the points of order 2 on A. The quotient Aft has 16 singular
l points, which are all ordinary double points. The minimal resolution
41 of the quotient is then а КЗ surface with 16 disjoint curves С with
'>) C2 = -2.
jince the canonical bundle of а КЗ surface X is trivial, pg(X) = 1 and in
ilct Pn(X) = 1 for all n, and moreover c^XJ = 0. By definition q(X) = 0.
rhus,x@x) = 2 and so c2(X) = 24. By the Hodge index theorem 6%(X) =
I. Since Kx is trivial and thus divisible by 2, the intersection form on Л' is of
jfype II, and in fact the intersection form on X is 1©/ф/ф(-.Е«)©(—Е&),
(rhere / is the hyperbolic plane and -Es is the unique unimodular negative
iefinite intersection form of rank 8 and Type II.
fft of the deeper properties of КЗ surfaces require looking at nonalge-
ff surfaces as well and studying the period map. For example, Kodaira
bowed that all КЗ surfaces fit into one 20-dimensional family of complex
lurfaces, and in particular they are all diffeomorphic. Thus, since by the
^e?schetz theorem on hyperplane sections a smooth surface in P3 is simply
Connected, all КЗ surfaces are simply connected. Using'the global Torelli
;heorem, one can show that, for every n > 1, the set of isomorphism classes
rf pairs (X, H), where X is а КЗ surface and H is a nef divisor on X with
p* = 2n which is primitive (not the multiple of another divisor), forms an
^reducible 19-dimensional family Tin- We have listed the first few cases
ibove.
,, The structure of ample and nef divisors on X is a topic for which we do
iot need the period map, and we want to say something about it here.
Lemma 23. Let X be а КЗ surface and Jet С be an irreducib/e curve on
X. Then OC(C) = шс. Thus, C2 > -2, C2 = -2 if and only if С is a
naooth rational curve, and C2 = 0 if and only ifpa(C) = 1. In al] other
cases, С is a nef and big divisor.
Proof. It follows from adjunction that Oc(C) = Uc, and thus C2 =
2pa(C) — 2. The other statements are clear. ?
By Lemma 23, the closure of the ample cone A(X) of X is exactly the
set of x in the positive cone such that x ¦ С > 0 for every smooth rational
curve С on X, or equivalently every curve of square —2. The set of walls
defined by such curves is locally finite, and the interior A(X) is exactly the
set of x in the positive cone of NumX with x ¦ С > 0 for all С irreducible
of square —2.

134 5. Some Examples of Surfaces
Lemma 24. If С is an irreducible curve on X with pa{C) > 1, then the
linear system \C\ is base point free.
Proof. Prom the exact sequence
0 -* Ox -> OX(C) -kjc -» 0,
and the fact that Hl{Ox) = 0, it follows that the map H°(OX(C)) -+
Н°(ыс) is surjective. By Proposition 6 of Chapter 1, uc has no base locus
if Pa(C) > 1. Thus, \C\ has no base locus either. D
By Bertini's theorem, if С is an irreducible curve on X with C2 > 0, then
the linear system \C\ contains smooth curves of genus equal to pa(C) >
2. So we may assume that С is itself smooth. Then Ox(C)\C = Kc.
Now either С is hyperelliptic or it is not. If it is not hyperelliptic, then
it is not difficult to show that show that the morphism <p defined by С
has degree 1 onto its image, and the image <p(X) is a normal surface of
degree 2g — 2 in P9, all of whose smooth hyperplane sections are curves
of genus g embedded by the canonical bundle (these are referred to as
canonical curves). The argument here is similar to the last part of the
proof of Theorem 16, but using the properties of canonical curves instead
of elliptic normal curves. The singularities of <p(X) correspond to the curves
D such that С • D = 0. By the Hodge index theorem, for such a curve D
we have D2 < 0, and thus D2 = -2 and D is smooth rational. Hence
<p(X) has just rational double point singularities (or no singularities, if С
is ample). Conversely, any surface in P9 with at worst rational double point
singularities, all of whose hyperplane sections are canonical curves, is а КЗ
surface. (See Exercise 12.)
Lemma 25. Let D be a divisor on X with D2 > -2. Then either D or
—D is effective.
Proof. Apply Riemann-Roch to OX(D): we have
h°(Ox(D))+h2(Ox(D)) > !?+x(Ox) > -1 + 2 > 1.
Thus, either h°(Ox(D)) ф 0, in which case D is effective, or h?(Ox(D)) Ф
0. By Serre duality, h2{Ox{D)) = h°(Ox(-D)) since Kx is trivial. Thus,
either D or — D is effective. ?
Lemma 26. If D is a nef and big divisor on the КЗ surface X, then
h°(Ox(D)) = 2 + D2/2.
Proof. Since D is nef, Я ¦ D > 0 for all ample divisors Я, and И ¦ D = 0
implies that D2 < 0, contradicting the assumption that D is big. Thus,

5. Some Examples of Surfaces 135
g. D > 0 and -D cannot be effective. Thus, h2(Ox(D)) = 0. Since D is
pef and big, hl(Ox(D)) = 0 as well by the Mumford vanishing theorem
(Theorem 26 of Chapter 1). So x(Ox(D)) = h°(Ox (D)), and by Riemann-
Roch, using the fact that Kx - 0, this is D2/2 + 2. ?
;> The following result of Mayer [88] gives a complete description of nef and
big divisors on X:
theorem 27. Let D be a nef and big divisor on the КЗ surface X. Then
,|jD| has a base point if and only if \D\ has a fixed1 curve if and only if
D = kE + R, where E is a smooth elliptic curve, R is a smooth rational
curve, R- E = 1, and к > 2. In this last case 2D has no base points. Thus,
every nef and big divisor on X is eventually base point free, and defines a
morphism from X onto a normal surface with only rational double point
angularities.
f. By Lemma 25, D is effective. Write D = Df + An, where Df is
'fixed curve of D and An = D — Df has no fixed curves. Clearly, by
;raction h°(Ox(Df)) = 1, h°(Ox(D)) = h°(Ox(Dm)), and Dm is nef.
Efasel. (А„J>0.
'By Lemma 26, h°{Ox(Dm)) = 2 + (DmJ/2 since Dm is nef and
t>jg. Likewise, h°((Ox(D)) = 2 + D2/2. From the assumption that
Ш°х(°)) = h°((Ox(Dm)) we see that D2m = D2 = (Df + AnJ =
[DjJ + 2Df ¦ An + (AnJ. Thus, (DfJ + 2Df Dm=0. Rewrite this as
Pf(Df+Dm)+DrDm = 0. As Df+Dm = D, wehave A/D+A/An = 0.
Since both D and An are nef, Df ¦ D > 0 and D/ • Dm > 0, and hence
DfD = DfDm = 0. Thus, Q = DrD = Dr (Df + Dm) = (A/J, since
A/An = 0. If Df Ф 0, then -Df is not effective, and so h2(Ox(Df)) = 0.
But then by Riemann-Roch h°(Ox(Df)) > 2 + (DfJ/2 = 2, contradicting
h°(Ox(Df)) = 1. It follows that D has no fixed curves. Using Exercise
11, the general element of \D\ is of the form ?^ А> where all the Di are
numerically equivalent irreducible curves. Since D is big, each A is an irre-
irreducible curve of positive square, and thus | A| is base point free by Lemma
24. It follows that \D\ is base point free as well.
Case II. (DroJ = 0.
Since A^ = 0, either the general element of An is irreducible, in which
case a general element of An equals E for a smooth elliptic curve E, or
An is composite with a pencil by Bertini's theorem [61, p. 280, Ex. 11.3].
In this last case An = kE again, where E is a smooth elliptic curve. If
An = E, then h°(Ox(D)) = h°(Ox(Dm)) = 2 + D2/2. From the exact

136 5. Some Examples of Surfaces
sequence
0-»Ox- OX(E) -+Oe-+0,
and using h}{Ox) = 0, there is an exact sequence
0 -» HX{OX{E)) -+ H\OE) -» H2{OX) - H2{<Dx(E)).
Since ? is effective and nonzero, h2(Ox{E)) = /i°(C>x(-?)) = 0. Thus,
H1(Oe) —* Д^(Ох) is surjective and so, since both spaces have dimension
1, an isomorphism. Hence Hl(Ox(^)) = 0 and so h°(Ox(Dm)) = 2 +
E?/2 = 2 = h°(Ox{D)). In this case D2 = 0 as well, contradicting the fact
that D was big.
So we may assume that к > 2. There exists a component R of Df such
that E ¦ R ф 0, since otherwise ? • D/ = 0 and so Dm ¦ Df = 0, and
thus Pj = (Dm + D/J = D2 > 0. But then by Riemann-Roch we would
have hr(Df) > 2, a contradiction. So we can find such an R, necessarily
a smooth rational curve with R2 = —2. Next we claim that E • R = 1: If
E ¦ R > 2, then (? + Д) • R > 0, so that E + Ria nef, and (? + RJ > 2.
Thus, E + Ria big, and h°(Ox(E + R)) = 2 + (E + RJ/2 > 2. It follows
that \E + R\ is strictly bigger than E, so that Я is not a fixed component
of \E + R\. But then Dm + R = (k - l)E + (E + R) does not have a fixed
component either, contradicting the choice of R С SuppE/. So E ¦ R = 1.
Now let Dx = kE + R and D2 = Df - R. Thus, Dt is still nef and D2 is
effective, and (kE + RJ = 2fc - 2 > 0, so that fc? + R is big. An argument
identical to the previous argument with the decomposition D = Dm + Df,
but applied to the decomposition D = Dx + D2, shows that if Д2 ф 0,
then DxDi = D% = 0. Thus, dim|P2| > 1 and |D2| С JZ^/J which has
dimension 0, a contradiction. It follows that D% — 0 and D = kE + R- We
leave it to the reader to check that h°(Ox(kE + R)) = к + 1, and thus
dim \kE + R\ = dim]kE\, so that Я is indeed a fixed curve of kE + R.
Finally, we must show that, if D = kE + R, then \2D\ has no base locus.
It suffices to show that \2D\ has no fixed components. The only possible
fixed component is R. But consider the exact sequence
0 -» OxBkE + R) -» OxBkE + 2Д) -> ORBkE + 2R) -* 0.
Since fc > 2, 2fc > 4, and 2kE+ R is nef and big. Thus, H1(OxBkE+R)) =
0 (we could also show this directly). So the map H°{OxBkE + 2R)) ->
H°{pRBkE + 2R)) is surjective. As OsBfcE + 2R) is a line bundle of
nonnegative degree on the smooth rational curve R, it has a nonzero section
which lifts to H°(OxBkE+2R)). Thus, R is not a fixed curve of }2D\. ?
Using arguments similar to the proof of Theorem 16, and in particular
the classical fact that a canonical curve is projectively normal, one can show
the following: if D is nef, big, and base point free, with D2 = 2g - 2 > 0,
then either every smooth curve in \D\ is nonhyperelliptic, and the morphism
from X to P9 is birational to its image, which is a normal surface with at

5. Some Examples of Surfaces 137
worst rational double point singularities, or every smooth curve in \D\ is
kyperelliptic and the image of X is a surface of minimal degree g — 1 in P9,
¦до] is thus a rational normal scroll (the image of Fn under a linear system
bf the form \a + kf\, k>n, от the Veronese surface in P5).
Ibcercises
.1. Show that, for n > 0, the negative section a on Fn is the unique
' irreducible curve with negative self-intersection. Likewise, for X a ge-
geometrically ruled surface with e(X) > 0, there is a unique irreducible
curve with negative self-intersection.
•i. Show that elementary transformations of a ruled surface X = P(V)
in the sense of this chapter correspond to elementary modifications of
the vector bundle V over C, in the sense of Chapter 2.
!8.' For which ruled surfaces X is — Kx effective? Show that, if A" is a
1 ruled surface such that Kx = —D, where D is irreducible, then X is
a rational ruled surface. (Show that pa(D) = 1, so that either X is
rational or X is ruled over an elliptic curve C, D is smooth, and the
morphism D-»Cisa covering space. Thus, #J(C>x) — #J(Oc) =
H^Od)- Use the long exact sequence associated to
0 _> Ox(-D) -»Ох-»Од-»0
to show that ^(Oxi-D)) = Hl(Kx) ~ 0, and derive a contradic-
contradiction.)
(|4* Let a and a' be two sections of a (geometrically) ruled surface. Show
that a2 = (<т'J mod 2. Show also that, for every rank 2 vector bundle
V over the curve C, we have ci(<9P(v)(l)J = —Ci(V). (We have seen
this for the case where V — ii°ir»Ox(^)v- Reduce the general case to
this case.)
; 6. Let A be an abelian surface, i.e., a complex torus of dimension 2.
Thus, Ka — 0. Show that there is no effective curve С on Л with
C2 < 0. Show moreover that if С is irreducible and C2 > 0, then С is
ample. More generally, use the Riemann-Roch theorem to show that
if D is a divisor on X with D2 > 0, then either D or — D is effective
and in addition ample. What can you say about the case C2 = 0, С
irreducible?
6. Let X — Sym2 E, the second symmetric product of an elliptic curve.
Thus, X is the quotient of the abelian surface E x E by the involution
(a, b) н-> (b, a). Using the group law, there is a natural map X —> E
defined by (a, b) i—> a + b. Show that the fibers of this map are P1 so
that X is geometrically ruled over E. Which ruled surface over E is
it (bearing in mind our classification in the example after the proof of
Proposition 12)? (Here is one approach. Note that the curves E x {p},
for p e E, give sections ap of X. Also, a2 = av ¦ aq = 1 for all p,q.
Now use the last two exercises.)

138 5. Some Examples of Surfaces
7.	Let ir: X —» С be a nonminimal ruled surface. Let t e С and let
i:~1(t) = Ui=iCi- Le* I c {l.--.^} be a proper set of indices
such that \Ji€l Ci is connected. Show that the lattice spanned by the
classes of the Ci, i € /, is negative definite and that \Jiej Ci con-
contracts to a rational singularity. (Note that, for a general fiber / of ir,
Hl(nf; Onf) = 0 for every n > 0 and thus the same must hold for all
fibers.)
Can you locate a ?4 rational double point configuration (the dual
graph is Z?4 and all curves have self-intersection —2) this way?
8.	Let X be a blowup of P2 at distinct points pi,..., pn lying on a smooth
cubic Do- Show that if there exists a smooth curve С disjoint from the
proper transform D of Do on X, then there is a nontrivial relation
3noh + Yji niPi = 0 in the divisor class group of Do, where the щ
are integers and h is the divisor class of degree three on Do which
corresponds to the restriction of ОрзA) to Do. Deduce that for general
choices of the p< there is no such curve С
9.	Let X be the blowup of P2 at nine points lying on a smooth cubic, and
let D be the proper transform of the cubic. Suppose that Ox{D)\D has
order exactly m. Show that \nD\ consists of one point for 0 < n < m
and that \mD\ defines a morphism to P1 whose general fiber is a
smooth elliptic curve and such that mD is a fiber.
10.	There exist elliptic surfaces which are the blowup of P2 at nine points
which are the base locus of a pencil of cubics, such that all fibers are
irreducible and such that the generic fiber has only a single ordinary
double point, i.e., is a nodal cubic. Show at least that there exist such
surfaces with all fibers irreducible as follows. The space of all cubics is
a projective space of dimension 9. What is the dimension of the space
of reducible cubics? Thus, show that a general line in the space of all
cubics misses the set of reducible ones.
We can show moreover that the general irreducible singular cubic С
is nodal as follows. If С is not nodal, then as pa(C) = 1, С has a
cusp. Show that all cuspidal cubics are protectively equivalent (since
all rational curves with a cusp are isomorphic and the automorphism
group acts transitively on the set of degree three line bundles on C)
and that the cubic y2z — x3 has a 1-parameter family of projective
automorphisms. The upshot is that the dimension of the space of cus-
cuspidal plane cubic curves is 7 = dimFGLC)-l, whereas the dimension
of the space of nodal plane cubic curves is 8 = dim PGLC). Thus, the
general singular cubic is a nodal curve.
Show that if X is a good generic surface and X —* X' is the contrac-
contraction of an exceptional curve, then X' is again a good generic surface.
Thus, again we see directly that good generic surfaces exist which are
blowups of P2 at n < 9 points.
11.	Let |D| be a linear system without fixed curves on an algebraic surface
X. Then the general element of \D\ is of the form J2i &г> where all

5. Some Examples of Surfaces 139
of the Di are numerically equivalent. (If \D\ has no base locus and
the general element of \D\ is reducible, then by [61, p. 280, Ex. 11.3],
|D| is composite with a pencil, i.e., the corresponding morphism to
P" has image a curve Co. If X -» С —* Co is the Stein factorization,
then each Di is a fiber oi X —* С and thus they are all numerically
equivalent. If the general element of \D\ is reducible but connected, it
must have a singular point. Such a point must be in the base locus,
by Bertini's theorem. After blowing up the base points of a general
pencil, we obtain a linear system without base points whose general
element is not connected, and we can reduce this case to the previous
case.)
Let X с Р9 be a smooth surface such that all hyperplane sections are
canonical curves. Then X is а КЗ surface. (Note that Kx • # = 0 for
every hyperplane section H. Choose a general pencil in \H\, such that
every member is irreducible, and blow up the base locus which consists
of distinct points p<. Let p: X —* X Ъъ the blowup, with exceptional
curves Ei, and let -к: X —» P1 be the morphism denned by the pencil.
Let С be a smooth fiber of ж. From
Kc = Kx ® OX(C)\C = КЯ\С = p*Kx ® Ox (?Ek)\C,
i
conclude that p*Kx\C is trivial for all smooth C. For every fiber C,
smooth or not, use semicontinuity and the fact that p*Kx\C and
(p'KxlC)^1 have sections for general С to conclude that p*Kx\C
is trivial. Thus, p*Kx = ж*ж*р*Кх must be pulled back from P1, so
is Ox(nC) for some n. Using p*Kx ¦ Ei = 0, conclude that n = 0 and
thus that Kx is trivial. Finally, argue that Hl(Ox) = 0 by using the
vanishing of Hl(Ox{H)) for an ample divisor H.)

6
Vector Bundles over Ruled Surfaces
Suitable ample divisors
In this chapter and in Chapter 8 we will discuss the structure of stable bun-
bundles on special classes of surfaces, ruled and elliptic surfaces. In general, it
seems a little difficult to obtain detailed information about moduli spaces
when stability is defined with respect to an arbitrary ample divisor. We
will need instead to consider ample divisors adapted to the geometry of the
surfaces at hand. The examples we have in mind are all given as fibrations
over a base curve, and the fibration itself will be the most interesting geo-
geometric feature of the surface..Thus, we will try to consider ample divisors
which reflect the fibration. The class of a fiber is not ample; it lives at the
boundary of the ample cone. Instead we shall consider ample divisors which
are sufficiently close to the class of a fiber, where how close will depend on
the particular choice of Chern classes of the problem we want to study. We
will further study bundles on the surface X by studying their restrictions
to the fibers of the fibration. This method of studying bundles on a surface
by looking at their restrictions to curves on the surface is one which has
been successfully applied in a wide variety of contexts.
We begin by discussing the general strategy for analyzing the meaning
of stability for a fibration X —» C, and then specialize to the case of ruled
surfaces. After an expository section describing the global and local struc-
structure of moduli spaces, we will find a Zariski open and dense subset of the
moduli space of bundles over a ruled surface. In the exercises, we shall
sketch how these ideas may be applied to prove that the moduli space of
rank 2 bundles over P2 is irreducible.
Fix the following notation for this section: X is a surface and 7r: X —+ С
is a morphism from X to the smooth curve С We denote by / the numerical
equivalence class of a general fiber. Next we recall the following terminology
from Chapter 4: For a rank 2 vector bundle V, we shall denote the mod
2 reduction of ci(V) by w(V) = w and the integer ci(VJ - 4c2(V) by
pi(adV) = p. If w is a numerical equivalence class mod 2 and p is an
integer, a class of type (w,p) is a class С such that p < С2 < О and such that

142 6. Vector Bundles over Ruled Surfaces
the mod 2 reduction of ? equals w. A wall of type (w, p) is the intersection
W^ of the hyperplane Q1- with A(X). The chambers of type (w,p) are the
connected components of the complement in A{X) of the set of walls of
type (w,p).
Definition 1. Let Д be a divisor on X, let с be an integer, and set w =
Д mod 2 and p = Д2 — Ac. An ample divisor Я is (tu, p)-suitable or (Д,с)-
suitable if Я does not lie on a wall of type (w,p) and, for all С of type
(tu,p) such that ? ¦ / Ф 0, we have sign/ ¦ ? = sign Я • С-
Clearly, Я is (iw,p)-suitable if and only if Я is not separated from / by
any wall, if and only if / lies in the closure of the chamber containing Я.
It is easy to see that, if H\ and Hi are both (u>,p)-suitable, then the only
walls of type (w,p) separating H\ and Hi must contain / in their closure.
If Я is (tu,p)-suitable, then clearly Я + tf € NumX®Q is (u>,p)-suitable
for all t 6 Q+, and by Proposition 22 of Chapter 4, a rank 2 bundle V is Я-
semistable if and only if it is H + t/-semistable. Finally, note the following
useful lemma:
Lemma 2. If H is an ample divisor lying on no wall of type (w,p) and V
is a strictly H-semistable bundle with ci(V) = Д and pi(adV) = p, then
there exists an exact sequence
0 -¦ OX(D) -* V -* OX{A - D) ® Iz -* 0
with 2D — Д numerically equivalent to zero.
Proof. By Proposition 21 of Chapter 4, there exists such an exact se-
sequence, with Ox(D) a destabilizing sub-line bundle. Thus, H-BD—Д) = 0.
By the Hodge index theorem BD — ДJ < 0, with equality if and only if
2D — Д is numerically trivial. On the other hand,
p = pi(adV) = Д2 - W ¦ (Д - D) - Щг) = BD - ДJ - U{Z),
so that BD - ДJ > p. Thus, either 2D - Д defines a wall of type (w,p)
or 2D — Д is numerically equivalent to zero. The first case cannot arise
since we have assumed that Я lies on no wall of type (w,p). So 2D — Д is
numerically equivalent to zero. ?
Lemma 3. For every w and p, {w,p)-suitable ample divisors exist.
Proof. Let Щ be an ample divisor. We may assume that Hq lies on no
wall of type (w,p). For n > 0, let Я„ = Щ <S Os(nf). It follows from
the Nakai-Moishezon criterion that Я„ is ample as well. We claim that if
n > —p(Ho ¦ /)/2, then Я„ is (iu,p)-suitable.
To see this, let С = 2F — Д be a class of type (w,p) with p < ?2 < 0 and
? ¦ / ф 0. We may assume that о = ? • / > 0, and must show that С • Hn > 0

6. Vector Bundles over Ruled Surfaces 143
as well. The class аЩ — (Яо ¦ /)C is perpendicular to /. Since f2 = 0, we
may apply the Hodge index theorem to conclude:
0 > (aH0 - (Ho ¦ /)CJ = a2Hl - 2a(H0 • f)(H0 ¦ C) + (Яо • /JB.
Using the fact that C2 > A2 - 4c = p, we find that
Thus,
Thus, Н„ ¦ С > 0, and to see that Hn is (w,p)-suitable, it will suffice to
show that Hn does not lie on a wall of type (w,p). We have just seen above
that, if С is a wall of type (w,p) such that / • С Ф 0, then Я„ ¦ С ф 0 as well.
On the other hand, if С is a wall of type (w,p) such that / • С = 0, then
Яп • ? = Яо ¦ С Ф 0 by assumption. In all cases, Я„ does not lie on a wall
of type (w,p), and so it is (u>,p)-suitable. ?
For example, suppose that X is a geometrically ruled surface with section
и and fiber /, and that the invariant of X is e. Since stability is unchanged
by taking positive multiples and only depends on the numerical equivalence
class of a divisor, we may as well work with elements in Num X ® Q. Thus,
: using the description of the ample cone in the last chapter, we may assume
that H — a + rf, where г e Q satisfies г > e if e > 0 and г > e/2
'. if e < 0. Suppose that d is a divisor class on С of degree d and that с
is a positive integer, and consider rank 2 vector bundles V over X with
,ci(V) = ?r*(d) and Сг(У) = с. (If с < 0, then there are no walls of type
\l(w,p).) Thus, w = df mod 2 and p = —4c. Although we shall not need the
; precise description of suitability, we include it for the reader's edification:
is
.Lemma 4. In the above notation, suppose that e + c = d mod 2. Then
;<r + rf is (w,p)-suitable if and only if r > max{e,(e+ c)/2} if e > 0 and
ir> (e + c)/2 He < 0.
|proof. The condition г > e if e > 0 is simply to insure that а + rf is
uple, which is automatic in case e < 0. Note that, if (o = 2<r + (e — c)/,
(o = —4e + 4(e — c) = —4c and ?o s d/ mod 2, so that Co is a class of
pe(w,p). Moreover, Co-/= 2 > 0 and Co• (<r + rf) = -2e+(e-c) + 2r =
fir - e - с Thus, if
sign/ • Co = sign(<7 + rf) ¦ Co,
[then г > (e + c)/2.

144 6. Vector Bundles over Ruled Surfaces
Conversely, suppose that r > (e + c)/2 and let С = 2ocr + (d + 2b) f be a
wall of type (w,p). We may assume that ? • / = a > 0. Now
-4c < C2 = -4o2e + 4a(d + 26) < 0,
and so, since a > 0, we have 0 < a(ae — (d + 2b)) < с Thus,
(a + rf) ¦ С = -2ae + (d + 2b) + 2or
= 2a(r - e) + (d + 2b) > 2o(c - e)/2 + (d + 26)
= ac - ae + (d + 26) >ac- c/a = c(a - I/a) > 0.
Thus, sign/-C = sign(<r + r/) ¦ ?. ?
Next we discuss the geometric meaning of suitability:
Theorem 5. Let тг: X —> С be a morphism from X to the smooth curve
С For given ш and p, iet H be a (w,p)-suitable divisor on X. Let V be a
rank 2 vector bundle on X with «^(V) = w and pi(ad V) = p.
(i) If У is H-semistable, then its restriction to almost all fibers / is
semistabJe.
(ii) If there exists a smooth fiber / such that the restriction of V to f is.
stable, then V is H-stable.	i
(iii) If there exists a smooth fiber f such that the restriction of V to f is:
semistable and V is not H-stable, then exactly one of the following,
holds:
(a)	There exists a sub-line bundle OX(D) such that / ¦ BD - Д) = 0,
and 2D - Д is a wall passing through f such that H ¦ {2D - Д) > 0.(;
In this case V is H-unstable.	J-
(b)	There exists a sub-line bundle OX(D) such that f ¦ BD - Д) = 0^
and 2D — A is numerically equivalent to a rationaJ multiple rf off,{
with r > 0. Moreover, V is H-semistable if r = 0 and H-unstable ifi
r>0.	|
Part of the proof. We begin with the easy part, the proofs of the secondl
and third statements. We shall prove the first statement in Chapter Щ
Suppose that there exists an / with V\f semistable. If У is not if-stable»
then there exists a divisor D on X such that OX(D) is a sub-line bundle^
of V and such that fi(L) > n(V). In other words, Я ¦ BD - Д) > 0. Wef
claim that / ¦ {2D - Д) > 0 as well. As we may assume that V/OX{D) is
torsion free, there is the usual exact sequence
0 -+ OX{D) -> V -> OX(A -
and thus as we saw in the proof of Lemma 2, B.О-ДJ > p. If BИ-ДJ>|
0, then since Я • BD - Д) > 0, it follows from Lemma 19 of Chapter;]
that / ¦ BD - Д) > 0 as well. Since V\f is semistable, / • BD - Д) = 0Ц

6. Vector Bundles over Ruled Surfaces 145
and we see that V\f is strictly semistable. In particular, if conversely V\f
is stable, then V is Я-stable. Now suppose that V is not Я-stable and that
V\f is semistable, so that we are in Case (iii). By the Hodge index theorem
BD—ДJ < 0, with equality if and only if ID—A is numerically equivalent
to a rational multiple of /. If BD- ДJ < 0, 2D - Д is a wall of type (ш,р).
By the definition of suitability, Я • BD - Д) / 0 and Я • BD - Д) > 0,
and so we are in Case (a) of (iii) in Theorem 5.
In the remaining case, BD — ДJ = 0, so that 2D — Д is numerically
equivalent to a rational multiple rf of /. If r = 0, then V is Я-semistable
by Lemma 6 of Chapter 4, and V is clearly unstable if г > 0. Finally,
Proposition 20 in Chapter 2, which says that the maximal destabilizing
sub-line bundle of V is essentially unique, shows that Cases (a) and (b) are
mutually exclusive.
The proof of Part (i) is harder, and I don't know a proof which does not
involve descent theory or something equivalent. We will discuss the main
idea in the general case and then give the proof in case the fibers have genus
0 or 1. Let us start with the general case. Suppose that V\f is unstable for
infinitely many fibers /. General properties of stability imply that in fact
• V\f is unstable for all smooth /. Thus, for every t such that ft = т^) is
smooth, there is a unique maximal destabilizing sub-line bundle Lt of V\ft.
(Here the only reason we restrict ourselves to smooth fibers is because we
have not bothered as yet to try and define stability for a singular curve.)
What we would like to say is the following:
Claim. There exists a sub-line bundle Ox (D)ofVonX whose restriction
to ft is Lt for a/most all t.
. Assuming the claim, let us finish the argument. We may assume that
Y/0x{D) is torsion free. Thus, we have the usual exact sequence
V	0-чОх{О)-> V -+OX{A-D) ®/z->0,
,and by a familiar calculation BD - ДJ > p. Moreover, since Ox(D)\f is
destabilizing, deg(BI> - Д)|/) = / • BD - Д) > 0. If BD - ДJ > 0, then
iy the Hodge index theorem Я • BD - Д) > 0 as well, and Я • BD - A) = 0
inly if 2D — A is numerically equivalent to 0, contradicting the fact that
?• BD - Д) > 0. Thus, Я • BD - Д) > 0, contradicting the assumption
inat V is Я-semistable.
} The remaining case is B?> - ДJ < 0, so that 2D - A is a wall of type
\ii,p). By the definition of suitability, Я • BD — A) > 0 in this case as well,
fence in all cases V is Я-unstable, contradicting the assumption on V.
|;How do we find 0x(Z?)? One method is via base change and descent
heory. However, in the case of ruled and elliptic surfaces we can find Ox (D)
iirectly, and this is what we shall proceed to do.

146 6. Vector Bundles over Ruled Surfaces
First assume that all of the fibers have genus 0, so that X is geometrically
ruled and there exists a section a of the ruling. In this case, we are allowed
to twist by Ox(kcr) for any k, so that we may assume that V\f has degree
either 0 or 1 for all /.
Lemma 6. Suppose that V is a rank 2 vector bundle on the ruled sur-
surface X with deg(V\f) = 0. Then there is a nonnegative integer a and a
nonempty Zariski open subset UofC such that V\f = Opi(a) © CPi(—a)
for all Gbers f lying over a point of U. A similar conclusion holds if
deg(V|/) = 1, in which case V\f = OPi(o) e Ori{l - a) for all fibers
/ lying over a point of U.
Proof. First assume that degV|/ = 0. For every fiber ft = n'1^) we
can write V\ft = Opi(at) ® 0pi(—at) for a uniquely specified nonnegative
integer at, and we need to show that the function a< is constant on a Zariski
open set. It suffices to show that the function at is upper semicontinuous in
the Zariski topology. In fact, consider the upper semicontinuous function
h°{V® Ox(-o)\h) = Я°(Ор.(о, - 1) ©Op,(-o, -1)) = at.
If the minimum value of h°(V ® Ox(— v)\}t) is h, then for a nonempty
Zariski open subset U of C, h = at. This proves the lemma in case
deg(V|/) = 0 and a very similar argument takes care of the case deg(V|/) =
1. ?
We return to the proof of Theorem 5 for the ruled surface X. In case
deg(V|/) = 0, the assertion is that if V is semistable, then the nonnegative
integer a of Lemma 6 is 0.
Lemma 7. In the situation of Lemma 6, suppose that deg(V|/) = 0 and
that a > 0. Тяеп there exists a unique sub-Jine bundle of V of the form:
Ox{oxr + тг*Ь), where b is a divisor on C, and such that the quotient is'
torsion free. If а = 0, there is aiso a sub-Jine bundle ОхGГ*'э) ofV, not
unique, with torsion free quotient. A similar conclusion holds if deg(V|/) =.
1, where in this case the sub-line bundle with torsion free quotient is always
unique.
Proof. First suppose either that deg(V|/) = 0 and a > 0 or that
deg(y|/) = 1, and consider the bundle V ® Ox{~<^cr). For a nonempty,
Zariski open subset U of С we have h°{Ox{-acr) ® V\ft) = 1 for all t ? U:
Thus, the torsion free sheaf Д°7г*(Ох(-ас) ®V) = L has rank 1 on С and
so is a line bundle, which we can write as Ос{Ъ). Thus,	J
L ® IT1 = Д0тг»(Ох(-аст) ® n'L-1 ® V) = fl°7r,(Cx(-<w - тг*Ь)
is trivial. Since H°(X;Ox(-aa - тг*Ь) ® V)) = H°(C;RPi:,(Ox{-aa 4
тг*Ь) ® V)), there is a section of H°(X; Ox{-aa - тг*Ь) ® V), or in othei!

6. Vector Bundles over Ruled Surfaces 147
words a homomorphism from Ox{<*a + 7r*b) to V. This homomorphism
factors through the inclusion of Ox(ocr + тг'Ь) in Ox(acr + тг*Ь + D),
where D is an effective divisor, necessarily of the form па + тг*Ь' for n > 0.
But clearly n = 0, for otherwise we would get a nonzero map Opi (a + n) —»
Opt (а) ф Opi (—a) for almost all fibers /, with n > 0, which is clearly
impossible. Thus, after changing notation we can assume that there is a
nonzero map
with torsion free quotient. We leave the uniqueness as an exercise. If
deg(V|/) = 0 and о = 0, choose a subbundle of rank 1 of the rank 2
bundle BPirtV = W and argue similarly. D
We have thus proved Theorem 5 in the case of a geometrically ruled
surface X. Note that, in case w ¦ f ф 0, or equivalently Д • / is odd,
the theorem says that there are no Я-semistable vector bundles V with
ci(V) = Д if H is (w, p)-suitable, since there are no semistable rank 2
vector bundles on P1 of odd degree.
Next consider an elliptic fibration тг: X —» С. We shall just give an
outline of the proof in this case. The main point is the following: if W is
an unstable rank 2 bundle on an elliptic curve C, then W splits into a
direct sum of line bundles L\ ® L2 of unequal degree, say deg L\ = d\ <
,degb2 = d2. Thus, Hom(W,W) = ОсФОсШСЧ"'®^©^!®-^)- Here
rthe two factors Oc correspond to diagonal maps from W to itself. Since
-degJLJ1 ® Li) < 0, the line bundle L^1 <& L\ has no sections, whereas
deg(Lfl ® L2) > 0 so that L^1 ® L2 does have sections. It follows that
every section of Hom(W, W) is strictly upper triangular, and that there
:fexist sections where the off-diagonal entry is nonzero.
If Let У be a rank 2 vector bundle on X such that V\f is unstable for
infinitely many smooth fibers /. Consider the bundle R°TrrHom(V, V). We
|have the functions det and Trace from Hom(V, V) to C. From the map
«7f*#°7r*#om(V,V) —» Hom(V,V), there are induced functions det and
ilrace on УД0тг*Яот(У, V), where YE?n*Hom(V,V) is the total space
tof the vector bundle Я°7Г*Яот(У, V). Now for the space of 2 x 2 up-
iper triangular matrices, the vanishing of det and Trace defines the linear
fsubspace of strictly upper triangular matrices. For infinitely many fibers
ift = ir-l(t) such that V\ft is unstable, the map R°ir*Hom(V, V)t -»
rH°(Hom(V\ft, V\ft)) is onto, and, for such t, the rank d of R°n,Hom{V, V)
:is equal to h°(Hom(V\ft,V\ft))¦ We consider the (homogeneous) subvari-
|ety Y of VR°TvtHom(V, V) defined by the vanishing of det and Trace,
>j,Vith its reduced structure. For infinitely many t corresponding to un-
etable Vt as above, the fiber of Y over t is a linear subspace of the
fiber of УД°тг» Hom(V,V) over (, of dimension d — 2. The same must
hold for a general fiber of Y. In this way we can find a subsheaf S of
S°(, V) such that the image of ir*S in Hom(V, V) via the natural

148 6. Vector Bundles over Ruled Surfaces
map ir*S —> 7Г*Я°7Г, Hom{V, V) —> Hom(V, V) consists of homomorphiams
from V to itself which are generically of rank 1. The image of k*S is thus
a rank 1 subsheaf of Hom(V, V). A local generator с of the image defines
a sub-line bundle Ox(D) = Kerer of V, which by construction restricts to
the destabilizing line bundle L2 on a fiber / such that V\f = Li Ф Li with
degZ/2 > degla. This proves the claim in the case of a fibration whose
fibers have genus 1. ?
We now give the general outline for attacking the study of if-stable
bundles on fibrations 7r: X —> С as above, when H is a (w,p)-suitable
ample divisor. Begin with a vector bundle V such that w\(V) — w and
pi(adV) = p. By Theorem 5, if V is if-semistable, then V\f is semistable
for almost all /. Now suppose that / is a smooth fiber such that V\f is
not semistable. Let j: f —* X be the inclusion. Then there is a maximal
destabilizing sub-line bundle L of V\f, necessarily a subbundle since / is a
curve, and thus an exact sequence
0 -> L -> V\f -> L' -* 0.
Here 2 deg V < deg V, and moreover L' is the unique quotient line bundle
of V\f with this property. Now we can define the elementary modification
of V with respect to the surjection V —> j+L':
0 _+ V' -> V -> uL' -> 0.
Let us make a definition formalizing the kinds of elementary modifications
that arise:
Definition 8. Let Vbea rank 2 vector bundle on X and j: f —* X the
inclusion of a fiber / in X such that V\f is unstable. There is a unique
maximal destabilizing subbundle L of V\f. Let V be the line bundle quo-
quotient (V\f)/L; thus fi(L') < fi(V\f). We call the corresponding elementary
modification V of V defined above allowable.
Lemma 9. If V is an allowable elementary modification of V, then
Pi{adV')>pi{adV) + 2.
Proof. By the formulas of Lemma 16 of Chapter 2,
-/J -4(c2(V) -ci(V) • /) + degl')
= Cl(VJ - 4c2(V) + 2ci{V) ¦ f - 4degL'
>Pl(adV)

6. Vector Bundles over Ruled Surfaces 149
where the last line follows since fi{V\f) — li(L') is a positive element of
\г. ?
This then is the general strategy: beginning with V, make all possible al-
allowable elementary modifications. At each stage pi(ad V) strictly increases.
There is an absolute bound onpi(ad V) by Bogomolov's inequality (we will
also see this directly in the cases described in this book), and so this pro-
procedure must stop. Of course, this idea is a little incomplete since we have
not done anything with the singular fibers. In Chapter 8, we shall give an
analysis of stability for singular fibers / which are irreducible curves with
just one singular point which is an ordinary double point. For a ruled sur-
surface, of course, we do not have to worry about singular fibers. So we will
reach a stage where V\f is semistable for all fibers /, smooth or not. At
this stage, of course, the work really begins. But in certain cases, namely
ruled surfaces or elliptic surfaces where degV|/ is odd, we will be able to
use this preliminary reduction to give a complete classification of stable
bundles.
Ruled surfaces
Prom now on in this chapter, X is a geometrically ruled surface. We shall
describe Я-stable vector bundles V on X for suitable ample divisors H.
After normalizing, we can assume that H = a+rf for some positive rational
number r. We need only consider the case where deg(V\f) = 0 or 1. By
Theorem 5, there are no Я-stable vector bundles V when deg(V|/) =
1. Thus, we may assume that deg(V|/) = 0, or in other words that the
determinant of V is pulled back from C. Fix a divisor d on С and an integer
c. We consider vector bundles V on X with c\{V) — 7r'(d), and let w and
p = —4c be the corresponding classes. Thus, ci(VJ = 0. First let us note
that we can give a direct proof of Bogomolov's inequality ci(VJ < 4c2(V)
in our case.
^Theorem 10. Let Я be an arbitrary ampie divisor on X. If V is H-
semistable, then Ci(V) > 0, and c^iV) — 0 if and only ifV is given as the
pullback of a semistable bundle W on C.
Proof. We may assume that Cb{V) < 0. Thus, there are no walls of type
.Gr*d,c), so that every ample H is suitable. By Theorem 5, the restriction
¦of V to almost every fiber / is О/ Ф Of. Using Lemma 7, there is an exact
^sequence
/Thus, c2(V) = t{Z) > 0, and c2{V) = 0 if and only if i(Z) = 0. In this
lease V is given as an extension of Ox(—7r*b +7r*d) by Ох{к*Ъ), and such

150 6. Vector Bundles over Ruled Surfaces
extensions are classified by Н1(ОхBт*Ъ — 7r*d). An application of the4!
Leray spectral sequence shows that Н1(ж*Ь) = ^(ЯРж^ж'Ь) = U4t)\
for every line bundle L on C. Thus, every extension of Ох(—ж*Ь + 7T*d)J
by Ох(ж*Ъ) is pulled back from an extension of 0c(—b + d) by Oc(b) onf
C. Thus, V = n*W for some W. Finally, we must decide when V = ir*W|
is Я-semistable. Assuming that H = a + rf, fi(V) = |d = | deg W. If L ia?
a sub-line bundle of W, then (a + rf) ¦ n*L = deg L, from which it follows!;
that, if ir'W is .ff-semistable, then W is semistable. Conversely, suppose-!;
that W is semistable. Since ci(VJ = 0 and e = 0, every ample divisor is'j
GT*d, 0)-suitable. The ample cone is a line segment and the walls are just |
certain points. In particular, there is no wall passing through the class of/.
By (iii) of Theorem 5, the only possible destabilizing sub-line bundles are
thus of the form Ox (D) where D is numerically equivalent to a rational;
multiple of /. It follows that Ox(D) — it*L for some line bundle L on-
X, by the description of PicX given in Chapter 5, Proposition 11. Since
HomGr*L, n*W) = Hom(L, W) and (i(n*L) = degL, the above arguments
show that, if W is semistable, then ir*W is semistable as well. ?
Note that the argument that сг(У) > О only used the fact that V\f =-;
О/ ф Of for almost all fibers /.
We now study the case where с > 0. By Theorem 5, we know that if i? is
a (w,p)-suitable divisor on X, then an //-semistable bundle on X restricts'
to Opi Ф Opi on almost all fibers. So we shall study such bundles without'
at first assuming that they are semistable. The following result is due to
Brosius [14]:
Proposition 11. Let У be a rank 2 bundJe on X such that V\f = Of®Of
for almost all fibers f. Then there is a natural exact sequence
where W = Я°7Г» V, the map ж' W -» V is the natural map ж*R°ir*V -» V,
and Q is a torsion sheaf supported in a union of fibers of ж. Finally, we
have H°(Q) = 0.
Proof. By assumption ж, V = W is a torsion free rank 2 sheaf on C, hence
is a rank 2 vector bundle. Moreover, the natural map ж*\? = 7Г*7Г«У —» V
is an isomorphism over almost all fibers and hence it is injective. Set Q =
To see that H°(Q) = 0, note that the projection formula gives
В.*Ж,Ж*\У = Л'7Г»7Г*7Г,У = Ж.У ® &Ж,ОХ
and thus 7r»7r*iy = W = тг.У and flV.7r*W = 0 for i > 0. Applying ЯЧ,
to the exact sequence
0 -* tt'W -» V -» Q -» 0,

6. Vector Bundles over Ruled Surfaces 151
we thus obtain
0 -» tt,V 9! 7Г„У -» n.Q -» 0.
Thus, -w,Q = 0, and therefore H°(Q) = H°(w*Q) = 0. П
Less canonically, we have the following description of V and W:
Lemma 12. With V as in Proposition 11, there exists a divisor Ъ on С
and an exact sequence
0 -> Ох(тт*Ъ) -» V -» 0x(T*(d - b)) ® /Z -» 0.
Thus, C2(V) = 1{Z) > 0. FinaiJy, a fiber / contains a point of the support
of Z if and only if the restriction V\f = Opi(a) ® Cpi(-a) with a > 0.
Proof. The existence of the exact sequence is a consequence of Lemma 7,
in the case a = 0. Restricting the exact sequence to a fiber / and noting
that Ox(ir*(d-b))\f is trivial gives a surjection V\f ~> Iz®Of -»IzOf.
Now IzOf is an ideal sheaf on /, which is equal to O/ if and only if
SuppZ П / = 0, and thus there is a surjection V\f —* Opi(— a), where
a > 0 and a = 0 if and only if / contains no points of the support of Z.
Since the kernel of the surjection must be 0pi (a), there is an exact sequence
0 -^ Opl(a) -^ V\f - Opi(-a) -^ 0,
which is split since tf^Opi^a)) = 0 if a > 0. Thus, V\f = OPi(a) в
Opi (—a) where о = 0 if / does not contain any point in the support of Z
and a > 0 otherwise. ?
We now describe more concretely how to obtain V from W. Suppose
that /0 is a fiber such that V\f0 = Opi(a) ® OPi(-a) with а > 0. The
surjection V\fo -* Opi (—a) is then unique mod scalars. Let j: /o —* X be
the inclusion. Then we can make the allowable elementary modification
,	0 -^ V -> V - j.Opl (-a) -^ 0.
By Lemma 16 of Chapter 2, ci(V') = Ci(V) - /0, and in fact detV =
ir*(b — po), where detV = b and po ё С is the point lying under /o.
Moreover, сг(У) = сг(У) — a, again by Lemma 16 of Chapter 2. Thus,
Ci(V) < c2(V). Clearly, the restriction of V to a general fiber / is Of®O}.
The restriction of V to /o fits into an exact sequence
0 -» Opi(-a) -» У'1/о — Op'(a) — 0.
Thus, it is easy to check that V'\f0 = Opi(a') ® OPi(-o') with 0 <
a' < a. Applying R°ir, to the defining exact sequence for V and using
iJ°7r,i»Opi (-a) = 0 since #°(Opi (-a)) = 0, we see that R°n.V = Д°тг»У.
So the bundle n*W is still a subbundle of V.

152 6. Vector Bundles over Ruled Surfaces
If V'\f = Of®Of for all /, then V = W by base change. Otherwise,
we may repeat the process with another allowable elementary modification.
This process must terminate, for example, because C2 drops each time and
C2 > 0 for a bundle with trivial restriction to almost all fibers. (We could
also compare ci at each stage with ir*Ci(W).)
Summarizing, then:
Theorem 13. Let V be a rank 2 bundle on X such that V\f = Of®0f
for almost all fibers /. Then Л°тг,У = W is a rank 2 vector bundle on C,
and ir'W is obtained from V by a finite series of allowable modifications.
Conversely, V is obtained from w*W, up to a twist by a line bundle of
the form rr*L, where L is a line bundle on C, by a sequence of elementary
modifications which are the duals of allowable modifications. ?
We must now analyze when such bundles are stable. We will have to
consider Case (iii) of Theorem 5, where the possible destabilizing sub-line
bundles are of the form Ox(rf), and semistability on the general fiber does
not necessarily imply that V is actually stable. i
Theorem 14. Let V be a rank 2 bundle on X with det V = Ox(n*d) fora,
divisor don С of degree d and with C2(V) = с and such that V\f — О/фО/
for almost aJJ /. Let W = R0/KtV be the corresponding rank 2 bundle on
С Suppose that H = сг + rf is a (w,p)-suitable ample Q-divisor, where w
is the mod 2 reduction of df and p = —4c. Then V is H-stable if and only
if, for all sub-line bundles L ofW, degL < d/2.
Proof. First suppose that V is /f-stable. If L is a sub-line bundle of W,
then 7r*L is a sub-line bundle of n*W and hence of V. Thus, H-n*L< d/2.
But H ¦ 7r*L = degL. Thus, degL < d/2.
Conversely, suppose that W satisfies the condition that deg L < d/2 for
all sub-line bundles L of W. We must show that V is ff-stable. Since there
are no walls passing through /, the argument of (iii) of Theorem 5 shows
that it suffices to check stability for all sub-line bundles Ox(D), where
2D — 7r*d is numerically equivalent to a rational multiple of /. This says
that D = 7r*b for some divisor b on С Applying Л°7г, to the nonzero map
OX(D) = Ox(ir*b) = тг*Ос(Ь) - V
gives a nonzero map Ос(Ъ) —» R?ir,V = W. Thus, by assumption 6 =
degb < d/2. But H ¦ тг*Ь = (a + rf) bf = b < d/2 as well. Thus, V is
Я-stable. D
Remarks. A) The sheaf Q of Proposition 11 can be shown to be Iz/Ib>
where Z is a O-dimensional local complete intersection subscheme on X
contained in E = ?\ щ/г, an effective divisor which is a sum of fibers. In a
certain sense Q records all of the elementary modifications simultaneously.

6. Vector Bundles over Ruled Surfaces 153
However, the sheaf Q can be quite complicated, as a result of performing
several elementary modifications on the same fiber.
B)	In Theorem 14, the bundle W has degree strictly less than d if
C2(V) > 0. In fact, if we need to perform к < C2(V) allowable elementary
modifications on V to get w*W, then degVK = d — k. Thus, the condi-
condition that every sub-line bundle of W have degree less than dji is always
weaker than assuming that W is semistable. However, one can show that
the generic such bundle W, in an appropriate sense, is stable. One conse-
consequence of Theorem 14 is that, if V is H-stable and V is obtained from
V by an allowable elementary modification, then V is not necessarily H-
semistable.
C)	In case g(C) = 0 and d - 0, a bundle W on P1 such that every
subbundle of W has degree less than 0 is of the form OPi (a) © OPi (b)
with both a,b < 0. To get from -n*W to V by elementary modifications
along fibers, if we want to have det V = 0 we must make exactly — (a +
b) elementary modifications which are the duals of allowable elementary
modifications. Each such modification will increase ci. Thus, if V is H-
stable for a suitable H, then ci(V) > 2, which is also easy to see directly.
Likewise, for an //-stable rank 2 bundle V on a surface X ruled over an
elliptic curve with detV = 0, one can show that сг(У) > 1. In all other
cases every value of c-}(V) > 0 is possible, and in fact Theorem 10 shows
that every such bundle V with c2(V) = 0 is the pullback n*W of a stable
bundle W on C.
A brief introduction to local and global moduli
The abstract point of view on moduli problems is that every moduli space
arises from a functor from the category of schemes or analytic spaces to the
category of sets. For example, the moduli space of stable vector bundles
V	over X with fixed Chern classes arises from the following functor M
from the category of schemes (over C, say) to the category of sets: Given a
scheme Г, let M(T) be the set of all equivalence classes of vector bundles
V	over X x T such that the restriction of V to each slice X x {t} is a stable
rank 2 bundle V over X ^ Xx{t} with det V = Д and c2(V) = с Here two
bundles V and V are equivalent if there exists a line bundle L on T such
that V ^ V ® n-J?. Ideally the moduli functor would be representabie. In
other words, there would be a universal parameter space ЯЛ and a universal
bundle U over X x ?01 such that given a bundle V over X x T as above,
there would exist a unique morphism /: T —* 9Я such that V = (Id x/)*W
up to a twist by a line bundle ir^C. Unfortunately, the moduli functor is not
in general representabie, for the simple reason that equivalence as defined
above is too weak to force universal bundles defined locally on an open
cover of the moduli space to agree on overlaps. What is unique is the space

154 6. Vector Bundles over Ruled Surfaces
ОТ and the morphism f:T-*M. More precisely, we call ЯЛ a coarse moduli
space for M(T) if:
1.	The points of ОТ are in 1-1 correspondence with stable rank 2 vector
bundles V over X with det V = A and ci{V) = c;
2.	For every scheme T and bundle V over XxT corresponding to an element
of M(T), there is a morphism f:T-* ОТ, functorial under pullback;
3.	Conversely, suppose that N is a scheme with the property that, for
every scheme T and bundle V over XxT corresponding to an element
of M(T), there is a morphism g: T —* N, functorial under pullback!
Then there is a unique morphism h: ОТ —» ЛГ so that g = ho f.
Then it is a theorem in geometric invariant theory due to Gieseker
in the case of surfaces [47] that there exists a quasiprojective variety
ОТ#(Д, с) = ?M parametrizing Я-semistable rank 2 vector bundles V on
X with ci(V) = A and сг(У) = e. (For the much simpler case of curves,
this theorem is due to Mumford. For proofs in this case, see, for example,
[48], [113], [140], and [40].) The space ОТ is a coarse fnoduli space, but the
universal bundle U only exists locally in the classical (or etale) topology
on ОТ. Moreover, ?M is almost never compact. To compactify it, we have
to consider equivalence classes of Gieseker semistable torsion free rank 2
sheaves. Here, if К is a Gieseker semistable torsion free sheaf, then there
exists a (noncanonical) filtration on V, the Jordan-Holder filtration, denned
in Chapter 4, Exercise 16 (see also Proposition 21 in Chapter 4 for the case
of a rank 2 bundle). Given such a filtration, there is the associated graded
sheaf gr V, which is canonical and is also Gieseker semistable. We say that
Vi and V2 are S-equivalent if grVi = gr Vi. The set of S-equivalence classes
of Gieseker semistable torsion free rank 2 sheaves on X with сг (V) = Д
and c2(V) = с can then be given the structure of a projective variety which
is a compactification Шн(А,с) of ОТя(А,с). This theorem was proved by
Gieseker in the case of a surface X and generalized to higher dimension
by Maruyama [85]. (See also Simpson [141] for a new approach to these
questions.)
Let us describe why geometric invariant theory plays a role in the con-
construction of the moduli space. First, fix Д and с Standard arguments show
that, if К is a Gieseker semistable torsion free sheaf or rank r with the
given invariants, then there is an integer N, depending only on г, А, с such
that h^V ® OX(NH)) = 0, г > 0, and V ® OX{NH) is generated by its
global sections. In particular h°(V®Ox{NH)) = К can be calculated from
Riemann-Roch. Since V ® Ox{NH) is generated by its global sections, it
is a quotient of 0$, and since an isomorphism from V\ to V2 induces an
isomorphism of global sections, given two surjections
Ф{ :?)?-¦ Vi®Ox{NH), * = 1,2,
inducing isomorphisms on global sections, V\ ^ V-j if and only if there is
an automorphism A of 0$ such that Ф2 = Ф1 о A. Now by Grothendieck's

6. Vector Bundles over Ruled Surfaces 155
itheory of the Quot scheme (which is a generalization of the Hilbert scheme),
there is a scheme Q parametrizing quotient maps Ф: O\ —» V ® Ox(NH)
as above, and Aut(O^) = GL(K, C) acts on Q. In general, however, the
«ction of GL(K, C) on Q is very bad, and to get a reasonable quotient we
have to restrict ourselves to the open subset Q° corresponding to quotients
V® Ox(NH) such that V is Gieaeker semistable. Even here, because of
the existence of strictly Gieseker semistable sheaves, if we want to take
a quotient by the action of GL(K, C) it is necessary to identify two S-
equivalent sheaves. However, aside from this technical problem, geometric
invariant theory constructs a quotient which is in fact a projective variety.
What is the local structure of QJl? In the crudest sense, we could ask for
some very basic information about OTt at a point x corresponding to V:
What is the dimension of ЯП at x? Is ЯЛ smooth at i? It is these kinds of
questions that deformation theory can sometimes answer. We run into some
trouble where there are strictly semistable torsion free sheaves, because the
notion of equivalence is not just that of isomorphism. So we will only worry
about this problem for V stable; this will usually take care of a Zariski open
and dense subset of 071. Deformation theory works in ideal circumstances
when the moduli space 07t is a fine moduli space. This means that there
is a universal sheaf V over X x Ш such that V\X x {x} — V under the
isomorphism X x {x} = X, where the point x € 07t corresponds to the sheaf
V. In general 07t is not a fine moduli space, but a universal sheaf does exist
locally in the classical (or etale) topology around every point corresponding
to a stable sheaf V, and this is enough for our purposes. Related to this
issue is the fact that stable torsion free sheaves are simple, by Corollary 8
(and Exercise 9) of Chapter 4. Thus, the automorphisms of a stable sheaf
cannot suddenly jump at a point. By contrast, given a strictly semistable
sheaf, it is S-equivalent to its associated graded object which always has at
least an extra C* of automorphisms, and these can create singularities in
the moduli space.
Instead of starting with a moduli space OTt, deformation theory takes a
completely local point of view: start with a family of, say vector bundles
over some parameter space T, or in other words a single vector bundle
V over X x T. We assume that Г has a distinguished point to and that
we are given a fixed isomorphism from the restriction of V to the slice
X x {to} to V. To start, we will also assume that T is smooth of dimension
1, with coordinate t. How do we describe V? Like all bundles, V is given
by transition functions. Let A,j be transition functions for V with respect
to some open cover {U,} of X. After possibly shrinking T, we may assume
that we have trivialized V on the open cover {Ui x T} of X x T. The
transition functions for V can then be taken to be of the form

156 6. Vector Bundles over Ruled Surfaces
The main idea of deformation theory is then to look at the linear term
d
¦ t, or more naively to take the derivative —
. Since Aij(t) is
a 1-cocycle, i.e., Aij{t) ¦ Ajk(t) = Aik(t) on UiHUjDUk, the B^ satisfy the
following condition: on Ui Л Uj DUk, we have
Using the cocycle condition for A^-, with a little manipulation we can
rewrite this as
ВЦА~} + Aij(BjkAjk1)A$ = BikA?.
This condition says exactly that B^A~^ transforms like a 1-cocycle for
Hom(V,V). It is not hard to show that different choices lead to a 1-
coboundary for Hom(V, V), so that we have intrinsically defined an ele-
element in H1 (X; Hom(V, V)). This element is the Kodadra-Spencer class of
the family V -+ XxT. We can partially reverse this procedure: given an el-
element Cij e Hl(X\ Hom(V, V)), we can define first-order terms in a power
series expansion for Aij(t), by the rule Bij = С^Ац.
More intrinsically we can start with any family V —¦ X x T, where T
can be any scheme or complex space; for example, T could be the dual
numbers SpecC[t]/(t2). Then this construction leads to a map from the
Zariski tangent space of T at t0 to Я1 (X; Hom(V, V)). Thus, we can think
of Hl(X; Hom(V, V)) as the space of all first-order deformations of V.
Given the first-order terms in a potential power series expansion for
Aij(t), what is the obstruction to extending this to second order? We would
like to find terms B'^ so that A{j + Btj ¦ t + B'^ ¦ t2 is a 1-cocycle mod t3.
So we must compute the t2 term in
(Лу + Вц ¦ t + В'ц ¦ t2){Aik + Bik ¦ t + B'jk ¦ t2)(Aik +Bik-t + B'ik ¦ tV.
Let us first work out this product in case B[^ = 0. In this case, since
{Aik + Bik ¦ t)-1 = (Id +A^Bik ¦ t)Arkl
= (Id-A"lBifc . t + (A^Bik)* • t2 + • ¦ ¦ )A7k\
a rather tedious calculation using the cocycle rules shows that the product
(Aij + Bh ¦ t)(Ajk + Bjk ¦ t)(Aik + Bik ¦ t)
is equal to Id -H^ijk, where
«у* = ВцВ,кА? = (BtjAtflAiiiBpAjftA;,1,
which is the product of the cocycle в = ВцА^У with itself under the natural
cup product map from Н1(Х\Нот{уУ)) to H2{X;Hom(V,V)) induced
by multiplication on Hom(V,V). We can also write this as ^[0,0], where
[•, •] is the Lie bracket on Hom(V, V), which also induces a cup product map.
(Since the Lie bracket is anticommutative, the associated cup product map

6. Vector Bundles over Ruled Surfaces 157
from Я1 to H2 is commutative. Thus, [в, в] will not in general be 0.) If the
element \{в,в] is 0 in cohomology, in other words is a 2-coboundary, then
this will solve the problem of finding B'^ so that A^ + Bij ¦ t + By ¦ t2 is a
1-cocycle mod t3, and conversely; we leave these calculations to the patient
reader.
The general principle is the same, although the formulas get progres-
progressively messier. At each stage, assuming that we have a 1-cocycle to or-
order n, we try to lift to order n + 1 and find an obstruction that lives in
Я2(Х; Hom(V, V)). Thus, for example if H2(X; Hom(V, V)) = 0, we can
always lift to order n + 1. The general formalism of deformation theory
is just some technical glue to put these calculations into more respectable
form.
For our applications, we shall just be interested in the case of deforma-
deformations of V where the determinant is held fixed, equal to Д, say. In this case
the differentiated form of det V = constant is Trace = 0. We can interpret
this as follows: there is a natural splitting Hom(V, V) = ad V Ф Ox, where
ad V is the kernel of the trace and we map Ox —* Hom(V, V) via the map
(r is the rank of V). In this case the obstruction space is
(in fact this is true in characteristic 0 even for the usual de-
deformation theory of V, where we don't fix the determinant). Globally one
can prove the following result about the local structure of 9Л:
Theorem 15. Suppose that x 6 ГО1 is a point corresponding to a sta-
stable bundle V. If H2(X;adV) = 0, then Ш1 is smooth at x of dimension
/i1(ady). In general, there is an analytic neighborhood ofxinffl which is
isomorphic to the zero set о/Л bolomorphic functions /i,..., /л defined in
a, neighborhood of the origin in Hl (X; ad V), where h = dim H2(X\ ad V).
Moreover, the /j have no constant or linear terms and thus the Zariski
tangent space of EOT at x may be identified with H1(X;adV). ?
We make the following definition:
Definition 16. A vector bundle V on X is good if H2(X;adV) = 0.
Equivalently, the moduli space is smooth at x (in the sense of schemes) of
dimension equal to h1 (X; ad V).
On a curve C, we can calculate hl(a.dV) for a rank 2 bundle V if V is
simple, i.e., /i°(ad V) — 0, by Riemann-Roch: the degree of ad V is the same
as the degree of Hom(V, V) = Vv ® V, namely — deg V + deg V = 0, and
the rank is 3, so x(ad V) = -/ix(ad V) = 3A - g). Thus, dim Я1 (ad V) =
3<? — 3. Likewise, for a surface X and a rank 2 vector bundle V on X, if
/i2(adV) = 0, then x(adV) = —^(adV) again, and by Riemann-Roch
and the splitting principle we find that
hx(ad V) = 4c2(V) - c2(V) - 3x(Ox) = -Pi(ad V) -

158 6. Vector Bundles over Ruled Surfaces
In general, if /i2(ad V) Ф 0, the expectation is that all of the possible h =¦
h? (ad V) equations to define the moduli space locally really are there. Thus,
the moduli space always has an expected dimension hx(ad V) — h2(ad V),
and we say that the moduli space has the expected dimension at x if indeed
its dimension is Л1(аЛ V) —/i2(ad V). The Riemann-Roch calculations above
show that the expected dimension for a surface is just — pi(ad V) — 3\(Ох).
We also see readily that the dimension of the moduli space at a point x is
always at least its expected dimension, and, if it is equal to the expected
dimension, then the moduli space is a local complete intersection at x, or
in other words is denned by exactly as many equations as the codimension
of its local embedding in the smooth space Я 1(ad V).
Next we shall give some criteria for the moduli space to be smooth:
Proposition 17.
(i) For a curve С of genus g, every vector bundle is good. IfV is a simple
bundle of rank 2, then the moduli space is always smopth of dimension
3p - 3.
(ii) If X is a surface and Kx = Ox(—D), where D is effective, then a
simple vector bundle is good.
(iii) (Maruyama.) Let X be a surface and Я an ampJe divisor on X such
that H ¦ Kx < 0. Then every H-semistable rank 2 vector bundle on
X is good.
Proof. Part (i) follows since, by dimension reasons, Я2(С; ad V) = 0.
For (ii), let V be simple. By Serre duality H2{X;&dV) is dual to
H°(X;(adV)v ® Kx). Now Hom(V,V)v = (Vv ® V)v ~ V ® Vv a
Hom(V, V), and similarly (ad V)v — ad V. (In fact, Trace defines a nonde-
generate pairing on Hom(V, V) for which ad V is the perpendicular space
to Ox ¦ Id.) Thus, we must show that H°(X; ad V ® Kx) = 0. But since
Kx = Ox{—D), there is an inclusion Kx Q Ox and therefore an inclusion
H°(X; ad V ® Kx) С H°(X; ad V). By assumption H°(X; ad V) = 0, and
so H°(X; ad V ® Kx) = 0 as well.
Finally, let us prove (iii). First note that H°(X;nKx) = 0 for all n >
0, since H ¦ (nKx) < 0. Let V be an й-semistable bundle and let ip G
H°(X; ad V ® Kx). Then det ip is a section of 2KX and so is 0. If у ф 0, it
follows that ip is a rank 1 map from V to V ® Kx ¦ Thus, the kernel of ip is
a torsion free rank 1 subsheaf of V and the cokernel of ip is a torsion free
rank 1 subsheaf of V ® Kx, necessarily of the form Ox (F1) ® Iz, where
Ox(F') is a line bundle. There is a surjection V -* OX(F') ® Iz and an
inclusion Ox(F') ® Iz —* V ® Kx- Now V ® Kx is #-semistable since V
is Я-semistable, and ci(V ® JOr) = ci(V) + 2JGr- So
Я • ci(V) < 2Я • F' < Я • (ci(V) + 2JGc).
Thus, 2Я • Kx > 0, contrary to hypothesis, and therefore <p = 0. D

6. Vector Bundles over Ruled Surfaces 159
Corollary 18. HX is a ruled surface and Я is a (w, p)-suitable ample divi-
divisor, then every H-semistable rank 2 vector bundle V on X with w^iV) = w
and pi (ad V) = p is good. Thus, the moduli space Ши (w, p) is everywhere
smooth of dimension —p — 3x(Ox) = —р+Зд — 3.
Proof. Recall from Lemma 14 of Chapter 5 that Kx = -2ст + тг* {Kc — e)
for some divisor e on C, where О„{а) = Ос(—ъ) and dege = e. If Я is
numerically equivalent to а + rf, then
Я • Kx = 2e + Bfl - 2 - e - 2r) = 2</ - 2 + e - 2r.
Thus, if r > 5 — 1 + e/2, then H ¦ Kx < 0. However, if Я is an arbitrary
(u/,p)-suitable divisor, then Я + tf is also suitable if ? > 0 and if V is H-
semistable, then it is (Я+ t/)-semistable. Thus, we may assume that V is
semistable with respect to a divisor of the formcr+r/, with r > g — l + e/2.
Applying (iii) of the previous proposition, we see that V is good. П
A Zaxiski open subset of the moduli space
Finally, we shall give a description of a Zariski open and dense subset of
the moduli space of Я-semistable bundles on a ruled surface. What should
the generic bundle V look like? It is natural to expect that most bundles V
have the following form: let с = c2(F). Then V\f = Of®Of for almost all
fibers /, and for the remaining fibers V\f = Opl A) e OPx (-1), the smallest
possible deviation from being trivial. If тг* W is obtained from V" by making
к elementary modifications, then by the formula for сг(V) given in Lemma
16 of Chapter 2, c2(V) = c2(n*W) + k = k and thus Jfc = с We obtain n*W
canonically from V as follows: there are с distinct fibers /i,..., /c such that
V\fi — Opi(l) Ф Of\(— 1). For each such fiber /, there is a canonical map
from V]fi to Ofi(— 1). Hence we have the exact sequence (where we leave
off the notation for the inclusion of /i in X):
с
0 -»-к* W -* V -» 0 Ou (-1) - 0.
Taking the dual elementary modification, we see that we can recover V as
follows:
с
0-» Vv-*7r*WrV-»0O/i(l) -0.
i=l
Now if detF = w*Oc(d), then det W = d — ?\pii which has degree
d — c, and det Wv = 5^ Pi — d. If for the sake of neatness we wish to
twist W by an appropriate line bundle so as to wind up with V at the
end, we can use the fact that V = Vv ® det V and take the elementary
modifications of ir*Wv ® det V = 7r*(Wv ® Oc(d)) corresponding to the

160 6. Vector Bundles over Ruled Surfaces
ones for tt*Wv given above. If we set Wo = Wv ® Oc(d), then det Wo =
det(Wv®Oc(d)) = d+^Pi and V is obtained from w*W0 by a canonical
sequence of elementary modifications.
Reversing this procedure, start with с points Pi,-¦¦ ,Pc 6 С For such
a choice, choose Wo a rank 2 vector bundle on С with det Wo = d +
Si Pi- We also need to impose a weak stability condition on Wo, coming
from the condition on W. Without writing this condition down, we expect
that the generic Wo will actually be stable. Finally, we need to choose
surjectivemaps7r*W0 -» 0/t(l)fori = 1,... ,c. Since n*W0\fi = O]t®Oft,
a map from tt*Wo to Oj{(l) is the same as the choice of two sections
of Of{(l) and the generic choice of sections corresponds to a surjection
from n*W0 to Of,(l). Thus, the set of surjections from tt'Wo to О/Д1)
is a Zariski open subset of HomGr*Wo,0/<(l)) = C4. Now given a map
it* Wq —> ф;=1 Oj, A), the kernel will be unchanged if we, compose with an
automorphism of ф°=1 0/Д1), namely an element of (C*)c. So we should
divide out by the set of such automorphisms. Of course, we should also
divide out by Aut Wo. But if Wo is simple, then Aut Wo — C, and we may
identify this C* with the diagonal of (C*)c. So we don't have to factor out
by this additional C*.
Let us now count the number of moduli involved in this construction.
For simplicity we will just consider here the case g > 2, and will say a few
words about the cases g = 0,1 later. First we choose с points p\,... ,pc of
C, using с moduli. Next, given the Pi, we choose Wo a stable bundle over
С with determinant equal to d + J^ p,-. This choice involves 3g — 3 moduli.
Finally, we need to choose a surjection tt*Wo —» O/,{1), modulo C*, for
every »"; this choice involves cD — 1) = 3c parameters. Tallying these all up,
we arrive at
с + 3p - 3 + 3c = Ac + 3p - 3 = -p - 3x(Ox)
parameters, which is the expected dimension of the moduli space. Moreover,
we have the following birational picture of the moduli space DJt: There is a
rational map fflt --¦* Symc C, whose fibers rationally look like a bundle over
3Jl(C), the moduli space of rank 2 bundles of a given fixed determinant over
C, and the fiber of the bundle over Wl(C) looks like a Zariski open subset
of (C4)c, modulo an action of (C*)c. The moduli space ЯЯ(С) is known to
be irreducible and unirational [113], [140]. (A variety Z is unirationa,! if
there is a dominant rational map PN --+ Z\ see Definition 32 in Chapter
10.) Thus, the moduli space Ш1 exhibits behavior very much like that of X
itself: it fibers over Symc C, and the fibers are unirational varieties.
How do we go about making the above heuristic discussion more pre-
precise, and showing that we really have described an open dense subset of
the moduli space in this way? The basic idea is to analyze the functorial
properties of the moduli space, and especially the fact that coarse moduli
spaces have a weak universal property. To identify an open subset of 9R, we
start by finding a space M and a bundle W over X x M with the following

6. Vector Bundles over Ruled Surfaces 161
property: the restriction of W to every slice X x {p} is stable. The bundle
U' induces a morphism /: M —» ЯЯ. For a good construction of M and
U\ j will be an embedding onto an open subset of ЯЯ. If ЯЯ is singular,
this may be rather difficult to show, in the sense that it may be very hard
to determine the exact scheme structure on ffl or to check that a given
morphism is an isomorphism. However, if we know, for example from de-
deformation theory, that 9Я is smooth we can often just use the easy case of
Zariski's main theorem (Theorem 8 of Chapter 3) to conclude that the map
M —» 9Л is an open embedding. We will not work through the technical
details of these constructions, but will just give an informal approach to
the description of the moduli space.
Having found a Zariski open subset of the moduli space, how do we then
conclude that it is dense? The problem, as should already be apparent in
the discussion for ruled surfaces above, is that it is rare to have a uniform
description of all stable bundles. Instead we tend to have a stratified de-
description of the moduli space, with a description of a set of presumably
"generic bundle" as well as more specialized descriptions of all of the oth-
others. One way to prove that the "generic" description really does give a
Zariski open and dense subset of the moduli space is to show that there
exist a finite number of schemes TJ and bundles Vj over X xT( parametriz-
parametrizing all of the "other" stable bundles, in other words such that, if the stable
bundle V is not in the image of /: M —> ЯИ, then V is isomorphic to the
restriction of V< to some slice X x {t} for some i and for some ( e Ij.
Suppose further that, for all i, dimTi is less than the expected dimension
of the moduli space. If so, letting fi: Ti —¦ 9H be the morphism induced
by Vi, the closure of LJ«/«№) w^ De contained in a subvariety of ЯЯ of
dimension strictly less than dimQJt. In particular the remaining bundles, in
other words the image of M, must be Zariski dense.
To see how this works in the case of a ruled surface, we shall make an
informal parameter count. To be nongeneric, we should take Wo to be an
unstable bundle, or allow some of the points Pi to coincide, or make an
elementary modification using 0/.(a) with a > 1. Let us argue that all of
these will lower the number of parameters in the construction. For the case
where Wo is not semistable, Wo is uniquely described by an extension using
the maximal destablizing subsheaf. We leave it as an exercise (Exercise 4)
to show that all such extensions are parametrized by a parameter space of
dimension at most 2p - 2. If g > 2, then 2g - 2 < 3g - 3, so we are done in
this case. Clearly, if two or more points coincide, then the points p, depend
on fewer moduli. Finally, what happens if we do an elementary modification
using Of^ui) with o,- > 1, possibly at a point where we have already done
an elementary modification? We shall just analyze what happens as we go
from V to a vector bundle V' by a single elementary modification as above.
As we noted in the discussion after Lemma 12, if V\f = Opi (a) © Opi (-a)
with a > 0, then V'\f = OF(a')®OP.(-a') with 0 < a' < a. Conversely,
to go from (V")v to Vy we must do an elementary modification coming

162 6. Vector Bundles over Ruled Surfaces
from a surjection from OPi (а') ф OPi (—a') to 0Pi (a). Now
dimHom@P. (a') в OP> (-a'), OP. (a)) = /i°(OPi (a - a') в Or (a + a'))
So the total number of parameters needed to make an elementary modifi-
modification dual to an allowable elementary modification is 2a+ 2. On the other
hand, the total change in the number of expected moduli is the amount by
which 4сг increases from V to V, and by the discussion after Lemma 12,
this change is 4a. Now 4a > 2a + 2 as long as a > 0, and equality holds if
and only if a = 1. So the only way we can get the full number Ac^ + 3g — 3
parameters is to make the generic construction described above. Summa-
Summarizing:
Theorem 19. Let X be a ruled surface over a curve С of genus д > 2, and
let d be a divisor on C. Let Я be a {ir*d, c)-suitable divisor on X. Then the
moduli space of H-stable rank 2 vector bundles V dn X with det V = 7r*d
and ci (V) = с is nonempty for all с > 0, and it is irreducible and smooth of
dimension Ac + 3p - 3. A Zariski open and dense subset fibers over Symc С
and the fibers are unirational varieties.
Variations of the above arguments prove similar results for g(C) = 0,1.
For example, in case g = 0, there is just one semistable bundle OPi (n) ф
Opi (n), but there are infinitely many unstable bundles OPi (n)®OPi (m) for
тфп, and we cannot simply argue that they have fewer moduli than the
semistable ones. In this case, however, Нот(тг* Wb,®* О/Д1)) is acted on
not only by Aut@i О/Д1)) = (C'Y, but also by AutGr*Wo) = Aut(Wo).
If Wo = Opi(n) Ф 6>Pi(n), then dimAutWb = dimG?B,C) = 4, but if
Wo = Opi (mi) Ф Opi (тг) for m-i j? m.2, тп\ + ТП2 — 2n, then
dimAut Wo = dim End Wo
= h°(OPi(mi -mj)fflOpi®OPi ©OPi(m2 -
= |mi — 1712I +3 > 4.
So we again get away with a smaller number of moduli to describe the
nongeneric bundles, those where Wo is not semistable. Using this circle of
ideas, one can then prove:
Theorem 20. Let X be a rationaJruJedsurface. Let H bean (ir*OPi (d),c)-
suitable divisor on X. Then the moduli space of H-stable rank 2 vector
bundles V on X with det V = 7r*OPi(d) and сг(У) = с is nonempty for
с > 2 if d is even, and for с > 1 if d is odd. It is irreducible, smooth, and
unirational of dimension Ac — 3.
These ideas may then be applied to the study of bundles over P2. For
example, Barth, Maruyama, and Hulek have proved that the space of stable

6. Vector Bundles over Ruled Surfaces 163
bundles over P2 is irreducible and unirational, and indeed rational if the
determinant is a line bundle of odd degree. Their proof uses the study of
bundles of P2 via monads. (See [117] for a discussion of these methods.)
Using Theorem 20, Qin [125] gave another proof, starting with the rational
ruled surface ?t and using the study of the way the moduli space changes
as we cross walls in the ample cone coming from chambers of type (w,p),
to pass from the chamber containing (tu,p)-suitable divisors to one more
suited to studying blowups. We outline this approach in the exercises.
Exercises
1.	Let X be an arbitrary surface and Я an ample divisor on X. Show
that a strictly Я-semistable rank 2 vector bundle V always satisfies
Bogomolov's inequality pi(ad V) < 0.
2.	Show that, if У is a semistable rank 2 bundle (for some divisor Я)
on a ruled surface X with deg(V|/) = 1, then V satisfies Bogomolov's
inequality. (Use the fact that there are no stable bundles for a suitable
ample divisor, so that every stable bundle is actually strictly semistable
for some ample divisor.)
3.	Let X = Ci x C2 be a product of two curves, let m: X —> d denote
the projection, and let W be a stable rank 2 vector bundle on C\.
Show that, for a suitable ample divisor on X, -k\W is a stable vector
bundle on X with c{ = c2 = 0. Moreover, if W has odd degree (which
is only possible if g(Ci) > 0), then -k^W is not of the form V ® L,
where V is a flat rank 2 vector bundle on X and L is a holomorphic
line bundle.
4.	Let С be a curve of genus g > 2. Show that the set of rank 2 vector
bundles V of fixed determinant such that there exists a nonsplit exact
sequence
with deg L > deg V/2 may be parametrized by a space of dimension at
most 2g — 2. (If L is fixed, use Clifford's theorem and Riemann-Roch
to estimate Hl(C; (L')~l ® L). Then use the fact that the space of
all L is parametrized by the Jacobian J(C), which has dimension g.)
What about an estimate for the space of strictly semistable bundles?
Our discussion of ruled surfaces concentrated on the case of bundles
with determinant the pullback of a line bundle on C, or equivalently
bundles V such that deg V\f is even. In the case where deg V\f is odd,
say det V = a, there are no iJ-stable bundles if Я is a (tu,p)-suitable
ample line bundle. Show that for a divisor Я lying in the chamber
C\ in A(X) next to the chamber Co containing (a, c)-suitable divisors,
we can construct stable bundles via extensions corresponding to the
wall separating C\ from Co as follows. We suppose that X = Fe is a
rational ruled surface. Show that the unique wall separating Co from
C\ is given by W^, where С = ° — 2c/ (argue as in the proof of Lemma

164 6. Vector Bundles over Ruled Surfaces
4). Show that for every H e C\, every Я-stable rank 2 vector bundle
V with det V — a and c2(V) = с is given as an extension
0 - 0F> - cf) -» V -> Op.(c/) -» 0
(since V must be Co-unstable, and necessarily a destabilizing sub-line
bundle Ox(D) satsifies ID —a = ?), and conversely by Proposition 24
of Chapter 4 such a bundle V is Я-stable. Conclude that the moduli:
space is a projective space PN, and show directly that N = —p - 3.
If X is not rational, argue informally that the moduli space is a ?N-
bundle over Pic0 X, where N = 4c + e + 2g - 3 = -p + 2# - 3, and
again recover the fact that the moduli space is smooth of dimension
-p-3X(Ox).
6. Let X = Fe be a rational ruled surface, and let ЯЯо be the moduli space
of stable bundles V with c\ (V) = 0 and сг(У) = с for the chamber
Co corresponding to @, c)-suitable ample divisors. In this exercise, we
outline a proof that, for every chamber С in A(X), the moduli space
JOT of stable bundles V for С is birational to 3JIq. In particular, it
is irreducible and unirational. Using Theorem 20 and Proposition 24
of Chapter 4, it is enough to show that, for every С a wall of type
@, c) and for every local complete intersection subscheme Z such that
-C2 + A?(Z) = -p = 4c, for D the divisor such that С = 2?», the set
of extensions of the form
0 - OX(D) -> V -> Ox(-D) ®Iz->0
may be parametrized by a scheme of dimension less that 4c—3. Argue
informally that Z 6 Hilb<(z)(X) depends on 2l(Z) moduli and that,
given Z, the set of all extensions depends on dim Ext1 (Ox (-D) ®
Iz, Ox(D)) moduli. Using the fact that ID is orthogonal to an am-
ample divisor and that Kx = — E for an effective divisor E, show
that h°BD) = h2BD) = 0 and thus that dimExt1(Ox(--D) ®
Iz,Ox(D)) = e(Z) + ^B0). Thus, the total number of moduli of
such an extension is hxBD) + 3?(Z) — 1, since if two given exten-
extension classes differ by a nonzero scalar, the corresponding bundles are
isomorphic.
Now since h°(-2D) - h?(-2D) = 0 as well, we have
\2
h\2D) < h1BD) + h1{-2D) = ~2^y~ - 2
(Riemann-Roch) and so h}BD) < —(? — 2, with equality only if
h1(-2D)=0. Thus,
h1 BD) + 3?{Z) - 1 < -C2 + M{Z) - 3 = Ac - 3 - ?(Z) < 4c - 3,
with equality if and only if tf{-2D) = 0 and 1{Z) = 0. In this case,
if D = сит + bf, show that a(ae — 26) = с and that (Riemann-Roch

6. Vector Bundles over Ruled Surfaces 165
and the canonical bundle formula) oe — 2b = 2o + 2c — 2 and derive a
contradiction, since с > 1. What if det V = df instead?
Show similarly that for det V = A with A • / = 1, then for every cham-
chamber C, the corresponding moduli space 9ЭТ is empty if С is the chamber
containing (Д, c)-suitable ample divisors, and (using the previous ex-
exercise) ОТ is birational to a protective space in all other cases.
For an explicit description of the blowups and blowdowns required to
get ОТ from ЯЛо, see [29], [44], and [86].
7. Let p: Fx —» P2 be the blowup of P2 at a point. If V is a stable
rank 2 vector bundle on P2, show that p*V is H-stahle for all ample
line bundles of the form p*Opi(N) ® Ov1 (-E), where E is the excep-
exceptional curve and N » 0, and that p* defines an open embedding from
?Dlps @, c) to fflt, where ®t is the moduli space of corresponding vector
bundles for an appropriate chamber of Fj. Conclude that 9Jt|«@,c) is
irreducible and unirational. Establish a similar result for rank 2 stable
vector bundles on P2 with determinant Opa A), where the moduli space
is in fact rational. (If, say, К is a stable bundle on P2 with c\(V) = 0
and C?Fi (D) is a sub-line bundle of p*V, use the fact that p»p*V = V
to show that Orx(D) = р*ОРэ(к) ® ?>F, (aE) with к < 0. To see that
p* defines an open immersion, look at Zariski tangent spaces.)
For more detailed analysis of stable bundles on blown up surfaces, see
Theorem 17 in Chapter 9 as well as [38] and [15].

An Introduction to Elliptic Surfaces
The goal of this chapter is to survey the classification of elliptic surfaces.
We begin with a preliminary section on general fibrations. Next we give
Kodaira's classification of singular fibers of elliptic fibrations, and describe
the canonical bundle and other basic invariants of an elliptic surface. We
give a complete discussion of elliptic surfaces with a section and discuss,
without proofs, the mechanism for obtaining all elliptic surfaces starting
with those which have a section. Finally, we describe some of the basic
topological invariants of an elliptic surface.
Singular fibers
Let n: X —» С be a (proper) morphism from a smooth surface X to a
smooth curve C, such that the general fiber of ж is connected. We shall
refer to such a map as a Sbration. Let / be a smooth fiber, and let g
be the genus of /. We shall primarily be concerned with the case where
0=1, but let us record here some general facts about such fibrations. For
much of this study, we will look at the local case where С = Д is a disk
in С and 7Г is smooth except possibly at 0 6 Д. In this local setting, if
Ci,..., CT are the components of ir-1@), then standard arguments show
that X deformation retracts onto 7г~г@) = \Jf C;. In particular H2(X; Z) =
$jZ • [Ci]. Intersection pairing on H2(X; Z) may then be defined in the
usual way, by first defining C, ¦ Cj = degOx(Ci)|C,- and extending by
linearity. Note that the pairing will never be nondegenerate, since if тг*0 =
Y^i п&, then QxE2i щСг) is a trivial line bundle on X and so ?V щС{ is
in the radical of the intersection pairing. However, up to a rational multiple,
this is the only element in the radical of the lattice spanned by the C^.
Lemma 1. In the above situation, Jet ir~l@) — Ui^i anc' ^ Ж*® —
$3; rijCj. Then (Ji Ci и connected, щ > 0 for all i, and the lattice spanned
by the classes of the Ci is negative semidefinite with a radical of rank 1.
Moreover, the radical is spanned over Z by an element ^ a^Cj = e with
Oj > 0 for aJJ г and gcd aj = 1 such that / = me for some integer m > 0.

168 7.- An Introduction to Elliptic Surfaces
Proof. That Ц C« is connected follows from the connectedness theorem,
and clearly щ > 0 for all i. That the lattice spanned by the Cj is negative
semidefinite with a rank 1 radical spanned over Q by / = X^n«C' follows,
in case X is projective, from the Hodge index theorem and the fact that
d ¦ f = 0 for all г. Let us show that this conclusion holds in the local case
as well, where С — Д and the fiber over 0 is the unique singular fiber.
First note that / = YU n*ci with / • C,- = 0 for all i. If i = 1, i.e., if the
reduction of тг~1(О) is connected, then / = nC\ and the statement about
the lattice is clear. Otherwise, by connectedness, for every i there exists a
j Ф i such that С, П Cj ф 0. Thus, we have
0 = / ¦ d = m(C?) + ?n,(C, ¦ d) > щ(С?).
So Cf < 0 for all i. Let w,- = щС,-, so that vf < 0 for all i as well. For every
i there exists a j such that i>j • u, > 0. Finally, (?V тл,) ¦ и< = 0 for all i.
A minor adaptation of the arguments of Theorem 21 in Chapter 3 shows
that ($2i AmJ < 0 for all choices of A< e R, and that (?V AiViJ = 0 if
and ony if Aj = Aj for all i,j. Thus, the lattice is negative semidefinite and
its radical is generated over Q by J^v* = ^ПгС,. If e = J3i a^Ci is the
primitive generator of the radical over Z such that / is a positive multiple
of e, then at > 0 for all i, gcdat = 1, and / = me for some positive integer
m. G
Corollary 2. In the above situation, if {C,- : i С A} is a proper subset
of the components of тг*{0}, then {C,- : i 6 .A} spans a negative definite
lattice. D
Next we show that in the above situation h°(Oe) = 1:
Lemma 3. Let it : X —» Д be a fihratioB with fiber /, and let e = ]TV OjCi
be the divisor over 0 which is the primitive generator for the radical of the
pairing. Then h°(Oc) = 1. In other words, the only holomorphic functions
on e are the constants.
Proof. First we claim that we can find e via a modified computation
sequence as in Exercise 9 in Chapter 3 as follows. Let Z\ = C,- be any
component of the fiber over 0. Thus, Z\ is contained in the support of e,
i.e., Z\ < e. Inductively suppose that we have found Zi and 2, < e. If
%i • Cj < 0 for all j, then in fact we must have Zi ¦ Cj = 0 for all j, for
otherwise the proof of Theorem 21 in Chapter 3 shows that the Cj span
a negative definite lattice (cf. Exercise 9 in Chapter 3), contradicting the
fact that the lattice has a rank 1 radical. So Zi lies in the radical and is
thus a positive multiple of e. As Zi < e, Zi = e. Conversely, if Zi Ф e,
then there exists a j such that Zi • Cj > 0. Set Zi+\ = Zi + Cj. Since
(e—Zi)-Cj — —(ZfCj) < 0, Cj is contained in the support of the effective

7. An Introduction to Elliptic Surfaces 169
divisor e — Zi. Thus, e — (Zi + Cj) is effective, so that Zi+\ < e as well.
This procedure must eventually terminate with Zk — e.
As in Exercise 10 of Chapter 3, using the exact sequence
0 - OCi{-Zi) -+ OZi+1 -¦ OZi -> 0
and induction, we see that h°{Ozi+1) = 1 for all i. In particular h°(Oe) —
Definition 4. If f — me, where e is the primitive generator of the radical
of intersection pairing and m > 1, then we call / = me a multiple fiber of
multiplicity m.
In fact, multiple fibers can exist only if \J{ Ci is not simply connected:
Lemma 5. Suppose that f = me is a multiple fiber of the Ebration
ir: X —+ С. ТАеп tie reduction of f has a covering space of order m.
Moreover, the normal bundle Ox(e}\e is a torsion line bundle of order ex-
exactly m on the possibly nonreduced scheme e.
Proof. It is enough to consider the local case where С = Д. Let t be
a coordinate on A. Locally on X, there exist coordinates x, у such that
7r*t = g(x,y), where g = hm is an mth power (and h is a local defining
equation for e). We have the map Д —» Д defined by t = sm. Let X' be
the pullback; thus X' is the subset of X x Д locally given by
{(x,y,s):sm = hm(x,y)}.
Let X be the normalization of X'. Since we can factor
sm-hm(x,y)= Y[(s-Ch(x,y)),
Cm=l
we see that locally X is the union of то smooth pieces. Clearly, X —> X is a
covering space. Since the general fiber of X —> Д is connected, the fiber ё
over 0 is connected as well. Thus, the fiber over 0 of X has a covering space
of order m (in fact the cover is Galois with group Z/mZ). Note that, if we
write ё as a sum of irreducible curves with positive coefficients, then the
coefficients of the components of ё are the same as the coefficients of the
components of e. In particular the gcd of the coefficients of the components
of t is 1. Thus, ё is a primitive generator of the radical of intersection pairing
for X, and it is not a multiple fiber.
We turn now to the statement concerning the normal bundle. Let N
be the normal bundle of e in X and let JV be the normal bundle of ё in
X. Since X —* X is a covering space, N is the pullback of ЛГ to e. Now
IV is trivial, and indeed an everywhere generating section is given by s.
The group of covering transformations Z/mZ acts on X fixing ё, and thus
acts on N and on its sections, as well as on various tensor powers of IV.

170 7. An Introduction to Elliptic Surfaces
Clearly, Ъ/тЪ acts on the section s by the roots of unity. Since N is trivial,
Lemma 3 applied to X shows that h°(N9k) = 1, and that a generator for
the Z/mZ-action acts on a nonzero element of H°(N®k) as C*\ where С is a
primitive mth root of unity. Now the line bundle N on e has order dividing
m since Oe(e)®m = Oe{f) = Oe- A section of JV®* would give a section of
N®k invariant under the Z/mZ-action. Clearly, this is only possible if m
divides k. Thus, the order of N is exactly m. D
Corollary 6. ForeveryBberfofir:X-*C, h°{Of) - 1 &ndhl{Of) = g.
Proof. Let / be a fiber, and suppose that / = me where the gcd of the
coefficients of e is 1. It follows from Lemma 3 that h°(Oe) = 1. For к > 1,
we have the exact sequence
0 -» Oe{-ke) -> O(k+i)e -» Oke -+ 0.
If 0 < к < m, the line bundle Oe(—ke) is a nontrivial torsion line bundle
on e. In particular its restriction to each component of e has degree 0 and
is either trivial or has no sections. It follows that a nonzero section ip of
Oe(-ke) restricts on each component С of e to a section of ?><.(— ke)\C
which is either identically 0 or nowhere 0. Now consider the computation
sequence used in the proof of Lemma 3, which we can start at an arbitrary
component С of e: we have
0 - Oct(~Zi - ke) -» OZi+1 (-ke) -* OZi{-ke) -+ 0,
where degOCi(-Zi-ke) = -(CjZi)-k(Cre) = -(Су2г) < 0 and Zx =
C. It follows that, for all i, we have #°(OZi+1(-*e)) Q H°(C;Oc(-ke)).
Choosing in particular Zi+i = e, we see that, for every component С of
e, H°(Oe(-ke)) С Н°(С;Ос{-ке)). Thus, if H°(Oe(-ke)) is nonzero,
then a nonzero section <p of H°(Oe(—ke)) restricts to a nonzero section of
H°(C; Oc(-kej) for every component С of e, which must be everywhere
generating since degOc(-ke) = 0. In particular Oc(-ke) = Oc for every
component С of e. But then the map Ce —» Oe(—ke) denned by (p generates
Oe(-ke) mod the maximal ideal at every point. Thus, Oe —¦ Oc(—ke)
is surjective and hence is an isomorphism. This contradicts the fact that
Oe(—ke) is not trivial. It follows that Oe{—ke) has no sections and by
induction /i°(O(fc+1)e) = h0{Oke) = 1 for all к < т. In particular, taking
к = m - 1 we see that h°(Of) = 1. Next we claim n is a flat map: this
follows from the local criterion of flatness [61, p. 276, Ex. 10.9], since X is a
smooth surface over a smooth base curve and all fibers of тг have dimension
1. Thus, x(C/) is independent of /. Since
it follows that hl(Of) is also independent of /, and is equal to its value on
a smooth fiber, namely д. D

7. An Introduction to Elliptic Surfaces 171
We may define relatively minimal models for fibrations by analogy with
Definition 16 of Chapter 3. We shall show that there is a unique relatively
minimal model, as long as the genus g of a smooth fiber is at least 1.
Definition 7. Let тг: X —> С be a fibration from the smooth surface X
to the smooth curve C. A relatively minimal model X' of X is a smooth
surface X' obtained by contracting exceptional curves lying in fibers of тг,
and such that X' has no exceptional curves lying in fibers of the induced
map тг': X' —> С. Equivalently, there is a commutative diagram
X 	> X'
4
= c,
such that the morphism X —> X' is birational and such that X' has no ex-
exceptional curves contained in fibers of тг'. Strong relatively minimal models
are similarly defined.
We then have the analogue of Theorem 19 of Chapter 3:
Theorem 8. With notation as above, relatively minimal models always
exist. If g > 1, then every relatively minimal model is a strong relatively
minimal model.
Proof. Arguing as in Lemma 17 of Chapter 3, we can keep contracting
exceptional curves in fibers until there are none left, and so we reach a
relatively minimal model. If it is not a strong relatively minimal model,
then the proof of Theorem 19 in Chapter 3 shows that there is a fibration
X —» C, birational to X over C, such that the fiber over a point contains
exceptional curves F and E with F ¦ E = n > 1. Since the intersection
matrix determined by F and E is negative semidefinite, 1 — n > 0. Thus,
n < 1 and so n = 1. Hence E+F has square 0, so it generates the radical of
the lattice spanned by the components of the fiber over Q. There can thus
be no other components of the fiber, which is then a multiple of E + F.
Since E + F is simply connected, the fiber must then be E + F, by Lemma
5. Since E + F has arithmetic genus 0, g = 0. Conversely, if the genus of
a general fiber is at least 1, relatively minimal models are strong relatively
minimal models. D
One way to characterize a (strong) minimal model in case g > 1 is by
nef properties of the canonical bundle:
Definition 9. Let тг: X —> С be a proper morphism from the smooth
surface X to a smooth (not necessarily compact) curve С A line bundle L

172 7. An Introduction to Elliptic Surfaces
on X is nef relative to тг or ж-nef if L ¦ D > 0 for every irreducible curve;
D contained in a fiber of тт.
Proposition 10. Let тг: X —> С be a Sbration and let g be the genus ol
a general Sber. If д > 1, then X is relatively minimal if and only if Kx is
v-nef.	-:
Proof. If X is not relatively minimal, then there exists an exceptional
curve E contained in a fiber of тг, and Kx ¦ E = — 1. Thus, Kx is not тг-nef.
Conversely, suppose that Kx is not тг-nef, and let D be an irreducible curve
contained in a fiber such that Kx • D < 0. Now D2 < 0, and D2 = 0 if and
only if D generates the radical of the intersection pairing on the components
of the fiber containing D. We have 2pa(D) - 2 = Kx ¦ D + D2 < 0, so that
D is a smooth rational curve. If D2 = 0, since D is in the radical of the
pairing, there can be no other component of the fiber. Since D is simply
connected, the fiber is not multiple, by Lemma 5. Thus, D is a smooth fiber
and д = 0. In the remaining case D2 < 0 and D is exceptional. In this cast.
X is not relatively minimal. D
Singular fibers of elliptic fibrations
Prom now on we shall only consider the case where the genus of all smooth
fibers is 1. We shall refer to such an X as an elliptic Gbration. We may as
well only consider the case where X is a (strong) relatively minimal model
as well. Our first goal will be to classify the possible singular fibers.
Lemma 11. Let ж: X —> С be a relatively minimal elliptic fibration, and
let D be an irreducible component of a fiber of w. Then Kx ¦ D — 0 and
either pa(D) = 1 and D is the only component of the fiber, or D is a smooth
rational curve and D2 = —2.
Proof. By Proposition 10, Kx • D > 0. If / is a smooth fiber of тг, then
Кx • / + /2 = 0, and so Kx • f = 0. If D = Di,..., Dr are the components
of the fiber containing D, then / is algebraically equivalent to 53» nt^*i f°r
some positive integers n,. Thus, ?V n*(Kx ¦ Di) = 0, and since Kx ¦ A > 0
for all i, we must have Kx ¦ A = 0 for all i. Hence Kx ¦ D - 0. If D2 = 0,
then, by an argument as in the proof of Proposition 10, D is the unique
component of the fiber in which it lies. In this case pa(D) = 1. Otherwise,
D2 < 0, and so 2pa(D) - 2 < 0. Hence pa(D) = 0, D is a smooth rational
curve, and D2 = -2. D
The irreducible curves D of arithmetic genus 1 lying on a smooth surface
are easy to classify, using Exercise 4 of Chapter 1: either D is a smooth
elliptic curve or the normalization of D is a smooth rational curve and D

7. An Introduction to Elliptic Surfaces 173
has exactly one singular point, which is either an ordinary double point or
a cusp. Moreover, D is simply connected if and only if it has a cusp, so this
case cannot correspond to a multiple fiber.
;», To handle the case of a reducible fiber, we define the dual graph Г of
the fiber as follows: if the fiber is of the form J2i «i A, consider the graph
Г whose vertices V{ correspond to the components D,-. We join Vi to Vj by
m(i,j) edges, where m(i,j) = Di ¦ Dj. There is also the lattice spanned by
the Di with intersection pairing. More generally, if Л is a free Z-module
with a bilinear form and v\,..., vn e Л, we can define the dual graph
Г associated to vi,..., vn in the analogous way. We then have a purely
algebraic lemma, whose proof is left to the exercises:
Lemma 12. Let A be a lattice spanned by V\,..., vn such that vj ~ —2
for all i. Then Л is negative semidefinite with a 1-dimensional radical and
every sublattice spanned by a proper subset of the Vi is negative definite if
and only if the dual graph Г determined by the Vi is the Dynkin diagram
of an extended root system An, Dn ,Ee,Et,Es- ?
Here the Dynkin diagrams of An,Dn,Ee,Er,Eg are depicted in Figure
3 on the next page, together with the coefficients of the primitive positive
generator of the radical. (In the notation, the number of vertices is n + 1
and Dn is defined only if n > 4.) Note that Ё6 = T3i3|3, Ё-, = T2,4i4, and
We make the following remarks in the case where Л is the lattice spanned
by the curves Di in a reducible fiber of an elliptic fibration:
1.	It follows from the description of the lattices (or by an easy direct ar-
argument) that the Di meet transversally in at most one point and that
no three pass through a single point except in the following cases: In
case the lattice is A\, there are two components D\ and Di and either
D\ and Da meet transversally in two points or they meet at one point
x and Di -x Di — 1 (the curves are simply tangent). In case the lattice
is Л2, there are three components Di,D2,D3, and each Д meets the
other two curves transversally in one point. However, it is possible for
all three curves to pass through the same point, as in three concurrent
lines in P2.
2.	The graphs DntEg, Et,Es are simply connected, and so (since the com-
components are smooth rational curves) the fibers are also simply connected
in this case. The graphs An are not simply connected. The fibers are also
not simply connected, except in the exceptional cases where there are
two components which are simply tangent at one point or three compo-
components which all pass through the same point.
3.	Direct inspection of the radical for the lattices corresponding to An, Dn,
Eg, E-j, Ев, shows that the radical is generated by a combination of the
basis vectors corresponding to the vertices with at least one coefficient

174 7. An Introduction to Elliptic Surfaces
equal to 1. Assume that the fiber is not multiple; for example, this if;
always the case if the lattice is not of type An. Then the fiber is ofi
the form ?»n»D» with at least one щ = 1. There is always a local!
holomorphic section passing through such a component. Thus, if the'
fiber is not multiple, there always exist local holomorphic cross sections
(and cross sections in the etale topology in the algebraic case). Of course,
for a multiple fiber of multiplicity m, the best we can hope for is a local
m-section, in other words a local holomorphic curve EcX such that
3 ¦ / = m, and in fact we can always find such an m-section locally
(either in the classical or etale topology).
1 2
¦ m
1 2
2 4
Figure 3

7. An Introduction to Elliptic Surfaces 175
We conclude by giving Kodaira's notation for the types of singulax fibers
We have described:
|{ mlo: A multiple fiber of multiplicity m, with smooth reduction. In case
Sni = 1, we simply write Io for this fiber.
Bmlni n > 1: A multiple fiber of multiplicity m, whose reduction is a
feycle of smooth rational curves meeting transversally. Here a "cycle" of
length 1 is an irreducible fiber with an ordinary double point and a cycle of
length 2 consists of two smooth curves meeting transversally at two points.
The corresponding Dynkin diagram is Лп_1. As before, in the case m = 1
we simply write In for this case.
';. II: An irreducible fiber with a cusp (whose normalization is then ra-
rational).
у III: Two smooth rational curves meeting at one point with local inter-
intersection number two. (The dual graph is A\.)
IV: Three smooth rational curves meeting at one point, with each pair
meeting transversally. (The dual graph is A^.)
Ц, n > 0; This corresponds to Dn+i.
П*: This corresponds to Ё%.
Ill*: This corresponds to Ej.
IV*: This corresponds to Ее-
We shall not show here that all of these types are realized, nor explain
the reason for the notation * or the local form of monodromy. General
references which deal with these questions are [7], [71], [72], and [94].
The list of singular fibers is rather complicated looking. However, by a
theorem of Moishezon, given an elliptic fibration ж: X —» С, after a small
perturbation of the complex structure of X and C, the only singular fibers
are either (nonmultiple) rational curves with an ordinary double point or
multiple fibers with smooth reduction [99], [40]. On the other hand, these
singular fibers cannot disappear under a deformation of complex structure,
and thus are called stable. Thus, in contrast to the case of ruled surfaces
singular fibers play an essential role in the theory of elliptic surfaces.
Invariants and the canonical bundle formula
In this section, we consider the global situation: ir: X —> С is an elliptic
fibration, where X and С are compact. Consider the sheaf Л'тг.Ох °n
C. Since hx{Oj) = 1 for every fiber of it, smooth or not, standard base
change results imply that Д1я-»С'х is a line bundle on C. We denote the
dual line bundle by L and set d = deg L. We will see shortly why it is more
convenient to work with L instead of its dual R1ir*Ox- The line bundle L
will reappear in a slightly different guise in the next section.

176 7. An Introduction to Elliptic Surfaces
One basic fact about L which we shall prove in Corollary 17 is:
Lemma 13. deg L > 0. D
In fact, if degL = 0, then L is a torsion line bundle on С of degree
1,2,3,4, or 6.
Lemma 14. Let d = degL and let g = g(C). Them
(i) If L is not trivia/, then g(X) = g and pg(X) = d + g — 1.
(ii) I/L is trivia], then q{X) = g + 1 and ps(X) = ff.
Jn aii cases we have x{®x) = d.
Proof. Apply the Leray spectral sequence to calculate H*(X;Ox)- By
dimension reasons, if eitherp or g is greater than 1, then HP(C; RqirmOx) =
0. It follows that the spectral sequence degenerates at the Ei term. Thus,
there is an exact sequence
0 - H\C; Oc) - H^X; Ox) - #°(C; Rlrr,Ox) -* 0,
and H2(X; Ox) = Hl(C; Rlir.Ox)- Hence
q(X) = h\X; Ox) = h\C; Oc) + ha(C; R1tt,Ox).
Now dimHl(C]Oc) = g. As for H°{C;RlirtOx) = H°(C;L~l), since L
has nonnegative degree, H°{C; L~l) =0 unless L is the trivial bundle on C,
in which case H°(C; L) has dimension 1. Thus, q(X) = g if L is not trivial
and q(X) = 5 + 1 if ? is trivial. As for H2(X; Ox) = Я1 (С; RlirtOx), we
must calculate /i1(C;L~1). By Riemann-Roch on C,
Thus, pg(X) = Лг(С; L) = d + g-lifL\s nontrivial, and pg(X) = g if
L is trivial. Since x(Ox) = 1 - g(A') +p9(X), in all cases x{Ox) = d. D
Note that the above proof uses the fact that deg L > 0 to calculate
hf{C; L'1). However, to calculate x(Ox), it is sufficient to know x{C\ L~l),
which follows from Riemann-Roch on C. So the proof that x{®x) — d does
not depend on knowing that d > 0.
Finally, we have Kodaira's formula for the canonical bundle of an elliptic
surface:
Theorem 15. Let ir: X —> С be a relatively minimal elliptic fibration.
Suppose that F\,..., F^ are the multiple fibers of ж and that the multiplic-
multiplicity of Fi is mj. Then:
Kx = ^{Kc ®L)®OX

7. An Introduction to Elliptic Surfaces 177
|Proof. First note that, for a smooth fiber /, the adjunction formula im-
implies that Kx\f is trivial. Moreover, we have seen that Kx ¦ D = 0 for
:ry component of a fiber of тг. Now it,Kx is a torsion free rank 1 sheaf
C, and thus is a line bundle A on С Moreover, we have a natural map
x —* Kx, which by general base change results is an isomorphism
г the smooth fibers. Thus, this map of line bundles vanishes along an
:tive divisor supported on the singular fibers of тг. We may thus write
тг*А®0х(Х!г aiDi), where the ?); are components of singular fibers.
|Moreover, ?V a^Di-D = Ofor every component of a fiber of тт. Thus, given a
ffiber of тг, which we may write as mF for some effective divisor F such that
jsthe gcd of the coefficients of F is 1, we have that YloiCF a'Di is a positive
iijntegral multiple of F, On the other hand, OximF) = ir*Oc(p), where
|p = ir(F). So after absorbing such factors into the term тг'А, we see that
¦>we can write Kx = тг'А' ® Ox{Yli а<^г), where the multiple fibers of тг are
[of the form rriiFi, and 0 < a< < m* — 1. By adjunction, Kx ® Ox{Fi)\Fi is
|ihe trivial line bundle on Fi. Thus, <DFi((oi+l)Fi) is trivial. But as OFi (Ft)
fhas order exactly m<, by Lemma 5, a< +1 = m,, so that a, = rrii — 1. Thus,
Ых = тг*А' ® Ox(?,i(mi - \)Fi) for some line bundle A' on C.
Next we must identify the line bundle A'. Now
irtKx = А'®7г,0
by the projection formula. By Exercise 2 below, тг»ОхE2г(т' ~
Oc- Thus, in fact our new A' is equal again to тт*Кх = A. To complete the
identification of A = тг*Кх, we need to recall a few easy facts concerning
relative duality (see, for example, [7, p. 98]). First let Kx/c — Kx ®
{v*Kc)~l¦ Thus, Kx = Kx/c ® ir'Kc, and so, again by the projection
formula, тг.АГх = (тг»ЛГх/с) ® Kc- There is a canonical adjunction map
Kx/c\F — шр (if we had not tensored by (тг'ЛГс)", this would no longer
be true). Now for each fiber F of тг, Serre duality says that H1(Of) and
Н°(шр) are canonically dual. Relative duality is the statement that the line
bundles Л1тг»Ох and R°n,KX/c are dual to each other. Thus, тг*Кх/с —
L, and so irmKx = L®Kc- Putting this together gives the canonical bundle
formula. ?
Corollary 16. For a relatively minimal elliptic surface X, Kx = 0. Thus,
by Noether's formula,
c2(X) = 12X(OX) = Ш.	D
Corollary 17. degL > 0, and degL = 0 if and only if the only singular
fibers are multiple Rbers with smooth reduction.
Proof. Using the remark after the proof of Lemma 14, we see that in
any case x(Ox) = d. Thus, the Euler characteristic of X, which is

178 7. An Introduction to Elliptic Surfaces
is equal to 12d and we must show that this number is nonnegative. Let|
Pi i • • • i P* G С be the points lying under singular fibers and let U be the!
open subset which is a union of sets of the form fi = 7Г~1(Д,), where Aj|
is a disk in С containing p, and no other singular point. Thus, #;([/;) =?|
•H»('r~1(Pi)) smce Ui deformation retracts onto тг^. Let AJ be a smaller!
disk in С centered at p{ and set V = ir-1(C - Ui Д*)- Tnen Wv) is Щ
open cover of X and V and [/ П V are fiber bundles with fiber a 2-torua|
Thus, the Euler characteristics х(У) and x(UC\V) are 0. A Mayer-VietoriaJ
argument shows that	Щ
1
Y,	1
¦3
An easy argument using the explicit description of the singular fibers shows]?
that xGr~1(Pi)) ^ 0, with equality if and only if тг~1(р^) has smooth re^j
duction. (In the remaining cases we leave it as Exercise 4 to calculate!
xi^iPi))-) Hence x(X) = Ых) ^ °. with equality if and only if the!
reduction of every fiber is smooth. ?	j|
¦")
The method of proof of the above lemma relates the Euler characteristic!
of X to the singular fibers. For each fiber F, singular or not, we define e(F)i
to be the Euler characteristic of F. Thus, if F is a smooth fiber e(F) = O.J
Then
For example, if the only singular fibers of X are rational curves with an
ordinary double point, then there are exactly 12d such singular fibers.
Elliptic surfaces with a section and Weierstrass models
Let 7Г: X —» С be a relatively minimal fibration with a section a. First note
that since a ¦ f = 1, 7r has no multiple fibers. If F is a reducible fiber of тг,
then let Fo be the unique component of F meeting a. The components of
F not meeting a all have self-intersection -2 and span a negative definite
lattice. In fact, by inspection of the possible components of F of multiplicity
1, the dual graph of this lattice is connected. Thus, the components of F
not meeting a can be contracted to rational double points, to obtain a new
surface n: X —» C. Here X has just rational double point singularities and
X —» X is its minimal resolution. The curve a induces an isomorphic curve
in X, which we shall also denote by a, and it isan effective Cartier divisor.
In fact a is contained in the smooth locus of X and meets each fiber in a
smooth point. Since X is a normal surface, it is Cohen-Macaulay. Thus, by
[61, III.10, Ex. 10.9], * is flat, and all fibers of n are reduced irreducible
curves of arithmetic genus 1. We have classified such curves in Exercise 4

7. An Introduction to Elliptic Surfaces 179
8f Chapter 1: they are either smooth elliptic curves or curves with either
l ordinary node or a cusp whose normalization is rational. All such curves
! isomorphic to a plane cubic, and hence their dualizing sheaves are the
rivial line bundle.
Definition 18. The surface X is the Weierstrass model of X.
To understand fibrations ж with these properties, we begin by reviewing
|he theory of embeddings of elliptic curves in the plane. Let D be a reduced
'irreducible curve of arithmetic genus 1. Choose a smooth point p G D. It
jfollows from Serre duality that ^((D; OD{np)) = /i°(wD ® OD{-np)) = 0
for all n > 0. By Riemann-Roch, h°(D; OD(np)) = n for all n > 1. There
jjs a natural inclusion H°(D; Oo(nip)) С H°(D; 0в(п2р)) for ni < n^.
jjlnder this inclusion H°(D; Od{p)) = С is just the space of constant mero-
ftaorphic functions on D. Moreover, there is a nonconstant meromorphic
jjt e H°{D\ OD{1p)) such that {1, x] is a basis otH°(D; ODBp)). The func-
ition x induces a morphism D —> P1 which is 2-1, for which p is a branch
point. Here more invariantly P1 = Р(Я°A>;С>дBр))*) and the morphism
jto P1 is given by the complete linear system |2p|. We note that the usual
[arguments show that |2p| has no base locus at p, and the inclusion of the
constants in H°(OpBp)) shows that |2p| has no base locus elsewhere (in-
(including the singular point if D is singular). Now h°(D; Од(Зр)) = 3 and so
there exists an element у of H°(D; O?>Cp)) having a triple pole at x. The
usual arguments show that j/2 is a linear combination of I,x,x2,x3,y,xy,
where the coefficient of ar3 is nonzero. First complete the square for y, by
replacing у by у + ex + d for unique c,d € С We may then assume that
y2 = f(x), where f(x) is a cubic polynomial in x. Such a choice of у is
unique up to a nonzero constant. Completing the cube in x, we see that
after replacing x by ax + b for unique a 6 C*, b e C, we can assume that
A9)	y2 = 4i3 - рга; - pa-
This choice of x will again be unique up to a nonzero constant. The
linear system defined by 3p embeds D in the projective space P2 =
P(H°(D;OdCp))*), and the image of D is equal to the cubic curve de-
defined by the homogeneous polynomial associated to A9). More precisely, if
p 6 D is a smooth point, the linear system |3p| defines a degree 1 morphism
from D to P2 whose image is a cubic curve D'. Since pa(D') — 1 = pa(D),
the morphism from D to D' is an isomorphism. We call A9) a Weierstrass
equation for D. Conversely, every curve in P2 defined by the affine equation
A9) is in fact a reduced irreducible curve of arithmetic genus 1. Note that
p e D С Р2 is an inflection point for ?), in other words the tangent line
through the smooth point p meets D to order 3 at p. Moreover, у t-» —у is
an involution t of D, fixing the singular point if D is singular, whose quo-
quotient is P1. Note also that у lies in the (—l)-eigenspace of the Z/2Z-action

180 7. An Introduction to Elliptic Surfaces
on H°(D;ODCp)) defined by i, and it is the unique nonzero element of
the (—l)-eigenspace up to a nonzero constant.
What are the possible choices in the Weierstrass equation? As we have
seen, у is unique up to multiplying by a nonzero constant. If we replace у
by a nonzero multiple, which for convenience we shall write as y' = A3j/,
then we have
(У1J = AV = 4AV - \6g2x - A6ff3
= 4(x'K-\4g2x'-\*g3,
where we have set x' = А2ж. Thus, x is replaced by A2s, у by X3y, g2 by
A4#2 and <7з by А6рз. Conversely, making these changes of variable gives us
a new Weierstrass equation isomorphic to the original one.
Now let us do the relative version of the above discussion. Let it : T> —> S
be a proper flat family such that every fiber is a reduced irreducible curve
of arithmetic genus 1, and let E С V be an effective Cartier divisor which
is a section of 7Г, i.e., such that the intersection of E with every fiber is
a reduced point, necessarily contained in the smooth locus of the fiber.
Here we can take S to be, for example, a reduced scheme of finite type
over C, or a complex analytic space, but any scheme such that 2 and 3 are
invertible will do. Since ж is flat, standard base change arguments show
that 7r»Ci)(nE) is a vector bundle ?n of rank n on S for every n > 1. Note
that E\ is the trivial bundle, since the natural map Os = КтО-р —» ?\
is an isomorphism. Moreover, there is an induced map T> —» ?(?2) which
realizes V as a double cover of the P1-bundle №(?%) over S, and there is an
embedding of V in №(?3), which is a P2-bundle over S.
Our goal now will be to identify the bundles ?2 and ?3 explicitly, and to
use these identifications to describe the embedding of V in Р(?з/). Begin
by setting С = Ox>(-E)|E, viewed as a line bundle on S = E. Prom the
exact sequence
0 -+ 0o(nE) -+ Or>{{n + l)E) -» Ot>((n + l)E)|E -» 0,
and the fact that Л^.ОиСпЕ) = 0 for all n > 0, we see that, for all n > 0,
there is an exact sequence
0 - ?n - ?n+1 - ?-("+1> - 0.
Note further that, applying R'v. to the above exact sequence for n = 0
and using the fact that the map тг»Ос -+ тг.Ои(Е) is an isomorphism, we
see that
In particular, the line bundle Л°7г»Ох)(Е)|Е = С'1 does not depend on the
choice of the section E.
Cover S with affirm open sets Щ such that С\Щ is trivial and such that
the exact sequence 0 -» ?n -+ Sn+\ -+ ?~(n+1) -+ 0 splits for n - 1,2.

7. An Introduction to Elliptic Surfaces 181
It follows that Г([/ь?п+1) = T(n-1(Ui),Ov((n + 1J)) is a free Н°(Ои<)-
module for n = 1,2. We may choose sections x, e rGr~1(J7i))C>i>B2)),
y, ? rGr~1(l/i),Oi)CS)) which restrict on each fiber D to generators of
the respective sheaves at D П S. Using the case of a single curve and
Nakayama's lemma, the sections l,Xi, (xjJ, {xif ,yi,Xiyi are a basis for
rGr(l/i),6'x)FS)) over На{Ои{)- Thus, we can write (y^2 as a combi-
combination of these elements, and then complete the square in yi and the cube
in X{. It follows that there is a local Weierstrass equation
y2 =4x? -gi^Xi -03,i,
with <72,i,ff3,i G H°(Oui)- Here the sections Xi, yi are well defined up to
multiplication by a section of Oy.. Now compare Weierstrass equations
over Ui П Uj. There must exist irivertible functions /iy, yy on Ui П Uj such
that Xj = fiijXj and у< = Vijyj. Let Ajj = j/y//uy. Note that Xj generates
ОцB2)) in some neighborhood of ЕП7Г~1(O;) С 7r~1(f7i), and similarly for
yi. Hence i/j/xj is a generating section for Ot>(E)) in some neighborhood of
Enir^). It follows that the Ay, viewed as functions on 7r~1(t/i П Uj),
are the transition functions for 0d(E))|I! = ?-1.
Take the local Weierstrass equation y2 = 4x? — ff2,tXi — <73,i and multiply
by f,^2. On t/j П IT,-, we then have
й,<а:< - ff3|1) = v~2y2 = y] = Ax) -
Thus, fy = /iy and so, as Ay = Vij/pij, we see that /iy = Ay and i/y = Afj-.
Furthermore ?2,i = Ayp2j and рз,; = Ayp3j. Thus, the p2,i and p3|i fit
together to give sections g2, дз of ?4 and ?e, respectively. Clearly, g2 and
рз are unique modulo invertible elements of TOs of the form A~4 and
A~6, respectively. Finally, using the basis 1,2* for ?2 in Ui shows that
the transition functions for ?2 can be taken to be in the form I .2 ).
\u xijJ
Thus, ?2& Oc® ?~2. Likewise, ?z a Oc © ?~2 © ?- It follows that
V{?z) =?(OC®C2®C3). Note that, viewing i as a local generator for the
factor ?3, у as a local generator for the factor C2, and z as the generator for
the trivial factor Oc, the homogeneous Weierstrass equation for V, namely
y2z — 4x3 — g2xz2 — <7з23, is then a naturally defined global section of
Sym3 ?3 <8> ?6, which is the relative homogeneous coordinate ring of ?(??(),
and the vanishing of this section defines a subscheme of №(?%) which is
isomorphic to V.
With this preliminary discussion, we can state the following result con-
concerning Weierstrass models of elliptic surfaces:
Theorem 20. Let n: X —» С be a relatively minimal elliptic fibration
with a section cr. Let L~x — R}-r*Qx — Ox{o)\o under the natural iden-
identification. Finally, let X be the Weierstrass model of X. Then there exist

182 7. An Introduction to Elliptic Surfaces
sections g2 e Н°{С;ЬА) and g3 € H°(C; Le) such that X is defined by tbj
equation y2z — Ax3 — g2xz2 — g3z3 inside P(Oc © L2 Ф L3}. The sections J|
and (ft are unique up to replacing p2 by A2p2 and рз by А3рз for A ? C||
In addition, the p* satisfy;	Д
1.	The section Д = p2 — 27pf o/ L12 is not identically 0.	?Я
2.	For all p e С, тт{3г1р(р2),2ир(рз)} < 12, where vp denotes the ordelj
o/ vanishing o/ the corresponding section at p.	3
Conversely, given a iine bundie L on С and two sections p2 6^ Я°(С;14|
and рз 6 H°(C; L6) satisfying A) and B) above, the surface X defined int
P(Oc ®I!e i3) by the equation j/2z - 4ж3 - p2a;^2 - рз-г3 is the equatio|
of a surface with at worst rational doubie points, such that the indue
morphism X —¦ С is a flat /amiiy o/ irreducible curves o/ arithmetic genus'
1, and the minimal resolution X is a relatively minimal elliptic fibratio||
with a section. Moreover, every section cr of X satisfies о1 = — deg L.
Proof. We shall give a proof of Theorem 20 modulo some basic fi
about rational double points. First suppose that X is an elliptic surfac||
with a section, and apply the preceding discussion to the family X -+ 0%
Here V = X, S = C, and E = a. We see that X is indeed denned by *
Weierstrass equation. Condition A) in the theorem is equivalent to saying;
that the general fiber of X, or equivalently X, is smooth. Condition B) te;
exactly the condition that X has rational double points. We will not prove,
this here, but will simply refer to [67] or [7, III.7 and II.8] for the necessary
results on singularities of double covers and simple curve singularities.
Conversely, given a line bundle L on С and two sections pa € H°(C; L4)
and <7з e H°(C;Le) satisfying A) and B), running the above argument1
backward constructs a surface X with at worst rational double points.
Note that X has a section a not passing through the singular points of X
or of any fiber of X, by looking at the point at infinity on each fiber (this
is the divisor defined locally by x = z = 0, у = 1). Let X be the minimal
resolution of X. The section a then induces a section of X. To see that
X is relatively minimal, it suffices to show that the canonical bundle Kx
is тг-nef. Let F be a fiber of X and let F be the corresponding fiber of
X. We may write F = F' + ?V щС{, where F' is the proper transform of
F and the C< are the components of the resolution of the rational double
point singularity. Thus, each C* is smooth rational with Cf = —2, so that
Kx ¦ Q = 0. As Kx ¦ F = 0, we must have Kx ¦ F' = 0, and Kx is тг-nef.
Finally, note that L~x = Oa{cr) and hence a2 = — degi. ?
To explain Condition B) further, note that if there exists a point peC
such that тт{3г>р(р2),2г>р(рз)} > 12, then ир(р2) > 4 and vp(g3) > 6.
Thus, 02 and рз induce sections р2,Рз of (I/L and {L'N, respectively, where
L' = L ® Oc{— p), and we can make a birational model of the Weierstrass
equation which is "better" in some sense by using the sections p2lp3.

7. An Introduction to Elliptic Surfaces 183
prollary 21. In the above notation, deg I, > 0. If deg Z, = 0, then L has
der either 1,2,3,4, or 6. Finally, L is trivial if and only if X is a product
proof. Since Д = <?f ~ 27<?з is not 0, at least one of gi and дз is nonzero.
P?hus, either L4 or L6 has a nonzero section. It follows that deg L > 0,
that if deg L = 0, then either L4 or L6 is trivial. Hence the order of
ptdivides either 4 or 6, and so the order of L is as claimed. If L is trivial,
|hen ft? and дз are constant, ?(Oc Ф L2 Ф L3) = P2 x C, and X = X is
^product. Conversely, if X is a product surface, then L~l = Л'тг.Ох =
~1(E;Oe) ® Cc i by the Runneth formula, and thus lrx = О с and L is
trivial. To see this last statement directly, note that we have shown that
Dx = Off(u), regardless of the choice of the section ст. For a product
surface E x C, we can choose the section а = {p} x C, and then it is clear
fhat Оа{а) is trivial. D
One can relate the various types of the singular fibers to the orders of
lyanishing of (fci fl3i and Д at a point p; see, for example, [93].
|More general elliptic surfaces
In this section, we wish to describe without proof how to construct all ellip-
elliptic surfaces starting from an elliptic surface with a section. This procedure
is not as elementary as the methods we have used in the preceding sections,
and requires either some knowledge of etale cohomology or (over C) work-
working with complex analytic surfaces which are not necessarily algebraic. For
a discussion of the first method, see [19] and for the second see [40, Chapter
1]-
We begin with the algebraic method. Given тг: X —* С an elliptic surface
over C, let к = k(C) be the function field of C. If we let 77 = Spec A; be
the generic point of C, then by base change the generic fiber Xn of X
over 7j is a smooth curve of genus 1 over the field k. Quite generally, let
к be an arbitrary field with algebraic closure к and let E be a smooth
curve of genus 1 defined over k. Then the Jacobian J(E) of divisors of
degree 0 on E is again a smooth curve defined over k, J(E) has a fc-rational
point (the trivial divisor), and there is a Л-morphism (given by translation)
ip: J{E) Xfc E —» E which realizes E as a principal homogeneous space over
J(E). Here we may identify E itself with the divisors of degree 1 on E and
if is then the usual sum of divisors. If there is a A;-rational point P € E,
then x i-* у>(ж, P) — x + P defines a ^-isomorphism from J(E) to E, and
conversely if E is A-isomorphic to J(E), then it clearly has a A;-rational
point.
The general mechanism for classifying principal homogeneous spaces E
over J(E) is as follows. If E is such a space, then there is a finite extension

184 7. An Introduction to Elliptic Surface»
of k, say A", over which E has a A'-rational point P. We may assume that'
К is Galois over к (in positive characteristic, this amounts to showing that;
we can assume that К is separable over k). Given an element p of the
Galois group Gk/h °f К over ^> define ap = p(P) — P- Thus, ap is a divisor
of degree 0 denned over K, in other words an element of J(E)(K), the set
of A'-rational points of J{E). It is easy to check that ap is a 1-cocycle for,
GK/k with values in J(E)(K). Changing P to some other point of J(E)(K)
replaces ap by a 1-coboundary. Thus, there is a well-defined element of the
group cohomology Hx(GK/k; J(E)(K)) associated to E, and one can show1
via descent theory that the principal homogeneous spaces E over J(E)
which have a A'-rational point are classified by H1{GK^k;J(E)(K)). To
deal with all possible extensions К at once, we consider the profinite group'
G = Gal(fc/fc), where к is an algebraic closure of к (in the case of positive
characteristic we would use instead the separable closure) and consider the
group cohomology (in the sense of profinite groups) Hl(G, J(E)(k)), which
we shall just denote by Hl(G, J(E)).
Returning to the case of тг: X —» С, let X^ be the generic fiber and let
J{X,,) be its Jacobian. We can complete J{XV) to some smooth elliptic
surface over C, which has a unique relatively minimal model. We denote
this relatively minimal model by J(X) and call it the Jacobian elliptic
surface associated to X. Clearly, X and J(X) are isomorphic if and only'
if X has a section. Our goal is to classify those elliptic surfaces X having
a fixed Jacobian elliptic surface В —» С, using the theory of the previous
section to take Jacobian elliptic surfaces as essentially known. Thus, we
must compute Hl{G, B) in the above notation.
Now given p e C, we can consider the completion of the local ring Oc,p.
Call this complete local ring Rp and let its function field be kp. There
are inclusions Specfcp —» SpecRp —» C, and thus we can consider the
restriction of X to SpecRp and to Specfcp. Here we should think of the
restriction to Spec Яр as corresponding to the analytic restriction of X
to a small tubular neighborhood of the fiber at p, and the restriction to
Spec kp as corresponding to the tubular neighborhood minus the singular
fiber. Hensel's lemma implies that if there is a component of the fiber Fp
at p of multiplicity 1 (which is equivalent in the elliptic surface case to
saying that Fp is not multiple) then there is a section of the morphism
X x с Spec Rp —» Spec Др. Now X x с Spec kp is a curve of genus 1 over kv,
and it is a principal homogeneous space over В Хс Spec kp = Bp. We may
then define HX(GP, Bp) by using the Galois group Gp of the local field kp,
and there is a homomorphism
pec
since Gp is the decomposition group of the point p and thus can be identified
with a subgroup of G. If f is an element of Я1 (G, B) corresponding to an
elliptic surface X —» С with Jacobian surface B, for each p € C, we let fp

7. An Introduction to Elliptic Surfaces 185
fenote the image of f in Hl{Gp, Bp). Thus, ?p = 0 if and only if the curve
Шс Spec A;p has a section, which by the above discussion is equivalent
|p saying that the fiber Fp of X over p is not multiple. This implies that
Щ\(Gp, -Bp) = 0 if the fiber of В at p is not either smooth or of Type In,
]S 1. More precisely, one can show that (at least in characteristic 0)
С (Q/ZJ, if the fiber is smooth,
H\GP, Bp) = i Q/Z, if the fiber is of Type 1„, n > 1,
\ 0,	otherwise.
all cases we can identify fp with an element of finite order in the Jacobian
"Щ Ep, the fiber of В over p. It turns out that the order of f p in Я1 (Gp, Bp)
щ exactly equal to the multiplicity of the corresponding multiple fiber of X,
;and that fp is a certain local invariant of the multiple fiber. The meaning
of ?p in the analytic case is described below.
|~Next we may ask when the map Hl(G,B) —» фре<7Я1(Ср,Вр) is sur-
jective. At least over the complex numbers, the answer is as follows: if В is
not isomorphic to a product elliptic surface E xC, then the map is always
surjective. If В isomorphic to a product elliptic surface E x C, then we
tan identify all of the fibers Ep with the fixed elliptic curve E, and there
is an algebraic elliptic surface with given local invariants ?p if and only if
к,
€
, Lastly we may ask about the structure of the kernel of the map
H\G,B) -» фр€С Hl{Gp,Bp). This kernel is called the Tate-SbafareWch
¦group of В and can be analyzed further via Galois cohomology or etale
cohomology. See, for example, [132] or [19].
, Now let us redo the above discussion in the complex analytic category.
Let v. X —» С be a holomorphic map from the smooth compact complex
surface X to the smooth curve C, such that the general fiber of ¦к is a smooth
elliptic curve. It is a straightforward argument in complex surface theory
that X is algebraic if and only if there exists a holomorphic multisection
of 7Г, in other words an irreducible curve E С X such that H ¦ / > 0, where
/ is a general fiber of ж (or equivalently such that ir(H) = C). The general
theory of fibrations and relatively minimal models works equally well in the
complex analytic case, and we may therefore assume that ir is relatively
minimal (no exceptional curves in the fibers). In this case the classification
of singular fibers is as in the algebraic case. Our first task is to define the
Jacobian elliptic surface J(X) in the complex analytic case. To do so, we
introduce three basic invariants of X:
1. The j-function j: С -» P1, defined by j(t) = jGr~1(i)). the j-invariant
of the smooth elliptic curve ж~1 (t), whenever t e С is such that 5r-1(i)
is constant. It turns out that j (which is defined on the complement of
the points of С lying under the singular fibers) has an extension to a
meromorphic function С —* С, or equivalently to a holomorphic function

186 7. An Introduction to Elliptic Surfaces
2.	The homological invariant G, defined as follows: let U С С be the open
subset of points lying under smooth fibers. Let тгц denote the restriction
of 7Г to 7Г" (U) and let г: U —¦ С be the inclusion. Then we set G to be
the sheaf i,Rx (tt?/).Z. The restriction G\V is a locally constant sheaf of
rank two Z-modules on U. In fact G = Д1тт,2 if there are no multiple
fibers, but, if there are multiple fibers, then there is a finite discrepancy
between G and Rlit,Z at the multiple fibers.
3.	The line bundle L on С such that L-1 =
There are various relations among the above invariants. For example, let
Uq С U be the open set consisting of smooth fibers whose j-invariants
are ф 0,1728, and assume for simplicity that Uq ф 0. Then j maps
Uq to Sjo/PSLB,Z), where $jo is the upper half plane Sj, minus the set
PSLB,Z) • {0,1728}, and PSLB,Z) acts freely on f)Q. Thus, by the the-
theory of covering spaces there is a homomorphism щ (Uo) —» PSLB,Z), well
defined up to conjugation. On the other hand, it is easy to see that the
locally constant sheaf R}(ttu),Z has monodromy contained in SLB,Z), so
that it is essentially equivalent to a representation n\(U) —¦ SLB,Z). The
compatibility between G and j is then that, after conjugation, the homo-
homomorphism 7Ti([/o) —¦ PSLB,Z) defined by j is equal to the composition
Ti(fo) -» 7Ti(?0 -+ SLB,Z) -» PSLB,Z). In general, for any curve C, we
may consider pairs (j, G) consisting of a holomorphic function j: С —» P1
and a sheaf G on С of the form i.Go, where Uq is the complement of a
discrete set of points of С contained in j~x (P1 - {0,1728, со}, and Go is a
locally constant sheaf on Uq with fibers ^ 7? associated to a representation
Т1(Уо) -» S?B,2). Such a pair (j, G) will be called compatible if the ho-
homomorphism 7Ti([/o) —* PSLB,Z) defined by j is equal up to conjugation
to the composition iri(U0) -» ^\{u) -» SLB,Z) -* PSLB,Z). We then
have the following theorem of Kodaira [71]:
Proposition 22. Let С be a curve. Given a compatible pair (j,G), there
exists an elliptic surface ip: В —» С with a section and with associated
invariants j and G, and В is unique up to isomorphism of elliptic surfaces.
Moreover, if (j,G) are also associated to the elliptic surface тг: X —» С,
then the line bundles (Л1тг,Ох)" end (Кхф*Ов)~х are isomorphic. ?
In the above situation, starting with the elliptic surface X, we call the
unique elliptic surface В with a section and with the same invariants (j, G)
as X the Jacobian elliptic surface J(X) of X. (In Kodaira's terminology
В is called the basic elliptic surface associated to X.) This definition of
J(X) coincides with the usual definition of the Jacobian surface in case X
is algebraic.
The next step is to classify those complex elliptic surfaces which have
the same Jacobian elliptic surface В and which do not have multiple fibers.
Equivalently, these are elliptic surfaces X with J(X) = В such that local

7. An Introduction to Elliptic Surfaces 187
pections exist above every point p e C. Using the proof of Proposition
j5S2, which is essentially a local argument, one shows the following: the set
jpf pairs (Х,ф), where X is an elliptic surface without multiple fibers and
ф: J(X) —» В is an isomorphism of elliptic surfaces from J{X) to B,
'modulo addition by a section in B, is classified by the sheaf cohomology
group Hl (C; B), where В is the sheaf of abelian groups given by the group
of local holomorphic cross-sections of В —» С. This group Hl(C; B) is the
analytic version of the Tate-Shafarevich group. It can be further analyzed
j.as follows: fix once and for all a holomorphic section E of B. Then if E' is
^another section over an open subset U of C, the divisor E' — E defines a
ffine bundle OT-i([/)(S' — E) on ir~1(U), of degree 0 on each smooth fiber,
:jand thus it defines an element of H1(n~1(U); O*_i(m)- Conversely, given
fa line bundle L on тг~1(Г/) such that С has degree 0 on each smooth fiber,
fthe line bundle L ® O^-i^^E) has degree 1 on each smooth fiber and so
й°7г» (С® Ож-\(щ{Т,)\ is a line bundle on U. Moreover, the natural map
fR°ir, (? ® O,-i(y)(E)j —» ?®C,r-i((y)(E) vanishes along a section E', and
! possibly also along the union of some components of reducible fibers. Thus,
?' can be recovered from ?. More precisely, there is a split exact sequence
of sheaves
0 -» В -» Rlir.O*B/S -. Z -» 0,
where 5 is the skyscraper subsheaf of Ц}ж*О*в generated by line bundles
of the form Ob{D), where D is a reducible component of a fiber of я
(note that ObU) defines the trivial element of Rl-n*O*B), Ъ is the constant
sheaf on C, the map RlntOB/S —• Ъ is given by taking the degree of a
line bundle on the general fiber, and the splitting is given by n e Z н->
Ов(пЕ). Prom this, the study of Hl(fi\B') is essentially reduced to the
study of Я*(С; Л17г»Од). Moreover, this group can be analyzed in detail
by applying R*tt, to the exponential exact sequence
and using the Leray spectral sequence. The upshot is that, if тг has a sin-
singular fiber, then the set of elliptic surfaces classified by H^iC-^) are all
deformation equivalent. For more details see [40] (see also [68] for the case
where the base is P1).
Finally, we must add multiple fibers to complex analytic surfaces. To do
so complex analytically, there is the method of logarithmic transformations
introduced by Kodaira. For simplicity we shall just consider the case of a
multiple fiber with smooth reduction. Let F be such a fiber, of multiplicity
m over p €. C, and let Д С С be a small analytic disk centered at p such
that тг~1(Д) has F as the only singular fiber. Let z be a coordinate on Д
and make the base change A —> Д defined by z = wm. In the following
discussion, we shall replace X by 7г~1(Д), so that we will assume that
X fibers over Д. Arguing as in the proof of Lemma 5, we see that, if X
is the normalization of the fiber product of X with Д with coordinate

188 7. An Introduction to Elliptic Surfaces
w, then X is smooth and the map X —+ X is a cyclic covering space
of order m. Let T be the generator of the covering group corresponding
to the automorphism го к-* e2w*/mw of A. The central fiber F of X is a
smooth curve, necessarily connected, and the restriction of T to F is a
fixed point free morphism with quotient F. Since F is an elliptic curve, F
is an elliptic curve and T\F is given by addition by an element ?p e F. One
can show that this element fp may be identified with the local invariant ?v
introduced above in the discussion of the algebraic classification of elliptic
surfaces. Kodaira's theory then classifies all such group actions T on X
and shows how to glue these into elliptic surfaces without multiple fibers
to obtain (complex analytic) surfaces with prescribed multiple fibers. Once
we have this construction in the complex analytic category, we can then go
back and try to determine when such surfaces are actually algebraic. For
details on this, we refer to [40] (see also Exercise 11).
The fundamental group
In this final section, we shall briery describe some of the "classical" topology
of an elliptic surface X. The most important invariant is the fundamental
group. Before we can describe the fundamental group of an elliptic surface,
we need to make the following definition:
Definition 23. Let С be a compact Riemann surface, and let pb ... ,pn
be a set of n distinct points on C. Suppose that for each г we are given
an integer m* > 2. We call the data of C, the pi, and the ттц a 2-orbifold.
We define the orbifold fundamental group тт°гЬ(С, *) as follows: the fun-
fundamental group of С — {p\,... ,pn} is generated by the usual generators
Ол,/?г,1 < * < 9, where д = д(С), together with additional generators
7ii • ¦ • >7n corresponding to loops enclosing each pi simply, not enclosing
any Pj,j^i, and which are homotopic to zero rel * on C. There is also the
relation [ai,/?i] • • • [ag,f3g]"fi ¦ ¦ -7„ = 1, where the ait Д are the standard
generators of 7ri(C,*) and [fti.ft] is the commutator of Qj and /?;. Define
тг°гЬ(С, *) to be the quotient of ni(C — {pi,-¦ ¦ ,Pn},*) by the smallest
normal subgroup containing 7,m' for all i. Thus, 7r°rb(C, *) is freely gener-
generated by the elements ct\, 0\,..., ag, /3g, 71,..., 7„, subject to the relations
[on, A] • • ¦ [a9,A,]7i ¦ • -7n = 1 and 7Г = 1 &* all i.
In general, the orbifold fundamental group is quite large. For example,
one can show that ir°rh(C,*) determines g(C) and the m* unless С has
genus 0 and there are at most two points Pi, i.e., n < 2. In case g(C) = 0 and
there are exactly two multiple fibers of multiplicities mi and ma, п°тЪ(С, *)
is cyclic of order gcd(mi,m2). In particular it is trivial if and only if mi
and m.2 are relatively prime. (In case g(C) = 0 and n = 1, 7rirb(C,*) is
always trivial.)

7. An Introduction to Elliptic Surfaces 189
The connection with elliptic surfaces is the following. Let тг: X —> С be
an elliptic surface. Then X defines a 2-orbifold structure on С as follows:
the points pi are the points lying under multiple fibers (not necessarily with
smooth reduction) and the integers m^ are the corresponding multiplicities.
With this said, there is the following result due to Kodaira [74], Moishezon
[99],andDolgachev [22]:
Theorem 24. Suppose that n: X —> С is a relatively minimal elliptic
surface such that тг has at least one fiber with singular reduction (or equiv-
alentJy X does not have Euler number 0). Then vi(X,*) Si тг?гЬ(С,*).
1л particular, X is simply connected if and only if С — P1, there are at
most two multiple fibers, and, if there are two multiple fibers, then their
multiplicities are relatively prime. ?
Let us use this to discuss the homotopy type of a simply connected elliptic
surface X, always assumed to be relatively minimal. As a 4-manifold, X
is then determined up to homotopy type, or equivalently homeomorphism,
by the intersection form on H2(X;Z). Let p9 = pg(X). Since c2(X) =
12A + pg), we have
Ъ2(Х) = 10 + I2pg.
Moreover, b2 PO = 2p9+l, by the Hodge index theorem. Thus, we know the
rank and signature of H2(X\ Z), and it remains to determine whether the
intersection form is even or odd (i.e., of Type II or Type I). The intersection
form is even if and only if Kx is divisible by 2, by the Wu formula. Now,
by Lemma 14, L is a line bundle over P1 of degree d = 1 + pg. Thus, since
Kpi = Opi(— 2), the canonical bundle formula for an elliptic surface over
P1 with multiple fibers F\ and F2 of multiplicities mi and m2 gives
Kx = Ox{(p3 - 1)/+ (mi - l)Fi + (ma - 1)F2).
Since mi and vn.2 are relatively prime, there exists a linear combination
ami + Ьгпз = 1 with a,b e Z. Taking к, = bF\ + aF2, we see that
= bmi.Fi + amiF-i = bf + A — bmi)F<i
Thus, milt = F2, and similarly ma/с = Fi,mim2K = /. We also have, by
[40, Chap. 2, Prop. 2.7]:
Proposition 25. Suppose that тг: X —+ С is a relatively minimal elliptic
surface such that n has at least one fiber with singular reduction, and let
m\,... ,m.k be the multiplicities of the multiple fibers. Let m be the least
common multiple of the т.{. Then there exists a class x 6 H2(X; Z) such
that x- f =m. Thus, if the m, are pairwise relatively prime, then m is the
exact order of divisibility of f in H2 (X; Z). ?

190 7. An Introduction to Elliptic Surfaces
Applying the above proposition, we see that the class к described above
is a primitive class, in other words it is not the multiple mx of a class
x e H2(X;Z) with m > 1. Thus,
Kx = (mim2(pfl -1) +m2(mi - 1) +mi(m2 - l))/c
= (mim2(pfl + 1) - mi - тп2)к.
If pfl is even,
Wiffl2(pj + l)-mi —m2 = mim2-mi — mj = (mi — I)(m2 — 1) — 1 mod 2.
Since mi and m2 are relatively prime, at least one of them is always odd,
so that (mi — l)(m2 — 1) is always even. It follows that mim2(p9 +1) -
mi — m2 = 1 mod 2 if pg is even, so that the intersection form on X is
always odd. If pg is odd,
mim2(pfl + l)-mi-m2 = —mi — m2 = mi + m2 mod 2.
Thus, if pg is odd, the intersection form on X is even if and only if mi +m2 =
0 mod 2.
Summarizing, then:
Proposition 26. Suppose that X is a minimal simply connected ellip-
elliptic surface, with two multiple fibers of multiplicities m\ < m2. Then the
intersection form on X is of Type I if pg is even or if pg is odd and
mi + m2 s 0 mod 2, and is of Type II otherwise. Similar statements hold
if X has less than two multiple fibers. ?
Let us conclude by discussing the possible torsion in H2(X;Z):	'
Proposition 27. Suppose that тг: X —t С is a relatively minimal elliptic
surface such that n has at Jeast one fiber with singular reduction, and let
mi	ti»t be the muJtipficities of the multiple fibers. Then the torsion
subgroup ofH2(X; Z) is isomorphic to the quotient of ®i=J Z/mjZ by the
image of Z embedded diagonally, or in other words by the image of the
subgroup generated by A,..., 1).
Proof. Since the torsion subgroup of Я2(Х;2) is isomorphic to the tor-
torsion subgroup of Hi(X;Z), the proof follows easily by working out the
abelianization of тг?гЬ(С, *). D
For example, if the m; are all pairwise relatively prime, then
is torsion free. At the other extreme, if k = 2 and mi = тг = т, then the
torsion subgroup of H2(X;Z) is isomorphic to Z/mZ.
Exercises
1. Generalize the proof of Corollary 6 as follows: recall (Exercise 13 of
Chapter 1) that an effective divisor D = Yli я«С« *s numericaiJy con-
connected if, whenever D = ?>i + D% with each Dj effective and nonzero,

7. An Introduction to Elliptic Surfaces 191
then D\ ¦ ?>2 > 1. For example, if D is nef and big, then it follows
by Exercise 14 of Chapter 1 that D is numerically connected. Show
that, if e is the primitive effective divisor supported in a fiber of a
fibration which generates the radical of intersection theory, then e is
numerically connected. (If e — Di + D2, then
and by assumption (-D1J < 0.) What about the fundamental cycle Zq
of the resolution of an isolated surface singularity? (I don't know the
answer to this question.)
Show that, if D = ]Г]{т^С; *s numerically connected and L is a
line bundle on D such that degZ|Ci < 0, then h°(D;L) < 1, and
h°(D;L) = 1 if and only if L is trivial (Ramanujam's lemma). In
particular h°(Oo) — 1. (First show that, if C; is an arbitrary compo-
component of D, then there exists a sequence Z\ = Ci,Z2,..-,Zn = D with
Zi+i = Zi + Cj for some j such that Zi ¦ Cj > 0, and then argue as in
the proof of Corollary 6.)
2.	Let ж: X —» С be a fibration, and let Fi be the multiple fibers of ir.
Suppose that a,- are integers such that 0 < a; < mj — 1, and that D
is a divisor on C. Show that vm(v*Oc{D) ® Ox(Ei аг-^)) = ®c(D).
Use this to determine H°(X; Ox(n*D + ?\ fl^)).
3.	Prove Lemma 12. (Suppose that every sublattice of Л spanned by a
proper subset of the vertices of the dual graph is negative definite. If
there is a cycle, show that we are in case An. If there are no cycles
but the dual graph is not a Tp qr graph, show that we are in case Dn.
Otherwise, the dual graph is a Tp<q>T graph such that the lattices of
type Tp_i,,,r, TPig_i,r, and TPi,ir_i are all negative definite. Show that,
in this case, the only possible choices for (p, q,r) up to permutation
are C,3,3), B,4,4), B,3,6).)
4.	Let Fbea singular fiber of an elliptic fibration. Define the invariant
e(F) = x(F)- Calculate e(F) for the various singular fibers of an
elliptic fibration.
6. Generalize the above exercise and Corollary 17 to genus g fibrations:
let тт. X —> С be a fibration with general fiber of genus g. Then
X(X) = B - 2g)X(C) + T,Fd(F), where d(F) = *(F) - 2 + 2g =
x(F)-2x(OF). Show d(F) > 0. (By Corollary 6, x(Op) = 1-p. First
show that pa(F) > pa(FTed) — h}(Fle^), where Fred is the reduction
of F, and thus — 2x(Op) > — 2x(&Frcd)- Thus, we may assume that
F is reduced. Now dime H2(F; C) is the number of components of F,
which is > 1, so it suffices to show that 2h1(OF) > dimH1(F;C). Do
this by considering the normalization v: F —» F and the commutative

192 7. An Introduction to Elliptic Surfaces
diagram
0 	> С
0 	> OF 	> utOp 	> S' 	> 0,
showing that the map S —> S' is an injective map of skyscraper
sheaves.) Show finally that d(F) = 0 implies that .Fred is smooth.
6.	Suppose that ? С |СраC)| is a pencil of plane cubics containing a
smooth member. Show that the resolution of indeterminacy of С de-
defines an elliptic fibration n: X —» P1, where X is the blowup of nine
points of P2, possibly infinitely near. The map тг has a section. The
general such pencil has 12 singular nodal fibers (see also Exercise 10
of Chapter 5). Without using infinitely near base points, show that we
can arrange singular fibers of Types In (n = 1,2,3), II, III, and IV.
Using infinitely near base points, show that we can arrange fibers of
type Di (take the pencil defined by a double line plus a reduced line,
together with a smooth cubic meeting each line transversally), Ёв (.
triple line together with a smooth cubic meeting it transversally at
three points), Ё? (a triple line together with a smooth cubic such that
the line is tangent to the cubic at one point), and Ё& (a triple line
together with a smooth cubic such that the line meets the cubic at an
inflection point).
7.	The numerical equivalence class of Kx '¦ Show that Kx is numerically
equivalent to rf for some r G Q. Show r = 0 if and only if either
g = 1, d = 0, and rrii = 0 for all i (X is a complex torus if L is trivial
and is a hyperelliptic surface otherwise), g = 0, d — 2, тгц = О for
all » (X is а КЗ surface), g = 0, d = 1, and m\ = m.2 = 2, no other
multiple fibers (X is an Enriques surface), or g = 0, d = 0, and (up to
permutation) (mbm2,...) = B,2,2,2), B,4,4), B,3,6), C,3,3) {X
is a hyperelliptic surface). Likewise, r < 0 if and only if g = 0,d = 1
and there is at most one multiple fiber, or g = 0, d = 0, there are less
than two multiple fibers of arbitrary multiplicity or there are three
multiple fibers and the possible multiplicities up to permutation are
B,2, m), m arbitrary, B,3,4), B,3,5). In all other cases r G Q+. (Not
all of these cases can arise for an algebraic surface; see Exercise 11.)
8.	If, in the notation of Exercise 6, г = 0, or equivalently Kx is numeri-
numerically trivial, then there is a finite unramified cover where Kx is trivial.
Thus, Kx is a torsion line bundle, and in fact the order of Kx is either
1,2,3,4, or 6. In all cases, \2KX = 0.
9.	An algebraic surface which is an elliptic surface in two different ways
satisfies: Kx is numerically equivalent to 0.
10. Let E be an elliptic curve with an automorphism ф of order d =
2,3,4,6 (here automorphism is understood as automorphism of the
group structure, so that ^@) = 0). Let С be a smooth curve and С а

7. An Introduction to Elliptic Surfaces 193
finite unramified cyclic cover of order d, with т: С —> С a generator
of the automorphisms of С over C. Define the surface X to be the
quotient of E x С by the automorphism group generated by (e,a:) i->
(</>(е),т(х)). Show that X is an elliptic surface over C, and that all
fibers of X are smooth elliptic curves isomorphic to С Show that the
sections of X —> С correspond to morphisms f:C—tE such that
11.	Let L be a line bundle on the curve С such that L? = Oc but L is not
trivial. Let </2 ? Н°(ЬА) = С and </3 G Я°(?6) й С be two sections
such that the curve with Weierstrass model y2 = Ax3 — g^x — рз is a
smooth elliptic curve E. (Here we use a fixed trivialization of I? to
identify L4 and Le with Oq.) Let X be the corresponding Weierstrass
model. Show that X is an elliptic surface over С with all fibers smooth,
such that the line bundle (R}n,Ox)~l is equal to L. If С —> С is
the unramified double cover of С corresponding to L, show that the
pullback of X to С is the product elliptic surface E x C, and that X
is the quotient of E x E by the involution (e,i) i-* (—e,r(a;J), where
т is the involution on С corresponding to the double cover С —> С.
Do the same for the case where L is a line bundle on С of order 3,4,6
and E is an elliptic curve with an automorphism of order 3,4,6. (Here
you will need to use the fact that such a curve always has either g%
or рз equal to 0 in order to define the appropriate Weierstrass model.)
Using this, show that, for the elliptic surfaces X constructed in the
previous problem, the line bundle L is always nontrivial.
12.	In the algebraic case there is a condition on the multiple fibers if L is
trivial. Suppose that 7r: X —» С is an algebraic surface with L trivial;
thus all fibers have smooth reduction. Since X is algebraic, there is a
holomorphic multisection ? of тт. Taking the normalization of X x с 2,
there is a finite cover тг': X' —> С" of X —> С such that X' has no
multiple fibers and has a section. We may assume that C" —> С is
Galois. By flat base change the line bundle L' = (Д^тг'^Ох') is
the pullback of L and is therefore trivial. Thus, X' - E x C" for
some smooth elliptic curve E and X is the quotient (E x C')/G, for
some finite group of automorphisms G acting faithfully on C" with
C'/G = C. Show that, after adjusting C", we may assume that G acts
faithfully on E. Next, using the previous two problems, show that the
assumption that L is trivial forces the image of G to be contained
in the translation subgroup of E. Thus, G is a finite subgroup of the
translation group E, and is therefore abelian. The multiple fibers of X
arise from fixed points for the action of G on С For a point p* € С
which is a ramification point of order m* for C" —> C, a small loop
7i enclosing Pi simply counterclockwise lifts to a unique element of G
(since G is abelian) and thus defines an element t.Pi of E, corresponding
to the local invariant of the multiple fiber. Show that ?\ fp. = 0 in E;
this is a formal consequence of the fact that the abelianization of the

194 7. An Introduction to Elliptic Surfaces
fundamental group of С — {p\,...,Pk) is
Hi (C; Z) © ( ф Z7i)/ZGi + ¦
i
For example, if L is trivial X cannot have just one multiple шег
of multiplicity m, or several multiple fibers whose multiplicities are
pairwise relatively prime.	j	:;
Show by a direct construction that the necessary condition Yli ?p( =0
is also sufficient. Working a little harder, show that if L has degree
0 but is not trivial, then we can arrange arbitrary preassigned multi-
multiplicities for X (this amounts to showing that, if т is an automorphism
of E of order d = 2,3,4,6, then Etc,т'(() = 0 for all ? G E.) Less
trivially, if X has a fiber whose reduction is singular, or equivalently
X has positive Euler number, or equivalently deg L > 0, then it fol-
follows from work of Shafarevich and Ogg that we can arrange arbitrary
multiplicities for the multiple fibers. Likewise, in the complex analytic
category, by using logarithmic transforms we can arrange arbitrary
multiplicities even when L is trivial.
13.	Let С —> С be a double cover branched at points Pi,... ,P2* and with
corresponding involution r: С —* С. Let E be an elliptic curve. Then
the surface X which is the minimal resolution of the quotient of E x С
by the involution (e, t) i-+ (— e, r(t)) is an elliptic surface with con-
constant j-invariant and with 2k singular fibers of type D*. Show that
the line bundle L satisfies I? = Oc(Pi + ¦ • • + P2k) and corresponds
to the line bundle defining the double cover, and that the sections
fl2 e H°(C;OcBpi + --+2p2k)) and g3 e H°(C; OcCPl + - ¦ -+3p2k))
are constant sections. Consider a cyclic cover С —> С of order 3, 4, or
6, such that т is a generator for the group of covering transformations
and take for E an elliptic curve with the appropriate complex multi-
multiplication, denoted Ьуен fi(e). Let X be the minimal resolution of
the quotient of E x С by the group generated by (e,t) >-* (fi(e),r(t)).
Show that at a point of total ramification the singular fiber of X -+ С
is of type Ё6 in case the order of fj, is 3, ?7 in case the order of fi is 4,
and Eg in case the order of ц is 6.
14.	Suppose that it: X —* С has a section a. Using adjunction for a give
an easier proof of the canonical bundle formula in this case.
15.	Suppose that X is an elliptic surface with д = 0 and d = 1 and either
no multiple fibers or a single multiple fiber of multiplicity m. Show
that Kx is either —/or —F, where mF is the multiple fiber. Note
that X is a good generic surface if and only if there are no reducible
fibers. Show that on every smooth blowdown X of X there exists an
effective smooth divisor in |—К% |- Show that if X blows down at all to
a rational ruled surface F^, then it blows down to P2, and that in this
case either / or F is the proper transform of a smooth cubic. Discuss
when each case happens. (See Exercise 9 in Chapter 5.)

7. An Introduction to Elliptic Surfaces 195
16. Let X be an elliptic surface with a section. Show that there is a mor-
phism X —> ?(?%) which realizes X as a double cover. In case X is an
elliptic surface over P1 and X = X, show that X is a double cover of
F2* with к = ps + 1. Show further that the branch locus of the map
X —> F2fe is of the form cr+B', where В is a smooth curve in |3<r+6A:/|
disjoint from a.

Rector Bundles over Elliptic
Surfaces
i chapter gives a variety of techniques for understanding stable rank 2
;tor bundles V over elliptic surfaces X. Because of constraints of space
time, we do not give complete proofs of all of the results, and this
pter is mainly intended as a sampler of various methods for studying
lies over X. For simplicity, we shall assume that X is simply connected,
i that the base curve is P1, and shall concentrate on the case where X has
a section a and the fibers are generic, i.e., smooth or nodal. After describing
allowable elementary modifications for singular fibers, we begin with the
technically much simpler case where the vector bundle V has degree 1 on
jBvery fiber. In this case, there is a unique stable bundle on each fiber, and
correspondingly there is a unique stable rank 2 vector bundle Vo on X, up to
twisting by the pullback of a line bundle on the base curve, which restricts
to a stable bundle on each fiber. The bundle Vo then generates all stable
bundles which have degree 1 on every fiber, via elementary modifications. A
second approach to the moduli space is via sub-line bundles and extensions.
We give a brief description of Donaldson invariants and some methods for
computing them, and apply this to calculate the 2-dimensional Donaldson
invariants coming from stable bundles of degree 1 on every fiber. Next we
turn to the case where the degree of V on every fiber is 0. Here the moduli
space of rank 2 semistable bundles of degree 0 on an elliptic curve is a
P1, and so the geometry of the moduli space when the fiber degree is 0 is
much more complicated. We have tried to outline most of the important
ideas in this case. Finally, we describe the general shape of the Donaldson
invariants and how this shape both reflects the geometry of the moduli
space and determines the smooth classification of elliptic surfaces.
Stable bundles on singular curves
We begin with more discussion of the program outlined in Chapter 6 of
making allowable elementary modifications. Here we will allow singular

200 8. Vector Bundles over Elliptic Surfaces
and use Exercise 6, j*L corresponds either to the Л-module R/xyR or to
R/xR ф R/yR. In the first case, R/xyR has the short free resolution
0-+й-»й-+ R/xyR -> 0,
where the first map is multiplication by xy, and, in the second case, there
is the resolution
0 -» R2 -* R2 -* R/xR © R/yR -> 0,
where the map R2 -* R2 corresponds to the matrix I „ I. Thus, in
both cases j»L has projective dimension 1.
Next we must verify the formula for pi(ad V')- Away from the singular-
singularities of C, we have ci(V') = Ci(V) - [C]. Thus, a similar formula holds
over all of X. To find Ci{V) we can use Riemann-Roch on X, applied to
the formula \{У') = \ty) — x(L). A brief calculation with Riemann-Roch
and the adjunction formula gives ca(V) =c^(V) — ci(V)-C + degL. Thus,
p1(ad V) - pi(adV) = 2(deg(F|C) - 2degL) + C2, as claimed. ?
Stable bundles of odd fiber degree over elliptic surfaces
For the rest of this chapter, unless otherwise noted, we shall let тг: X -* P1
denote a regular elliptic surface over P1 with a section a (although much
of what follows will work under more general circumstances). We shall
also assume that all fibers of тг are irreducible, with at worst ordinary
double points. Let p3 = pg{X). By the canonical bundle formula Kx =
Ox{{pg - 1)/), where / is a general fiber of ir. Thus, Kx • <r = p9 — 1, so
that
a2 = -2 - Kx • о = -pg - 1.
In this section, we shall consider rank 2 vector bundles V on X such that
deg V\f = 1, or more precisely such that det V — a + kf. Here since we
will make allowable elementary modifications along the fibers we will not
try to specify к in advance, since an allowable elementary modification will
replace det V by det V — f. Doing two such modifications replaces det V
by det V - 2/ = det(V ® Ox (-/)), and so it is natural to allow ourselves
to identify two bundles which differ by a twist of the form Ox(nf). Note
that (<r + fe/J = a2 + 2fc = -pg - 1 mod 2 and 3x(Ox) = 3(p3 + 1) =
-pg - 1 mod 2 as well. Thus, since Pi(adV) = Д2 mod 4, we see that
-pi(adF)-3x(Ox) = 0 mod 2 for our choices. In particular the expected
dimension of the moduli space is always an even integer It.
Since det(V]/) = Of{p), where p = a ¦ /, it follows by Theorem 9 of
Chapter 4 that there is a unique stable bundle on / with determinant
equal to O/(p). A similar result holds for the singular fibers:

8. Vector Bundles over Elliptic Surfaces 201
7. Let С be an irreducible nodal curve of arithmetic genus 1, and
be a smooth point of С Then there is a unique stable rank 2 bundle
f/on С with det W = Of(p).
jf. If W is a stable rank 2 bundle on С with degW = 1, then by
ft Riemann-Roch theorem there is a section of W, and thus a nonzero
Of —> W. By stability the cokernel must be torsion free, and thus
a torsion free rank 1 sheaf L with degL = 1. A local calculation
ercise 7) shows that, if S = C{x,y}/(xy), then there are no nontrivial
sions of S by 5. It follows that, in the above situation, since W is
lly free, L is locally free as well. Thus, ? is a line bundle of degree 1 on
i'and so by Riemann-Roch L = Of(p) for some smooth point p, where
B/(p) = det W. Hence V is given as an extension of 0/(p) by Of. The
ension cannot be split if W is stable. Now Hl{Of(-p)) has dimension
that there is a unique nonsplit extension of 0/(p) by Of. An easy
at along the lines of the smooth case (Theorem 9 in Chapter 4)
that this extension is stable. Thus, there exists a unique stable W
det W = Of(p), as claimed. ?
we can begin the analysis of stable bundles on X. Let Д = а + kf
let w be the mod 2 reduction of Д. For an integer p, we consider
t/,p)-suitable ample divisors H, and shall always understand stability to
with respect to such a divisor. By Theorem 5 of Chapter 6 and the
that there are no strictly semistable bundles on the fibers, a rank 2
ctor bundle V is Я-stable if and only if its restriction to some fiber / is
ble, if and only if its restriction to almost all fibers / is stable. Moreover,
the proof of Theorem 5 in Chapter 6 shows that, in case V is not stable,
a destabilizing sub-line bundle of V is destabilizing on every fiber. Note
finally that there are no strictly semistable bundles in this case.
Our first result says that every Я-stable bundle with det V = а + kf is
good.
Lemma 8. IfVis H-stable for a suitable ample divisor H, then V is good.
Hence the moduli space corresponding to V is always smooth of dimension
equal to -pi(ad V) - 3x(Ox)-
Proof. Suppose that ip € H°(X; ad V ® Kx)- Then tp defines a map from
V to V (8> Kx of trace 0. Restricting to a fiber /, and using Kx\f = Of,
<p\f: v\f -* V\f is a map of trace 0. Since V\f is simple for almost all /,
ip\f = 0 for almost all /. But then <p = 0. Thus, H°(X; ad V ® Kx) = 0,
so that V is good. ?
Next let us show that there is, up to a twist, a single rank 2 vector bundle
Vq on X such that V0\f is stable for every fiber /. This bundle will turn
out to be the ancestor of all stable rank 2 bundles on X.

202 8. Vector Bundles over Elliptic Surfaces
Proposition 9. With notation as above, there exists a rank 2 vector
die Vq on X such that Vo\f is stable for every fiber f. Moreover, we
construct Vq as follows: start with an arbitrary rank 2 vector bundle V q
X such that there exists a liber / with V\f stable, and successively perform
aJIowabie elementary modifications on V. Then this procedure terminate
with a Vo as desired. Finally, Vo is unique in the following sense: ifVg afaj
has the property that det Vo' = а 4- k'f and V0'\f is stable for all f, tbti
Vq = Vo ® Ox(nf) for some n.
Proof. By Lemma 8, there is an absolute bound pi(adV) < -Sx(Ox)'
for every bundle V such that V\f is stable for almost all /. Thus, if щ
begin with a fixed V and make allowable elementary modifications, then
since this procedure increases pi (by Proposition 6) it must terminate.
Clearly, the only way this can happen is when we reach a Vo for which no
more allowable elementary modifications are possible, i.e., Vo|/ is stable for'
every fiber /. Thus, it suffices to find a V such that V/ is stable for almost'
all /, or equivalently for one /.	.-'A
Fix a fiber /, and let p = а П /. The unique stable bundle W on / with
detW = Of(p) is given as an extension of Of(p) by Of. This suggests
that we try to construct V as an extension of Ox(a) by Ox, such that the
restriction of V to / is not split. While this may not always be possible, we
are free to modify a by adding Nf for some unspecified N. The extensions
of Ox(p + Nf) by Ox are classified by Hl(X; Ox{-o - Nf)) and the
extensions of Of(p) by Of are classified by H1(f;Of(—p)). Moreover, the
restriction of an extension of Ох(о + Nf) by Ox to / corresponds to
looking at the restriction map Hx(X;C>x(-<7 - Nf)) -» Hl(f\Os(-p))
which arises from the short exact sequence
0 - Ox{-o -(N+ 1)/) - Ox{-a - Nf) - Of(-p) -» 0.
Thus, the cokernel of the map on the Hl's lies in HP(X;Ox(-o -
(N + 1)/)). By Serre duality, Н*(Х;Ох(-<т - (N + 1)/)) is dual to
H°(X; Ox(a + (N + 1)/ + (pg - 1)/)). Since /, a, and pg are fixed, this
group is zero for all N <C 0 (since, for example, we can arrange that the
divisor a+(N+l)f + (pg — l)f has negative intersection with a given ample
divisor). Thus, for a suitable N we can find an extension of Ox{a + Nf)
by Ox such that the induced extension on / is not split, and is therefore
stable.
Finally, we must establish the uniqueness property of Vo- If Vq is an-
another bundle on X with the property that V0'\f is stable and det Vq \f =
detVbl/ = Of(p) for all /, then dimHom(V0|/, V0'\f) = 1 for every /,
and every nonzero element of Hom(Vo|/, V0'\f) is an isomorphism. Thus,
H?KtHom(V0, Vq) = L is a line bundle on P1. Moreover, there is a nat-
natural section of i?°ir»Hom(Vb, Vq) ® L~l = Opi, and thus there is a nat-
natural section of Hom(Vo, Vo' ® ir*L~l). The corresponding homomorphism
Vo —* Vo ® it*L~x is an isomorphism on every fiber and thus an isomor-

8. Vector Bundles over Elliptic Surfaces 203
. Thus, up to a twist by ж*Ь~х = Ox(nf) for some n, Vb and Vo' are
jomorphic. ?
%¦
Jlprollary 10. For p = -3x@x) and w such that иг = p mod 4, the
duli space of Я-stabJe rank 2 vector bundles V with the invariants w
I p consists of a single reduced point. О
i we shall see, Corollary 10 computes a Donaldson invariant: it implies
the 0-dimensional Donaldson invariant corresponding to the choices
; and p above is always 1. We will describe how to compute more
sting invariants below.
Zariski open subset of the moduli space
5 Proposition 9, we can give a description of a Zariski open subset of the
(luli space of Я-stable rank 2 bundles V with px(ad V) = ~3x(Ox) - 2t
nonnegative integer t. As with ruled surfaces, we first describe the
behavior of a stable bundle V with given invariants w and p. The
lies on an elliptic curve / with determinant 0/(p) which are the "least"
able are those of the form Of(q) ®Of(p — q) for some point p € /, and
i maximal destabilizing quotient is 0/(p - q). Reversing this procedure,
8 should start with the stable bundle Tv on / which is the nontrivial
sion of Of(p) by Of and look for surjections Tv —> Of(q).
11. For an elliptic curve f and a point q e /, there ex-
,a surjection Tv —> Of(q), and it is unique mod scalars. Moreover,
^(f;Hom(Fp,Of(q))) = 0. A similar result holds if instead f is an irre-
dble nodal curve.
jf. We shall just write out the proof in the case where / is smooth;
: case of a singular / is similar. There is an exact sequence
I	0 - Of(q - p) - Hom(Fp, Of(q)) - Of(q) - 0.
if q ф p, then H°(f;Of(q-p)) = H\f; Of(q-p)) = 0 and Hl(f;Of(q)) =
0. Thus, H1{f;Hom(J:p,Of(q))) — 0 and the unique nonzero section of
H°(f;Of(q)) mod scalars lifts to give a map Tv —* Of(q). If this map is
not surjective, then its image lies in Of(q — d), where d is an effective
nonzero divisor. Thus, deg(g - d) < 0, contradicting the stability of Tv. So
the map Tv —* Of(q) is surjective. This concludes the proof iiq^p.
In case q = p, we are given a surjection Tv —> O/(p) and need to
show that it is unique mod scalars. If there were two linearly indepen-
independent maps from Tv to 0/(p), then some linear combination of them would
have to vanish at any given point. But as we observed above, by stabil-
stability every map from Tv to a line bundle of degree 1 is surjective. Thus,

204 8. Vector Bundles over Elliptic Surfaces
p, Of(p)) = 1, and then it follows from Riemann-Roch or from
the exact cohomology sequence associated to
0 — Of -> Hom(Fp, Of(p)) -> Of(p) -> 0
that #!(/;Hom{Tv,Os(p))) = 0. D
Now choose points ?i,...,zt € P1 lying under distinct smooth fibers
Д,..., ft, and choose a line bundle of degree 1 on each /i. Since a line
bundle of degree 1 on /i can be written in the form 0/Д<&) for a unique
point q, € /i, the choice of the t fibers and t line bundles amounts to a
choice of t points qi,.. ¦, qt 6 X such that for different г and j, qt and qt
lie in different fibers. In this way we find a Zariski open subset of Syme X.
Let ji- fi —* X be the inclusion. By Lemma 11, Hom(Vo, ji*O/t(qi)) = С
and up to a nonzero scalar (corresponding to Aut Од(ф) = С*) there is a
unique surjection from Vq to Of^Qi)- Define V by the exact sequence
It follows that V is uniquely specified by the choice of points q,. We see
that V is obtained from Vq by elementary modifications which are dual to
allowable elementary modifications, as we would expect.
We still must check that this construction gives a dense open subset
of the moduli space, or in other words that the bundles obtained with a
repeated choice of fiber, or using destabilizing sub-line bundles on the fiber
of higher degree, or using singular fibers, form a subset of the moduli space
of dimension less that —p — 3x(Ox) = 24. The argument follows along
lines similar to those outlined in Chapter 6 for ruled surfaces, although the
details are more complicated (see [36] for details). Summarizing, we have:
Theorem 12. There exists a Zariski open and dense subset of the moduli
space of stable bundles V on X with ci(V) — а + kf and pi(adV) =
—Ъх{Ох) — It which consists of t points in X lying in smooth distinct
fibers. Thus, there is a rationaJ fibration ЯЛ —¦ Sym'P1 = Pe, and the
fibers are a product of t elliptic curves. ?
Once again, we see the structure of X reflected in the birational structure
of the moduli space. Note that, instead of describing ГО1 as a fibration, we
could give another description as follows: the choice of t points lying in t
distinct fibers is the same as choosing t general points on X. Thus, the
moduli space is birational to the Hilbert scheme Hilb* X.

8. Vector Bundles over Elliptic Surfaces 205
in overview of Donaldson invariants
ffe will give a very brief discussion of Donaldson invariants for a smooth
(dented 4-manifold M in this section, and discuss how to compute them
inder special circumstances in case M is an algebraic surface. General
inferences are [26], [27], and [40].
Let X be an algebraic surface and let Я an ample divisor on X. Let
tibe the moduli space of Я-stable rank 2 vector bundles with invariants
Djjj), for a fixed choice of w and p. For our purposes, we will need to make
le following very special assumptions:
I (i)	Every bundle V corresponding to a point of Ш1 is good. Thus, Ш1 is
i\'	smooth of dimension —p — 3x(Ox) = d.
ji(ii)	The space Ш1 is compact.
f Hi)	There exists a universal vector bundle V —* X x ЗЛ.
|pte that the bundle V is only well defined up to twisting by a line bundle of
«form ir^L, where L is an arbitrary line bundle on Ш1. However, the vector
nidle ad V is well defined, independent of the twisting. In fact, we can
ae ad V even if V is not defined. To do so, one can show that, following
! discussion in Chapter 6, there is always a universal Px-bundle ir: P —+
x 9Л, which equals P(V) in case there is a universal bundle V. Thus,
fktet can define the rank 3 bundle ir.Tp/xxon, where Tp/xxm is tne relative
/tangent sheaf, and this rank 3 bundle equals ad V in case V exists. In any
iptee, we have the class p\ (ad V) € H4(X x Ш). Here of course pi (ad V) =
^rica(ad V) by definition in case there is a universal bundle V. Now given a
'class ? € H4(X x Ш1), we have the "slant product map" from H2{X) to
Я^ЯИ), defined roughly as follows: take the class f, and look at the part of it
which lies in Я2 (X)<g>H2 (Ш) under the Runneth decomposition. Neglecting
torsion, H2(X)<8H2(m) =* Нот(Я2(Л"),Я2(Ш1)), and the homomorphism
corresponding to ? is by definition slant product with ?. In particular, we
can take slant product with - \pi (ad V). Here the factor — \ is chosen since,
in case w = 0 and a universal bundle exists, one can check that p\ (ad V)
is always divisible by 4. Indeed, in this case Pi(ad V) is just -4c2(V). The
end result is the д-тар
Using the //-map, we can then define the DonaJdson polynomial D =
as follows: it is a multilinear Q-valued form on Яг(.Х') given by
/an
where the integral means evaluation of fi(a)d on the fundamental class of
Ш1 (with its natural orientation). Of course, a priori the value of D*iP also
depends on Я, the ample divisor which we used to define stability. We will
discuss the dependence on Я shortly.

206 8. Vector Bundles over Elliptic Surfaces
For example, if d = 0, then under our assumptions on Wl, D*p simply
counts the number of points in 3Jt. More generally, D*p(H) can be inter-
interpreted as the degree of ОТ in a suitable projective embedding. Thus, it ц
an important numerical invariant of the moduli space. These numbers have
been calculated for certain classes of surfaces, for example, K2> and elliptic
surfaces. For P2, let H be the class of a line. Then there are two sequences
of numbers Dqc(H) and D^p(H), where w is the nontrivial element of
//^(P2; Z/2Z). Assuming certain conjectures on the transition functione
of Donaldson polynomials, Gottsche has shown that these sequences are
denned by a very nonobvious generating function involving modular func-
functions; see [53].
We have already seen that Assumption (iii) above is not necessary in
order to define the /x-map, and thus it is not necessary in order to define
D*p. In fact, it is easy to dispense with (i) as well. For example, if ОТ is
generically reduced of the expected dimension, or in other words if each
component of ЯЛ contains a good bundle, then the above discussion goes
over word-for-word to define an element fi(a) € Н2(Ш). Now every com-
compact projective variety has a fundamental class, and so D*p can be defined
as before. It might happen that SOT is not reduced, or worse still, has possi-
possibly nonreduced components of the wrong dimension. In these cases, there
is still a procedure for defining D*p [40], [16].
The most serious assumption above is (ii), that ffl is compact. Typically
9Л is not compact, and it must be compactified by adding in Gieseker
semistable torsion free sheaves. In this case, one can prove that the classes
fi(a) extend in a natural way to the compactification [115], [100], [80], and
then D*p may be defined as before.
The above construction can be made for a general smooth oriented 4-
manifold M, together with a Riemannian metric g. Let P be a principal
SUB)-bundle over M, with the unique characteristic class сг(Р) = с 6
H4(M; Z) = Z. More generally, we can consider principal S0C)-bundles P,
with characteristic classes w2(P) € H2(M;Z/2Z) and p^P) € H4(M;Z) ?
Z. However, for simplicity we shall just stick to the case of 51/B). Let A
be a connection on the principal SC/B)-bundle P. Then the curvature Fa
is a 2-form with values in the vector bundle ad P. Using the metric g,
there is an associated Hodge ^-operator from Ak(M) to A4~k(M), where,
in the notation of Chapter 4, Ak(M) is the bundle of C°° 2-forms on M.
In particular, ¦: A2(M) —> A2(M) satisfies *2 = Id, and thus as in the last
section of Chapter 4 we can take the decomposition of п2м into the +1 and
—1 eigenspaces for ¦:
There is an induced decomposition of the space А2(М)(вАР). We call A
anti-self-dual if FA e fli(M)(adP). If b?(M) > 0, for a generic metric g,
the set of all anti-self-dual connections A on P, modulo the action of the
group of C°° bundle automorphisms of P, is a finite-dimensional manifold

8. Vector Bundles over Elliptic Surfaces 207
= M, and morally speaking there is a universal SC/B)-bundle
jijjover M x M. (Just as in the holomorphic case, we might have to use
associated SOC)-bundle adT' instead, which always exists even if the
ndle V does not.) Thus, we can again use slant product with сг^), or
AV) if no universal St/B)-bundle exists, to define a /i-map
^	ц:Н2(М)^Н2(М).
«^case M is a complex surface and g is the Hodge metric associated to
q ample divisor H, we have identified M. with the moduli space Ш1 of
! rank 2 bundles V with ci(V) = 0 and c2(V) = c2(P) in Theorem
Jjof Chapter 4. While a Kahler metric need not be generic in the sense
that M. need not be a manifold, it turns out that M always can be locally
tjjodeled on a real analytic space and that in this sense the spaces M and
jjtare isomorphic. Moreover, one can identify the corresponding //-maps.
"'Just as with Ш1, the manifold M is typically noncompact, even for a
metric. One can construct a natural compactification X(P, g), the
compactification, for M. The space X(P,g) is not in general
old, but it is naturally a stratified space and carries a fundamen-
class [X(P,g)\. Furthermore, the classes ц(а),а е H2(M), extend in
mique way to classes in H2(X(P,g)), also denoted by ц(а). Thus, we
, define the Donaldson polynomials D^c for a general smooth oriented
lifold M as before, by evaluating fi(a)d over [X(P,g)]. (To be com-
comely precise, we also need to choose an orientation for M, but shall not
cribe the procedure for doing so here.) In order for the above evaluation
febe meaningful, d should be one-half the real dimension of X(P, g), or
^uivalently of the manifold M. It follows from the Atiyah-Singer index
гт„огет that dimR.M is even exactly when b%(M) — bi(M) is odd, and
thus Donaldson polynomials can be defined in this case. Of course, in case
M is an algebraic surface, then bi(M) is even and, by the Hodge index
theorem, b%(M) is odd. Thus, the Donaldson polynomials are defined in
this case, and one natural choice of orientation is to always choose the com-
complex orientation on M. = Ш1. Similar constructions define the more general
Donaldson polynomials D^p.
In case M is an algebraic surface and g is a Hodge metric, there are two
potentially different definitions of the Donaldson polynomial, corresponding
to the choice of either the Gieseker or the Uhlenbeck compactification.
These two compactifications are quite different, but the resulting definitions
of the Donaldson polynomial agree, as has been shown by Morgan [100]
and Li [80]. Both [100] and [80] investigate the relationship between the
two compactifications, and in [80] there is an algebro-geometric definition
of the Uhlenbeck compactification, which is a generalized blowdown of the
Gieseker compactification.
To produce actual invariants of the C°° structure on the 4-manifold
M, we need to analyze the dependence of the moduli space M{P, g) and
its compactification X(P,g) on the metric g. The moduli space M(P, g)

208 8. Vector Bundles over Elliptic Surfaces
acquires singularities when the bundle P admits reducible anti-self-dual
connections. By Hodge theory, this happens when there exists a complex
line bundle L on M such that the complex 2-plane bundle V over M asso-
associated to the standard representation of SUB) is isomorphic to L © L~x
and such that ci (L) is orthogonal to every self-dual harmonic 2-form with
respect to the metric g. The first condition is simply the condition that
с = c2(P) = -C2, where С = ci(L) 6 H2(M;Z). The second condition
is vacuous if the intersection pairing on H2(M) is negative definite, and
indeed the singularities forced on the moduli space in this case for с = 1
are a crucial ingredient in Donaldson's famous theorem that every smooth
definite 4-manifold has a diagonalizable intersection form. In general, the
condition that a metric g admit reducible anti-self-dual connections has
codimension b%{M) in the (infinite-dimensional) space of all metrics on
M. Thus, as we noted above, these singularities do not appear in case
Ь$(М) > 0 and g is generic.
Since M is noncompact, we also have to consider what happens to the
(singular) space X(P,g). The smaller-dimensional strata of X(P,g) in-
involve moduli spaces M{P,g), with (^(P1) < Oi(P). In case P1 admits
reducible anti-self-dual connections, these will contribute "extra" singu-
singularities to X(P,g). By the above discussion, such singularities will arise
whenever there are classes ? 6 H2(M;Z) orthogonal to every self-dual
harmonic 2-form and satisfying
-c < C2 < 0.
By the Hodge theory arguments of Chapter 4, if M is a Kahler surface and
u) is the Kahler form of a Kahler metric, an integral class orthogonal to
every self-dual harmonic 2-form on M is an integral A, l)-class orthogonal
to ш, and in particular it is the first Chern class of a holomorphic line
bundle. Now if V is a strictly w-semistable rank 2 bundle on M, then there
is an exact sequence
0 -» OM(D) ^V-+ Ou(-D)
with w D = 0, and V is S-equivalent to the torsion free sheaf Om{D) ©
Om{-D) ® Iz- The double dual of this sheaf is a vector bundle of Chern
class -С2 < с, with a reducible anti-self-dual connection. Thus, singularities
of the moduli space are related to strictly semistable bundles.
Returning to the case of a general 4-manifold M, if frj"(M) > 1, a generic
path of metrics linking two generic metrics will not contain any metrics
admitting reducible connections, for M(P,g) or for any of the boundary
pieces of X(P,g) which involve moduli spaces Л<(/*',</) for smaller values
of c2(P')- One can then show that the value of the Donaldson polynomial
is unchanged along this path, and thus that there is a well-defined Don-
Donaldson polynomial Dj?c. Note that this picture is quite different from the
one described in Chapter 4 for the behavior of the moduli space under a
change of polarization. In that case, a generic path of metrics linking two

8. Vector Bundles over Elliptic Surfaces 209
dodge metrics on an algebraic surface X with b^iX) > 3 will not involve
reducible connections, except possibly at the endpoints. However, every
path of metrics linking the two Hodge metrics which itself consists entirely
>f Hodge metrics may well involve metrics which admit reducible connec-
connections- In particular, in the algebro-geometric case, there is no distinction
between the case Ь%(Х) - 1 (i.e., pg(X) = 0) and fcj"(X) > 1 (pg(X) ф 0).
Thus, it is not a priori clear that algebro-geometric definition of the Don-
Donaldson invariant sketched above, in the case X is an algebraic surface with
ц,ф О, via moduli spaces of stable bundles, is independent of the ample
line bundle used to define stability. In fact, aside from a few special cases
jiich as КЗ surfaces, it is a challenging open problem to see this via alge-
algebraic geometry! However, the equivalent C°° version does not depend on
the choice of metric provided that b?(M) > 1, or equivalently pg(M) > 0.
i In case 6j(M) = 1, a generic path linking two generic metrics will usu-
usually contain metrics admitting reducible connections. Thus, the Donaldson
polynomial is not independent of the metric, but depends on a "chamber
Sbructure" in the positive cone
!	С = {x e H2(M;R): x2 > 0}.
In this case, the induced chamber structure on the ample cone is the one
described in Chapter 4. The change in the Donaldson polynomial as we
Cross a wall of the chamber is surprisingly complicated [30].
The 2-dimensional invariant
>¦,
Let us use the discussion of vector bundles on the elliptic surface X to
calculate its 2-dimensional Donaldson invariant. In this case, the moduli
space Ш birationally consists of the choice of one point p lying in a fiber
of X. Of course, this last condition is automatically satisfied since every
point of X lies in a (unique) fiber, and as we shall see the moduli space
is exactly X itself. In particular it is compact. We could also see that
the moduli space is compact as follows: as there are no strictly semistable
bundles, the moduli space can only be compactified by adding in torsion
free, non-locally free sheaves whose double duals are stable. If Ш1 were not
compact, boundary points would correspond to torsion free sheaves V with
Pi(Vvv) > pi(V)+4. This would imply that Vvv lives in a moduli space of
expected dimension at most —2, contradicting the fact that for the bundles
we consider the moduli space is always smooth of the expected dimension.
The recipe for computing the Donaldson invariant in this case is as fol-
follows: let V be a universal sheaf over X x X. Here the first factor X is
to be viewed as the original surface and the second is the moduli space,
and V is a vector bundle over X x X. The universal property of V may be
expressed as follows. Let Vo be the unique stable vector bundle on X for
which pi(adVb) = ЩОХ) and detV0 • / = 1. Given pe X, let / be the

210 8. Vector Bundles over Elliptic Surfaces
fiber through p and let i: f —> X be the inclusion. Then, if p does not lie
on a singular point of a fiber, V\X x {p} = Vp, where by definition there is
an exact sequence
and the map Vb —> i*Of(p) is unique mod scalars. If p is the singular
point of its fiber, then we replace Of(p) by n,O,-, where n: / —> / is the
normalization map. We also have n*Oj = Hom(mp,Of) (Exercise 7). By
Lemma 11 in the singular case, there is again a unique map V& —> n,Of,
and it is surjective. Now take pi(ad V) G H*(X x X) (all cohomology with
rational coefficients). Using the projection H*(X xX) -> H2(X)®H2(X),
and given a e Нъ{Х), we can then define /j,(a) e H2(X) by taking slant
product with ^pi(adV). By definition, then,
D{a) = m(«J G H*(X) S Q.
To carry out this program we will have to construct the bundle V. First
we must fit together the sheaves i»0/(p) as p ranges over X. To do so, let
D be the fiber product X Xpi X С X x X. Then D is smooth except at a
point (p, p), where p is a singular point of a singular fiber. Near such a point,
D has the local equation xy = zw, which is a hypersurface singularity, in
fact an ordinary double point in dimension 3. Let D be the diagonal inside
X x X. Then ВсД and we consider the sheaf Ор(Щ. In the complement
of the singular points of D, D is a Cartier divisor in D and the notation
Оо(Щ is meaningful. Moreover, if i: D —> X x X is the inclusion, we
clearly have г*Оо(Щ\Х x {p} = i»O/(p), provided that p is not a singular
point of a singular fiber. Thus, Od(P) has accomplished the goal of fitting
the sheaves uOf(p).
In case p is the singular point of a singular fiber, we must still assign
meaning to i*Of(p). First we should interpret O/(p) as the unique torsion
free rank 1 sheaf on / of degree 1 which is not locally free at p. Thus,
C/(p) = Hom(vcip, Of) = n,Opi. One way to fit this sheaf into the family
constructed above is to take the well-defined sheaf Id/Id and then set
Od(D>) = Hom(In/lD,OD). Local calculations (Exercise 12) show that
Od(D>) is locally of the form Ie/Id, where E is locally defined in coordinates
by the ideal (x—w, y—z) С Id- Moreover, again by Exercise 12, if we dualize
the inclusion Id/Id —* Od, we obtain (locally) an exact sequence
0 -+ OD -» OD{W) -» Оц -» 0.
In particular Od(P) is flat over the second factor X m X x X since O-q is
flat over X (the projection is an isomorphism) and Od is flat over X by
the local criterion of flatness, since D is a hypersurface inside the smooth
family X x X -»X, and all fibers of the morphism D —* X have dimension
1. For p a point in a smooth fiber /, or a smooth point of a singular fiber /,
we clearly have Оо(Щ\Х x {p} = O/(p), viewed as a sheaf on X. \n case p
is the singular point of a singular fiber /, Od(P)\X x {p} = Hom(mp, Of).

8. Vector Bundles over Elliptic Surfaces 211
ndeed, such a statement is true locally since Ie/Id\X x {p} has the local
orm mp = Of, so that Оо(Щ\Х x {p} is the push-forward of a non-
ocally free rank 1 sheaf on /. Thus, it is specified by its degree, and it
mffices to show that the degree is 1. But by flatness deg(Oo(^)\X x {p})
s independent of p, and this degree is 1 if p is a smooth point in its fiber.
Thus, it is 1 in the case where p is the singular point as well.
Next, let 7Г! and тг2 be the projections from X x X to the first and second
actors. We seek a surjective map 7r*Vb —> 0?>(D). If such a map existed, it
vould induce on every slice X x {p} a surjection Vb —> i»O/(p) (with the
ippropriate modifications if p is a singular point of a singular fiber) and
ю we could take V to be the kernel of the surjection. The existence of the
jundle V, by the properties of a coarse moduli space, implies that there is
i morphism X —> 9Jt, where 9Jt is the coarse moduli space corresponding
,o the 2-dimensional problem. By the classification results of the preceding
lection, the map X —> 9Jt is a bijection and 9Jt is smooth. Thus, the map
К —> 9Jt is an isomorphism, identifying X with the moduli space.
Unfortunately, the map n\V0 —> Od(D) need not exist! Instead, for every
j G X, it follows from Lemma 11 that
НтЕот(п{У0,00(Щ\Х х {p}) = dimtf0(KVb)v®OD(D)|A'x {p}) = 1.
Thus, by general properties of cohomology and base change, the coherent
sheaf
в a line bundle on X. The trivial section of Д07г2»ЯотGГ1 Vb, Od(D)) ® С
;hen gives a global section of Hom(it\ Vb, Od(D) ® я^-С), and thus a map
This map is surjective because its restriction to each fiber is surjective.
Thus, we may finally define V via the kernel of this map:
0-»V-> n^Vo -> 0D(D) ® n$C -> 0.
To calculate D(a), we must therefore calculate Pi(ad V). Now V is given
by an elementary modification, and so the general formulas of Chapter 2
tell us in principle how to calculate Pi(ad V): using the formulas for c\(V)
and c2(V) in Lemma 16 of Chapter 2, we find that
Pi(ad V) = Pi(adirJVb) + 2a(n{V0) ¦ D + D2 - 4itCl(n*C) - 4D.
Strictly speaking this formula holds away from the singular locus of D,
where the sheaf Оо(Щ ®f^ 1S no* necessarily a line bundle. However, the
singular locus of D has high codimension inXxX, so that the formula
above, which holds о priori on X x X - Sing D, is actually valid on X x X.
So we must calculate these terms. Also, a term of the form x ® 1 or 1 ® x,
for x G H*(X), will not affect slant product as far as H2(X) is concerned
(although we would need to keep track of these terms if we were interested

212 8. Vector Bundles over Elliptic Surfaces
in the four-dimensional class) so we will omit such terms as needed. Fbr
example, pi (ad n* Vo) = n^pi (ad Vo) is of the form x ® 1 and will not affect
slant product.
Next we have the following lemma:
Lemma 13. The cohomoJogy class of D is given by
[D] = f ® 1 + 1 ® /.
Proof. It suffices to show that, if a 6 H2(X), then тг^а U [D] = / • a, and
similarly for ttJo (note that X is simply connected so that HX(X) = 0).
Dually, for a general point x 6 X, if a is dual to the homology clan
represented by a smoothly embedded 2-manifold C, we must compute the
intersection number (C x {x}) • D, which is clearly equal to С ¦ f. О
Thus, we see that D2 = 2/ ® / and that, since by assumption cj(Vo) =
a + kf for some к,
2ci(ttI Vo) ¦ D = 2[(<r + kf) ® 1] • [/ ® 1 + 1 ® /] = 2(<r + kf) ® /.
We come now to the main term of interest,
4г,С1(тг5?) = 4г,гХС1(?) = 4[?>] • A ® Л)
= (/ ® 1 + 1 ® /) - A ® Л),
where Л = Ci(?) 6 H2(X). Up to a term not affecting slant product this
is just 4/ ® Л. Thus, it suffices to compute Л. Now
-A = ci^) = c^Tra.tfomKV&.CbODO)).
Thus, we must compute Ci of a direct image sheaf. Moreover, in this case it
follows from Lemma 11 that В}к2*Нот{-к\Уо,Оо(р>)) = О. Since тг2 has
relative dimension 1, all other Д*7Г2»'з are 0 also, and we are set up to use
the Grothendieck-Riemann-Roch theorem to find
Lemma 14. We have the following formula for —A:
Before we give the proof of Lemma 14, which will be a lengthy if standard
calculation, let us complete the calculation of D(a). If we collect all the

8. Vector Bundles over Elliptic Surfaces 213
arms for —pi (ad V) which will affect slant product, we see that —4д(а) is
by slant product with
2(ct + kf) ® / + 2/ ® / - 4/ ® Л - 4B
2
=2<x ® / - 4/ ® <t + Bk + 2 - 2pg - 2fc - 2)/ ® / - 4D
=2cr ® /- 4/ ® a - 2pgf ® /- 4D.
.bking the slant product with a gives our formula for ц(а):
Proposition 15. -4д(а) = 2(ст • a)/ - 4(/ • a)a - 2p9(/ • a)/ - 4a. ?
•: Squaring this term, we find that
+ Wpg(f ¦ aJ + 32(/ • a)(«r • a) + 16pg(f ¦ aJ + 16a2
= - 16(p9 + 1)(/ ¦ aJ + 32pg(f ¦ aJ + 16a2
=16a2 + 16(p9 - 1)(/ • aJ.
s, we have the desired formula for the degree 2 Donaldson polynomial:
theorem 16. D(a) = a2 + (pg - 1)(/ • aJ. D
Biroof of Lemma 14. Recall the Grothendieck-Riemann-Roch formula,
tnich says that ch(Gr2)i((jrf V0)v ® Or»(D))) ToddA" is equal to
тг2, [сЬ(тг*V0)v ® Od(D)) Todd(A" x X)].
iere ch is the Chern character, so that ch is additive over direct sums and
nultiplicative over tensor products, provided that one of the factors is a
rector bundle. For a line bundle L, chL = 1 + c\(L) + ct(LJ/2 + • ¦•.
For a rank г bundle V, ch V = г + ci(V) + (cj(VJ - 2c2(V))/2 + • ¦ •.
Moreover, Tbdd V is the Todd class of V, which is again multiplicative,
and Todd X is by definition the Todd class of the tangent bundle of X. In
general Todd К = 1 + c1(V)/2 + (c1(VJ + c2(K))/12 + --- .Finally, (тг2),.Г
denotes the formal sum of coherent sheaves 53Д—1)*Я*7г2*^. In our case
the only such term which is nonzero is the Д°тг2» term, and it is a line
bundle. Thus, the formula says that ci(.R°7r2»Gr*Vo)v®C>?)(D)) is equal to
the degree 1 term in
7г2, [ch((irj V0)v ® Od(D)) • Todd(X x X)) (To
я-2, [ch(«V0)v ® OD(B)) ¦ Todd(X x Л") ¦ TrJ
7г2» [ch(GrJK0)v ® Od(D)) • < ToddX • тг* ToddX
< ToddX]

214 8. Vector Bundles over Elliptic Surfaces
= 7T2, [chGr*Vb)v ¦ сЪ(Оо(Щ) • < ToddX]
= 7Г2, [7rr(ch(Vb)v • Todd X) • ch(OD(D))].
We proceed to calculate these terms. First note that we have supposed that
ci(Vo) = a + kf, where к is well-defined mod 2. Thus,
cl(V0J = -(l+pg)+2k,
and moreover pi(ad Vo) = ci(VbJ - 4c2(Vo). Since -pi(ad Vo) - 3A + pt)
is the expected dimension of the moduli space for Vo, which is smooth of
dimension 0, we have pi(ad Vo) = —3A + pg). Thus,
-A + pg) +2k- 4c2(Vo) = -3A + pg),
so that solving for c2(Vo) we find that
It follows that
= 2 - (or + */) + i(-(l +P9) +2k-(pg + l + *))pt
Moreover, using the fact that ci(X) = —Kx = — (pfl - 1)/ and Noether's
formula, that ci(XJ + c2{X) = Ux(Ox) = 12(p9 + 1), we find that
Todd X = 1 -
Putting these two calculations together, we see that
¦ Todd X) = 2-(a + {pg-l + k)f) ® 1 + Npt
where JV = Cp9 + 1 + fc)/2.
Next we need a formula for ch@?)(D)):
Lemma 17. Let i: D —> X x X be the inclusion, and simiJarJy for j: D
X x X. Then up through complex codimension 3,
сЪ(пОо(Щ) = D ——
Proof. We have an exact sequence
0 -» OD -> 0r>(O) -> Od(D) -» 0.

8. Vector Bundles over Elliptic Surfaces 215
?hus, ch(i»0D(D)) = ch(i,OD) + ch(j,0i>(D)), where j:D-.XxXis
the inclusion. Now from the exact sequence
0 - Oxxx(-D) -» Oxxx -» Or. -» 0,
,ye see that
= 1 - A - D + D2/2) = D- D2/2,
jfrhere we have used the fact that D3 = 0. As for the term
jjtre ignore the fact that D has singularities since they will occur in high
codimension and our arguments will work as if D were a Cartier divisor on
lp. Applying Grothendieck-Riemann-Roch to the embedding j, we obtain
D|D +
we have used the fact that the normal bundle of D in X x X is the
t bundle of X to calculate Todd^n/xxx), fcnd the term (D)d refers
i;the self-intersection of D on D, which is meaningful in the complement
the finitely many singular points of D. To calculate the term (Щ2О, we
calculate the normal bundle to D in D, at least at the smooth points,
sing the exact sequence
0 -» Nq/d -+ ND/xxx -» ND/Xxx\® -» 0.
•fire see that
(DJD = ci(JVD/D) = Cl(X) - (f ® 1 + 1 ® /)|D,
again using the fact that ND/Xxx is the tangent bundle to X = D>. Thus,
(D)d = -0»e - 1)/ - 2/ = -(ps + I)/-
Putting this together we see that
= D+|?*_:_ (pe +1I j./eD-
Thus,
ch(i,OD(D)) = D-^-+D- №^Ь;/+---.	П
To complete the proof, we need to apply тг2» to the degree 3 term in

216 8. Vector Bundles over Elliptic Surfaces
We have D = /® 1 +1®/ and D2 = 2/®/. Also, (a® 1)D = j,a. Thus,
the degree 3 term in the above product is
-(pg + 3)j,/ + pt ® / - j,(<x + {jpg - 1 + k)f) + Npt ® /.
Applying тг2» gives
-(pg + 3)/ + / - a - (pg - 1 + k)f + Nf = -a + (-2pg -l-k + N)f
Thus, we have established the formula for — Л. ?
Finally, we describe briefly how to extend these results in case there are
multiple fibers. For simplicity we assume that X is an elliptic surface over
P1 with just two multiple fibers, with relatively prime multiplicities mi and
ТП2- We shall consider bundles V over X such that ci(V) • / is odd, where /
is a general fiber of X. Since / = лцтг/с, where к is a primitive cohomology
class, we see that ci(K) ¦ / is odd only if both mi and m^ are odd. Much of
the analysis of the first part of this chapter goes through to show that the
moduli space is always smooth and irreducible of the expected dimension
(which is even), and thus that the O-dimensional moduli space is a single
reduced point. The formula for the 2-dimensional Donaldson invariant is
as follows [36]:
Theorem 18. Let X be a simply connected elliptic surface over P1 with
two multiple fibers with relatively prime multiplicities mi and Гог. Let D
be the Donaldson invariant corresponding to a 2 dimensional moduli space
of vector bundles V on X with а (V) ¦ f odd, where f is the class of a
genera/ fiber on X. Let к be the cohomology class such that f =
Then
D(a) = a2 + ((mim2J(p9 + 1) - m?m^)(/t • aJ.	?
The idea of the proof is to imitate the calculation in the case of a section
as far as possible, using an approximate moduli space. This determines
D(a) up to correction terms which only depend on an analytic neighbor-
neighborhood of the multiple fibers. Now a simply connected elliptic surface with
p9 — 0 and just one multiple fiber of multiplicity m is a rational surface, and
its Donaldson polynomial is therefore the same as the corresponding poly-
polynomial for a rational elliptic surface with a section. But we have computed
these polynomials above. Using this information, we can then determine
the correction terms and complete the calculation for Theorem 18.

8. Vector Bundles over Elliptic Surfaces 217
Moduli spaces via extensions
In this section we shall describe a different approach to constructing moduli
ipacee. While this approach only works in case X has a section, it is still
Instructive to redo the calculations in this way. We begin with a lemma on
yarious cohomology groups:
mma 19. Let X be an elliptic surface over P1 with a section a and let
pg(X) = pg. Then, for ail n 6 Z,
end
h2(X;Ox(-<x + (pg + l-n)f)) =
2,~ - / . / . - w- , n - 1, if n > 2,
ifn < 1.
iProof. Clearly, ЯРж»Ох(-сг + (pg + 1 - n)f) = 0 for all n and
;Д2тг»С>х(-о" + (Pg + 1 — n)/) = 0 for all n since тг has relative dimen-
ion 1. Thus,
цг the Leray spectral sequence. So we must determine the line bundle
,	Rlit*Ox{-o + (pg + 1 - n)f)
jnP1. Note that
R1^Ox(~ar + (jpg + 1 - n)f) = Д'тг^ОхС-о-) ® Opi(p9 + 1 - n)
by the projection formula. Thus, it suffices to determine R1ntOx(—(r).
Now apply R*nt to the exact sequence
0 -> Ox{-a) -.Ox-»O,-0.
Since R°n,Ox = R°Tr.Oa - Oa and R}n<.Oa = 0, we see that
R^.Oxi-a) а Д'тг.Ох = CV(-0»e + 1)),
by the formulas in the last chapter. Thus, Rln*Ox(-o + (p9 +1 -n)f) =
Opi(—n), and the statement of the lemma follows from the formulas for
the cohomology of P1. D
Corollary 20. With notation as above, there is a unique bundle Vq on X
such that Vq is a nonsplit extension
0 -¦ Ox -* Vo - Ox(a - (p9 + 1)/) - 0.
Moreover, Vo has stable restriction to every fiber. Finally, ci(Vb) — а -
iPg + 1)/, c2(Vb) = 0, and -Pl(adVb) = 3(p9 + 1).

218 8. Vector Bundles over Elliptic Surfaces
Proof. The set of all possible extensions as above is classified by
® Ox) = Hl(Ox(-a + (pg + 1)/)).
By the above lemma, this group has dimension 1, proving that there is a
unique nonsplit extension. To see what happens on a fiber /, the restriction
of Vo to / is an extension of O/(p) by Of, where p = а П / is a smooth
point of /. Thus, Vo|/ is stable provided this extension does not split. Tb
analyze the restriction of the extension, we consider the map
H\X-tOx{-a + (ps + 1)/)) -» ff*(/;Of(-p)).
It is easy to see that the induced map on cohomology groups corresponds
to restricting extensions. Thus, we must show that the image of a nonzero
element of Hl(X\ Ox(-a + (pg + 1)/)) is nonzero in ЯХ(/; Of(-p)). But
the kernel of the restriction map is
Again by the lemma, this last group is zero (it corresponds to n = 1).
Clearly, ci(Vo) and c2(Vo) are as claimed. Thus,
Pi(adV0) = Cl(VoJ = <r2 - 2(p9 + 1) = -3(p9 + 1).
This concludes the proof of the corollary. D
The bundle Vo above is described in [36] as well as by Kametani and
Sato [66] (with slightly different normalizations).
Now let us describe the bundles corresponding to the 2-dimensional mod-
moduli space. If V is such a bundle, then up to a twist V is obtained from Vo
by an elementary modification along a single fiber. Thus, we may assume
that C!(V) = a - (pg + 2)/ and that -px(ad V) = 3(p9 + 1) + 2.
Proposition 21. Let V be a stable rank 2 bundle on X with ci(V) =
a — iPg + 2)/ and -pi(ad V) = 3(p9 + 1) + 2. Then either V is given as an
extension
0 -» Ox(-f) -» V -» Ox(° - (p9 + 1)/) ® m, -»0,
where tnx is the ideal of a point on X, or V is given as an extension
0 -» Ox -» V -» Ox(cr - (pe + 2)/) -» 0.
Proof. By the proof of Proposition 9, we obtain V from Vo by making a
single allowable elementary modification. Thus, there is an exact sequence
at least if / is a smooth fiber, with a similar result holding if / is singular.
In particular, we have the map Ox —* Vo > and either its image is contained
in V, or there is at least an induced map Ox(—f) —> V. If the induced

8. Vector Bundles over Elliptic Surfaces 219
inap Ox(—f) —* У vanishes along a curve, it must do so along a curve
jupported in fibers, for otherwise the map Ox —* Vo would have to vanish
Jong a curve. Thus, if the map Ox(-f) —» V vanishes along a curve C,
the only possibility is that С = / and the map Ox —» Vo has image in V
torsion free quotient. Thus, either we can find an exact sequence
some O-dimensional subscheme Z, or we can find an exact sequence
Keeping track of the Chern classes shows that Z is a reduced point in the
(irst case and that Z = 0 in the second case. D
V, Next we must determine when extensions as described above are stable:
sposition 22.
If V is an extension of the form
0 -» Ox(-f) -» V - Ox{° - (pg + 1)/) ® mx -» 0,
then V is locally free if and only if the extension is nonsplit, and there
is a unique such V. Moreover, V is stabie if and only ifx $ or. If V is
not stable, then there is an exact sequence
1 where the inclusion Ox (p — (Pg + 2)/) —» V gives the maximal desta,-
а bilizing sub-line bundle of V.
An extension of the form
O^Ox^V^ Ox{° - (pg + 2)f) -» 0
is stable if and only if it is nonsplit. Moreover, the set of all such
nonsplit extensions is parametrized by a P1 which we may identify
with a.
iProof. (i) Suppose that V is an extension as in (i) above. We have an
exact sequence
0 -» H\Ox{-o + paf)) -» Ext^Oxfa - (p, + 1)/) ® mx, Ox(-f))
¦	^H°(Cx)^H2(Ox(-<r+Pgf)).
By Lemma 19, H\Ox(-o + Pgf)) = tf2@x(-er + pgf)) = 0. Thus,
Ext1 (Ox{o - (jpg +1)/) ® mx, Ox(-/)) maps isomorphically onto #°(CX)
and so has dimension 1. Thus, up to isomorphism there is a unique nonsplit
extension and it is locally free.
¦ Next suppose that V as above is not stable. Then a destabilizing sub-line
bundle of V would restrict to a destabilizing sub-line bundle on the generic

220 8. Vector Bundles over Elliptic Surfaces
fiber. Thus, it must be of the form Ox(ff + af) for some a. Since there
is a nonzero map from Ox (c + af) to V, whose image clearly cannot be
contained in the sub-line bundle Ox(-f), there is an induced nonzero map
Ox{a + af) ~» Ox{<? - (pg + 1)/) ® m». This is only possible if there is a
sectionofC>x(-(Pg + l+a)/)®niI, or in other words if a < -(p9 + l).Now
the maximal destabilizing sub-line bundle of V has torsion free quotient,
so that there is an exact sequence
0 - Ox(<r + af)->V-> Ox(-{pg + 2 + a)f) ® Iz - 0
for some O-dimensional subscheme Z. A calculation gives
-Pi(ad V) = -(<r + (p9 + 2 + 2a)/J + U{Z) > 3(p9 + 1) + 2 + U(Z),
with equality holding if and only if a = — (pg +2) and Z = 0. But for the
V under consideration, we have assumed that — pi(ad V) = 3(p9 + 1) + 2.
Thus, we must have Z = 0 and the maximal destabilizing sub-line bundle
is Ox{o — (j>g + 2)/). Note that, in this case, there is an exact sequence
0 -» Ox{a - (pg + 2)/) -» К -¦ Ox - 0,
which is the reverse of the extensions in the second possibility (ii) for V
above.
By the above argument, an extension of the form
0 -» Ox(-f) -*V-* Ox^- (pg + 1)/) ® m, - 0
is unstable if the natural map Ox (°r-(Pg + 2)/) —» Ox (o — (jpg +1)/) ®xnx
lifts to a map Ox{o< - (pg + 2)/) —» V. Equivalently, we ask if the nonzero
section of Ox(f) ® m, lifts to a section of V ® Ox{~o + (jpg + 2)/). Now
V®Ox(-<r+(p9+2)/) is an extension of Ox(f)®mx by Ox(-ar+(jpg+l)f).
Furthermore, Ext1 (Ox(/) ® m,,Ox(-<r + (p9 + 1)/)) и equal to
Ext1(Ox(oi " 0»e + l)/)®m«,0x(-/)),
which as we have seen has dimension 1. The induced map Ox ~* Ox(f) ®
nii induces a commutative diagram
Hom(Ox(/)®mI,Ox(/)®mI) — Ext1(Ox(/)®mI,Ox(-<r + (pg
I	I
Hom(Ox,Ox(/)®mx) —¦ Ext1 (Ox, Ox (-<r + ipg + 1)/)),
and the kernel of the lower map
Hom(Ox,Ox(/) ® m.) -» Ext1(Ox,Ox(-<r + (pg + 1)/))
is the image of
Hom(Ox, V ® Ox(-ar + (pg + 2)/)) = Hom(Ox(<r - (j>9 + 2)/), V).
By definition Id 6 Hom(Ox(/) ® шх,Ох(/) ® nij) maps to the exten-
extension class ? corresponding to V. Clearly, Id restricts to a nonzero ele-

8. Vector Bundles over Elliptic Surfaces 221
inent of Hom(Ox,Ox(f) ® xnx), which is mod scalars the same as the
unique nonzero section of Ox(/) ® mx. Chasing through the diagram we
see that the nonzero section of Ox(f) ® vcix lifts to give a nonzero map
Ox(& - iPg + 2)/) —» V if and only if the extension class ? corresponding
to V maps to 0 in Ext1(Ox,Оx(-<x + (pg + 1)/))- Now, in case x is a
Smooth point of /, the quotient of Ox (/) ® xnx by Ox is easily seen to be
О/(-х), where / is the unique fiber containing x, as we see by looking at
the exact sequence
Similarly, if x is the singular point of a singular fiber, then (Ox(f) ®
щ»)/Ох is the ideal sheaf of x in /. Assuming for simplicity that x is a
smooth point of /, the restriction map
into the exact sequence
{-a + (pg +1)/)) - Ех^@,(-х), Ox(-a + (pg + 1)/)) -
(pg + 1)/)) - Exbl(OXtOx(-<r + (pg + 1)/))
Rirthermore
!r Hom(Ox, Ox(-tr + (pg + 1)/)) = H°(OX (-a + (j>g + 1)/)) = 0.
Йу Chapter 2, Exercise 15, we see that
1	-a + (p, + 1)/)) = H°(Of(x -p)),
where p = а П /, with a similar result holding if x is a singular point of /.
Thus, this group has dimension 0 if x ф p and has dimension 1 if x = p.
Since Ext^OxC/) ®mi,C>x(-o" + (pg + 1)/)) also has dimension 1, we see
that the map
Ext1 (Ox(/) ® mX) Ox(-<r + (p, +1)/)) - Ext1 (Ox, Ox(-<r + (pg +1)/))
is injective if x ^ cr, and is the zero map if x 6 or. Thus, there is no possible
destabilizing sub-line bundle if x ^ a, whereas V has the destabilizing
sub-line bundle Ox{<? ~(pg + 2)/) if x 6 a.
(ii) In this case Ext1(C7x(<r-(p9 + 2)/), Ox) = Hl(Ox(-<r+(pg+2)f)).
By Lemma 19, the cohomology group has dimension 2, and the set of
nonsplit extensions is therefore P1. Arguing as in the proof of (i) above, a
destabilizing sub-line bundle would have to be of the form Ox(<r+af) with
a < — (pg + 2), and the case a = — (pg + 2) would split the exact sequence.
Assuming that the destabilizing sub-line bundle has torsion free quotient,
we reach a contradiction by working out pi(adK) using the destabilizing
sub-line bundle along the lines of the proof of the corresponding fact for
(i) above; the details are left to the reader. Thus, the bundle V is stable if
and only if the extension is not split.

222 8. Vector Bundles over Elliptic Surfaces
The proof of Lemma 19 identifies Hl(Ox(-0 + (pg + 2)/)) „эд,
Я°(Р\С>р1A)), where we can think of the P1 as the base of the ellip.
tic surface, identified with a. A section s 6 H°(P1,Opi(l)) vanishes at
a unique point p of P1. Running through the identifications, this shows
that, if V is the extension of Ox (a — ipg + 2)/) by Ox corresponding to
s € H°(?1,Opi(l)), then the restriction of the extension У to a fiber /
splits if and only if / lies above p. Note that V\f is given by the extension
0 - Of -+ V\f -* Of{jp) - 0,
where we identify p б P1 with p = а П /, the point of a lying over p. Thus,
if s vanishes at p and / is the fiber over p, then V\f = Of Ф 0/(p), and
otherwise V|/ is a nonsplit extension. ?
We will refer to the stable bundles satisfying the hypotheses of (i) of
Proposition 22 as bundles of Type A), and those satisfying the hypotheses
of (ii) as Type B) bundles. We can interpret Proposition 22 as follows:
the 2-dimensional moduli space ЯИ we are considering breaks up into two
pieces: one piece corresponds to the extensions of Type A) in Proposition
22, where x ? a, and the other piece corresponds to Type B). Now given
x 6 X, there is a unique Type A) extension V corresponding to x, so that
the first piece looks like X — a. As for nonsplit Type B) extensions, we
have seen that they are naturally identifierj with a. Our goal now will be
to describe how to fit together the two pieces X — a and a so as to obtain
a compact moduli space isomorphic to X.
We begin by constructing a moduli space for all Type A) extensions.
The natural way to fit together extensions of the form
0 -» Ox(-f) ^V-* Ox{a - (pg + 1)/) ® mx - 0
is to work on X x X, where we view the first factor as the surface and the
second as the moduli space. We try to find an extension of the form
0 -* <OX(-/) -*W- -n\Ox(p ~(pg + 1)/) ® /D - 0,
where as usual D is the diagonal. Such an extension would restrict to give
a Type A) extension on every slice X x {x}. To carry out the construction,
we would need to find an everywhere generating section of
- (pa + 1)/) ®
and then lift this section to an element of
V	- (pg + 1)/) ®
Now as we have seen in Chapter 2, Exercise 16, Extl(v\Ox(.<? - (pg +
1)/) ® 1о,я$Ох(—/)) is isomorphic to

8. Vector Bundles over Elliptic Surfaces 223
is just the tangent bundle on X, under the natural identi-
tion of D with X, we see that the Ext sheaf is just the line bundle
bus, we instead try to find W as an extension
irJ0x(-/) ® *lOx(° - f) -» W -»irJOx(<r - (p, + 1)/) ® /d -» 0.
! point of adding the extra factor тг^ОхС"" — /) to the first term in the
sequence is to cancel out the Ext sheaf, and indeed we now have
~(Pg + 1)/) <8> /d,i
which has an everywhere generating section, unique mod scalars. The
^Obstruction to lifting this section back to Ext1GrjOx(o' — (Pg + 1)/) ®
%^iOx{-f) ® it%Ox{cr - /)) lives in
|	Ox((T - (pg + 1)/)
I.	=H2{-K*lOx{-
Ш.п easy application of the Leray spectral sequence (or the Kiinneth for-
jnula) and Lemma 19 shows that this group is zero. Thus, we may construct
the extension W.
? Unfortunately, W is not the right universal bundle, since on every slice
'X x {x} where x € ff the bundle W\X x {x} is unstable. However, let us
Record the Chern classes of W anyway:
ci(W) = irj(a - (pg + 2)/) + тг2> - /),
c2(W) = тг*(<т - (pg + 1)/) ¦ «fa - f) - wJpt + D,
Pi(ad W) = -2<(<т -pgf) ¦ тг2*(<т _ /) _ 4D + • • • ,
where as usual the omitted terms do not affect slant product.
We now deal with the problem that the restriction of W to the slice
X x {p} is unstable for every p 6 a. In fact, if Wp is this restriction, then
by Proposition 22 there is an exact sequence
0 -» Ox (<r - (ps + 2)/) -» Wp -» Ox -> 0.
Since the destabilizing quotient bundle is unique, we have Hom(Wp, Ox) —
С for allp € a. Thus, ifp2: X x a —»cr is the projection, then
is a line bundle on a. We will not have to know what ? is, though! It follows
that the induced map
ЩХ xa-* plC-1
is surjective. If j: X x a —* X x X is the inclusion, we can therefore make
the elementary modification defined by

224 8. Vector Bundles over Elliptic Surfaces
For each p 6 <x, there is thus an exact sequence for Vp, the restriction of V
to the slice X x {p}, reversing the destabilizing sequence for Wp:
0-+Ox-+Vp-+Ox(cr-(pg+ 2)/) -»0.
Thus, if this sequence does not split, then Vp is a Type B) extension, and
we have solved the problem of how to fit together the two different types
of extensions. We shall just state the result we need, referring to [36] for
the proof:
Proposition 23. In the above notation, the restriction of V to every slice
X x {x} is stable, and V identifies the second factor X with the correspond-
corresponding moduli space of stable bundles. ?
Now let us calculate the /i-map. By general formulas for elementary mod-
modifications from Chapter 2, Lemma 16, we find that
Pl(adV) = pi(adW) + 2сх{Щ ¦ [X x a) + [X x aJ - А^
Omitting terms that do not affect slant product, we obtain
-2?rJ(o- - pgf) • тг^((т - /) - 4B + 2*1@ - (pg + 2)/) •
and so — 4/i(a) is equal to
[-2(a • a) + 2pg(f ¦ a))(ar - f) - 4a + 2{<x ¦ a)a - 2(pg + 2)(/ • a)o
=2(<r • a)/ - 4(/ ¦ a)a - 2pg(f ¦ a)f - 4a.
This agrees with our previous calculation for —Ац(а) in Proposition 15 of
the last section.
Similar arguments will identify a Zariski open subset of the moduli space
in general. We have the following straightforward generalization of the ar-
arguments of this section:
Theorem 24. Let Z consist of t points on X lying in distinct Bbers of
7Г, such that Z Па = 0. Then up to isomorphism there is a unique vector
bundle V given as an extension
0 -¦ Ox(-tf) ^V^ Ox(cr -(pa + 1)/) ® Iz -» 0.
Moreover, V is stable. Finally, h°(V ® Ox(tf)) = 1, so that Z is uniquely
determined by V. ?
Theorem 24 gives a second identification of a Zariski open subset of the
moduli space with a Zariski open subset of Hilb' X, and it is easy to check
that this identification agrees with the one given by Theorem 12. For t - 2,
the moduli space is in fact isomorphic to Hilb2 X [36], but this is no longer
the case for t > 2.

8. Vector Bundles over Elliptic Surfaces 225
Vector bundles with trivial determinant
Jn this section, we begin the description of stable rank 2 vector bundles
iV over X with det V = O\, or more generally such that det V is pulled
iback from the base curve. For simplicity, we return to the convention that
pie base curve of the elliptic surface X is P1, that all fibers of тг are either
Smooth or nodal, and that X has a section a. For more general results
j|J6ng these lines, we refer to [42]. In the next section, we will discuss what
happens when there are multiple fibers.
' If V is stable with respect to a suitable ample divisor, then V\f is
semistable for almost all /. After a sequence of allowable elementary modi-
modifications, we can assume that V\f is semistable for all /. Note however that
У\f is a rank 2 vector bundle of degree 0 on /, and hence is never actually
stable. In fact, V\f also fails to be simple, by Theorem 25 below. Thus, the
(eemistability of V\f on one or all fibers does not guarantee the stability or
|jjemistability of V. It is also possible that we begin with a stable V and
Йпаке a sequence of allowable elementary modifications, and the final result
\b actually unstable. We shall give a sufficient condition for V and all of its
elementary modifications to be stable in Theorem 30 below. Finally and
most interestingly, in the case of fiber degree 1, the set of stable bundles
on a fixed fiber / with a given determinant of degree 1 is a single point,
and this led to the bundle Vo of Proposition 9, which was unique up to
¦	twisting by the pullback of a line bundle of the base. In the case of degree
0, the moduli space of semistable bundles is positive-dimensional (in fact it
¦	is isomorphic to P1), and this allows for much more complicated behavior.
Following the above comments, we begin by analyzing the moduli space
' of semistable bundles of determinant zero on a fiber /. If / is smooth, the
answer is given by Theorem 6 of Chapter 2:
Theorem 25. Let E be a smooth elliptic curve and Jet V be a semistable
rank 2 vector bundle over E. Then either V = Л ® A, where A is a line
bundle of degree 0 on E, or V is of the form € ® A, where ? is the unique
nonsplit extension of Oe and A is one of the four line bundles on E with
A®2 = Ое- Moreover, as C-algebras,
!CxC,
СЩ/t2, ifV^?®\, A®2 = OE,
M2(C), ifK^AAA A1
Proof. All but the last statement is contained in the statement of Theorem
6 of Chapter 2 (see also Theorem 9 in Chapter 4). To prove the statement
about the homomorphisms of V, note that, if А Ф- A, then as deg A = 0,
every homomorphism from A to A" is 0, and the same is true for every

226 8. Vector Bundles over Elliptic Surfaces
homomorphism from A to A. Thus, (as algebras)
For the other statements, we can assume that A = Ob and must calculate
Hom(?,?) and Нот(Ов ф Ое,Ое Ф Oe)- Clearly, the last group is ieo-
morphic to Мг(С), the algebra of 2 x 2 complex matrices, and it suffices
to calculate Нош(?,?). There is a rank 1 endomorphism tp: ? -+ ? de.
fined by ? -» Of —> ?, i.e., take the defining quotient map to Of, followed
by the defining inclusion of Of in ?. Clearly, tp is a nontrivial element of
Hom(? ,?) of square 0, and it suffices to show that every endomorphism
can be uniquely written as a Id +Ыр. Since the coboundary map д arising
from the long exact sequence
0 - H°{O}) — H°(?) — H°(Of) Л H\Of)
is nonzero (its image is the extension class denning ?), h°(?) = 1. Fix a
nonzero section s of ? and let ф € Hom(?,?). Then ф(в) = as for some
о e C, and thus (ф - ald)(s) = 0. It follows that ф - aid factors through
the quotient map and defines a morphism ?/Oj =* Of —+ ?. Necessarily
the image of this map is contained in the distinguished subbundle Of of ?,
and from this it is clear that ф — a Id is some multiple btp of <p. Thus, we
have shown that Hom(?, ?) S C[t]/(t2), as claimed, a
Note that the bundles ? ® A and А Ф A are S-equivalent in the sense of
Chapter 6. Thus, they must be identified as points of the moduli space of
semistable bundles on /. After this identification, the moduli space is then
Pic0 // ± 1 S P1, where Pic0 /, the set of line bundles on / of degree 0, is
identified with / via the choice of an origin p. As we shall see shortly, if
we try to make a universal bundle over / xP1, we must use the bundles
? ® A and not А Ф A. One reason for this is that dim Hom(? ® A, ? ® A) = 2,
which is the same as the dimension of the algebra of homomorphisms for
the generic bundle. Another and related reason is that, in a certain sense,
the bundles А ф A are more special than ? ® A: the bundle А Ф A has a small
deformation which is isomorphic to ? ® A, but not the other way around.
Before we discuss the problem of trying to find a universal bundle for
a smooth elliptic curve /, let us give the analogue of Theorem 25 in the
singular case, whose proof is left as an exercise:
Theorem 26. Let f be a nodal curve of Arithmetic genus 1. Then the
semistable rank 2 vector bundles V on / with det V — Of are the bundles
of the form:
(i) А ф A, where A is a line bundle on f of degree 0;
(ii) ? ® A, where ? is the unique nontrivial extension of Of by Of and A
is one of the two line bundles on / with A®2 = Of,

8. Vector Bundles over Elliptic Surfaces 227
i(iii) The unique nontrivial locally free extension G with det Q = 0/ of the
form
where F is the unique torsion free rank 1 sheaf of degree 0 which is
not locally free.
Moreover, in cases (i) and (ii) the algebra Hom(V, V) is as described in
'Theorem 25. In case (iii), Hom@,0) 9* C[t]/(t2). ?
.,'
1 Note that, in case / is nodal, Theorem 26 says that up to S-equivalence
the moduli space of semistable bundles on / with trivial determinant is just
Pic0 //±1 plus the remaining bundle Q. Now Pic0 / ^ C*, and Pic0 //±1 <*
C. Adding in the remaining bundle Q then compactifies С to a P1 as in the
smooth case.
f For both smooth and nodal fibers /, the bundles ? ® A are distinguished
within a given S-equivalence class by the requirement that dimHom(Vr, V)
]s as small as possible. We make this a definition:
Definition 27. Let / be a smooth elliptic curve or a nodal curve of arith-
arithmetic genus 1, and let V be a rank 2 semistable vector bundle on / with
det V = Of. Then V is regular if dimHom(V, V) = 2.
Next we want to reinterpret the moduli space P1, in both the smooth and
the nodal case, to make clear that it behaves like a coarse moduli space.
One way to do so, given the choice of origin p 6 /, is the following: if A
is a line bundle of degree 0 on /, then we can write A = Of(q - p) for a
unique q 6 /, and similarly A = Of(r — p). The condition that the line
bundles Of(q — p) and O/(r — p) are inverse to each other is exactly the
condition that q + r is linearly equivalent to 2p. Moreover, the map which
associates to the vector bundle А ф A the point q + r € |2p| = P1 defines
an isomorphism from Pic0// ±1 to |2p|, in case / is smooth. To handle
the singular case, as well as to make clear that the map in question really
is a morphism, we have the following:
Theorem 28. Let f be a reduced irreducible curve of arithmetic genus 1,
and fix a smooth point p € /. Let V be a semistable vector bundle on / of
rank 2 with det V = Of. Then h°(V ® Of(p)) = 2, and the natural map
<p: O) = H°(V ® O,(p)) ®cOf^V® 0,0»)
is an isomorphism on a Zarisii open subset of /. Thus, det^ is a well-
defined nonzero section of O,Bp), and so V defines an element of |2p|
agreeing with the one given above in case V = А ф A. More generally,
let S be a reduced scheme and let V —> / x 5 be a vector bundle such
that V|/ x {s} is a semistable rank 2 vector bundle over / with trivial

228 8. Vector Bundles over Elliptic Surfaces
determinant for alls e S. Then there is an induced morphism Ф: S -+ |2p| 1
such that Ф(з) is the point denned by V|/ x {s} for every s e S.	'
Proof. By Riemann-Roch, ft°(V cg> O/(p)) - ft^V ® O/(p)) = 2. On the |
other hand, using Serre duality, ftJ(V ® O/(p)) = ft°(V ® O/(-p)) - q j
since by semistability there are no nonzero таре O/(p) -* V. Thus, Л°(Уф I
O/(p)) = 2. Now suppose that the image of <p is contained in a rank l'i
subsheaf / of V ® Of(p). By semistability, degl < 1, and by construction
ft°(I) = h°(V ® O/(p)) = 2. But by definition deg/ = ft°G) - ft1 (I) < lf
and using Serre duality ft1 (I) = dimHom(I,O/) = 0, by (ii) of Lemma 3.
Hence h°(I) < 1. This contradicts the assumption that ft°(/) = 2. It follows
that the image of ip is a rank 2 subsheaf of V (not necessarily locally free),
as claimed, and thus that det <p is a nonzero section of det(Op-1 ®det(V®
O()) OB)
/	/
Incase V = АфА~\ we can write V®O/(p) =O/(?)®O/(r), and the
map tp fails to be an isomorphism exactly at q and r. Thus, det </? vanishes
at q + r, as claimed. (It is straightforward to check that, if V = € ® A with
A®2 = Cfy and A = Of(q — p), then det tp vanishes at 2q. If / is nodal and
V — G, then det <p is the unique element of |2p| supported at the singular
point.)
In the relative case, since det V is trivial on every fiber / x {a}, det V =
7г5Л( for some line bundle M on S. It follows from base change that
7T2.(V ® K\Of(p)) is a rank 2 vector bundle on S, and that the natural
map 7Г2?Г2,(У ® it\Oj(p)) —у V ® tt*O/(p) restricts on every fiber to give
the map tp defined above. Thus, the determinant of the above map is a
section of ttIjC ® irJAI ® ir*O/Bp), where ? is the determinant of the
rank 2 bundle 7T2»(V® 7гГ<9/(р)) on S. There is then an induced morphism
Ф: 5 —> |2p|, which agrees with the above construction on every fiber. D
Suppose now that ir: X -* V1 is an elliptic surface with a section <r, and
such that all fibers are smooth or nodal, and that V is a rank 2 vector
bundle over X such that det У is pulled back from the base P1. Assume
further that the restriction of V to every fiber / of 7r is semistable. For
each /, we have the space #"(/; OfBp)) and the corresponding projective
space |2p|. Globally over X, there is the rank 2 vector bundle ж*Ох(<т) and
the associated P1-bundle РGг» Ox (<?¦))• In the last chapter, we identified
ir»Ox(<r) with Ofi ®L~2, where L - OPi (p9 +1). Thus, setting Jfc = pg +1,
we see that ?(irtOx(ff)) is the rational ruled surface Fjfe. By restriction,
the bundle V defines a point in я" (я) for every x € P1. In fact, we have
the more precise result:
Theorem 29. Suppose that тг: X -* P1 is an elliptic surface as above, and
that У is a rani 2 vector bundle over X such that det V — v*M is pulled
back from the base P1 and such that the restriction of V to every fiber /

8. Vector Bundles over Elliptic Surfaces 229
is semistable. Then V defines a section A = A(V) of the rational ruled
jf. Arguing as in the last part of the proof of Theorem 28, the de-
dent of the natural map 7г*тг,(У ® Ох((т)) -* V ® Ox(cr) defines a
i of det V ® Ox B<x) whose restriction to every fiber is nonzero. The
Identifications
#°(det V ® Ox{2c)) = H°(n*M ® ОхBа)) = #°(Af ® я-»С>хB<т))
define a nowhere zero section of M ® 7r,C?xBcr) and thus a section of
^ The surface A" is a double cover of the rational ruled surface Ргм, by
JSxercise 16 of the last chapter. For example, we can identify F2* with the
— otient X/l, where t is the involution corresponding to taking -1 in every
r. The scheme-theoretic inverse image С of the section A = A(V) in X
i a double cover of A, possibly reducible or even non-reduced, and thus it
a double cover of the base P1, which we will call the spectral cover
Jof P1. In fact, if V is merely assumed to have semistable restriction to the
generic fiber /, we can still define the spectral cover of V as follows: the
.method of Theorem 29 defines a meromorphic section of Ргм, which then
completes to an actual section since the base is a curve. Of course, the
spectral cover of V, with this definition, is the same as the spectral cover
of every elementary modification of V along a fiber.
In general, as we remarked above, it is hard to give a complete char-
characterization of the stability of V, but we do have the following sufficient
condition in terms of the spectral cover C:
Theorem 30. Let V be a rank 2 vector bundle over X such tiat det V =
7г* M is pulled back from the base P1 and such that the restriction of V to
a generic fiber f of w is semistable. Suppose that the spectra/ cover С of
A(V) is reduced and irreducible. If(w,p) are the invariants of V, then V
is stable with respect to every (w,p)-suitable ample divisor H.
Proof. By (iii) of Theorem 5 in Chapter 6, if V is not Я-stable, then there
exists a divisor DonX and an exact sequence
0 -» OX(D) -*V-> Ox(-D + af) ® Iz -* 0,
such that D ¦ f = 0. The divisor D defines a section Q of X such that D is
linearly equivalent to Q — a on the generic fiber, by looking at the support
of the natural map n*ir*Ox(D + a) ~* Ox(D + a), and similarly —D
defines a section R. (Note that Q = R if and only if D has order 2 on the
generic fiber.) By construction С = Q + R. Thus, С is either nonreduced
or reducible. Conversely, if С is reduced and irreducible, then V must be
Я-stable. ?

230 8. Vector Bundles over Elliptic Surfaces
We now turn to the problem of constructing bundles over X. As in the
above discussion, we begin with the problem of finding a universal bundle
U over / x P1. In fact, we will find not one universal bundle but rather an
infinite family of such. The reason is the following: suppose that we were
given a moduli space ОТ of bundles on /, all of which were simple, and two
different universal bundles U, W over / x Ш1, whose restrictions to every
slice /x{i} were isomorphic. By simplicity, E?X2+Hom(U,W) — Lisa line
bundle on ФГ, and it is then easy to see that there is a map U ® ttJL —> Ц'
which is an isomorphism on every fiber, and thus an isomorphism. Thus,
two universal bundles can only differ by twisting by the pullback of a line
bundle from the Wl factor. In our situation, however, the bundles V corre-
corresponding to points of P1 are not simple. Thus, given two universal bundles
U, W as above, the most we can say is that Д°7Г2» #om(W,W) is a rank
2 locally free sheaf over P1. In fact, it is easy to see that this bundle has
rank 1 over the sheaf of algebras Л = i2°7r2» Hom(W,U), which is itself a
sheaf of rank 2 commutative algebras over P1. The sheaf Л defines a double
cover.of P1, which is unbranched where the fiber is СфС and branched at
the four points of P1 where the fiber is C[t]/(t2). Thus, the double cover is
a copy of / again. (Special care must be taken when / is a nodal curve.)
We will now give two methods for constructing universal bundles, first by
pushing forward line bundles on the double cover / x / —> / x P1, and
second by taking universal extensions of O/(p) by Oj{—p).
To describe the first method, we assume that / is smooth. Let u: f —»P1
be the quotient of / under the involution v = — Id. Consider the line bundle
С = O/x/(A - {p} x /). The restriction ?|/ x {q} = Of(q - p). The
pushforward (Id xi/)»? is then a rank 2 bundle Uo on / x P1. If x e P1
is a point such that v~1(x) — {q,r} consists of two distinct points, then
i/*?|/ x {x} = O/(q-p)® Of(r - p). Thus, away from the branch points
of i/, Uo fits together the line bundles AfflA. Of course, we will have
to analyze what happens at the branch points. More generally, let d be a
divisor on /. Then we can consider Via = (Id Xi/)»(?®7r20/(d)). It is clear
that Uo and Ua have isomorphic restrictions to every slice / x {a;} of / x P1.
To describe what happens at the branch points of u, we have the following
result, for whose proof we refer to [42]:
Proposition 31. For all x e P1, tie restriction Ua\f x {x} is a regular
semistable bundle. It is isomorphic toO/(g-p)®O/(r—p) ifi/~'1(x) = {q,r}
has two distinct points, and isomorphic to ? ® Oj(q — p) if x is a branch
point of v with preimage q. ?
Although Uo and U& have isomorphic restrictions to every slice, they have
different Chern classes in general:

8. Vector Bundles over Elliptic Surfaces 231
lamina 32. With notation as above, suppose that degd = d. Then
c2(Wd) = 1.
oof. Fixing d, let D be the divisor Д - ({p} x /) + Trjd. Then by
proposition 27 of Chapter 2, Ci(Z/a) is represented by the divisor (i/ x
JM)*-D — 2/ x {я}, where x is an arbitrary point of P1. (Here the branch
divisor of v x Id is linearly equivalent to 4/ x {a:}.) Now (i/ x Id), Д = Г„
!is' the graph of v: f -» P1. Since Pic(/ x P1) ^ 7rJ Pic/ e Z, Г„ is linearly
equivalent to a(/ x {ж}) + я^е for some divisor e on /. Clearly, Г„ П (/ x
{x}) = v~1(x) x {a;} and Г„ П (p x P1) = (p,i/(p)) (transverse intersections
if a; is not a branch point of v). Thus, Г„ = (/ x {x}) + 2({p} x P1).
Moreover, (i/ x Id)»({p} x /) = 2({p} x P1), since v has degree 2, and
^ld)(^d) = d(/x M)- Combining, we see that (i/xId),D-2(/x {x})
to
(/ x {x}) + 2({p} x P1) - 2({p} x P1) + d(/ x {x}) - 2(/ x {x})
as claimed. As for ca(?/d), by Proposition 28 of Chapter 2 and the fact that
(i/»DJ = v*D ¦ В = 0, where В is the branch locus of у х Id, we see that
c2(Wd) = -\{v x Id).(D2).
Since Д2 = ({p} x /J = (Tr^dJ = 0,
D2 = -2 - 2d + 2d = -2.
Thus, Oi(UA) = 1. П
The Chern class ег(?/а) is defined more generally inside the Chow group
A2(f x P1). Since rational equivalence is essentially trivial on the fibers
P1, it is elementary to show that A2(f x P1) 2 / x Z. Working inside
A2(/ x P1) ®Z[i], the proof of Lemma 32 shows that oi(Ud) is represented
by the 0-cycle -7r*(d - (d-f- l)p) • ^(x). In fact, this equality already holds
in A2(f x P1). The main interest in this more refined calculation is that the
universal bundle U& detects not only the integer d but also the full linear
equivalence class of the divisor d. In this sense, twisting by line bundles on
the spectral cover has led to different universal bundles.
Given V, we have an associated section A = A(V) С Рг*, or equivalently
the spectral cover С С X. Conversely, a section A of F2* leads to a bisec-
bisection С of X, not necessarily reduced or irreducible, and we can use С to
construct vector bundles on X whose restriction to every fiber is regular
and semistable. To do so, we assume for simplicity that С is smooth and
does not pass through the singular points of any singular fiber (although
neither of these assumptions is necessary). Let T = С Xpi X be the pulled

232 8. Vector Bundles over Elliptic Surfaces
back elliptic surface and let p: T —* С be the natural projection. There is
a degree 2 morphism v: T —* X. The diagonal D с X xPi X then pulls
back to a Cartier divisor on T, which we denote by E. Clearly, E is the
section of Г —» С corresponding to the embedding С —> С хр1 X given by
taking the identity on the first factor and the inclusion of С in X in the
second factor. Note also that the preimage of С С X under the degree 2
morphism. v:T—*X splits into E and another component E', and E and
E' meet transversally along the intersection of the branch locus of v with
E. The section a ol ж pulls back to a section ?o of T. The analogue of
the construction of the universal bundle Wo on a single fiber is then the
following: let /x be a line bundle on C, and define
The proof of the following may then be found in [42]:
Lemma 33. The restriction of tie bundle V(A,n) to every fiber / is
regular and semistable with trivial determinant. Moreover, if V is a rank
2 vector bundle on X such that the restriction of V to every fiber / is
regular and semistable with trivial determinant, and A = A(V), then there
exists a unique fine bundle ц on С such that V — V(A, fi). D
In the situation of Lemma 33, suppose that V is a rank 2 vector bundle
on X such that the restriction of V to every fiber / is semistable with trivial
determinant. We would like to find a criterion for when the restriction to
each fiber is actually regular. The following result is proved in [35] and [40]:
Theorem 34. Let V be a rank 2 vector bundle on X such that the re-
restriction of V to every fiber / is semistable with trivial determinant. Let
A = A( V) be the corresponding section ofW2k- If A meets the branch locus
В of the double cover i: X —» F2t transversally, then the restriction ofV
to every fiber / is regular, and hence V = V(A, /x) for a unique line bundle
/x on С ?
Our next goal is to compute the Chern classes of V(A, u). Before we do
so, however, let us record some basic facts about the geometry of W^ and
the double cover map rj: X —» W-^k-
Lemma 35. Let «To be the negative section ofWik, let ? be a fiber of the
ruled surface W^, and let A e |<т0 + Bfc + r)?\ be a section with А ф его,
i.e., г > 0.
(i) A2=2fc+2randdim|A| = 2Jfc + 2r+l;
(ii) If В is the branch locus of the double cover map tj: X —* Fa/t, then В
is linearly equivalent to 4<To + 6A:? and A • В = 6fc + 4r;
(iii) If С = rj*A is the preimage of a generic section A, so that С is smooth,
then g(C) = 3* + 2r - 1.

8. Vector Bundles over Elliptic Surfaces 233
jProot Parts (i) and (ii) are routine calculations, and (iii) follows from the
Riemann-Hurwitz formula. ?
Theorem 36. With V(A,n) defined as above, let m = deg/x. Then we
have the following formulas for the Cbern classes ofV(A,fj):
j,(i) detV(A,/x) = Ox((-k - r + m)f);
Proof. The map С -> P1 5! A is ramified at the points В(Л A by construc-
construction. The branch locus in X of the double cover map p: T —* X is linearly
equivalent to Ffc+4r)/ = (AB)f, by (ii) of Lemma 35 above. Thus, using
Proposition 27 in Chapter 2, det V(A, /x) corresponds to the divisor
i/.(E) - i/,(E0) + m/ - Cfc + 2r)f.
Clearly, i/*E = С and i/*Eo = v»v*<j — 2a. To determine the divisor class
of C, note that (as divisor classes)
С = rfA = jjVo + BA; + r)j]4 = 2a + Bk + r)f.
Combining, we see that
-3k-2r)f = (~k-r+m)f,
as claimed in (i). In particular, f»(<C — Eo) + mf is linearly equivalent to a
multiple of /.
To see (ii), we use Proposition 28 in Chapter 2: if D = ? - So + p*m,
where m is a divisor on С corresponding to /x, then
c2(V(A, /x)) = i {(u.DJ - v.(D2) - (i/,D) • Cfc + 2r)/).
Since vtD is a multiple of /, (v*DJ = (i/»D) ¦ / = 0, and so C2(V(A, /x)) =
-|i/,(D2). Since ? and Eo are both sections of T, f ¦ (E - Eo) = 0 (here
we also use / to denote a fiber of p), and since /2 = 0 as well, we see that
D2 = (E - EoJ = E2 + Eg - 2(E • Eo).
As both E and Eo are sections of T, E2 = Eg = (v*aJ - -2k, and thus
E2 + Eg = -4k. Moreover,
2(E • Eo) = 2(E • v*a) = 2(i/»E • a)
= 2(C -a) = (C- 2a) = {rfA- jjV0)
= 2(cr0 + BA; + r)e) ¦ a0 = 2r.
Thus,
c2(V(A,n)) = -i(-4ft - 2r) = 2ft + r,
as claimed. D

234 8. Vector Bundles over Elliptic Surfaces
We can now describe the birational structure of the moduli space of
stable bundles on X whose determinant is pulled back from the base. The
proof is given in [35]:
Theorem 37. Let A = af be a multiple of the Bber on X and let с =
2k + r, where к = pg(X) + 1 and г > 0. Let (w,p) be the corresponding
invariants. Then, for every (w,p)-suitable ample divisor H, the moduli
space ofH-stable rank 2 vector bundles V with invariants (w,p) contains a
Zariski open and dense subset M which Gbers over a nonempty Zariski open
subset U of \<t0 + Bk + r)t\ =* p2fc+2r+i The fiber over a pofnt AeU is
isomorphic to the Jacobian J(C), where С is the double cover ij-1(A). Q
Thus, the elliptic fibration X —* P1 has become a fibration over a pro-
jective space, whose fibers are the Jacobians of the hyperelliptic curves
C.
The second method for constructing bundles is via extensions. As before,
we begin with the case of a single curve /, possibly singular. We shall give
a construction of bundles corresponding to points of P1, more in keeping
with the interpretation of the moduli space P1 as the linear system |2p|.
Lemma 38. Let V be an extension
0 -» Of(-p) -*V-* Of(p) -* 0.
(i) V is semistable if and only if the extension is nonsplit.
(ii) If V is an extension А ф A, then А ф \~х.
(iii) Conversely, the bundles A 0 А, А ф A, S ® A, A®2 = Of, and (in
case f is singular) the bundle Q of Theorem 26 are all extensions of
Of(p)byOf(-p).
Proof, (i) If V is unstable, then the only possible destabilizing line bundle
must be O/(p), but in this case the extension would split.
(ii) It suffices to show that, for all A of degree 0, Л°(У ® A) < 1. But this
is clear from the defining exact sequence for V, since h°(O/(-p) ® A) = 0
&ndh°(Of(p)®\) = l.
(iii) It suffices to show that, with V any of the bundles enumerated in
(iii), there exists a subbundle of V isomorphic to Of(-p), since then the
quotient V/Of(-p) is automatically isomorphic to O/(p). For example, if
V = AeA, then
Hom(O/(-p), V) Si H°(Of(p) ® A) 0 H°(Of(p) ® A).
Now Of(p)<8>\ = Of(q) for a unique point q g /, and likewise Of(p)®\~1 =
O/(r). The hypothesis that А ф A means exactly that q ф г. Every
nonzero section oiOj(q) vanishes just at q, and likewise every nonzero sec-

8. Vector Bundles over Elliptic Surfaces 235
tion of Of(r) vanishes just at r. Thus, a general element of Hom(Of(-p), V)
does not vanish anywhere and so its image is a subbundle of V.
Next consider the case V = ? ® A. We shall just write out the case where
A is trivial. Using the exact sequence
0 -» Of(p) -* S ® Os{p) -¦ Of(p) -> 0,
and the fact that hl(O/(p)) = 0, we see that there exists a section s of
? ® Of{p) which projects onto a nonzero section of Oj(p). Clearly, s can
only vanish at p, and if s is nonvanishing it defines a subbundle of ? ® Of (p)
isomorphic to Of and thus a subbundle of ? isomorphic to Of(—p). In this
case, the quotient is necessarily O/(p). So it suffices to show that s does
not vanish at p. But if a vanishes at p, the inclusion Of —* ? ®O/(p) factors
through a map Of[p) -* ?®Of(p), such that the composition
is nonzero and thus is an isomorphism. It follows that there is a nonzero
map Of -~* ? splitting the extension, contrary to hypothesis.
The case of Q is similar and will be left to the reader. D
Suppose that V corresponds to a nonzero extension class
How do we decide what bundle V corresponds to in terms of the classifi-
classification of Theorem 25? The answer is given by the following lemma, where
for simplicity we will not work out the case of Q for a singular /:
Lemma 39. Let V correspond to a nonzero extension class ? €
^(/;О/(-2р)). Then V is of the form A e A or S ® A, where A =
®f(.4 ~ P)i if &nd only if ? = v(q) under the natural morpbism v. f —>
V(H°(OfBPy) = 1
Proof. V is of the form А ф A or ? ® A, where A = Of(q - p), if and
only if h°(V ® A) = 1, if and only if a nonzero section of Of(q) lifts to a
section of V ® A, if and only if the image of H°(Of(q)) in Hl(Of(q - 2p))
is zero under the coboundary map coming from the exact sequence
0 -> Of{q-2p)~*V®\-* Of(q) -> 0.
Now the commutative diagram of exact sequences
0 	> Of(-2p) 	> V®Of(-p) 	> Of
I	I	1
0 	> Of(q-2p) 	> У®А 	> Of{q)

236 8. Vector Bundles over Elliptic Surfaces
gives rise to a commutative diagram
H\Oj{-2p))
I	1
H°(Of(q)) —^ Hl(Of(q-2p)),
where the vertical таре are the natural inclusions, and the image of 1 e
H°(Oj) is the extension class ?. Since the left-hand vertical map is an
isomorphism, we see that d(H°(Oj(q))) = 0 if and only if the image of (
under the natural map Я1(О/(-2р)) -* Я1(О/(? - 2p)) is 0, if and only
if $ is dual to the hyperplane which is the image of Я°(О/Bр - q)) in
Я°(О/Bр)), if and only if f = u(q) under the natural morphism u: f -+
P(H°(O/Bp)v). ?
Corollary 40. Iffi,f2 € Я1(О/(-2р)) define isomorphic vector bundles,
then & and ?2 differ by a nonzero scalar. О
We can then make a universal extension of O/(p) by Oj(— p) which fits
into an exact sequence
0 -> 7rJO/(-p) ® ^OPi(l)"-* U -v n^Of(p) -* 0.
Note that such extensions are classified by
by the Runneth formula. Now canonically the P1 is РЯ1(/;О/(-2р)),
and thus Я°(Р1;С>р»A)) = Я1(/;О/(-2р))у = X, say, where X is a 2-
dimensional vector space. We can then take the element in
corresponding to the identity, to define an extension U which restricts on
every slice / x ? to the extension defined by f. (Note that if we had not
twisted the subbundle by 7T2<!VA), the possible extensions of 7r*O/(p) by
n*Of(— p) would then all be pulled back from a single fixed extension of
Of(p) by Of(-p).)
Computing Chern classes of U, we see that detW = 7rJOpi(l) and that,
in A2(f x P1), C2(U) = 7rJ(p) -TTaC1)- From this, one can show that U = U%^
in the notation of Lemma 32.
Similar but more involved constructions yield a universal relative exten-
extension over X. However, without using line bundles on the spectral cover, we
cannot construct a Zariski open subset of the moduli space in this way, but
only a section of the fibration described in Theorem 37.

8. Vector Bundles over Elliptic Surfaces 237
Even fiber degree and multiple fibers
[n this section, we want to generalize the results of the last section to the
zaae where X does not necessarily have a section. For simplicity, we shall
just limit ourselves to the case where X is simply connected. Thus, the
base curve of X is P1 and X has at most two multiple fibers, of relatively
prime multiplicities mi and m^. We shall also assume that the multiple
libers have smooth reduction and that all other singular fibers are irre-
irreducible nodal curves. As usual, stability is assumed to be with respect to
a suitable polarization. We would like to outline the description of stable
vector bundles V over X with ci(V) • / = 0 (mod 2) for the general fiber
/. More detailed arguments can be found in [35], [40], and [36].
As in the last section, we shall just discuss V under the assumption that
V\f is semistable for all fibers /. Working on a fixed smooth fiber /, if E
is a semistable rank 2 bundle on / with det E = 6, where deg б = 2d, then
either
for a line bundle A with deg A = deg 6/2, от Е is given as a nonsplit exten-
extension
where A is one of the four line bundles on / such that A®2 = 6. Moreover,
such E are identified in the moduli space with the direct sum А ф A. Thus,
'the moduli space of semistable bundles is isomorphic to Jd(/)/t, where
Jd(f) is the group of line bundles of degree d on /, which is a principal
homogeneous space over J°(/), and t is the involution of Jd(f) denned
by А ь+ 6 ® A. The involution t has four fixed points, and the quotient
Jd(/)/t is isomorphic to P1.
Now we can fit this picture together over X: define the surface Jd(X) to
be the elliptic surface over P1 whose fiber over t is naturally Jd(tr~l(t)).
In terms of the classification scheme of the last chapter, if X corresponds
to (, e tf^G.B), then Jd(X) corresponds to the class d? € Hl{G,B).
For example, if d = 0, then J°(X) = J{X) is the Jacobian surface of X.
Thus, if X has a multiple fiber of multiplicity m at t, corresponding to
the image of ? in H1(Gt,Bt) = (Q/ZJ having order m, then the image
of df has order m/ gcd(d, m) in (Q/ZJ. In our case, we assume that there
exists a divisor AonX with Д • / = 2d (here Д will be the determinant
of our rank 2 bundles V). Thus, mim2 divides 2d, since / is divisible by
пнтг in NumX, and furthermore mi and mi are relatively prime. We see
that there are two cases: either пнтг divides d, in which case Jd(X) has
no multiple fibers, or тп\тпг does not divide d. In the second case, one of
тьт2 is odd, since they are relatively prime. Say тг is odd. Since miin^
divides 2d but т^ does not divide d, mi is even, say mi = 2m'lt and
2mi divides 2d so that m\ divides d and d/m^ is odd. Clearly, the gcd of

238 8. Vector Bundles over Elliptic Surfaces
mi = 2mi and d is т'х, so that mi/gcd(d,mi) = 2. It follows that in thfa
case Jd(X) has a multiple fiber of multiplicity 2 at the point correspondinj
to mi and no other multiple fibers. Finally, if к is the primitive class sucl
that / = пцтгк, then as A • / = 2d,
л	2d
A • к —	.
In particular, Jd(X) has no multiple fibers if A • к is even and has a single
multiple fiber of multiplicity 2 if A ¦ к is odd.
Now a stable rank 2 bundle V gives a bisection С of Jd{X) (possibly
reducible or of multiplicity 2), defined as follows: for each fiber / ovei
t 6 P1, we associate to t the pair of line bundles {A, 6 ® A} of degree d
such that F|/ = Ae^A), in case F|/isadirect sum(and6 = A|/),and
if V\f is a nonsplit extension of A by A we assign to t the line bundle A with
multiplicity 2. Clearly, С is invariant under the natural involution of Jd(X)
corresponding to the involution t on each fiber. Thus, С is the pullback o(
a curve A on the quotient Jd(X)/L. Note that Jd(X)/i is birational to a
ruled surface F over P1 and Л is a section of the ruling. In general, as we
vary the bundles V, the curves A will move in a linear system on some
surface birational to a ruled surface, and we will denote this linear system
by |A|. Conversely, a section A defines a bisection С of Jd(X) via pullback,
where С may be reducible or multiple with multiplicity 2. Note however
that if Jd(X) has a multiple fiber of multiplicity 2, then for every bisection
С of Jd(X), the induced map С -* P1 must be branched at the point of
С meeting the multiple fiber, whereas this phenomenon does not happen
if Jd(X) has no multiple fibers.
Given the bisection C, how close is it to determining V? Assume that
С is smooth and irreducible, and let Y = С xPi X —» С be the induced
elliptic surface. We assume that С —» P1 is generic (not branched over the
points corresponding to singular or multiple fibers of X) so that У is a
smooth elliptic surface. On X, since С is irreducible, there is no way to
distinguish A from (S ® A). But if v: Y -» X is the double cover map,
then one shows that there is a divisor D on Y such that, for a given fiber /
on X, if V\f = A$ F® A), where 6 - A|/, if /' 9i f is a fiber above / on
Y, then Oy(D)\f' is either A or F® A) under the natural identification.
Thus, on the double cover we can coherently fit together the line bundles
A to obtain a well-defined line bundle Oy(D). It then follows easily that
there is an exact sequence
0 -+ OY{D) -» v*V -» Oy(i/'A - D) -» 0.
(Here we use the assumption that V\f is semistable for all / to conclude
that the quotient v*V/OY(D) is actually a line bundle.) The most difficult
part of the analysis is to show the following [40], [35]:

8. Vector Bundles over Elliptic Surfaces 239
Theorem 41. Suppose that V\f is semistabJe for all f and that the cor-
corresponding bisection С is generic. Let F be the branch locus of the map
Y -+ X. Then V = v»Oy(D + F). Conversely, starting with a generic bi-
bisection С of Jd(X) and letting g: Y —> X be the induced double cover, if
D is a divisor on Y such that det vtOy(D + F) = Д, then the rank 2 vector
bundle V = u,OY(D + F) is a stabte rank 2 bundJe on X with ci(V) = A.
*', Thus, we must analyze the condition that det i/»Oy (D + F) = Д. Let G
be the divisor class on X such that 1G is linearly equivalent to the branch
locus on X. In other words, v*G = F. Using the formulas of Chapter 2,
Proposition 27, we see that deti/»Oy(D + F) = v»D + v»F - G, where
v*G = F and thus i/,F = i/,i/*G = 2G. It follows that
det i/,Oy (D + F) = i/»D + G = A.
fJow the set of all divisor classes D such that v,D — Д — G is a prin-
principal homogeneous space over the set of all divisor classes EonY such
that v+E = 0. A straightforward argument identifies this group with an
Extension of Pic°(C) by the torsion in #2(У; Z).
i; To describe the torsion in #2(Y;Z), note that Y has two multiple fibers
<F{ and F'i of multiplicity mi and two multiple fibers F^ and F? of mul-
multiplicity ТП2, if С is not branched over either multiple fiber. In this case,
F[ — F" tea torsion element of order mi, and similarly F? — F% is a torsion
element of order тпг- In fact, it is easy to show, using Proposition 27 of
the last chapter, that these two elements generate the torsion subgroup of
H^YjZ), which is isomorphic to Z/miZ x Z/m2Z = Z/m1m2Z. In this
case, we arrive at the following picture for a Zariski open subset of the mod-
moduli space (at least for large enough сг): it fibers over a Zariski open subset
of the projective space |A|, and the fibers are mima copies of a complex
torus Pic°(C) of dimension g = g(C). Once again, we see the ghost of the
elliptic structure on X in the birational structure of the moduli space.
In case mi = 2mi and Д • к is odd, the picture is slightly different: the
map С —¦ Pl is always branched at the point corresponding to the multiple
fiber Fi. Thus, Y has a single multiple fiber of multiplicity m'1 and two
multiple fibers of multiplicity m-2 lying over F^. The torsion subgroup of
H2(Y\ Z) is then just isomorphic to Z/m2Z. Otherwise, the picture is the
same: the moduli space fibers birationally over a projective space, and the
fibers are m-2 copies of a complex torus.
Using the above and an analysis of the case of odd fiber degree, one can
determine the "leading coefficient" on of the Donaldson polynomial D*p
of a simply connected elliptic surface. Here it follows from very general
arguments [40] that we can write

240 8. Vector Bundles over Elliptic Surfaces
where qx is the intersection form on X and the above expression lives in the
symmetric algebra on H2(X). We assume that n is chosen so that an ф о
and refer to an as the leading coefficient of D*p. The final answer is the
following:
Theorem 42. Let X be a simply connected elliptic surface with pg(X) =
pg > 0 and multiple fibers of multiplicities mi and год. Letan be the leading
coefficient of the Donaldson polynomial corresponding to invariants w,p.
Up to universal combinatorial factors,
f(mim2)p», ifw- к = 0 mod 2,
(mim2)p»m2, i/w • к = 1 mod 2 and mi = 2m'! is even,
1,	i/w • к = 1 mod 2 and mi and m2 are odd.
Roughly speaking, the idea of the calculation is as follows. We have
defined the map /x: Нз(Х) -* ff^JW), where ffl is the appropriate moduli
space. On the Zariski open subset of 971 that we have described above,
one shows that /x(/) is represented by the pullback of a hyperplane in
the projective space \A\. Thus, if dim|./4| = a, /x(/)a is represented by а
number of copies of the Jacobian J(C) ofLthe corresponding bisection C,
and /x(/N = 0 if 6 > a. In fact, this statement holds over the full moduli
space and its Uhlenbeck compactification, not just on the dense open subset
described above. Now if X has a section cr, then /x(/)° is represented by
a single copy of J(C), and one shows that /x(<r) restricts on J(C) to the
theta divisor в of J(C). In this way, we can calculate /x(/N • n(cr)d~b for
all b > a: it is 0 for b > a and (d - a)! for 6 = a. If X has multiple fibers,
there are no sections, but one can show that, for every multisection т, if
t = т ¦ f, then /x(r) restricts to t ¦ в on each copy of J(C), and so we can
make the calculation as before.
Combining Theorem 42 and Theorem 18, it is an algebraic exercise to
show that we can recover the integers m,\ and m2 from the Donaldson
polynomials of X in case p9 > 0. A more detailed analysis, which involves
working out the four-dimensional invariant in the case where mi and m%
are odd, also handles the case where pg = 0.
Beyond the leading coefficient, the second coefficient of the Donaldson
polynomial has been determined in the case pg = 1 by Bauer [8] and Morgan
and O'Grady [102], and this information has been used to determine the
second coefficient in the case of trivial determinant for all pg > 1 [101],
[83], [142]. Independently, Fintushel and Stern have worked out the entire
Donaldson series for all pg > 1, without using algebraic geometry [33].
Exercises
1. If 0—* L' —> L —>t—>0isan exact sequence of sheaves on C, where
V and L are torsion free rank 1 sheaves on С and r is supported on a

8. Vector Bundles over Elliptic Surfaces 241
finite number of points, then deg L' < deg L, with equality holding if
and only if I! = L.
2.	If L' and L are torsion free rank 1 sheaves on С and Hom(Z', L) ф 0,
then deg L' < deg L with equality only if L = L'.
3.	If V is a rank 2 vector bundle and
is exact, where U and Z are torsion free rank 1 sheaves, then deg L' +
degLTdegV.
4.	If n: E -* С is the normalization and Z is a line bundle on C, show
that degn»L = deg L+S.
5.	Suppose that A is a line bundle on C. Show that deg(Z ® A) = deg L +
deg A.
8. Let С be a curve with only one singular point, which is an ordinary
double point, and let n: С —» С be the normalization. Show that every
torsion free rank 1 sheaf on С is either a line bundle or of the form n*L,
where I is a line bundle on C. (Let R = C{x, y} and let 5 = R/(xy)R.
If L is a torsion free rank 1 5-module, then let M = xL®yL and let
N = L/xL ф L/yL. The hypothesis that L is torsion free implies that
there are inclusions M С L С N. Both M and TV are 5-modules, where
5 = C{i}®C{j}, and since they are torsion free we may assume that
M = S and M = m • 5 = m, where m is the maximal ideal of 5 and
also of 5. Thus, either L^SorL^m^S (locally), or S/L has
length 1. Show that we can then find an isomorphism L = S.)
T. Let 5 = R/(xy)R as in the previous exercise, and let 5 = R/xR ф
R/yR. Show that
is a resolution of the 5-module 5, where the maps alternate between
(a, b) <-* (xa,yb) and (a, b) ь+ (ya,xb). Conclude that Ext*sE,5) = 0
for all t > 0. Verify that HomE,5) = m and that Hom(m,5) = S
(canonically). Finally, show that Ext 1(S, S) ^ S/S as 5- and 5-
modules. Show that, in P Ext1 E,5) = P1, there are two points cor-
corresponding to non-free 5-modules, both isomorphic to 5 ф 5, that 5
acts transitively on the remaining C*, and that all of the points of this
C* correspond to free 5-modules = 5 ф 5.
S. Let С be irreducible with only nodes as singularities. Let L' and L
be two torsion free rank 1 sheaves on C. Show that Hom(L'', L) is
again a torsion free rank 1 sheaf on C, and that deg L — deg V <
deg Hom(L', L) < deg L — deg V + 6. When does equality at either
end hold? When is Hom(L', L) locally free?
9. Let С be a reduced irreducible curve with normalization С and let A,
В be rank 1 torsion free sheaves on С of degrees a and b, respectively.
If Hom(A,B) Ф 0, then a < b, and in this case 6imUom(A,B) <
b-a+1. (If Нот(Л, В) ф 0, then there is a nonzero map, necessarily

242 8. Vector Bundles over Elliptic Surfaces
an injection, from A to В and thus an exact sequence
0-+A-+B-+Q-+0.
Thus, b = a + ?(Q) > a. To prove the second statement, argue by
induction on b - a. In case b - a = 0, either dim Hom(.A, B) = 0 or В ^
A, the claim is that dimHom(.A, A) = 1. In this case, Hom(A, A) is a
coherent sheaf of commutative torsion free rank 1 Ьс-algebras, which
agrees with Oc at the smooth points of C, and thus is a subalgebra of
Oc. Thus, dimHom(A,A) = h°(C;Hom(A,A)) < H°(O6) = 1. For
the inductive step, assume that the result has been shown for all B'
such that degB' — a < n. Given В such that degВ — a = b -a = n,
let x be a smooth point of С and define B' by the exact sequence
0 -* B' -+ В -v Ox -* 0.
Thus, deg B' = b — 1 and there is an exact sequence
0 -* Hom(A, B') -+ Hom(A, B) -+ Hom(A, Ox).
By assumption dimHom(A, ?') <b-l-a+l = b — a. Since я is a
smooth point of C, A is a line bundle in a neighborhood of x and so
dimHom(i4, Ox) = 1. It follows that dimHom(i4, B) < b - a + 1.)
10.	Let С be an irreducible curve with only ordinary double points such
that pa(C) = 1. Let L be a torsion free rank 1 sheaf on С Show that if
degZ > 0 or if deg L = Oand L is not equal to Oc, then h°(L) =degL
and h^L) = 0. What happens if L - Oc or if deg L < 0?
11.	Show that, if V is an unstable rank 2 vector bundle on the irreducible
curve C, then there is a unique maximal destabilizing torsion free rank
1 subsheaf L in the sense of Proposition 20 of Chapter 4.
12.	Let 5 = C{x, y, z, w} be the ring of convergent power series in four
variables and let R = S/(xy—zw). Thus, R is the analytic local ring of
an ordinary double point of dimension 3. Let / = IoR = (x—z, y-w)R.
We seek a free resolution of the Д-module I. Let A — I	1. where
\z xj>
we view the entries as lying in R, and let В = ( X W ). Note that
\—z у J
det A = det В = 0 in R and that AB = BA = 0 as well. Show that
... Д д2 Д Д2 Д R2 Д Д2 J, j _^ 0
is exact, where /(a, b) = а(х - z) + b(y - w). Dualizing the above, we
find that
0 -v Нотд(/, R) -* R2 -^ Я2 -^ Я2 -* • • •
is exact. Conclude that ExtlfiG, Д) = 0, г > 0, and that Нотд(/, R) =
IeR = (x -w,y - z)R. Arguing similarly, show that Extl(R/I,R) =
0,г ф 1, and that Ext^fl//, R) = Д/7. Show also that J is reflexive.

8. Vector Bundles over Elliptic Surfaces 243
Is it true that the Д-module I = /dR(x - z,y - w)R is isomorphic to
the Д-module IeR(x -w,y - z)R?
13. Let С be an irreducible curve with only ordinary double points such
that pa(C) = 1. Let L\ and L^ be torsion free, non-locally free rank
1 sheaves on C, with degZ* = dj. Show that, if d\ > da, then there
exist locally free extensions
0 -* Fi ~* V -» F2 -» 0,
with det V any line bundle of degree d\ + d^. (First show that one such
V exists, using Exercise 7, and then tensor with an appropriate line
bundle of degree 0.)

9
Bogomolov's Inequality and
Applications
Statement of the theorem
Dur main goal in this chapter will be to give a proof of the following:
Theorem 1 (Bogomolov). Let X be an algebraic surface and let H be
an ample divisor on X. Suppose tiat V is an H-stable rank 2 vector bundle
oh X. Then c?(V) < Aoz(V). Equivalent^, Pi(adV) < 0.
;f The argument given here can be adapted, with a few technical modifica-
modifications, to prove the general theorem: if, in the above, V is assumed to have
jjank r, then
\	(r - l)c?(V) < 2rc2(V).
f here is a further generalization to higher-dimensional varieties: if X is
a smooth projective variety of dimension n and V is an Д'-stable vector
bundle of rank r on X, then
(r - l)c?(V) • Я" < 2rC2(V) ¦ Я".
Bogomolov's original proof of Theorem 1 appears in [10]; see also [129]
for an account of the proof in the case of rank 2. Gieseker [46] gave a proof
which involves studying the restriction of V to a general hyperplane section
and reducing mod p. The proof we give here is a combination of the two
proofs and is essentially the same as Miyaoka's proof [97]. A deep approach
to the theorem, as discussed in Chapter 4, is via the Donaldson-Uhlenbeck-
Yau theorem that a stable V carries a Hermitian-Einstein connection, in
which case the theorem becomes an exercise in differential geometry. More-
Moreover, this proof shows that, in case equality holds and c\(V) = 0, then V
is a unitary flat vector bundle, in other words a bundle associated to a
unitary representation of ni(X). Finally, Fernandez del Busto [32] has re-
recently given a proof in the rank 2 case based on the Kawamata-Viehweg
vanishing theorem.

246 9. Bogomolov's Inequality and Applications
Next we give some easy applications in the rank 2 case.
Corollary 2. Suppose that X is an algebraic surface and that V is a rank
2 vector bundle such tiat c\{V) > 4сг(^). Then there is a unique sub-line
bundle OX(D) of V and an exact sequence
0 -¦ OX(D) -+V^ OX(D') ® IZ -¦ 0,
wiere Z is a codimension 2 Jocai complete intersection subscbeme of X,
such that (D - D'J > 0 and (D - 1У) ¦ H > 0 for every ample divisor H.
Proof. Fix an ample divisor Ho. Theorem 1 implies that V is not #0-
stable, and in fact V cannot be Яо-semistable, since a strictly #o-semistable
vector bundle V satisfies Bogomolov's inequality (Exercise 1 of Chapter 6).
Thus, V is Яо-unstable, and there exists an exact sequence
0 -» OX(D) -» V -» OX(D')
where (D - D') • Ho > 0. Note moreover that
c\{V) - 4c2(V) = {D - ГУJ - U{Z) > 0,
and thus (D - D'J > 0. But now Lemma 19 of Chapter 1 implies that
(D — D') ¦ H > 0 for every ample divisor H. The uniqueness is clear from
the uniqueness of the destabilizing sub-line bundle for H. D
The above says that D - D' is in the cone dual to the ample cone, which
is exactly the cone spanned by the effective divisors. In fact, by Lemma 12
in Chapter 1, some multiple of D - D' is effective.
The next corollary, due to Bogomolov, is a strong version of a restric-
restriction theorem for rank 2 vector bundles due to Mumford and Mehta and
Ramanathan, which says that for large к and a generic curve С € |А;Я|,
the restriction V\C is semistable. In fact, in the course of the proof of
Bogomolov's inequality, we shall prove a weak version of the theorem of
Mumford and Mehta and Ramanathan. The final inequality then allows us
to prove the following much more precise restriction theorem:
Corollary 3. Let X be an algebraic surface, let H be an ample divisor on
X, and let V be an H-stable rank 1 vector bundle on X with p\ (ad V) = p.
Suppose that к > —p. Then for every smooth curve С € \kH\, the vector
bundle V\C is stable.
Proof. Suppose instead that V\C is not stable. Then there exists a quo-
quotient line bundle L of V\C such that 2degZ < deg(V|C) = ka(V) ¦ H. If
j: С —> X is the inclusion, there is a surjection V -* j»L. Let V be the
kernel of this surjection, so that there is an exact sequence
0 _* V' - V -»jtL -» 0,

9. Bogomolov's Inequality and Applications 247
and V is an elementary modification of V. By Lemma 16 in Chapter 2,
Cl(V) = a(V) - kH and o>(V) = c2(V) - a(V) ¦ (kH) + degL. A calcu-
calculation gives (cf. also Lemma 9 in Chapter 6):
Pl(adF') = Pl(adV) + 2Cl(V) ¦ (kH) + (kHJ - 4degZ
>р+А;2(Я2)>р + р2>0.
Thus, V does not satisfy Bogomolov's inequality and is therefore unstable.
Let Ox(D) —* V be the maximal destabilizing sub-line bundle, so that
there is an exact sequence
0 -¦ OX(D) -*V-* OX(D')
with H ¦ (D - D') > 0. In particular
0 < p + А;2(Я2) < Pl(ad V) = (D - D'f - U(Z) <(D- D'J.
Now V is a subsheaf of V, and so there is a nonzero map from Ox(D) to
V. Since V is Я-stable, 2DH < cx(V)H. Thus, if m = -BD-ci(V))-H,
then m > 0. Moreover, the Hodge index theorem (Exercise 10 of Chapter
1) implies that BD - c^V))^2 < ™?.
Since D+V = ci(V') = cx(V)-kH, we have D-D1 = 2D-d(V)+kH.
Using H(D-D')> 0, we have m = -BD - a(V)) ¦ H < kH2. Thus,
p + А;2(Я2) < (D - Г/J = BD - d(V)J + 2kBD - a(V)) ¦ H + k2H2
< -Jin ~ 2km + к Н ,
so that 2km < m2/H2 - p. Using 0 < m < kH2 and -p < A;, we finally
have
H2 m	m
a contradiction. Thus, V\C is stable. П
The proof of Corollary 3 goes over unchanged in case С is just assumed
to be reduced and irreducible.
We remark that, if we take к > —p, the argument above shows that
the bundles V\C become progressively "more stable," in the sense that for
every quotient line bundle L of V\C, ft(L) — fj,(V\C) is bounded away from
0 by a fixed positive rational number (depending on the choice of k).
The last easy corollary of Bogomolov's inequality is the following ele-
elegant proof (due to Mumford) of Mumford's generalization of the Kodaira
vanishing theorem on a surface (Theorem 26 in Chapter 1):
Corollary 4. Let X be a smooth surface and let L be a net and big line
bundle on X. Then Hl(X; L'1) = 0.

248 9. Bogomolov's Inequality and Applications
Proof. Suppose instead that Я1^;!-) Ф 0. Then there is a rank 2
vector bundle V which is a nonsplit extension of L by Ox', in other words
there is an exact sequence
By the Whitney product formula, C\{V) = L and &г(У) = 0. As L is big,
d(VJ = I2 > c2(V) = 0. Thus, V is Я-unstable for every ample Я. Let
Ox(D) be the maximal destabilizing sub-line bundle. As usual we have an
exact sequence
0 -¦ Ox(D) -+ V -* Ox(-D) ®L®Iz-*0.
The induced map Ox(D) —> V —> L must be nonzero, for otherwise the
map Ox(D) -* V would factor through Ox, so that D = — E where E
is effective. In this case Я • D = -(Я ¦ E) < 0, so that OX(D) is not
destabilizing.
Thus, Ox(D) = L ® Ox(-E) for some effective divisor E. Since L is
nef, L ¦ E > 0. We can write V as an extension
0 - L 9 Ox(-E) ->V-> OX(E) ® Iz -+ 0,
and thus
Rewrite this as E2 = (Z, • S) + *(Z) > Z, • E > 0. Now 2Я • D > H ¦ L
for every ample Я. Since L is nef, it is a limit of ample divisors, and thus
2LD = 21? - 2(L ¦ E) > L2. So I? > 2(L ¦ E). Thus, {L2)(E2) >
2(L • E)(E2) > 2(L ¦ EJ. On the other hand, by the Hodge index theorem
we have (L2)(E2) < (L ¦ EJ. So 2(L ¦ EJ < (L ¦ EJ, which is only possible
if L E = 0. By the Hodge index theorem, we would have E2 < 0, and since
we have seen that E2 >0,E2= 0, and E is numerically equivalent to 0. As
E is effective, E = 0 and since there is a nontrivial map L —* L®IZ, Z = 0.
But now there is a map L —* V projecting to a nonzero map L —* L, which
after adjusting by a scalar we can assume to be the identity. In particular
the defining exact sequence for V is split, contrary to assumption. It follows
that Hl{L~l) = 0. D
In the next section we shall generalize the methods of the above proof
to give Reider's analysis of linear systems of the form \D + Kx\, where D
is nef and big.
The theorems of Bombieri and Reider
Bombieri's famous theorem on the multiple of the canonical bundle
needed to define a birational morphism for a surface of general type is
the following:

9. Bogomolov's Inequality and Applications 249
Theorem 5. Let X be an algebraic surface, and suppose that Kx is nef
ind big. Let y>n: X --* ?N be the rational map corresponding to the linear
fystem |n.K"x|, if this is meaningful. Then:
(i) For all n > 5, v?n is a birational morphism to its image, which is the
normal surface X obtained by contracting the smooth rational curves
1 С onX with C2 = -2.
(ii) (fit is always a morphism (i.e., \4KX\ has no base points), and y>\ is
birational provided that K\ > 2.
[Hi) f3 is a morphism if Kx > 2 and is birational provided that K\ > 3.
(iv) v?2 is a morphism ifK\ > 5 and is birational provided that K\ > 10,
unless there is a pencil of curves on X of genus 2.
. (Prior to Bombieri's theorem, Moishezon had shown in [132] that </?g is
birational to its image, and Kodaira [73] established a similar result for
P6-)
* We will deduce Bombieri's theorem from the following theorem:
Theorem 6 (Reider). Let X be an algebraic surface, and let D be a nef
and big divisor on X. Then:
(i) If D2 > 5 and x is a base point of \KX + D\, then there exists a curve
E on X with x e Supp? such that either D ¦ E = 0 and E2 = -1 or
D ¦ E = 1 and E2 = 0.
(ii) If D2 > 10 and x, у € X are two points, possibly infinitely near, such
that \Kx + D\ does not separate x and y, then there exists a curve E
on X such that x, у € SuppS such that either D ¦ E = 0 and 252 = — 1
or -2 or D ¦ E - 1 and .Ё2=0ог1ог?)-.Е = 2 and fj2 = 0.
Proof that Reider's theorem implies Bombieri's theorem. For n >
2, the divisor D = (n — l)A"x is nef and big. Note that if E is a curve
with E ¦ D = 0, then ?? is even, by the Wu formula, and, if n > 2, then
DE = (n-l)(JGr •#) which can never be 1. If n = 2 and ?•? = KXE = 1,
then again by the Wu formula E2 is odd, and so we can never have E2 — 0.
Thus, we can apply (i) of Reider's theorem as soon as (n — lJ#x ^ 5.
This inequality is automatic if n > 4, and it is satisfied for n = 3 as soon
as Kx > 2, and for n = 2 as soon as Kx > 5.
Next we consider when tpn is birational. First note that (n— 1JKX > 10
as long as n > 5, or n = 4 and K\ > 2, or 71 = 3 and K\ > 3, or 71 = 2 and
¦f^x > 10. Leaving aside the question of the existence of smooth rational
curves of self-intersection —2 on X for the moment, we see that the case
D -E = 0, E2 = -1 is impossible, as well as the case D -E = 1, E2 = 0.
Moreover, if n > 3, we cannot have D ¦ E = 1, since D • S is divisible by
71-I. Likewise, if 71 > 3, we cannot have D ¦ E = 2 and E2 = 0: either
71 > 4, in which case (n-1)|2, which is impossible, or n = 3, D E = 2, and

250 9. Bogomolov's Inequality and Applications
so Kx • E - 1. But then by the Wu formula E2 is odd. Thus, Kx fails to
separate p and q if and only if there exists an effective divisor E containing
p and q such that E2 = -2, E Kx = 0, or n = 2, Kx -E = 1, and E2 = i,
or n = 2, A"x -25 = 2, and E2 = 0. There are only finitely many curves E
such that Kx ¦ E = 0, and so <?>„ is always birational away from these. In
the remaining exceptional cases, n = 2 and either Kx E = 1 and E2 = l(
or A"x • S = 2 and E2 = 0. In both cases pa(E) = 2. Either there are
only finitely many such curves E, or X has a (possibly irrational) pencil of
curves of genus 2.
The remaining point to check is that, if n > 5, then <pn(X) really is
the normal surface X obtained by contracting the smooth rational curves
of self-intersection -2 on X. This would lead us too far afield into the
geometry of rational double points, and so for the proof (as well as for the
proof that фп(Х) is projectively normal if n > 5), we refer to [11] and
[7]. ?
Proof of Reider's theorem. First suppose that D is nef and big, with
D2 > 5, and that x is a base point for \Kx + D\. We seek a rank 2 vector
bundle V which is given as an extension
0 -¦ Ox -> V -> Ox (D) ® mx -¦ 0,
where mx is the ideal sheaf of the reduced point x. By Theorem 12 in
Chapter 2, a locally free extension V exists if and only if every section of
Kx + D vanishes at x, i.e., x is a base point for \Kx + D\. By assumption,
then, we can find such a V, and clearly ci(V) = D and c2(V) = 1. If
D2 > 5, Bogomolov's inequality is violated and so V is unstable. Thus,
there exists a sub-line bundle Ox(F) of V such that V/Ox(F) is torsion
free and 1H-F > H-D > 0 for every ample divisor H. Since D is a limit of
ample divisors, 2DF>D2. Now the induced map Ox(F) -> Ox(D)®mx
is nonzero, for otherwise F = —E, where E is effective, and thus 2H-F < 0
for every ample H. Thus, F = D-E, where E is effective and x € Supp E.
In particular E is not numerically trivial. The inequality 2D ¦ F > D2
becomes D2 >2D ¦ E. Since there is an exact sequence
0 -¦ OX(D - E) -> V -> OX(E) ® Iz -* 0
for some 0-dimensional subscheme Z, 1 = c2(V) = D- E- E2 + i(Z), and
thus D E - E2 < 1. Thus, we have:
2DE<D2, DE-E2<1,
DE>0,	D2E2 < (D ¦ Ef,
where the third inequality holds since D is nef and E is effective, and the
fourth follows from the Hodge index theorem (Exercise 10 in Chapter 1).
Putting this together,

9. Bogomolov's Inequality and Applications 251
and so D ¦ E < 2, with equality if and only if D and E are proportional.
In this case E2 > 1 and so D2 < 4, a case excluded by the hypotheses
of Reider's theorem. Thus, either D-E = 0orD-E= 1. If D ¦ E = 0,
then E2 < 0 by the Hodge index theorem and E2 = 0 if and only if E
is numerically trivial, which does not hold since E is a nonzero effective
divisor. Since -E2 < 1, in fact E2 = -1.
;. The remaining possibility is that D -E = 1. In this case, E2 < 1/D2 <
1/5, so that E2 < 0. But as E2 > 1 - (D ¦ E) = 0, E2 = 0 in this case.
j Now assume that D2 > 10 and that x, у are distinct points of X such
that \Kx +D\ does not separate x and y, in other words that every section
pf Ox(Kx + D) which vanishes at x also vanishes at y. Again by Theorem
12 of Chapter 2, there exists a vector bundle V and an exact sequence
0-*Ox-+V-+ OX(D) ® I{XiV} -* 0,
where I{x,y} is the ideal sheaf of the reduced scheme supported on {x,y}.
fa this case, ci(V) = D and C2(V) = 2, and since D2 > 10, we have
tfi(V02 > 4c2(V) = 8. Arguing as in the previous case, there is an effective
divisor E with x, у e Supp E such that
2DE<D2, DE-E2<2,
DE>0,	D2E2 < (D ¦ EJ.
Thus,
and so D ¦ E < 4, with equality if and only if D and E are proportional.
If D • E = 4 and D = XE {от some positive rational number A, then
10 < D2 = 16/S2, so that E2 = 1, contradicting D ¦ E - E2 < 2. If
D • S = 3, then E2 > 1, and so D2 < 9/E2 < 9, contradicting D2 > 10.
Thus, D ¦ E is either 0,1, or 2.
If D ¦ E = 0, then 2J2 < 0 by the Hodge index theorem (note that
E2 = 0 is impossible since E is effective and nonzero). On the other hand,
since D -E - E2 < 2, E2 = -2 or -1. If D -E = 1, then E2 > -1 and
E2 < 1/D2 < 1/10, so that E2 < 0. The only possibilities are E2 = -1,0.
Lastly, if D ¦ E = 2, then E2 > 0 and E2 < 4/10, so that E2 = 0.
Finally, we consider the case where x is a point of X and у is an infinitely
near point, in other words a tangent direction at x. In this case there is
a O-dimensional local complete intersection subscheme Z supported at x
corresponding to the tangent direction y, with ?(Z) = 2. To say that the
sections of Kx + D do not separate the tangent directions at x is to say
that every section of Kx + D vanishing at x also vanishes on the direction
y. A local argument shows that there is an everywhere generating section of
Extl(Ox{D)®Iz, Ox) which lifts to an element of Ext^OxCD)®/^, Ox)-
Thus, we may construct a locally free extension V as before, and reach a
similar conclusion. D

252 9. Bogomolov's Inequality and Applications
The proof of Bogomolov's theorem
Our goal in this section is to prove the following theorem:
Theorem 7. Let V be a rank 2 vector bundle on X. Suppose that there
exist two smooth curves Ci and Сг on X with C\ • C2 > 0 and C| > 0 such
tnat V\d is semist&ble for i = 1,2. Then cx{VJ < ^{V)
In the theorem, C\ is allowed to equal C2, in which case the statement
simply reads C\ > 0. We will first prove the theorem, and then show in
the next section that if V is stable it fulfills the hypotheses of the theorem.
We will also postpone the proof of the following basic result to the next
section:
Theorem 8. Let С be a smooth curve and Jet W be a semistabJe vector
bundle of rank 2 on С such th&t deg W = 0. If L is a line subbundle of
Symn W, then deg L < 0. If W is a semistabJe rank 2 vector bundJe with
deg W arbitrary, and Lisa line subbmdle of Sym2" W ® (det W)~n, then
degL<0.
We note that the theorem is an easier special case of the following feet:
if W is semistable, then so is Sym" W. We shall discuss this more fully in
the next section.
Next we have the following easy lemma:
Lemma 9. Let С be a smooth curve, and let W be a semistabJe vector
bundle of rank 2 on С such that deg W = 0. Then ft0 (Sym" W) < n +
1. If W is a semistable rank 2 vector bundle with deg И^ arbitrary, tien
ft°(Sym2" W <g> (det W)~n) < 2n+ 1.
Proof. We shall just give the proof for the case where deg det W = 0; the
other case is similar. We shall prove the following: if 5 is an r-dimensional
subspace of ff°(Symn W), then the corresponding map (Tc -* Sym" W
is injective. In particular, taking г = n + 1, if there is a subspace of
#°(Sym" W) of dimension n +1, then the induced map 0?+1 -> Sym" W
is injective, and its determinant, which is a section of det(Sym" W), is
nonzero. Now it follows from the next lemma (and is easy to check di-
directly) that degdet(Sym" W) = 0. Thus, if there is a nonzero section of
det(Symn W), it is nowhere vanishing, and the corresponding homomor-
phism Oq+1 —* Sym" W is an isomorphism. In particular ft0 (Sym" W) =
n + 1. Hence, if ft°(Sym" W) > n + 1, then ft°(Sym" W) = n + 1, and in
all cases ft°(Sym" W) < n+ 1.
To see the claim on r-dimensional subspaces of iJ°(Sym" W), we argue
by induction on r. The case r = 1 is clear. Now suppose that 5 is an
(r + l)-dimensional subspace of ft°(Sym" W) such that the induced map

9. Bogomolov's Inequality and Applications 253
Oc+1 ~* Sym" W is not injective. Denote its image by U, so that U is a
torsion free subsheaf of Sym" W and is therefore a vector bundle of rank at
most r. Choosing an r-dimensional subspace of 5 and applying induction,
there is an injection Orc -* Sym" W whose image is contained in U. Thus,
U has rank r and properly contains the image of 0?. There exists a point
t € С such that the induced map on fibers Cr = {prc)t -» Ut fails to be
injective. If и is in the kernel of this map, there is a global section s of
Orc such that s(t) = v, and so the induced map Oc -* Sym" W defined
' by s vanishes at t. It follows that there is an effective nonzero divisor d
on С such that the map Oc -» Sym" W factors through the inclusion
Oc -> Oc(d). But then Sym" W has the line subbundle Oc(d), which has
positive degree, contradicting the previous theorem. ?
More generally, if W is a semistable bundle of rank r and degree 0 on
a smooth curve C, then arguments very similar to those given above show
that h°(W) < r +1, and the lemma would follow from this if we had proved
directly that, starting out with a semistable rank 2 vector bundle W, then
Sym" W is also semistable.
We turn now to the proof of Theorem 7. The basic idea of the proof is
as follows. Fix a vector bundle V and two smooth curves C\ and C2 with
Ci-C2 > 0. Suppose that V|Ci is semistable but that ci(VJ > 4c2(V).
First suppose that detV = Ox, and that 0 = ci(VJ > 4сг(К), in other
words that c2(V) < 0. We shall show that, for all n > 1, ft°(Sym" V) >
n3/12 + O(n2). Thus, for n » 0, ft°(Sym" V) > n+ 1. While Sym" V
has many sections, it follows from Lemma 9 that ft°(Sym" V\Ci) < n + 1.
Thus, we can eventually find a nonzero section of Sym" V vanishing on
Ci, and then it is easy to see that V\C2 cannot be semistable. There is a
standard device for handling the case where det V is not necessarily trivial:
we consider instead the vector bundle Sym2" V®(det V)~n, which has rank
2n + 1 and (as we shall see) trivial determinant. Note that, if we replace
V by a twist V ® L, where L is a line bundle, then Sym2" V ® (det V)~n
is unchanged, and in particular it is unchanged if we replace V by Vv. In
this case, we shall show that ft°(Sym2" V ® (det V)"n) > n3/6 + O(n2).
Thus, for n » 0, ft°(Sym2n V ® (det V)"") > 2n + 1, and we can reach a
contradiction as before.
To prove these statements, we begin with the following Chern class cal-
calculations:
Lemma 10. Let V be a rank 2 vector bundle on X.
(i) // det V is trivial and сз( V) = с, tien for all n> 1,

254 9. Bogomolov's Inequality and Applications
c2(Symn V) =
i=n (mod 2)
(ii) In general, if pi (ad V) = p, then for all n> 1,
ci(Sym2n V ® (det V)~n) = 0,
t2n^®(det^)-n) =
i=O
Proof, (i) By the splitting principle, we may assume that V is a direct
sum of line bundles L\ and L2 with ci(Li) = a and ci(Z«2) = -a. Thus,
c?(V) = -a2. In this case, Sym" V is a direct sum of the line bundles
L\ ® 1%~\ with cxiLi ® ti^1) = m + (n - i)(-ot) = Bt - n)a. Thus,
Cl(Svm" V) = X,B« "«)«=( L Va = °-
t=0	-n<i<n
i=n (mod 2)
Likewise,
У ^ Bг - n)Bj — n) la .
0<«J<n
We see that it suffices to prove the identity
i-n)Bj-n) = - ]T
i2.
0<<<n
i=n (mod 2)
In the sum on the left, consider those i,j with г < j and i + j — n. Such
terms contribute
Bi - n)B(n - г) - n) = -(n - 2iJ
to the sum. Clearly, such terms exist for i = 0,..., [n/2] and their sum is
exactly the right-hand side of the claimed equality above. Thus, it suffices
to show that the sum of the remaining terms is 0. Given i < j with г+j Ф n,
we have n-j^=i.Un-j=j, then 2j-n = 0 and the corresponding term
in the sum is 0. Otherwise, either i <n — j отп — j < i, but in any case the
corresponding term in the sum is Bi - n)B(n - j) - n) = -B* — n)Bj - n)
which cancels the term Bi - n)Bj - n). Thus, the nonzero terms of this
form cancel each other off in pairs, and so the sum of all the terms with
г + j ф п is indeed 0.
The proof of (ii) is similar and will be left to the reader. ?
Corollary 11. Let V be a ranJc 2 vector bundle on X.
(i) Ifci(V) = 0 and c2(V) = c, tien x(Symn V) = -cn3/6 + O(n2).

9. Bogomolov's Inequality and Applications 255
;(ii) In generai, if pi (ad V) = p, then x(Sym2n V ® (det V)~n) = pn3/3 +
Proof. To see (i), note by the Riemann-Roch formula that
x(Symn V) = -c2(Sym" V) + (n + l)x(Ox).
Thus, by (i) of Lemma 10, it suffices to see that
z2)=n3/6 + O(n2),
0<t<n
t=n (mod 2)
which is an elementary exercise left to the reader. The argument for (ii) is
similar. ?
Lemma 12. Let V be a rank 2 vector bundle on X.
(i) IfCl{V) = 0 and ca(V) = c, then ft°(Symn V) > -en3/12 + O(n2).
(u) In general, if Pl(ad K) = p, then ft°(Sym2n V^ ® (det V)"n) > pn3/6 +
O(n2).
Proof. As usual we shall only write down the case (i). Note that
(Symn V)v = Symn(Vv) S Symn V. Using Serre duality and Corollary
ft°(Symn V) + ft°(Symn К ® Kx) > -en3/6 + O(n2).
Thus, it suffices to show that h°(Symn V) = ft°(Symn V^ ® й"х) + O(n2).
More generally, let Z, be an arbitrary line bundle on X. We claim that
ft°(Symn V) = ft°(Symn V ® Z,) + O(n2). Write Z, = OX{DX - D2), where
Di and D2 are smooth curves on X. We shall show that A°(Symn V®Z-) <
h°(Symn V) + O(n2); the proof of the other inequality is similar. Prom the
exact sequence
0 -»Symn V ® Ox(-D2) -» Symn V^ ® Ox(Di - D2)
-* Symn V\DX в ODl (Dx - D2) -» 0,
we see that
ft°(Symn V®L) < ft°(Symn V®Ox(-D2))+ft°(Symn VIDx^Oo^Di
and using the inclusion Sym" V ® ©^(-Ог) Q Symn V^, we see that
ft°(Symn V ® Ox(-?>2)) < fc°(Symn V). Thus, it suffices to show that
ft°(Symn V|Di ® 00,@! - D2)) = O[n2).
Now on D! we can write V as an extension of two line bundles:
0 -¦ Lx -* V\Di -+L2~+0,

256 9. Bogomolov's Inequality and Applications
withdegLi =1= -degZ,2, say. It follows that Symn2)
has a filtration whose successive quotients are Z,"~* ®L2® Od,{D\ — D2).
Thus,
П
ft°(Symn V\DX ® ODl(Dx - ?>2)) < ?Л°(^Г* ® 4 ® ODl{Dx - D2)).
t=0
If deg0Dl(?>i - D2) = d, then degZ,?"' ®L2® ODl(Di - D2) = (n -
2i)< + d. If Z,"~* ® Z,2 ® ODl (Di - ?>2) is a nonspecial line bundle (i.e., if
ft1^""'®^®^,^! -D2)) = °)'then ^*y RJe111^111-^0^ on the smooth
curve D\,
Л°(^-' ® 4 ® ODl(Di - D2)) = (n - 2z)^ + d + A - <?),
where 3 = ff(Di). On the other hand, if Z,""' ® Z-2 ® ODl{D\ - D2) is
effective and special, then by Clifford's theorem
Ь°(Ц-1 ® L2 ® ODl (Di - ?>2)) < A((n - 2г)< + d) + 1.
In both cases we have a bound which is linear in n - 2г, with the im-
implied constants independent of n. Thus, summing over г, we see that
ft°(Symn V\Di 9 ODl (Di - D2) < O{n2). П
Now we can complete the proof of Theorem 7. For simplicity we shall
just write down the case where det V = 0. Suppose that c2(V^) = с < 0, and
so ft°(Symn V) = -cn3/12 + O(n2). Choose n so that ft°(Symn V) > n + 1.
By Lemma 9, since V\Ci is a semistable vector bundle of rank 2 and degree
0,
ft°(Symn V|Ci) <n + l.
It follows that there exists a nonzero section of Symn V vanishing on Ci,
in other words that fc°(Symn V ® Ox{-C\)) Ф 0. This is equivalent to the
existence of a nonzero map
Ox(C1)-*SymnV.
There is an effective curve D such that the map Ox(C\) —* Symn V van-
vanishes along D, and such that the induced map Ox(C\ + D) -» Symn V
vanishes at only finitely many points. In particular, the restricted map
Ox(Ci + D)\C2-+SymnV\C2
is injective. Note that C2 is nef, since C2 > 0, and hence C2 • D > 0.
Thus, Symn V\C2 has the sub-line bundle Ox(Ci+D)\C2, which has degree
(Ci • C2) + (D ¦ C2) > 0. By Theorem 8, this contradicts the semistability

9. Bogomolov's Inequality and Applications 257
Symmetric powers of vector bundles on curves
In this section, we study Sym" W, where W is a vector bundle on the
smooth curve C. We begin with the following lemmas, due to Gieseker
[46]:
Lemma 13. Let p: С -* С be a finite morphism between two smooth
curves С and C, and let W be a rank 2 vector bundle on C. Tben W is
semistabJe if and only if p*W is semistabJe.
Proof. First assume that W is unstable, and let V be a destabilizing line
subbundle. Thus, fi(L') > fi(W). Now n(p*W) = (degp)fi(W) and likewise
ц(р*Ь') = (degp)fi(L'). Thus, p*U is a destabilizing line subbundle of
p*W, which is therefore unstable.
Conversely, assume that p*W is unstable. After passing to a finite cover,
we may assume that p: С —> С is a Galois cover, with Galois group G. The
first part of the proof shows that, after passing to the Galois cover, the
pulled back bundle (which we continue to denote by p*W) is still unstable.
Let L be the maximal destabilizing line subbundle. Then for all а e G,
<x*L is the maximal destabilizing subsheaf of o*p*W — W. It follows that
o*L = L for all а € G. By Theorem 28 (to be proved in the Appendix),
there exists a line subbundle V of W such that p*V — L. Again using
ptfW) = (degp)MW) and fi(L) = fi(p*L') = (degp)Qi(?'). we see that
U is destabilizing. Thus, W is unstable as well. D
In the above situation, it is possible for W to be stable and p*W to be
strictly semistable. For a typical example, suppose that p: С —¦ С is an
etale double cover with covering involution t, and let L be a line bundle of
degree 0 on С such that i*L ф L. If W is the rank 2 vector bundle on С
denned by p.Z,, then we leave it as an exercise to show that W = p*L is
stable but that p*W = L ф i*L, which is strictly semistable.
We can interpret the example above in terms of unitary representations of
tti(C) (compare the discussion of the theorem of Narasimhan and Seshadri
in the last section of Chapter 4). If W is stable, corresponding to a unitary
representation p from tti(C) to 1/B), then the homomorphism tti(C) —>
tti(C) induces a unitary representation p: Ж\(С) —* 1/B), whose associated
vector bundle is just p*W. However, even if p is irreducible, it does not
necessarily follow that p is irreducible. Thus, p*W may split as a direct
sum of line bundles.
We now relate the stability of a rank 2 vector bundle W over the curve
С to the geometry of the corresponding ruled surface P(W).
Lemma 14. Let W be a semistabJe rani 2 vector bundle over the smooth
curve C, and let тг: X = Р(И^) —* С be the associated ruled surface. Let
Ox(l) be the tautological quotient line bundle ofn*Ww on X. IfdegW =

». Dogomoiov s inequality and Applications
Ox(l) й aef and c\(Ox(l)J — 0. More generaJJy, if degW в
y, let L = K^/c = 0xB) ® 7r* det W. Then L is nefand L2 = o.
0, then
arbitrary,
Proof. We shall just consider the general case. Let D be an irreducible
curve on X. If D = f is a fiber of тг, then L ¦ f = 2. If D ф /, then the
restriction of p to D defines a finite morphism p: D -* C, where D is the
normalization of D. Let v: D —* X be the induced map, so that p = •л- о v
By the previous lemma, p*W is semistable on P, and thus p*Wv is also
semistable. Since Ox(l) is a quotient of 7r*Wv onX, i>*Ox(l) is a quotient
of p'H^v. Moreover, м(^'ОхA)) = degOx(l)|D = LD> /i(p'Wv). This
says that
2deg@x(l)|O) > deg(p'^v) = - degGr* det W\D),
and so deg(Z,|D) = 2deg(Ox(l)P) + degfr* det W\D) > 0.
To prove the statements about C!(Cx(l)J and L2, note that
Thus, it suffices to show that, for an arbitrary rank 2 bundle W on C,
we have ci(C?x(l)J = -degW. This is a very special case of a general
formula about the tautological bundle on the projectivization of a vector
bundle [55], [45]. In this special case, we can also use Exercise 4 of Chapter
5. D
Note that the line bundle L = K^\c defined above is unchanged if
we replace W by a twist, and in particular if we replace W by Wy. The
converse to Lemma 14 is an easy exercise: let W be a rank 2 vector bundle
on С and let L be defined as in Lemma 14. If L is nef, then W is semistable.
Now we recall the correspondence between curves D on X, not containing
any fibers in their support, and with D • f — n > 0 and line subbundles of
Symn W*. Suppose that D is an irreducible curve on X with ?> • / = n >0.
Then we can write Ox{D) = Ox(n) <g> тг*А for a uniquely determined line
bundle A on С The exact sequence
0 -> Ox -» OX(D) -» OD(D) -* 0
together with the isomorphisms #°7г»0х = Ос, Д1тг»Ох = 0, and
HPn.Ox(D) = (Д°я-»С»х(п)) ® A = Symn W* ® A, shows that there is
an exact sequence
0-*Oc~* Sym" W* ® A -»7r,0D(D) -> 0.
Thus, D gives a line subbundle A of Sym" Wy. Conversely, a line sub-
bundle of Sym" Ww, which we can write as A for some line bundle A,
gives a nowhere vanishing section of Symn Wv ® A and thus a section of
Ox(n) ® тг*А, which, as is easy to check, does not vanish along any fibers
(but may be reducible).

9. Bogomolov's Inequality and Applications 259
Theorem 15. If W is semistable and deg W - 0, then Symn Wv has no
tine subbundles of positive degree. If deg W is arbitrary, then Sym2n W ®
(det W)~n has no line subbundles of positive degree.
Proof. First consider the case where deg W = 0. Note that
T deg(Sym" W*) = --L-**2±ll deg W = 0,
by the splitting principle (as applied in Lemma 10). Now suppose given a
line subbundle of Sym" Wv, which we write as A. We claim that
deg(A-1) = -degA<0,
or in other words that deg A > 0. Let s be the section of Ox (n) <S> тг* A cor-
corresponding to the inclusion A -¦ Sym" Wy, and suppose that s vanishes
along the effective curve D. Then Ox(D) — Ox(n)<g>n*\, and since Cx(l)
is nef,
0 < C!(C?XA)) • D = nci{Ox(I)J + degA = 0 + degA,
so that deg A > 0.
The case where deg W is arbitrary is similar, noting that a line subbun-
subbundle A of Sym2" W ® (det W)~n corresponds to a section of Sym2" W ®
(det W)~n ® A. Moreover,
Sym2" W ® (det W)~n = Sym2n(lVv ® det W) ® (det W)~n
= Sym2n(W*)®(detW)n.
Thus, the line subbundle A gives a section of Sym2n(Wv)®(det W)n® A,
and thus a section of Ln <g> A vanishing along a curve D. In this case we
have
0 < L D = nL2 +2degA = 2degA,
and so again deg A > 0. D
A much deeper fact is that, if W is semistable, then Sym" W is semistable
for all n, and indeed all tensor powers of W are semistable. More generally
still, if Wi and Wjj are two semistable bundles on C, then W\ ® W2 is again
semistable. Note that it suffices to prove this statement when W\ and W2
are actually stable. In case degWi = 0, г = 1,2, this follows easily from
the fact that Wi is associated to an irreducible unitary representation p< of
n\(C), and thus Wi ® Wjj to associated to the (not necessarily irreducible)
unitary representation p\ ®/>2- A purely algebraic proof (in characteristic 0)
is given by Gieseker in [46], based on Hartshorne's theory of ample vector
aundles [60].

260 9. Bogomolov's Inequality and Applications
Restriction theorems
Our goal in this section is to prove the following weak restriction theorem:
Theorem 16. Let H be a very ample divisor on the smooth surface X and
let V be an H-stable rank 2 vector bundle with pi(ad V) = p. Then either
p < 0 or there exists a smooth curve С € \H\ such that V\C is semistable.
As a corollary, we see that Bogomolov's theorem holds for V, either
automatically because p < 0 or by Theorem 7 (taking C\ = Ci = C).
Proof of Theorem 16. Choose a pencil contained in \H\ whose general
member is smooth. Blowing up the base points, we obtain a smooth surface
X which is a blowup of X and a morphism тг: X —¦ P1 whose general fiber is
a smooth curve С with С € \H\. Let p: X —» X be the birational morphism
from X to X. To prove Theorem 16, we argue as follows: first we shall show
that p*V is stable for some ample divisor Щ on X. On the other hand,
we have denned suitable ample divisors in Chapter 6 (with respect to the
morphism тг: X -* P1). Let Hi be a (ui,p)-suitable ample divisor, where
w = иъ(р*У) and p = pi(p* ad V). Either p*V is also stable with respect
to Hi, or by Proposition 22 of Chapter 4 p*V is strictly semistable with
respect to some ample divisor Я which is a convex rational combination of
Ho and Hi. But a strictly semistable vector bundle is easily seen to satisfy
Bogomolov's inequality, by Exercise 1 in Chapter 6, and so in this case
p < 0. Otherwise, we can assume that p*V is in fact stable with respect
to a suitable ample divisor. In this case, by (i) of Theorem 5 in Chapter
6 (for which we shall give a complete proof below), the restriction of p*V
to almost all fibers of тг is semistable. In particular, the restriction of V to
a general smooth fiber С of тг is semistable, which concludes the proof of
Theorem 16. ?
The two points which remain to be proved are first, that p*V is stable
with respect to some ample divisor on X, and second, the proof of (i) of
Theorem 5 in Chapter 6. To deal with the first point, since p is a sequence
of blowups, it will suffice to handle the case of one blowup at a time. In fact,
we have defined stability with respect to nef and big divisors in Chapter 4,
and the proof of Proposition 22 in Chapter 4 works equally well in case Hq
is simply assumed to be nef and big. Take Ho = p*H. We must show that
p*V is р'Я-stable. But цР'н{р*V) = Мя(V). If ®x№) '"*a sub-line bundle
of p*V, then writing D = p'D+аЕ, we see that OX{D) = p»Ojf(D)vv is
a sub-line bundle of p»p*V = V. Thus,
p'HD = HD

9. Bogomolov's Inequality and Applications 261
and so p*V is р'Я-stable. In fact, there is the following more precise result
which says that p* V is actually stable with respect to an appropriate ample
divisor on X:
Theorem 17. Let У be a smooth surface and let H be an ample divisor
on Y. Suppose that p: Y -* Y is the blowup of Y at a point x, and let
E be tie exceptional divisor on Y. Let V be an H-stable rank 2 vector
bundle on Y with det V = Д and pijadV) = p. Then for all N such that
Np*H — E is ample on Y and N > y/\p\/2, the vector bundle p*V is stable
with respect to Np*H - E.
Proof. Suppose that OY(D) is a sub-line bundle of p*V. We must show
that (Np'H - E) ¦ D < 0. We may assume that p*V/OY(D) is torsion free,
so that there is an exact sequence
0 -> Oy(D) -> p'V -* OY(p*A - D) ® Iz -> 0.
Let D = p*D'+aE for a uniquely specified divisor class D' on Y. There is an
inclusion p,Oy(D) с p*p*V, and by the projection formula p»p*V = V ®
p»OY = V. Moreover, p,OY(D) is either of the form ОуAУ) or Oy{D') ®
m" for some positive integer n, and so its double dual, which is just Oy(D'),
includes into V. The quotient V/Oy(D') can have torsion only at x, since
it is isomorphic to p*V/OY(D) away from x, and so it is torsion free by
Proposition 5 of Chapter 2. Thus, there is an exact sequence
0 -¦ Oy(D') -* V -> OY(A - D') ®Iw-+0,
where W is a O-dimensional subscheme. By the stability of V, H ¦ BD' -
A) < 0. Since pi(adV^) = pi(adp'V) = p, we have:
p = B?> - ДJ - U(Z) < {ID - ДJ = B?)' - ДJ - 4a2 < B?)' - ДJ.
Moreover, {Np'H - E) ¦ D = N(H ¦ BD' - Д)) + a. Thus, if a < 0, then
automatically
(Np*H -E)D = N(H ¦ B?)' - Д)) + a < N(H ¦ BD - Д)) < 0.
So we may assume that о > 0. If B?>' - ДJ < 0, then
4а2<B?>'-ДJ-р<-р<|р|,
and so а < y/\p\/2. Thus,
(Np'H -E)D = N(H ¦ BD' -A))+a<a-N < у/Щ/2 - v1p172 < 0.
The remaining possibility is BD' - AJ > 0. Since, as we have just seen
above, (Np*H - E) ¦ D<a- N, and ЛГ > ^/|p|/2 by hypothesis, we may
also assume that а > ^/|p|/2. Then 2o2 > |p|, and so
BD' - ДJ > p + 4o2 > 4o2 - \p\ > 2a2.

262 9. Bogomolov's Inequality and Applications
Now by the Hodge index theorem (Exercise 10 in Chapter 1),
H2BD' - ДJ < (Я • B?>' - Д)J,
and so, since Я2 > 1 and Я • BD' - Д) < 0,
Я • B?>' - Д) < - [B?>' - ДJ]1/2.
Thus, since N > 1, we have
(Np*H -?)•?) = ЛГ(Я • BD' - Д)) + a
Thus, in all cases (Np*H - E) • D < 0 and so p*V is stable with respect
toNp'H-E. D
Theorem 17 says that p*V is stable for every ample divisor in every
chamber of type (tu,p) on Y containing p*H in its closure.
The final point is the following (Theorem 5 in Chapter 6):
Theorem 18. Let тг: X -* С be a morpnism from the smooth surface X
to a smooth curve C, let V be a rank 2 vector bundle on X with invariants
(w,p), and Jet Я be a (w,p)-suitabie ampie divisor on X. Suppose that
V is H-semistable. Then for all but finitely many fibers / of ж, V\f is
semistabk.
Proof. Suppose instead that V\f is unstable for infinitely many fibers /.
We have seen in the discussion of the proof of Theorem 5 in Chapter 6 that
it is enough to show that there exists a sub-line bundle L of V such that
the sub-line bundle L\f of V\f is destabilizing for a single smooth fiber /,
for then V is Я-unstable. To find L, we shall first find a finite Galois cover
q: С —* С and a sub-line bundle L' of q*X = X with the property that, if
p: X —* X is the induced Galois morphism, then L'\f destabilizes p*V\f
for infinitely many fibers of q*X —* C. We can then argue as in Lemma 13
that V descends to a line bundle L on X with the same property.
Let d = deg(V|/) = det V ¦ f. If / is a smooth fiber such that V\f is
unstable, then there exists a sub-line bundle of V\f with degree greater
than d/2. On the other hand, there is an upper bound independent of the
choice of / for the degree of a sub-line bundle of V\f. In fact, we can always
write У as an extension
0 - OX(D) -*V-* Ox(Д - D) ® Iz -» 0
for some divisor D and O-dimensional subscheme Z (as we have seen in
Chapter 2). Let ex = D ¦ f and e2 = (Д - D) ¦ f. For all but the finitely
many fibers / for which / П Supp Z ф 0, there is an exact sequence
0 -* Lx -> V\f -* L2 -> 0,

9. Bogomolov's Inequality and Applications 263
where deg?i = e,. Thus, any sub-line bundle of V\f has degree at most
> = max{ei,e2}. The cases where /DSuppZ ф 0 can then be handled
lirectly (or we can ignore these finitely many fibers in what follows).
Let T be any Zariski open subset of С such that тг is smooth over T.
We will for the moment replace X by rr^T), so that we can assume
that тг: X -* T is smooth. We would like to take the Picard varieties
Pic" /, for each smooth fiber / of тг, and fit them together into a variety
Picn(X/T) mapping to T. The main property that we need is the existence
»f a Poincare line bundle PonIxT Picn(X/T), with the property that,
if ? e Picn(A"/T) corresponds to a line bundle L on the fiber / lying
over the point of T corresponding to ?, then V\f xT {?} is naturally just
L. Unfortunately, such a construction is not always possible. However, if
тг: X -* T has a section, then the scheme Picn(X/T) and the line bundle V
exist (see, for example, [58] and [91]). Moreover, the fibers of the morphism
Picn(X/T) —» T are the Picard varieties of the fibers, and the morphism
Picn(.Y/T) —* T is smooth and proper. Note that, if T is a smooth curve,
$hen there is always a finite morphism f —*T, where T is again a smooth
curve, such that, if X = X хт T, then X is smooth (since тг is smooth)
and тг: X -* f has a section. In this case, the fibers of тг may be identified
with the corresponding fibers of тт. Let p: X —¦ X be the induced finite
morphism.
For a smooth fiber / of тг or тг, V\f is unstable if and only if V\f has
a sub-line bundle L of degree greater than d/2, where d = deg(Vr|/), and
every sub-line bundle L of V\f has degree at most e for some fixed integer
e. Let n be an integer with d/2 < n < e. Over X xf Picn(X/f), we
have the Poincare line bundle V. Let тп: X xf Picn(X/f) -t X and
тг2: X Xf Picn(X/f) -* Picn{Xft) be the projections to the first and
second factors. Note that тг2: X xf Picn(X/f) -* Picn(X/f) is smooth
and proper, and therefore flat. For a fiber / of тг, p*V\f is unstable if and
only if there exists an n with d/2 < n < e and a ? e Picn(^/T) lying over
the same point as / such that, if L is the line bundle corresponding to f,
theniy\!)®L~x hasasection.inotherwordsftVlp'F^P-1!^1^)) > 1-
Let Zn С Picn(Ayf) be defined by
Zn = {? e Picn(*/f): tfOrJp'VeP-Va X@) > 1}.
By the semicontinuity theorem, Zn is a closed subvariety ofPicn(X/f), and
since Picn(X/f) -* f is proper, the image Yn of Zn in f is also closed.
Thus, since f is a curve, either Yn is finite or Yn = T. Clearly, Ud/a<n<e ^"
is the set of t e f such that р*К|#-1(*) is unstable.
Suppose that Yn is finite for every n such that d/2 < n < e. Since there
are only finitely many such n, there are only finitely many t € f such
that р*К|тг~1(<) is unstable, and hence only finitely many fibers / of тг
such that V\f is unstable. Conversely, if V\f is unstable for infinitely many
fibers / of тг, then Yn = f for some n with d/2 < n < e. Choose such

264 9. Bogomolov's Inequality and Applications
ал n. Since Zn —* T is surjective, there exists a smooth curve Г in Z
mapping onto T. Since we axe free to replace T by the finite cover T, we
may assume that the morphism Zn -* T has a section a. Using the section
a to identify T with a subvariety of Picn(X/f), we may then identify X
with a subvariety olX *f Picn(X/T). The line bundle V then restricts to
a line bundle over X. After passing to a finite cover we can assume that
f —* T is Galois. By assumption Я°тг»(р*У ® V~l) is a nonzero torsion
free sheaf T onT. If ? is a sub-line bundle of T, then there is a section of
7®{C)~X and thus of p'V®^®**/:). Since we are free to replace V by
V® тг*?, we may assume that there is a nonzero map V —* p*V with V\f a
destabilizing sub-line bundle of V\f for every fiber / for which the induced
map is nonzero. After replacing VhyV® 0% (E) for some effective divisor
E, we may further assume that the quotient p*V/V is torsion free. Note
that this will not affect the condition that V\f is destabilizing, since the
only problem is when E contains some multiple of a fiber / as a component
Let us recapitulate the situation. Starting with the smooth morphism
¦к: X —> T, where T is a smooth curve, we find a Galois cover T —> T,
where T is again a smooth curve, with the following property: Making the
base changep: X = X x.TT -* X, there exists a line bundle V on X such
that V is a sub-line bundle of p*V and p*V/V is torsion free. It follows
that, for a generic fiber /, (V\f)/(V\f) is torsion free. Thus, for a generic
fiber /, V\f is the maximal destabilizing sub-line bundle of V\f.
Let G be the Galois group of T over T, let i\ be the generic point of T,
so that 17 = Spec fc, where fc is the function field of T, and let fj = Spec ife
be the generic point of T. The surface X -* T restricts to a curve Xv over
77, in other words a curve defined over the field fc, and likewise X restricts
to a curve X<j, the pullback of Xv to r). Likewise, we can restrict V to a
vector bundle Vv over Xv and p'V to a bundle Vy. The sub-line bundle
V restricts to a sub-line bundle С of Vfj, such that Vfj/C is torsion free,
and С is the maximal destabilizing subbundle of V^. It follows that, for all
a eG, a*C = С as a subsheaf of V% = <r*V^. By Proposition 23 of the
Appendix, there is a line bundle С defined over q and an inclusion С С Vv,
such that С pulls back to С In particular С is destabilizing, in the sense
that deg? > d/2.
We now consider the global case, where T is an open subset of С and X
is defined over all of C, not just over T. The coherent subsheaf С denned
on Xv extends to a coherent subsheaf Lu of V\U defined on some open
subscheme U of X. For example, taking an affine open subset of T of the
form Spec R, where fc is the quotient field of Д, it is easy to see that the
sheaf С involves only finitely many denominators in the ring R, and thus
there is a single / € R such that С can be defined over 7r"(Spec Rf). The
coherent subsheaf Lv of V\U then extends to a coherent subsheaf L of V
defined on all of X (see for example [61, p. 126, ex. 5.15(c)]). Here L is a
torsion free rank 1 sheaf, and after replacing it by its double dual we can

9. Bogomolov's Inequality and Applications 265
assume that it is a line bundle. By construction Lf = deg? > d/2, so that
L is the desired destabilizing line bundle. This then concludes the proof of
Theorem 18. ?
Appendix: Galois descent theory
We begin by considering the following situation: fc is a field, and К is
a finite Galois extension of fc with Galois group G. Let V be a fc-vector
epace, not necessarily finite-dimensional over fc, and suppose that W =
V О* K. In this case, the fc-linear action of a e G on К extends to a
fc-linear automorphism <pa: W —^ W, such that ipa{v ® a) = v® <r(a).
In particular, <?и = Id, <pa о y>T = <paT, and, for all a € A" and w e W,
(fa{a • w) = ar(a)ipa(w). More generally, we make the following definition:
Definition 19. Let W be а К-vector space. A twisted G-representation is
a homomorphism from G to Aut*(W), the group of fc-linear automorphisms
of W, such that, if ipa is the image of a, then for all a e К and w € W,
tpcr(a ¦ w) = <т(а)^<т(и;). In case W — V <8>k К for some fc-vector space
V, and ific is as defined above, then we will call <p the standard twisted
G-representation associated to V.
For a twisted G-representation W, we let
W° - {w e W : 4>a{w) = w for all a € G}.
Thus, WG is a fc-vector subspace of W.
We define homomorphisms and isomorphisms of twisted G-representa-
tions in the obvious way: Given W\ and Wjj two twisted G-representations,
a morphism of twisted G-representations from W\ to W2 is a K"-linear
map F such that F о <pa = у>„ о F for all <r € G. Equivalently
F € Нот/с(И^ьИ^)°, where F *-* ipa о F о ^~x defines a twisted G-
representation on
Example 1. Suppose W is a 1-dimensional twisted G-representation, with
basis element w. Thus, (since ч>а(ги) Ф 0) 4>a(w) = a(a)vi for a unique
a(a) € A*. Now
o((tt)u; = Variw) = <р„ о 4>T(w) = ipa(a{j)w) = а{а{т))а{а)%и.
Hence а{(тт) = <r(o(r))a(or), so that a(<r) is a 1-cocycle for G. The element
w is well-defined up to the choice of A € K*. Replacing w by Aw; gives
V?<,(Ati;) = ar(X)a(ar)w = ar(X)X~1a(ar)
so that the 1-cocycle a(a) is multiplied by the coboundary ar(X)X~l.
Thus, we may identify isomorphism classes of 1-dimensional twisted G-
representations with the Galois cohomology group H1(G, A"*). If W =

266 9. Bogomolov's Inequality and Applications
V ®fc K, then we can choose w = v ® 1 and 4>tr(w) = w for all a, so that
a(a) = 1 for all a.
Example 2. Let W be an arbitrary #"-vector space, and, for a e G, let
W be the ^-vector space whose underlying abelian group is W, but with
scalar multiplication given by the formula
a •„ w = <r(a)«;.
Note that (W)T = WT. The identity шар W -> W is fc-linear. There
is an induced K-linear homomorphism Ф: W ®* К —* ф^о W" defined
as follows: thinking of the elements of ф^о W" as functions w: G -* W,
define Ф(т ® a)(a) — ar(a)w. It is easy to check that Ф is well-defined
and A"-linear. In fact, we claim that Ф is an isomorphism, which fol-
follows by reducing to the case where diniK W = 1 and using the Ga-
Galois theory isomorphism К ®* К S фст€<3 К". Note that Ф carries the
standard twisted G-representation of G on W ®* К to the twisted G-
representation on ф^с W" defined as follows: thinking of the elements
of ф„еС W as functions w: G -> W, define <^T(w(<r)) = w(<tt). Since
by definition a ¦ w(cr) — cr(a)w(cr), v?r(« • w) = т(а) • w (so that we
have indeed defined a twisted G-representation on ®a€G W) and that
( (а)) = (pT ¦ Ф(ы ® а).
We remark that a fc-linear isomorphism tpa: W —* W such that
ipa(aw) = <x(a)w is the same thing as a if-linear isomorphism W —* W.
For all т € G, ipg also induces a ЯЧшеаг isomorphism WT -* WT, and
the condition that <p defines a twisted G-representation is just the condition
that, for all а, т e G, v?a о ipT = у>ат as A"-linear isomorphisms W —* WT
Lemma 20. Let V Ъе а k-vector space. For the standard twisted G-
representation on V ®k K, we have
Let V\ and V2 be k-vector spaces and let Wi = Уф^К, viewed as a standard
twisted G-representation. If F: W\ —* W2 is a morphism of twisted G-
representations, then there is a unique fc-linear map /: Vi —¦ V2 such that
Proof. To see the first statement, let v € V. Then ^„((fc • v) ®fc K) =
(fc-v)®fc К for all a e G. Thus, if we choose a basis for V, say V — ф^ k-Vi,
then V ®fc К — ф4 К ¦ Vi and ч>а preserves the direct sum. So it is enough
to consider the case where V is 1-dimensional. In this case V = к and
V®k К ^ K, with the natural G-action. So we are reduced to the statement
that KG = к, which is clear by Galois theory.
As for the second assertion, let / = F\(Wi)G = F\VX. Clearly, F{WG) С
(Wb)G = V2, and we let /: V\ —» V2 be the induced map. Since W\ =

9. Bogomolov's Inequality and Applications 267
V\ ®k K, it is easy to see that / ® Id = F and that / is the unique map
with this property. ?
The main result of this Appendix is a converse to Lemma 20:
Theorem 21. Let W be a twisted G-representation, where W is a K-
vector space. Then the natural map from WG ®kK toW isan isomorphism
of twisted G-representations. In particular, every twisted G-representation
is isomorpbic to a standard one.
We note that Theorem 21 is a generalization of the fact that the Galois
cohomology group H1^, K*) = 0.
Proof of Theorem 21. There is а ЙЧтеаг тар from WG ®k К to
W, denned by w <8> a >-> aw, and it is a homomorphism of twisted G-
representations, since for а € G and w € WG,
w ® <т(а)!-» o(ct)w = аг(а)ф„(ь>) = 4>a(aw).
Next, considering W as a fc-vector space, note that the twisted G-
representation on W induces a A"-linear representation of G on the K-
vector space W ®fc K, by letting G act on the first factor and К on the
second.
Claim. The bilinear map (w, a) ¦-» YiaeG afa-1 (w)'a induces a K-linear
isomorphism
p: W ®k К-+W ®K K[G] = 0 W ¦ а
a€G
which is equivaiiant with respect to the A"-Jinear action of G.
Proof of the claim. By definition, p is A"-linear. To check that it is G-
equivariant, let t € G. It is enough to check that, on generators w®a, we
have р(фт(у)) <8>a) = r(p(w ® a)). Now
))	])Г	T(w) ¦ or.
Make the change of variable by replacing a € G by таг:
J2 atpr-iriw) or=J2 aup(Ta)-iT(w) ¦ та
Thus, p{4>T(w) g) a) = t(p(w ® a)).
To see that p is an isomorphism, we must show that, given an element
Y^cr wa • <r e. ф,, W ¦ a, then there exists a unique element of W ®/t K,

268 9. Bogomolov's Inequality and Applications
necessarily of the form ]?"=i Wi®ait such that p($^7=i Ш1®">) = ?„ wa-a.
In other words, given a collection of elements wa e W indexed by a e G, we
must find c*i e K, Wi e W, for i = 1,... ,n such that iua = ?< оцРа-^щ)
for every (r. Applying ipo to both sides, we see that, given the collection
wa, we must find w, and ац such that
and such that the element ?^ a. ® w. is uniquely defined in W ®* K.
It suffices to show that the map тг: W ®* К —> ©ff€G W • a defined by
w®a *-* Yla o{a)w-(j is an isomorphism, for then the element ^2a <Р<г{ч>а)чт
is the image of J2i wi ® a< f°r a unique element ?4 *"* ® ^i € W ®jt K. On
the other hand, the map тг: W ®fc ^ —> ©,reG W ' CT given by u; ® a >-¦
J2a v(a)w ¦ a can be defined for every К -vector space W, regardless of
whether W has a twisted G-representation, and тг clearly commutes with
taking (possibly infinite) direct sums. So it is enough to prove that тг is an
isomorphism in case W is 1-dimensional over K, say W = К. In this case
we are reduced to considering the map К ®* if —» ®aeQ К ¦ a defined
by 0 ® a >-» 5]^ ет(а)/3 • (т. By standard Galois theory, this map is an
isomorphism. Thus, p is an isomorphism. ?
Returning to the proof of Theorem 21, the claim shows that there is an
isomorphism
Clearly, (фаеС W ¦ a) = W, by taking the diagonal embedding of W in
®oeG W-a. On the other hand, we claim that the natural map WG®kK ~*
(W ®* K)G is an isomorphism: consider the exact sequence
where the first map is the inclusion and the second is w >-* (<pa(w)-'w)oeG-
This exact sequence of fc-vector spaces remains exact when we tensor with
K, so that
Д W®kK
is exact. Thus, WG ®k К is identified with the kernel of W ®* К -»
n<reG W®fc K, namely (W®k K)G. Putting this together, we have showed
that the map p identifies WG ®к К with the diagonal embedding of W in
®aeG W ¦ a. For w ® а 6 WG ®fc K, p{w ® a) = Y!,Aaw)' °- Thus'the
map WG ®fc К —» W defined by w ® а •-> aw is an isomorphism, which
concludes the proof of Theorem 21. D

9. Bogomolov's Inequality and Applications 269
Corollary 22. Let К be a finite Galois extension of к and let U be a
k-vector space. Suppose that W is a K-vector subspace ofU®kK such
that, for all a e G, <pa{W) = W, where tpa is the standard twisted G-
representation on U ®* K. Then W° is a к-vector subspace of U and the
map WG ®* К -» W is an isomorphism.
Proof. By hypothesis the standard twisted G-representation on U ®* К
restricts to a twisted G-representation on W, for which WG С {U<S>k K)G =
U. Thus, the corollary is immediate. ?
Next we want to find circumstances where we can apply Theorem 21 to
sheaves over schemes defined over a subfield of an algebraically closed field.
Suppose that X is a (separated) scheme over Spec k, and that К is a finite
Galois extension of k. Let X' =¦ X xSpeck Specif, and let p: X' —> X be
the natural morphism. Locally X = SpeciZ, where Д is a fc-algebra, and
thus locally X' = Spec Д ®jt K, with the morphism p corresponding to the
inclusion R -» R ®* К. An element uSG defines a morphism а: X' —у Х'
with от = та. Let T be a coherent sheaf on X'. Locally T corresponds
to an (R ®jt Jf)-module M. We can form the pulled back sheaf o*T, and
a*f* = (fa)* = (of)*. If T locally corresponds to the (Д®* if)-module
Af, then it is eaasy to check that a*T corresponds to the (R ®jt iQ-module
M"'1. Note that the analogue of W ®fc К = фа W" is the isomorphism
Р*р.^^ф„а*^.
Suppose that we are given a coherent sheaf б on X, locally corresponding
to the Д-module N. Then we have the sheaf p*Q on X', and it corresponds
locally to N®R(R®kK) = N®kK. We define a twisted G-representation on
T to be a collection of isomorphisms ipa: d*T -» T, such that <ры = И, and
such that, for all а, г е G, if we also denote by tpT the induced isomorphism
(ot)*T = a'f'T -> a*T,
then <pa о <рт = ^„т. Equivalently we seek a homomorphism <p: G —»
Autp.^1" such that, for every local section a e p»Ox' and a e p..F,
^„(as) = cr(a)s under the natural action of the sheaf of algebras p.Ox1
on р„Т. In the local case X = Spec R and X' = Spec(fl ®* K), if .F cor-
corresponds to the (R ®fc iQ-module M, then <?>„ is the same thing as an
Д-module homomorphism M" —» M, or equivalently Af —¦ M°', and
V? is equivalent to a twisted G-representation on M commuting with the
Д-module structure.
Proposition 23. Suppose that X is a scheme over Spec к and that X' =
X У-Speck Spec К.
(i) Let T' be a sheaf on X' with a twisted G-representation on T' ¦ Then
there is a sheaf T on X such that p*T is isomorphic as a coherent sheaf
with a twisted G-representation to P'.

270 9. Bogomolov's Inequality and Applications
(ii) Suppose that T is a coherent sheaf on X and that Q' is a coherent
subsheaf of p'T such that o*Q' = Q', as a subsheafof p*T, for every
а € G. Then there exists a coherent subeheaf Q of T such that Q' щ
the subsbeaf p*Q ofp'T.
Proof. We shall just prove (i); the proof of (ii) is similar. First suppose
that X = Spec R and that X' = Spec(fl ®k К). Then M has a twisted G-
representation commuting with the Д-module structure, and thus N = Mа
is an Д-module satisfying N 8>R (R <8>k К) = N ®k К 9? М as twisted
G-representations. In this case, we can take Q to be the Ox-module cor-
corresponding to N. We leave it as an exercise to show that N is finitely
generated.
In general, take an affine open cover {Ui} of X, where Ui = SpecRj,
Щ = UiD Uj = SpecRij, and Uijk = ^П Uj П Uk = Spec Rijk. Over
each Ui, we have found a coherent sheaf ? and an isomorphism щ: p*Qt a
T\Ui as sheaves with twisted G-representations. Thus, over l/y, we have
isomorphisms
Фа = (VilUij)-1 о ЫЩ): p'GipySifiUij ^p'GjWij.
By Lemma 20, there exist isomorphisms ф^: Si\Uij —» Gi\Uij inducing ipij.
Moreover, over Uijk,
Фзк о Фа = Vk
Thus, by the uniqueness part of Lemma 20, if/jk о ф^ = if>ik on Uijk, and
so the sheaves Qt glue together to give a sheaf Q, such that p*Q =T. D
We next make some remarks which we will not need in the applications.
In practice, except under the circumstances of (ii) of Proposition 23 above,
it is hard to find an explicit twisted G-representation on a sheaf T. Instead
we consider the following situation. Suppose that X and X' are as above
and that T is a coherent sheaf on X' with Aut.F = K*. For example, if X
is proper over к and geometrically integral, then every line bundle on X'
satsifies this condition. Suppose further that a*T = T for every а € G, and
let фа: дТ —> T be such an isomorphism. By hypothesis фа is unique up
to multiplication by an element of K*. In particular, фаофтоф~*: T —* T
is multiplication by an element с(ст,т) е К*.
Lemma 24. The element c(d, r) defines a 2-cocycie for G with values in
K", and the associated cohomology class in H2(G,K*) is independent of
the choice of the фа. Moreover, the cohomology class is trivial if and only
if there is a choice of фа such that фа°Фг =Фот, m other words for which
фа defines a twisted G-representation.

9. Bogomolov's Inequality and Applications 271
Proof. We compute: c(a,r) is a 2-cocycle if and only if, for all <т,т,ре G,
we have
c((x,r) = с(<т,тр)сг(с(т, p))c(trr, p)~l,
which we can write as с(а,т)~1с(а,тр)с(ат,р)~1о[с(т,p)) = 1. Multipli-
Multiplication on T by the term с(а,т)~1 с{а,тр)с{ат, p)~1\s the same as the iso-
isomorphism T —> T given by
Фат О Ф~1 О ф-1 О^О фтр О ф~)р О фатр О ф-1 О ф~*
= фат о ф-1 о фтр о ф-1 о ф~}.
Now multiplication by (i(c(r,p)) on a*T is the isomorphism a*T -» o*F
defined by фт о фр о ф~р. We leave to the reader the verification that
c((t, r)~1c(G,rp)c(<rr,p)~1(r(c(r,p)) corresponds to the automorphism of
T defined by
Фат офг1оФт°Фр° Ф7р °Фтр°Фр1О Фат = И •
Thus, с((т, г) is a 2-cocycle. If we replace фа by Хафа for some A e K*, then
we multiply c(<r,r) by the coboundary A^(r(AT)A<r. Thus, the cohomology
class is independent of the choice of фа, and the class is trivial if and only
if there is a choice of фа for which c(a,r) = 1, in other words фа о фт =
Фат- ?
Corollary 25. Suppose that IP(G,K*) = 0 and that T is a coherent
sheaf on X' with Aut.F ? Jf* and such that ff-.F = .F for all а е G. Then
there exists a coherent sheaf Q on X such that T = p*G- Io particular, if
X is a geometrically integral scheme proper over к and L' is a line bundle
on X' such that o'U ^ L' for all a e G, then L' = p'L for a fine bundie
L on X. D
We turn now to finding circumstances under which H2(G,K*) = 0. In
general the groups lP(G,K*) are extremely complicated. For example, if
к is a local field or a number field, then the determination of H2(G,K*)
is the basic information in local or global class field theory. However, in
certain cases we can say that H2(G, K") = 0:
Definition 26. An algebraic function Beld к in one variable is the field of
meromorphic functions on an algebraic curve over an algebraically closed
field ко. Equivalently, A; is a finite separable extension of fco(x), the field of
rational functions in one variable over the algebraically closed field ко. If
fc is an algebraic function field in one variable over ко and L is a subfield
of к containing k0 such that [k : L] is finite, then L is also an algebraic
function field in one variable over k0, and likewise every finite extension
of an algebraic function field in one variable over ко is again an algebraic
function field in one variable over ко.

272 9. Bogomolov's Inequality and Applications
Theorem 27. If к is a finite fieid or an algebraic function fieid in one
variable, and К is a finite Galois extension of к with Galois group G, then
H2(G,K*)=0.
Proof. By standard Galois theory, Я1 (G, K*) = 0. (This also follows from
Theorem 21 and the discussion in Example 1 following Definition 19.) In
case к is a. finite field or an algebraic function field in one variable, the
norm map N: K* —» k* is surjective; we shall outline a proof of this fact in
Exercise 11. The same also holds when we replace G by a subgroup H of G
and к by the fixed field of H, or G by G/H, where Я is a normal subgroup
of G, and К by the fixed field of H. It is then a standard argument in
the cohomology theory of finite groups that H2(G,K") = 0. Indeed, if
G is cyclic H2(G,K*) is isomorphic to k*/N(K*), by [137, p. 141], and
thus is trivial. If G is solvable, then using the result for cyclic groups and
induction, it follows from the inflation-restriction sequence and the fact
that H^CK*) = 0 (see [137, p. 126, Prop. 5]) that H2(G,K*) = 0.
In particular, if G is a p-group, then H^iGjK") = 0. Now, for a general
Galois group G and a prime p, let Gp be the p-Sylow subgroup of G. Then
by [137, Cor. on p. 148], since H2(GP, К*) = О for every p, H2(G, К*) = О
as well. ?
Lastly we give a corollary of Theorem 21 which was used in the proof
of Lemma 13. Let p: X -» Y be a finite morphism of schemes. Suppose
that G is a finite group of automorphisms of Y with poo = p for all
a e G and such that (p,Ox)G = Oy In this case we shall refer to p as a
Galois morphism of schemes with Galois group G. Locally, if U = Spec R
is an affine open subset of Y, then p~1(U) = Spec R' where R' is a finite
Д-algebra with an action of G such that (R')° = R.
For example, if p: С —> С is a morphism between two smooth projective
curves such that the induced map on function fields is a Galois extension,
then p is a Galois morphism.
Theorem 28. Letp: X —> Y be a finite flat Galois morphism with Galois
group G, and suppose that X and Y are both integral. Let T be a locally
free sheaf on Y and suppose that Q' is a coherent subsheaf of p*T such
that:
(i) For allot. G, o*Q' = Q' as a subsheaf ofp*T;
(ii) p*T/G' is torsion free.
Then there exists a coherent subsheaf Q of T such that Q' = p*Q as a
subsheaf ofp'T.
Proof. Suppose first that У = SpecR, X = SpecR', T corresponds to
the Д-module M, and Q' corresponds to an Д'-submodule N' of M ®д R'
which is closed under the action of G. We may assume that Y = SpecR

9. Bogomolov's Inequality and Applications 273
is chosen small enough so that M is free. Let N = (N')G. Then N is an
Д-module, and it is a submodule of (M ®R R')G. Since M = Д" for some
n, (M®R R')G » ((R')n)G = Д" = M, and so ЛГ is an Д-submodule of M.
We shall show that the natural map N ® д R' —> TV' is an isomorphism.
Since R' is flat over R, the induced map N ®д R' —> M is injective, and
thus so is the map N ®д Rf -* N'. We claim first that the cokernel of the
map N ®я Д' -» TV' is a torsion module. Let к be the quotient field of R
and let К be the quotient field of Д'. Then G acts on A" and K° = fc, and
so if is a finite Galois extension of k. Indeed we have the following lemma,
whose proof is left as an exercise (compare also [5, p. 68, Ex. 12]):
Lemma 29. Let R' be a commutative ring, let G be a finite group of
automorphisms ofR', and let R = (R')°. Suppose that 5' is a multiplicative
subset ofR' such that <r(S') = 5' for all a e G, and set S = (S')G. Let N'
be ал R'-module with an action ofG such that o(rn) = o(r)o(n) for all
r G R' and n € N'. Then there is a natural isomorphism
In particular, if R' is an integral domain with fraction field K, taking
S' = R' -{0} and S = R- {0}, thenKG is the quotient Geld к of R. U
Let V = M <S>r k, so that (M ®r R') ®r> К = V ®k K. Let W =
(N1) ®R> К CV ®* K. Thus, W is a K-vecbor subspace of V ®fc К such
that <r(W) = W for all ex e G. By Corollary 22, Ж = (W')G is a ifc-vector
subspace of V such that W ®* Л" = W. By Lemma 29, since W is the
localization of N' with respect to the multiplicative subset R' - {0}, W is
the localization of N with respect to R — {0}, in other words W = N ®д fc.
Thus, if we tensor the inclusion N ®д R' -» W with if, it becomes the
isomorphism W ®* A" = W. So the cokernel T of the map N®RR' -* N'
is a torsion Д'-module. Now by assumption Q' = (M ®R R')/N' is torsion
free. Applying the functor ()G to the exact sequence
0 -* N' -> M ®д Rf -> Q' -* 0,
we obtain an exact sequence
0 -f N -> M -»(g')°-
Since Q' is torsion free, (Q')G is a torsion free Д-module. Thus, so is Q =
M/N. Since R' is flat over R, there is an exact sequence
So Q®RR! ? (M®rR')/(N®rR'). In particular, Q®RR' has a submodule
N'/(N®rR') — T, which as we have seen is a torsion module. On the other
hand, we claim that Q ®д R' is torsion free: since Q is torsion free, it is
a submodule of a free Д-module, by Proposition 20 of Chapter 2. Thus,
Q ®д Д' is a submodule of a free Д'-module, and so it is torsion free.

274 9. Bogomolov's Inequality and Applications
Hence N'/(N ®я R') is a torsion submodule of a torsion free module, and
is therefore 0. Thus, N' = N®RR'.
This concludes the proof in the affine case (where T is trivialized), and
in the general case it is easy to check that the isomorphisms over an affine
cover which trivializes T piece together as in the proof of Theorem 23. О
Exercises
1.	Using Reider's theorem, show that, if X is а К3 surface and D is a nef
and big divisor on X, then \2D\ has no base points and |3?>| defines a
birational morphism from X to P^ for some N.
2.	Let X be an algebraic surface and let Я be an ample divisor on X.
Show that, for n > 4, nH + K\ is very ample on X.
3.	Let X be an algebraic surface and let V be a vector bundle of rank
r on X, or more generally a torsion free sheaf of rank r. Define the
Bogomolov number B(V) = 2гсг(Г) - (r - l)ci(VJ. Thus, if V is
stable with respect to some ample divisor, then Bogomolov's inequality
is the statement B(V) > 0. Show that B(V) = B(V®L) for every line
bundle L, B(V) = B(VV) if Via locally free, and B(V) > B(\T) =
2?(VVV) in general. Now suppose that there is an exact sequence of
torsion free sheaves
0 -» Wi -> W -* W2 -> 0,
with ci(Wj) = Д*. Show that
B(W) = -B(Wi) + -B(W2) - —(r2Ai - г,Да)а.
4.	Let X be an algebraic surface and let V be a vector bundle of rank
r on X. Suppose that Bogomolov's inequality holds for every stable
bundle on X of rank less than r, and that V is strictly semistable
with respect to an ample line bundle H. Using the previous exercise,
show that V also satisfies Bogomolov's inequality (compare Exercise
1 in Chapter 6). Working a little harder, show that, if V is Яо-stable
and Hi -unstable for two ample divisors Ho, Hi, then Щ and Hi are
separated by a wall W( with
r!
4
where B(V) is the Bogomolov number of V. (Using the first part and
the previous exercise, it is enough to show that V is strictly semistable
with respect to some convex combination Ht = A - t)H0 + tH\. Using
Exercise 14 in Chapter 4, show that the set {t e [0,1]: V is ^-stable}
is open, and likewise {t e [0,1] : V is #t-unstable} is open. Thus, there
is a point t0 e [0,1] for which V is Hto-strictly semistable.)
5.	Let p: С -» С be an etale double cover with covering involution i,
and let L be a line bundle of degree 0 on С such that i*L Ф L. If
- --b(v) < c2 < o,

9. Bogomolov's Inequality and Applications 275
W = p,L, show that W = p,L is stable but that p'W = L ф l*L is
strictly semistable.
6.	Let С be a smooth curve and let W be a rank 2 vector bundle over
С Let X = P(W) and L = OxB) ® тг* det W as in Lemma 14. If ? is
nef, show that W is semistable.
7.	Let С be a smooth curve and let V a rank 2 vector bundle over С хТ,
where Г is a scheme. For each teT, let Vt be the vector bundle on С
corresponding to V\C x {t}. Arguing as in the proof of Theorem 18,
show that the set of t e T such that Vt is semistable (or stable) is an
open subset of T. (Note that instead of Picn(C x T/T) we can work
directly with ttJ Pic" С in this case.)
8.	Let Я be a commutative ring and R' an Д-algebra. We say that R' is
faithfully flat over Л if it is flat over R and if, for every Д-module M,
M ®д R' = 0 if and only if M = 0.
(a)	Show that, if к is a field, К is an extension field of k, and Я is a
A-algebra, then R' = R ®jt К is a faithfully flat Д-algebra.
(b)	Let R' be a faithfully flat Д-algebra and M an Д-module. Show
that M is a finitely generated Д-module if and only if M ® д Rf is a
finitely generated Д'-module.
9.	Prove Lemma 29. (Suppose that n/s € {S')~lN'. Then
So every element of (S')~1N' can be written as n/s with o(s) = a for
all ex e G.)
10.	Let p: С —¦ С be a degree 2 morphism between two smooth curves,
and let x be a branch point of p. Suppose that i is the involution
corresponding to p. Show that l*Oq(—x) is a subsheaf of Oq = p*Oc,
invariant under the Galois group, but that there is no subsheaf Q of Oc
withp*5 = Oq(—x). Thus, Hypothesis (ii) in Theorem 28 is necessary.
11.	Let A; be a finite field or an algebraic function field in one variable,
and let К be a finite Galois extension of к with Galois group G. Then
the norm map N: K* -* k* is surjective, where N(ct) = Y\aeG <r(a)-
More generally, let F be a homogeneous polynomial of degree d in
k[xi,...,xn]. If n > d, then F has a nontrivial zero in kn.
(To begin, the second statement implies the first, by taking d = [К : к]
and F(xi,...,xd+i) = N(J2i=ix«a«) ~ *xd+i> where ai,...,ad is a
fc-basis for K, noting that N(a) is nonzero if а Ф 0. The second state-
statement is due to Chevalley in case к is finite and may be found as an
exercise in Lang [78, p. 213]. Of course, it is easy to check directly that
the norm map is surjective for finite fields. For a function field к in one
variable, the second statement is Tsen's theorem. Prove Tsen's theo-
theorem first in case к is the function field of P1, in other words, к = ko(t)
where fco is algebraically closed. In this case F € ko{t)[xi,..., xn], and

276 9. Bogomolov's Inequality and Applications
after clearing denominators we can assume that F G fco[<][xi,.. .,xn].
Let N be the largest degree of a coefficient of F. For a given natural
number m, consider F(pi,... ,pn), where each p, is a polynomial of
degree < m. The space of all such vectors of polynomials is An(m+1)t
and since F is homogeneous F = 0 is well-defined in P"(m+i)-i. Now
F(pi,... ,pn) is a polynomial of degree < dm + N whose coefficients
are homogeneous of degree d in the coefficients of the Pi. Thus, to say
F = 0 is dm + N equations in pnCm+i)-^ which will have a solution
provided dm + N < ran + n — 1, or in other words if we choose
^ N-n+l
m>	— •
n — d
More generally, if Fb...,FT € ko(t)[xi,...,xn] are homogeneous of
degree di, then there is a common zero of the F* provided that n >
Ей*.
In case к is a general function field, then we can write к as a finite ex-
extension of ko(t) for ко algebraically closed. Let r = [к : ko(t)\. Choosing
a basis <*i,..., ar for к over ko(t), we may write F = ?V FjOj, where
each Fi is a function on kn = (ko(t))rn, which is homogeneous of de-
degree d. Since rd > m, we may apply the above to find a common zero

10
Classification of Algebraic Surfaces
and of Stable Bundles
In this chapter, we outline the major results in the classification theory of
surfaces, and then proceed to fill in the details of the proofs. While the
proofs given here do not rely on Mori theory, we give a brief description of
the corresponding results for threefolds, whose proofs rely heavily on Mori
theory. In the last section, we survey some of the known results on the
structure of the moduli space of stable bundles over a surface, and try to
relate these results to the geometry of the original surface.
Outline of the classification of surfaces
In this section we shall list the various results that go under the general
heading of classification of algebraic surfaces. The proofs will be given in
the following sections. We begin with the following definition:
Definition 1. Let X be an algebraic surface. Define the Kodaira dimen-
dimension k(X) of X as follows:
k(X) = min{fc e Z : Pn(X)/nk is a bounded function of n > 1}.
For example, it follows formally that if Pn(X) = 0 for all n, then к(Х) =
-oo. If Р„(Х) ф 0 for some n, then since Pnm{X) > Pn(X), it follows that
k(X) > 0. It is easy to see that к(Х) < 2 (and we shall prove much more
precise statements later). Thus, the possible values for к(Х) are —oo, 0,1,2.
By Corollary 5 of Chapter 3, к(Х) is a birational invariant.
We can similarly define the Kodaira dimension к(X) of any smooth vari-
variety X of dimension n, and show that it is either — oo or an integer between
0 and n. For example, if С is a smooth curve, then к(С) = — oo if g(C) — 0,
k(C) = 0 if g(C) = 1, and к(С) = 1 if g(C) > 2. A variety X is of general
typeifn(X) =

278 10. Classification of Algebraic Surfaces and of Stable Bundles
The idea that the asymptotic behavior of the plurigenera has a deep in-
influence on the structure of the surface X was already known to the Italians
(see, for example, Enriques' book [31]), and was used systematically by
Kodaira in the "Kodaira classification" of surfaces. However, the notation
k(X) was first introduced in the Shafarevich seminar [132].
Let us now list some of the major results in the classification theory. One
of the first major results is the intrinsic characterization of those surfaces
which are rational, in other words birational to P2. If С is a smooth curve,
then С is rational, in other words its function field is C(t) for some tran-
transcendental element t, if and only if its genus g(C) is 0. For surfaces X, the
analogous result is:
Theorem 2 (Castelnuovo). Let X bean algebraic surface with Pi(X) =
q(X) = 0. Then X is rational.
We note that it does not suffice to assume that pg(X) = q(X) — 0.
There are examples of algebraic surfaces with torsion in Я2 for which pg =
q = 0. The first such example was constructed by Enriques: there exist КЗ
surfaces Y with a fixed point free holomorphic automorphism i of order
2, and the quotient X = Y/i is an algebraic surface with q = pg = 0
and Hi(X; Z) = Z/2Z. Such a surface is called an Enriques surface. Later
Godeaux found certain quintic surfaces in IP3 which have a fixed point
free automorphism of order 5. The quotients are then surfaces X with
q = pg = 0 and Hi(X;Z) = Z/5Z. Thus, none of these surfaces can
be rational. Dolgachev [22] showed that the logarithmic transform of a
rational elliptic surface at two fibers with relatively prime multiplicities is
a nonrational algebraic surface X with q = pg = 0 and Hi(X;Z) = 0. By
the canonical bundle formula, it follows that X is not rational (for example,
Pn(X) Ф 0 for some n). These examples are in fact simply connected, and
thus homeomorphic to rational surfaces, by Freedman's classification of
topological 4-manifolds [34]. By Donaldson theory, it can be shown that
they are not diffeomorphic to rational surfaces [24], [38], [119]. Barlow [6]
constructed a simply connected surface of general type В with q = pg = 0.
Again using Donaldson theory, Kotschick [76] and also Okonek and Van de
Ven [120] showed that В is not diffeomorphic to a rational surface. After a
considerable amount of work in this direction [122], [127], [128], [124], the
author and Qin [43], as well as Pidstrigach [123], showed that an algebraic
surface diffeomorphic to a rational surface is necessarily rational. These
arguments were greatly simplified by the advent of Seiberg-Witten theory,
and could then be used to show that the plurigenera are smooth invariants
[41], [16] (see also [118]).
The arguments used to prove Castelnuovo's theorem also determine the
minimal models of a rational surface, a result due to Vaccaro and Andreotti:

10. Classification of Algebraic Surfaces and of Stable Bundles 279
Theorem 3. A minimal rational surface is either P2 or Fn for some n Ф 1.
Thus, every rational surface X is the blowup either of P2 or of?n for some
пф1.
A problem related to the characterization of rational surfaces is the char-
characterization of ruled surfaces, in other words those surfaces birational to
С x P1 for some curve C. One basic result in this direction is:
Theorem 4. Let X be a minimal surface such that Kx is not nef. Then
X is rational or ruled, i.e., X is either P2 or a geometrically ruled surface.
Mori has introduced a series of new ideas in the classification theory of
surfaces, threefolds, and higher-dimensional algebraic varieties. From the
viewpoint of Mori's theory, Theorem 4 says the following: let X be an
algebraic surface, and suppose that Kx is not nef. Then either there exists
an exceptional curve on X (X is not minimal), or there exists a morphism
from X to a smooth curve С with all fibers P1 (X is geometrically ruled), or
X = P2. We shall discuss the analogous classification results for threefolds
later in this chapter.
As an immediate corollary to Theorem 4, a minimal surface X such that
Kx < 0 is a ruled surface X over a curve С of genus at least 2.
The following theorem and its higher-dimensional analogues go by the
name of the abundance theorem:
Theorem 5. Let X be & minimal algebraic surface such that Kx is nef.
Then K% > 0. Moreover:
(i) k(X) = 0 if and only if Kx is numerically equivalent to 0.
(ii) k(X) = 1 if and only if K\ = 0 but Kx is not numerically equivalent
toO.
(iii) k(X) = 2, i.e., X is of general type, if and only if K\ > 0.
In all cases, either P4(X) or Рб(Х) ф 0.
Theorem 5 is really quite a surprising statement. For example, it says
that, if Kx is numerically trivial, then it has finite order in Pic X. Likewise,
if K\ = 0 but Kx is not numerically trivial, then some multiple of Kx has
at least two sections. Thus, for example Kx cannot be linearly equivalent
to an irreducible curve D with D2 = 0 but such that the normal bundle
of D has infinite order in Pic0 D. (The remaining statement, that if Kx is
nef and big then X is of general type, is much easier.)
Let X be a surface, not necessarily regular. Then X is the blowup of a
minimal surface X. Now either Kx is nef, in which case Pa{X) or Рб(Х) ф
0, or X is rational or geometrically ruled. Since the plurigenera are invariant
under blowup, and a blowup of a geometrically ruled surface is a ruled

280 10. Classification of Algebraic Surfaces and of Stable Bundles
surface, we have the following numerical characterization of rational or
ruled, which is akin to Castelnuovo's theorem:
Corollary 8 (Enriques). Let X be an algebraic surface with either
Pi(X) = 0 or Pe(X) = 0. Then X is rational or ruled. In particular,
X is rational or ruled if and only if к(Х) = —oo.
Since the least common multiple of 4 and 6 is 12, we could make the
slightly stronger assumption that Pi2{X) = 0 in the above corollary.
Using the above corollary, we have the following characterization of al-
algebraic surfaces X with к(Х) > О:
Proposition 7. Let X be an algebraic surface. Then the following are
equivalent:
(i) к(Х) > 0.
(ii) X has a unique minimal model
(iii) There exists a surface X' birational to X such that K\> is nef.
(iv) There exists a minimal model X' ofX for which Kx> is nef.
(v) For every minimal model X' ofX, Kx1 is nef.
(vi) X is not rational or ruled.
Proof. The implication (i) <=>¦ (vi) is Corollary 6. We have seen that (i)
=>• (ii) by Theorem 19 of Chapter 3. Conversely, if к(Х) = -со, then X
is either rational or ruled and so does not have a unique minimal model.
Thus, (ii) =» (i).
Next we show that (i) => (v). Suppose that n(X) > 0. Thus, there
exists an n > 1 and a section J^ °Ч^г G ln^x|, where the Ci are distinct
irreducible curves and the a* are positive integers. Clearly, Kx • С > 0
for every irreducible curve С on X which is not one of the d. Thus, У
Kx ¦ С < 0, then C=Ci for some i. Now
K* • Ci = ? ( E №i	)
Since ajiCyd) > 0 for all j ф i, and a< > 0, if KxCi< 0, then (CiJ < 0
In this case as Kx C, < 0 and (CiJ < 0, it follows by Lemma 11 of Chaptei
3 that Ci is an exceptional curve, so that X is not minimal. Conversely
if X is minimal, then Kx is nef. Thus, (i) => (v). The implications (v
=^ (iv) and (iv) => (iii) are trivial.
Finally, we show that (iii) => (i). If (i) does not hold, then X is rations
or ruled, and the discussion of the minimal models of such surfaces show
that there exists a smooth rational curve С on X with C2 > 0 and thu
Kx ¦ С = -2 - С2 < -2. Thus, Kx is not nef. D

10. Classification of Algebraic Surfaces and of Stable Bundles 281
For minimal surfaces with к = -оо, the classification scheme is the same
is that for rank 2 vector bundles over a curve C, up to twisting by a line
jundle. The unstable bundles have a very explicit description in terms of
extensions, and the semistable bundles at least fit together into a coarse
noduli space. There is also a fine classification in the cases к = 0,1. To
leal with the case к = 0, recall that an Enriques surface is the quotient
>f а КЗ surface by an fixed point free involution of order 2. An Enriques
iurface is also elliptic. In fact every Enriques surface is an elliptic surface
jver P1 with invariant d — degZ = I and two multiple fibers, both of
multiplicity 2, and conversely. There are also quotients of abelian surfaces
by fixed point free automorphisms, called bypereliiptic surfaces. All such
surfaces are elliptic, in two different ways, and have been described in the
sxercises to Chapter 7.
Theorem 8. Let X be a minimal algebraic surface with к(Х) = О. Then
X is а КЗ surface, an abelian surface, an Enriques surface, or a byperelliptic
Surface.
In particular, every minimal algebraic surface with к = 0 has a finite
covering space with trivial canonical bundle.
Theorem 9. Let X be a. minimal algebraic surface with к(Х) = 1. Then
X is an elliptic surface.
For every minimal elliptic surface X, the canonical bundle Kx is numer-
numerically equivalent to r/ as a divisor with rational coefficients, where r e Q.
Clearly, Kx is nef if and only if r > 0, and Kx is numerically equivalent
to 0 if and only if r = 0. Thus, the elliptic surfaces with к = -oo corre-
correspond to the case r < 0 and those with к = 0 correspond to r = 0. The
possibilities for such surfaces have been listed in Exercise 7 of Chapter 7.
It follows from this list that the elliptic surfaces with к = —oo are either
rational surfaces or else are ruled surfaces over an elliptic base (although
not every geometrically ruled surface over an elliptic curve is actually an
elliptic surface). The elliptic surfaces with к = 0 are the Enriques surfaces,
the hyperelliptic surfaces, and certain КЗ and abelian surfaces. Thus, with
few exceptions, an elliptic surface has Kodaira dimension 1. Sometimes the
surfaces with Kodaira dimension 1 are thus called the properly elliptic sur-
surfaces. In any case, all elliptic surfaces can be fairly explicitly described by
the classification scheme of Chapter 7.
No such fine classification theory exists for surfaces of general type. There
are restrictions on the Chern numbers of minimal surfaces of general type.
Classically, there is Noether's inequality
K2x>2pg(X)~4.

282 10. Classification of Algebraic Surfaces and of Stable Bundles
The cases of equality have been analyzed by Horikawa [64] and surfaces
where equality is attained are called Horikawa surfaces.
A much deeper result is the Bogomolov-Miyaoka-Yau inequality
K\ < 3c2(X) = 3X(X),
with equality holding if and only if X is the quotient of the unit ball in C2
by a discrete group of holomorphic automorphisms acting freely and with
compact quotient [95], [96], [147].
Beyond these numerical restrictions, the study of surfaces of general type
largely consists in studying examples and goes under the name "geography"
(for deciding which Chern numbers or other topological invariants arise
as the invariants of a minimal surface of general type), and "botany" (for
describing all of the deformation types of surfaces within a fixed topological
type). For a survey of some of these results, see [121].
Let us record the following useful result on surfaces of general type:
Proposition 10. Let X be an algebraic surface such that K\ is net and
big. Then X is a minimal surface of general type, and the plurigenera Pn
of X, for n > 2, are given by the formula
Conversely, ifX is a surface of general type, the following are equivalent:
(i) X is minimal.
(ii) Kx is nef.
(Hi) For all n > 2, Pn(X) = П("~ *V*J + x(Px)-
(iv) There exists an n > 2 such that Pn(X) = -^П~1\кх? + x(Ox)-
(v) For all n > 2, НЦХ; nKx) = 0.
(vi) There exists an n > 2 such that Я1 (X; nKx) = 0.
Proof. Let X be an algebraic surface such that Kx is nef and big.
Since Kx is nef, X is minimal. For n > 2, H1(X;nKx) is Serre dual
to H1(X; A - n)Kx). Since A - n)Kx is the negative of the nef and big
divisor (n-l)Kx, the Mumford vanishing theorem implies that HX(X; A-
n)Kx) = 0. Moreover, H2(X;nKx) is Serre dual to H°(X;(l - n)Kx)-
As Kx is big, Kx ¦ (A - n)Kx) < 0. Since Kx is nef, A - n)Kx cannot
be effective, and so H°(X; A - n)Kx) = 0. It follows that, for n > 2,
Pn(X) = h°(nKx) = x(Ox(nKx)).
Applying Riemann-Roch, we have x(9x(nKx)) = [n(n - 1)/2}(KXJ +
x(Ox)- Thus, we have established the formula for Pn, and clearly к(Х) —
2 since Pn is a quadratic polynomial in n. Thus, X is of general type.
(Another way to see that X is of general type is to note that Kx induces

10. Classification of Algebraic Surfaces and of Stable Bundles 283
ah ample divisor on the surface X obtained by contracting the — 2-curves
orthogonal to Kx)
To see the equivalences in the remaining statement of the theorem, note
that Kx is nef implies that X is minimal, so that (ii) ==>¦ (i). Conversely,
for a minimal surface with к > 0, we know by Proposition 7 that Kx is
lief. Thus, (i) => (ii). Moreover, if X is minimal, then the formula in (iii)
Jds by the first statement of Proposition 10, so (i) =>¦ (iii). We have
seen that (i) =*¦ (v) as well. The implications (iii) => (iv) and (v)
(vi) are trivial. Next, for all n, the dimension of H°(X; A - n)Kx) is
a birational invariant, and the argument above shows that this dimension
is 0 if X is minimal and n > 2. Thus, Я°(А"; A - n)Kx) is always 0 if
n > 2. Let Xo be the minimal model of X. Thus, Pn(X) = Pn{X0) and
KJt < KXg, with equality holding if and only if X = Xo- Let us show that
(iv) ==>¦ (i). Assuming (iv), we have
since Pn(X) is a birational invariant. As x(Px) = x(®x0) and n > 2, it
follows that (KxJ = (Kx0J and hence that X = Xo is minimal. Thus,
(iv) => (i). Finally, the implication (vi) =» (iv) is an easy consequence
of the Riemann-Roch formula and the fact that H°(X; A - n)Kx) = 0 and
so H2(X; nKx) = 0. This concludes the proof. ?
>>
>' One final result concerning surfaces with к > 0 goes under the general
jname of the Castelnuovo-deFranchis theorem:
Theorem 11. Let X be an algebraic surfece with к(Х) > 0. Then C2(JC) >
0, with equality holding if and only ifX is a minimal elliptic surface whose
only singular fibers are multiple fibers with smooth reduction or X is an
abelian surface.
Corollary 12. Let X be an algebraic surface with к(Х) > 0. Then
x(Ox) •> 0, with equality holding if and only if X is an elliptic surface
whose only singular fibers are multiple fibers with smooth reduction or X
is an abelian surface.
Proof of the corollary. It suffices to prove the statement under the
assumption that X is minimal, since x(®x) is a birational invariant. In this
case, by Noether's formula, 12x(Ox) = cx(XJ + c2(X) with c^XJ > 0
since Kx is nef and сэ( X) > 0 by the above theorem. Thus, x(®x) > 0, and
if equality holds Ci(XJ = C2(X) = 0. Hence X is a minimal elliptic surface
whose only singular fibers are multiple fibers with smooth reduction, or X
is an abelian surface. Conversely, if X is a minimal elliptic surface whose

284 10. Classification of Algebraic Surfaces and of Stable Bundles
only singular fibers are multiple fibers with smooth reduction or A' is an
abelian surface, then Ci(XJ = сгСХ") =0. ?
Proof of Castelnuovo's theorem
Let A" be a surface with q(X) = P2(X) = 0. Note that pg(X) = 0 as well,
so that Hl(Ox) = H2{Ox) = 0. Prom the exponential sheaf sequence,
PicX = I^iX-jZ). In this case, we shall prove:
Theorem 13. Let X be a surface with q(X) = P2(X) = 0. Then there
exists a smooth rational curve С on X with C2 > 0.
It is not even a priori obvious that a rational surface has this property!
In fact, if X --* P2 is a birational isomorphism, then the strict transform
of a line in P2 on X may well be singular. Thus, we cannot just use the
strict transform of a line to find the curve C.
Proof that Theorem 13 implies Castelnuovo's theorem. Let X be
a surface with q(X) — P2(X) — 0 and let С be a smooth rational curve on
X with C2 > 0. From the exact sequence
0 _» ox -¦ OX{C) -» OC(C) -» 0,
we see that dim \C\ > 1 and that \C\ has no fixed components. Choose a
pencil inside \C\. After blowing up the base locus, there is a morphism X —»
P1 whose general fiber is a smooth rational curve. Thus, there is by Lemma
8 of Chapter 5 a blowdown X —> Y, where Y -»P1 is a geometrically ruled
surface over P1. Hence У is birational to P1 x P1 and thus to P2. (Of course,
by Theorem 9 of Chapter 5 and the classification of vector bundles over
P1, Y = Fn for some n.) Thus, Y and therefore X are rational. ?
Proof of Theorem 13. We begin the proof with the famous "termination
of adjunction" lemma:
Lemma 14. Let X be a minimal surface such that q(X) = Рг(-^) = 0.
Then for every divisor D on X, and for all n » 0, \D + nKx | = 0-
Proof. First suppose that Kx > 0. By the Riemann-Roch theorem,
x(Ox(-Kx)) = \(-Kx)(-2Kx) + 1 > 1,
and thus either h°(-Kx) or h2(-Kx) is nonzero. But h2(-Kx) =
h°BKx) = P2(X) = 0, so that h°(-Kx) ф 0. Since h°(Kx) = 0, -Kx
is not the trivial divisor, and thus — Kx = С for some nonzero effective
curve С Thus, for every divisor D, if Я is an ample divisor and n » 0,

10. Classification of Algebraic Surfaces and of Stable Bundles 285
H(D + nKx) = (HD)- n(H • C) < 0. It follows that D + nKx is not
effective if n » 0.
If Kx < 0 and D is a fixed divisor, suppose that D + mKx is effective.
fJote that (D + nKx) ¦ Kx < 0 for all n » 0. In particular D + nKx is
not the trivial divisor for n » 0. If m is sufficiently large and D + mKx is
linearly equivalent to the effective nonzero divisor J^t фС») where the Ci
are irreducible curves and a* > 0, then as (D + mKx) • Kx < 0 we must
have Ci ¦ Kx < 0 for some i. As X is minimal, it follows that Cf > 0 and
thus that Ci is nef. In particular, Ct ¦ (D + nKx) > 0 for all n such that
D + nKx is effective and (D + nKx) • Kx < 0. But since Q • Kx < 0,
C* • (?> + пКх) < 0 for all n > 0, contradicting d ¦ (D + пЯ*) > 0. Thus,
for n 3> 0 the divisor D -f- nKx is not linearly equivalent to an effective
divisor. ?
Returning to the proof of Castemuovo's theorem, we claim that there
exists a very ample divisor Я such that Я is not an integer multiple of
Kx- In fact, if every very ample divisor is an integer multiple of Ajc,
then as every divisor is linearly equivalent to a difference of very ample
divisors it would follow that Pic X = Z[KX ] ¦ But we have seen that Pic X =
H\X; Z), so that H\X\ Z) <* Z[KX] as well. By Poincare duality, K%=1.
Moreover, ЩХ) = 1 and bi(X) = Ъ3(Х) = 0, so that сг(А") = 3. On the
other hand, from Noether's formula we have
c*(X) + c2(X) = 1 + 3 = 12X(Ox) = 12,
which is absurd.
Thus, we may choose a very ample H which is not an integer multiple
of Kx, so that h°(Ox(H)) ф 0 or equivalently \H\ ф 0. Using Lemma 14,
there exists an n > 0 such that \H + nKx\ ф 0 but \H + (n + 1)KX\ = 0-
Since H is not an integer multiple otKx, H+nKx is not the trivial divisor.
Thus, there exists JV OjCj e \H+nKx \, where the Ci are irreducible curves
and Oi > 0. We shall show that all of the C, are smooth rational curves
and that Cf > 0 for some i.
Since |?.aid + Kx\ = 0 and | ?4 а& + Kx\ contains \Ct + Kx\ for
every i, we must have \d + Kx\ = 0 for every i. On the other hand, by
applying adjunction to d, we have the exact sequence
0 -» Ох(АГл) -» Ox(Ci + Kx) -»wCj - 0.
As ЛЧ^х) = ЛЧОх) = q(X) = 0, the map H°(Ox(d+Kx))
is surjective. But since H°(Ox{d + Kx)) = 0, Я°(шС() = 0 as well, in
other words po(Cj) = 0. It follows that Ci is a smooth rational curve for
every i.
Finally, we must show that Cf > 0 for some i. If Cf < 0, then, since X
is minimal, Kx-Ct> 0. Thus, if Cf < 0 for all i, then Kx ¦ (?» a^d) > 0.

286 10. Classification of Algebraic Surfaces and of Stable Bundles
Let D = ?\OtCi. Then D = H + nKx, KxD>0,and
Kx ¦ D + D2 = Kx ¦ D + H ¦ D + n(Kx • D) > 0,
since H is ample and D ф 0. By assumption h°(Kx + D) = 0, and
thus h?(-D) = 0. Since D is effective and nonzero, h°(-D) = 0. Thus,
X(Ox(-D)) < 0. On the other hand, applying the Riemann-Roch theo-
theorem, we find that
X@x(-?>)) = i ((-?>J + (KX ¦ D)) + 1 = I {KX ¦ D + D2) + 1 > 0,
a contradiction. Thus, for some i the curve Ci is a smooth rational curve
and Cf > 0. D
The above argument also shows the following:
Corollary 15. The minimal models of P2 are exactly the surfaces P2 and
Proof. Clearly, the surfaces P2 and Fn,n ф 1, are in fact minimal. Con-
Conversely, suppose that X is a minimal rational surface. By Castelnuovo's
theorem, X contains a smooth rational curve С with C2 > 0. Considering
the exact sequence
and using h^Ox) = 0, we see that h°(Ox(C)) > 2. Choose С е \С\, С ф
С. Then С and С" span a pencil inside \C\ with no fixed curves. First
suppose that every element in the pencil is a smooth rational curve. Then
there is a blowup of X, say X, and a morphism /: X —» P1 such that all of
the fibers of / are smooth. In fact, we can take the P1 to be the parameter
space of the pencil and take X to be the incidence correspondence
where Ct is the curve in \C\ corresponding to t 6 P1. The projections of
X to the first and second factors of X x P1 induce a birational morphism
X —> X and a morphism тг: X —» P1 such that ir~1(t) — Ct. Since Ct is
smooth for every t, a local calculation shows that X is smooth, and so X
is a blowup of X by the factorization of birational morphisms. Since all
fibers of тг are smooth rational curves, X —> P1 is a geometrically ruled
surface, and thus is Fn for some n. If X = X, then we are done: We must
have X = Fn, n ф 1, since X is minimal. If X ф X, then, as X is a blowup
of X, it contains an exceptional curve E. Since the only curve of negative
self-intersection on Fn is the negative section a, with a2 = -n, necessarily
n = 1, and there is an induced birational morphism from the contraction
of E, namely P2, to X. As P2 is already minimal, X = P2.

10. Classification of Algebraic Surfaces and of Stable Bundles 287
Now suppose that every pencil in \C\ contains a reducible element
J2i <HCi. From the exact sequence
0 -» OX(KX) -» OX(KX + C) -»wc - 0,
we see that \KX+C\ = 0, and thus, as in the proof of Theorem 13, it follows
that \KX +d\, which is a subeeries of \KX +?,¦ <hd\ = \Kx +C|» » етР*У
for every i. As in the proof of Theorem 13, C< must be a smooth rational
curve for every i. We claim that Cf > 0 for some i. The argument parallels
that in Theorem 13: if Cf < 0, then since X is minimal, d ¦ Kx > 0.
Thus, if C? < 0 for every i, then 0 < Kx • (^VoiCi) = Kx ¦ C. But
tfx • С = -2 - С2 < -2, which is a contradiction. Thus, Cf > 0 for some
i. Now C2 = ?\ Oj(C ¦ Cj) > <ч{С ¦ C<), since С is nef, and
«ц(С • CO	Y
Thus, C2 < C2, with equality if and only if d meets no other Cj and
<ц = 1. By the connectedness theorem, Ct meets no other Cj if and only if
С = о<С<, and, if Oj = 1, then Ct = С is smooth, which we have ruled out.
Thus, 0 < Cf < C2. If there exists a pencil in \С,\ such that every member
is smooth, we are done. Otherwise, we may continue this procedure, noting
that at each stage the self-intersection is nonnegative and strictly decreases.
Thus, the procedure must eventually terminate. By the first part of the
argument, we then have X = P2 or Fn,n ф 1. ?
The following corollary to Castelnuovo's theorem is the affirmative an-
answer to the Liiroth problem for an algebraically closed field of characteristic
0. (It fails over algebraically closed fields of positive characteristic as well
as over fields of characteristic 0 which are not algebraically closed.)
Corollary 16. Let X be & rational surface, and let f: X —* Y be a
genericaHy finite moiphism to tie smooth surface Y. Then Y is rational.
In terms of function fields, let ц and хг be algebraically independent over
C. Let к be a. subfield of С(х1,хг) containing С and such that
[C(x,,x2):fc]<oo.
Then к is a pure transcendental extension of С of transcendence degree 2.
Proof. Since / is generically finite, its differential is an isomorphism on
a Zariski open subset of Y. Thus, /*: Я°(У;П^) -> H°(X;Slx) is injec-
tive. It follows that /* defines inclusions HX{Y;OY) Q Hl{X;Ox) and
H0(Y;OYBKY)) С H°(X;OxBKx)). Thus, if q(X) = P2(X) = 0, then
q(Y) = P2(Y) = 0 as well, and we can apply Castelnuovo's theorem to
Y. D

288 10. Classification of Algebraic Surfaces and of Stable Bundles
The Albanese map
The main tool which we need to handle the classification for irregular sur-
surfaces is the Albanese variety of X.
Definition 17. Let X be an algebraic surface (or more generally a com-
compact Kahler manifold). Define the Albanese variety AlbX to be the com-
compact complex torus #°(X;fl^)*/#i(X;Z). More precisely, there is a nat-
natural map
defined by integration: if 7 G Hi(X;Z), then, since holomorphic forms are
closed on a compact Kahler manifold, for cp G H°(X; fix),
(p I—»
is a well-defined linear function on H°(X; fix). By Hodge theory, the kernel
of the induced map Hi(X;Z) —* #°(X;fix)* is the torsion subgroup of
Hi (X; Z), and its image is a discrete subgroup of H°(X; Ях )* with compact
quotient. By definition this quotient is then AlbX. One can show that it
is in fact an abelian variety (in other words, a complex torus which is also
an algebraic variety).
Proposition 18. for every algebraic surface or compact Kahler manifold
X and for every choice of a base point p€ X, there is a morphism a: X -»
Alb X with the following properties:
(i) a*: H°(AlbX;fi\lbx) -> tf°(X;fix) is an isomorphism.
(ii) For every complex torus T, iff: X —> T is a morphism such that f{j>) =
0 € T, then there is a unique morphism of complex tori д: Alb X —* T
such that / = joa.
Proof. Define the morphism a via integration: given x € X, choose a
piecewise C°° path o: [0,1] —> X such that <r@) = p and <r(l) = x, and set
Q(x)= (v1-* / v).
well defined up to the choice of the path a. Since two choices of a differ
by a piecewise C°° closed curve 7, q(x) is well defined up to the image of
Hi(X;Z), and thus as an element of AlbX. By the fundamental theorem
of calculus, a is holomorphic and its complex derivative at x is the natural
map TXiI -> TAlbX]Ct(l) = tf°(fix)* defined by

10. Classification of Algebraic Surfaces and of Stable Bundles 289
Thus, a* is given by the natural map
<Uibx,«W = Н°(А1ЪХ;п\1ЪХ) = H°(X;UX) -» Пх,х.
In particular, a*: #°(AlbX; П^1ЬХ) -> H°(X; ftx) is an isomorphism.
Now suppose that T is a complex torus and /: X —* T is a holomorphic
map. Then /* induces a complex linear map Н°{Т;п\) -* Н°(Х;пх),
and thus a linear map /„: (H°(X;UX))* -* (Я°(Г;П^))*. Moreover,
ft(Hi(X;Z)) С #i(T;Z) and so there is an induced morphism g: AlbX -»
(H°(T;Q^))*/Hi(T;Z) = T. By construction /(p) =0 = joa(p) and
f* = (goa)*. Thus, the difference map /-(joo) has differential equal to
0, and so it is constant with image 0. It follows that / = joa. We leave
the uniqueness of g as an exercise. D
The morphism a: X —* AlbX defined above, which is unique up to the
choice of a base point, will be called the Albanese map.
Let X be a surface. Two basic properties of the Albanese map a: X —*
; KVaX which we shall need are:
Proposition 19. Let X be & surface and let a: X -» AlbX be the Al-
Albanese map.
(i) Ifa(X) has dimension 2, then pg(X) ф 0.
(ii) Ifa(X) is a curve C, then С is smooth and a: X -± С has connected
fibers. In this case Alb(X) = J(C), the Jacobian of the curve C.
Proof. If a(X) has dimension 2, there exists a point x such that a*
has rank 2. In particular, there exists an ша(х) 6 ^Aibx,o(i) sucn tnat
«*(^a(z)) Ф 0- Since П^1ЬХ is trivial, Пдшх ** trivial as well, and so we
can find a global holomorphic 2-form ш on Alb X such that ш induces u>a(x).
Thus, а*ш is a nonzero holomorphic 2-form on X so that pg(X) Ф 0.
If a(X) is a curve C, let С be the normalization of С Note that g(C) > 1.
Since X is normal, the map X —> С factors through a morphism X-»G,
and we have the composite map X —> С —» J(C), where ./(E) is the
Jacobian of C. By the universal property of Alb X, there is an induced
map AlbX -¦ J(C). The map С-*С-*А1ЪХ -*J(C) then agrees with
the map С -* J(C), and so the image of С in J(C) is C. Since the map
from a curve of genus at least 1 to its Jacobian is an embedding, С = C,
and in particular С is smooth. There is the map g: А\ЪХ —» J(C), and the
morphism С —» Alb X induces a morphism /: J(C) —» Alb X (essentially
because J(C) is the Albanese variety of С and by an argument similar to the
proof of Proposition 18). The composite map С -* J(C) -Д AlbX -^ J(C)
is by construction the natural map С —> J(C), and thus by the universal
property of J(C) we must have fog = Id. Likewise, the map X —* Alb X —»
J(C) -A AlbX is the map a: X -> AlbX, and so gof = Id as well. Thus,

290 10. Classification of Algebraic Surfaces and of Stable Bundles
МЪХ = J(C). Finally, let us show that the fibers of a are connected. Let
X -* D -> С be the Stein factorization of X -» C. Thus, the fibers of
X —> D are connected and D —> С is finite; moreover D is normal since
С is normal. Applying the above argument with С replaced by D, we see
that the map D —* AlbX induces an isomorphism J(D) —> AlbX, and
thus that the map J(D) —* J(C) is an isomorphism. Thus, D = С and the
fibers of q are connected. D
Proofs of the classification theorems for surfaces
Throughout the rest of this chapter we shall always take X to be a minimal
algebraic surface unless otherwise noted. The first main result of this section
shows that, for a minimal surface X, Kx is nef if and only if X is not
rational or ruled.
Theorem 20. If X is a minimal algebraic surface and Kx is not nef, then
X is rational or ruied.
Proof. First we claim that, if X is minimal and Kx is not nef, then
Pn(X) = 0 for all n > 1. This follows from the proof of the implication (i)
=* (v) of Proposition 7 (the proof did not require any of the classification
results).
Next suppose that q(X) = 0. Since Pz(^) = 0, X is rational by Castel-
nuovo's theorem. Thus, we may assume that q(X) > 0. Let a: X —» AlbX
be the Albanese map. If a(X) is a surface, then pg(X) ф 0, by Proposition
19. Thus, a(X) = С is a smooth curve and the Albanese map is a fibration
n:X-*C. Moreover, Alb X = J(C) and thus g{C) = q{X). We also may
assume that Kx < 0: Suppose that K\ > 0. By Noether's formula, we
have Kx + B - Aq + bj(X)) = 12A - q). Thus,
8q + ba(X) = 10 - K% < 9.
It follows that (since we have assumed that q > 1) q = 1 and b^(X) = 1.
But X maps onto its Albanese, which is a curve, and, for a general fiber
/, /2 = 0. Since K\ > 0, Kx is not a multiple of /, and so the classes
[/] and [Kx] span a rank 2 subgroup of H2(X;Z). But then b^Jt) > 1, a
contradiction. Hence we may assume that K\ < 0. Our goal now will be
to show that the fibers of the Albanese map are smooth rational curves, so
that X is ruled.
Claim 1. Let тг: X -* С be the Albanese fibration ofX, and let D be an
irreducible curve on X such that Kx-D < 0 and \Kx +D| = 0. Then either
X is ruled or D is smooth with g(D) = q, n\D: D —* С is unramified, and
it\D is an isomorphism itq > 1.

10. Classification of Algebraic Surfaces and of Stable Bundles 291
Proof of Claim 1. From the exact sequence
0 -» Ox(Kx) -» OX(KX +D)^ljd^0,
and the fact that H°(Ox(Kx + D)) = 0 by assumption, it follows that the
map H°(lod) -> ^(Kx) is injective. However, hl(Kx) = q and /i°(u>D) =
pa(D), so that pa(D) < q — g(C). Thus, if D is the normalization of D,
then as g(D) <_pa(D), g(D) < g(C) as well. On the other hand, there
is a morphism D —> C. If this morphism were constant, then D would
be contained in a fiber of тг and so D2 < 0. But since X is minimal and
Kx ¦ D < 0, ?>2 > 0, so that D2 = 0 and nD is a fiber of тг for some n. In
this case Kx • / < 0 for a general fiber / of тг, and thus Kx • f + f2 < 0-
It follows that / is a smooth rational curve, and so X is ruled.
Thus, if X is not ruled, the morphism D —* С is onto. But then g(D) >
g(C), by the Riemann-Hurwitz formula, since g(C) > 1, and so g(D) =
pa(D) = g(C). Moreover, since g(C) ф 0, again by the Riemann-Hurwitz
formula either the morphism D —* С has degree 1 or g(C) = 1 and D -*C
is unramified.
Claim 2. IfXis not ruled, there exists an irreducible curve D on X such
that Kx D < -1 and \KX + D\ = 0.
Proof of Claim 2. Since Kx is not nef, there exists an irreducible curve
С on X with Kx • С < -1. As X is minimal, C2 > 0 and thus С is nef.
By assumption K% < 0. Thus, for all n > 0, BC + nKx) ¦ Kx < -2. Now
|2C| ^ 0, but for n » 0, |2C+n#x| = 0 since BC+n#x)-C < 0 for n » 0
and С is nef. Thus, there exists an n > 0 such that |2C + nKxl Ф 0 and
|2C + (n + l)tfx| = 0. Let ?> = ?\ щСг 6 |2C+nA"x|. Then D is effective,
Kx D < -1, and \D + Kx\ = 0. If 0 < D' < D, then \D' + Kx\ = 0 as
well. Thus, we may assume that Q • Kx < 0 for all i. By Claim 1, Cj is
smooth and g(Ci) = q for all i. We shall show in this case that D = Ct for
some i, and thus that D is irreducible.
First we claim that n» = 1 for all i. Indeed, if n» > 2 for some i, then
\2C, + Kx\ = 0, and thus /i2(Ox(-2Ci)) = 0. From the exact sequence
0 -» Ox(-2C4) -» Ox -» O2Ci -» 0,
it follows that the map from Hl(Ox) to H1(O2ci) is surjective, and thus
that ft1 (Огс^) ^ 9- On the other hand, using the exact sequence
0 -¦ OCi{-Ci) -+ O2Ci -» OCi -» 0,
and the fact that Я°(О2с4) -¦ Н°{Осг) = С is surjective, it follows that
h\O2Ci) = h\OCi) + hl(OCt(-d)) = 9 +
Thus, ^(Oc.i-Ci)) = 0. But pa(d) > 1, and so Kx ¦ Сг + С? > 0. As
Kx-Ci < O,Cf >0. SoOci(-Cj) is a line bundle on Ci of strictly negative

292 10. Classification of Algebraic Surfaces and of Stable Bundles
degree, say degOct(-Ci) — -d with d > 0. By Riemann-Roch on
h\OCi{-Ci)) = h°(OCi(-Ci)) +d + g(Ci) - 1 > d > 0,
contradicting the fact that Л ^Ос, (-С*)) = 0. It follows that nt = 1 for all
i.
Finally, we show that there is just one curve C{. If, say, D contains the
curve C\ + C2, then again \C\ +C2+ Kx\ = 0. Arguing as above with the
exact sequence
0 -> Ox(-Ci - C2) - Ox - OCl+Ci -» 0,
we find that h}(Od+ct) < Q = 9(Ci),i = 1,2. From the exact sequence
0 -» ОсЛ-Сз) -* OCl+c2 -* Qc, -* 0,
and the fact that H°(OCl+c2) -> H°(OC2) is surjective we have
h}(OCl(-C2)) = 0. But degOCl(-C2) = -d < 0, with equality holding
only if Ocx (-Ca) is the trivial line bundle and in this case h°(Od (-C2)) =
1. Thus, by Riemann-Roch again
We again contradict Л1(Ос1(—С2)) = 0. It follows that there is just one
curve d, and that D = Ci is an irreducible curve with #х • D < — 1 and
Completion of the proof of Theorem 20. Choose a curve D as in
Claim 2. By Claim 1, if X is not ruled, either D is a section of the fibration
7Г or q = 1 and D —> С is unramified. First suppose that D is a section
of 7Г, so that D • / = 1 for a general fiber / of 7r. In all cases, by the
Riemann-Roch theorem,
h°(D) + h\D) > 1 - q + I (D2 - D • KX)
= -q + pa(D) -(D
Since ti*(D) = h°(Kx - D) and /i°(Kx) = 0, h?(D) = 0. Thus, |D| has
dimension at least 1, and has no fixed curves since it contains the reduced
irreducible element D. Thus, the base locus of \D\ is finite. On the other
hand, choose a general fiber / of тг such that / П D is not a base point for
|D|. Then the image of \D\ defines a morphism from / to P1 of degree 1,
so that / is rational and X is ruled.
In the remaining case, q = 1, pg = 0, and so x(Ox) = 0. Take X to be
the fiber product X xc D. Then /: X —* X is an unramified covering of X
of degree d, say, with a section D corresponding to the morphism D —> X.
Since f*Kx = Kx,
KkD = f*Kx -D = Kx-f.D = Kx-D<-l,

10. Classification of Algebraic Surfaces and of Stable Bundles 293
and thus Kx is not nef. Moreover, X is still minimal: an exceptional curve
EonX must lie in a fiber of X —* D, since it is rational, and thus f\E: E —*
f(E) is an isomorphism. Moreover, f(EJ = E2 = -1. It follows that f(E)
is a smooth rational curve on X, necessarily exceptional, contradicting the
minimality of X. Since X is a minimal surface such that Kx is not nef, we
have seen that pg(X) = 0 and so h?(D) = h°(Kx -D) = 0. On the other
hand, it is easily checked from Noether's formula that x(@x) — dx{Ox) —
0 and the Riemann-Roch argument given above in the case where D is a
section implies that h°(Ox(D)) > 2. It then follows by the above arguments
that the general fiber of X —* D is a smooth rational curve, and thus the
same is true for the fiber of X —> C. Hence X is ruled. D
We have now dealt with the case where Kx is not nef. We may thus
assume that Kx is nef. In particular it follows that Kx > 0. Let us restate
the abundance theorem in a more precise form:
Theorem 21. Let X be a minimal algebraic surface with Kx nef. Then
K\ > 0. Moreover:
(i) k(X) — 0 if and only if Kx is numerically equivalent to 0. In this
case X is a if 3 surface, an abeiian surface, an Enriques surface, or a
byperelliptic surface.
(ii) k(X) = 1 if and only ifK\ = 0 but Kx is not numerically equivalent
to 0. In this case X is an eUiptic surface.
(iii) k(X) = 2, i.e., X is of general type, if and only if Kx > 0.
In all cases, either Рл{Х) or P6(X) ф 0.
Proof. We consider the various possibilities, starting with (iii):
Case I: Kx is nef and K\ > 0.
In this case, X is of general type by Proposition 10. We show that Рл{Х)
or Рб(Х) is nonzero. If рд(Х) ф 0, we are done. If pg(X) = 0, we claim
that in fact Р^{Х) Ф 0. First we use the following slightly more general
lemma:
Lemma 22. Let X be a surface such that Kx is nefandpg(X) = 0. Then
q(X) < 1. IfKx isnefandpg(X) = 1, then q(X) < 2.
Proof. By Noether's formula,
c\ + c2 = K\ + 2 - 4q + 2Pg + ft1'1 = 12A - q + pg)
and so 8д + Л1Д +K% = 10 + 10p9. As Kx is nef, K\ > 0. Thus, if pg = 0,
then q < 1, and if pg = 1, then q < 2. ?

294 10. Classification of Algebraic Surfaces and of Stable Bundles
Now by Proposition 10, if Kx is nef and big, then P2{X) = K% +x{Ox).
If moreover pg(X) = 0, then q < 1, \(OX) > 0, and thus P2 > K\ > 1. In
particular, P2 ф 0 in this case.
Case II: Kx is numerically equivalent to 0 and q(X) = 0.
By Castelnuovo's theorem, Р2(Х) ф 0. Thus, either Kx is trivial or Kx
is not trivial but 2Kx is trivial. In the first case X is а КЪ surface. In the
second case there is an unramified double cover X for which Kx is trivial.
Moreover, съ(Х) ¦= 12. We shall shortly show that X is а КЪ surface, so
that by definition X is an Enriques surface.
Case III: Kx is numerically equivalent to 0 and q{X) > 0.
In this case, consider the Albanese map a: X —> Alb A". If the image is
not a curve, then pg(X) ф 0. It follows that Kx is trivial and that pg = 1.
By Lemma 22, q(X) < 2, and hence q = 2 since dim Alb X > 2. Thus,
Alb X is a complex torus of dimension 2. Since the pullback of a nonzero
2-form on AlbA' is nonzero, a is unramified. It follows that X is a complex
torus and that a is an isomorphism.
Otherwise, the Albanese map is a fibration X —* C. If / is a general
fiber, then 2g(f) - 2 = /2 4- Kx • / = 0. Thus, / is an elliptic curve and X
is an elliptic surface. The elliptic surfaces X for which Kx is numerically
trivial have been classified in Exercises 7 and 8 of Chapter 7. In all cases
either 4Kx or 6Kx is trivial, and if q(X) > 0, then X is a hyperelliptic
surface or a complex torus.
Finally, we have seen that if Kx is trivial, then X is а КЗ surface or a
complex torus. If X is a complex torus, then bz (X) — 6. If Y is a quotient of
X by a finite group, with a: X —*Y the quotient map, then ^(У) < ^(X),
because the map a*a* is multiplication by the degree of the cover and thus
the kernel of a* is torsion. Hence the case q = pg = 0, P2 Ф 0 above
corresponds to Enriques surfaces (since in this case 62 (Y) = 12 > 6).
Case IV: Kx is not numerically equivalent to 0, Kx = 0 and q(X) = 0.
In this case we shall prove that X is elliptic. We begin with two lemmas:
Lemma 23. Let X be a surface such that Kx is nef and K\ is not
numerically equivalent to 0. Then for all n > 1, h°(-nKx) = 0 and for
all n > 2, h2(nKx) = 0. More generally, suppose that D is a divisor on X
numerically equivalent to nKx + D', where D' is effective. If n > 2 or if
n = 1 and D' is not 0, then h2(D) = 0.
Proof. Since Kx is not numerically trivial, there must exist an ample
divisor H such that Kx • H ф 0, since every divisor is a difference of ample
divisors. Moreover, since Kx is nef, we must have Kx • H > 0. Thus,
H ¦ (-nKx) < 0 for every n > 1 and so -nKx is not effective. If n > 2,
h2(nKx) = Л°(A - n)Kx) and A - n)Kx is a negative multiple of KX-

10. Classification of Algebraic Surfaces and of Stable Bundles 295
Thus, A - n)Kx cannot be effective for n > 1 and so h2(nKx) = 0. The
case of D numerically equivalent to nKx + ?У is similar. ?
Lemma 24. Let X be a surface such that Kx is nef, Kx is not numerically
equivalent to 0, and Kx = 0. If Pn(X) > 2 for some n, then X is elliptic.
Proof. Choose an n such that dim|n.ft'x| > 1. Quite generally, if
Ei <hCi e IniiTxl, then Kx • Q > 0 since Kx is nef and Kx ¦ (Ei a<C<) =
nKx = 0. Thus, Kx ¦ Ci = 0 for all i. Since Kx is not numerically equiv-
equivalent to 0, it follows from the Hodge index theorem that Cf < 0 for all
t, with Cf = 0 if and only if Cj is numerically equivalent to a rational
multiple of Kx-
Let E be the fixed component of \nKx\- Thus, we can write nKx =
D + E, where |?>| has no fixed components and D is effective. It follows
that D2 > 0. On the other hand, since Kx ¦ D = 0, D2 < 0, so that
D2 = 0 and D is numerically a rational multiple of Kx- In particular \D\
has no base points. Let <p: X —> P^ be the morphism defined by \D\. Since
D2 = 0, the image of tp is a curve. Taking the Stein factorization, there is
a morphism it: X -* С such that all fibers f of n satisfy: / • Kx = 0. It
follows that a smooth fiber / is elliptic, and thus that X is elliptic. D
Returning to the study of Case IV, first suppose that pg(X) ф 0. Then
for all n > 0, h°(nKx) + h2(nKx) > 1 + p9 > 2. On the other hand, for
n > 2, h?(nKx) = 0 by Lemma 23. It follows that h°(nKx) > 2, and so,
by Lemma 24, X is elliptic.
Thus, we can assume that pg(X) = 0 and either that X is elliptic or that
Pn(X) < 1 for every n > 2. Since h2(nKx) = 0 for n > 2, h°(nKx) > 1,
and thus in fact Pn(X) = 1 for all n > 2. Let Е4а*С, е \2KX\ and
let ?<Ь»С< 6 \3KX\, where аиЬг > 0. Then 3^а,С( = 2^Ь^ in
leA'xl, and so there exists c^ > 0 with at = 2c< and b, = 3ci. Hence
Ei&«C» - Et0»^» = Ei6"^' ^ an effective curve in Kx, contradicting
the fact that h°(Kx) = 0. So X is elliptic, and moreover we have shown
directly (without appealing to Castelnuovo's theorem) that РгС-^О = 1>
and in particular that Рг(Х) ф 0.
Case V: Kx is not numerically equivalent to 0, K\ — 0 and q(X) > 0.
We shall show in this case that X is elliptic as well. First, if Pn > 2 for
some n, then X is elliptic by Lemma 24. Thus, we may as well assume that
Pn < 1 for all n. In particular pg < 1. Moreover, we have Л2(пА"х) = 0
for all n > 2 and thus h°(nKx) > xi^x)- Thus, we can assume that
x(Ox) < 1. Thus, if p9 = 1, then q > 2, and in fact, by Lemma, 22 q = 2.
In this case we have:
Lemma 25. If X is a surface with Kx nef, pg{X) = 1, q(X) = 2, and
Kx = 0 but Kx is not numerically equivalent to 0, then X is elliptic.

296 10. Classification of Algebraic Surfaces and of Stable Bundles
Proof. As in the proof of Lemma 24, if Kx = J^i °<С<, then Kx • Cj = о
and the span of the Ci is negative semidefinite with a radical spanned over
Q by ifx. In particular, Cf < 0, and either Cf = 0 and C< • Cj = 0 for
i ф j or Cf < 0, Ci is smooth rational, and all of the curves G, which lie
on the same connected component of Y^i °<С» as C< are smooth rational
curves which span a negative semidefinite lattice whose radical is a rational
multiple of Kx ¦
Consider the Albanese map a: X —t AlbX. If one of the curves d is
smooth rational, then a(Ci) = 0 and in fact the entire component through
Ci must be mapped to a point. By the Hodge index theorem, the image
of q is a curve. Hence the Albanese map is a fibration тг: X —> С (over a
curve of genus 2) and, if / is a general fiber, then Kx • / = 0. Thus, / is
an elliptic curve and X is elliptic.
We may therefore assume that all of the curves Ci are numerically equiv-
equivalent to a rational multiple of Kx and that Cf = 0. If <*(Cj) is a point
for some i, for example if some Ci is singular, then the above argument
shows that the Albanese map is an elliptic fibration. Thus, all of the C, are
elliptic curves and a(Ci) is a curve for every i. Since every morphism from
one complex torus to another is a homomorphism followed by a translation,
we can assume that a(d) = E is a smooth elliptic curve passing through
the origin of AlbX. There is thus an induced morphism X —» AlbX/E for
which the image of Cj is a point. Let тг: X —> С be the Stein factorization.
Then тг is a fibration, and there exists a fiber containing d. Since Cf = 0,
the complete fiber through Ci is of the form ad for some positive integer
a. It follows that, if / is a general fiber of тг, then Kx • / = 0 and so / is
elliptic. Once again, we see that X is elliptic. D
The remaining case is where Kx is not numerically equivalent to 0,
K% = 0, q(X) > 0, and pg(X) = 0. If pg(X) = 0, then we have q < 1 by
Lemma 22, and so (since we are assuming q > 0) q = 1. In this case, the
most difficult case in the classification, we have the following:
Theorem 26. If X is a minimal surface with pg{X) = 0, q(X) = 1, and
Kx = 0, then X is either ruled over an elliptic curve or X is elliptic.
Proof. Since c\ = \(Ox) — 0, сг(Х) = 0 as well, and thus, as h{X) = 2,
62 (X) = 2. We may assume that Kx is not numerically equivalent to
0, since hyperelliptic surfaces were shown to be elliptic previously. Let
тг: X —* E be the Albanese fibration, where E is a smooth elliptic curve.
Since Ьг{Х) = 2, all fibers of тг are irreducible. Using the last statement of
Exercise 5 in Chapter 7, if тг is the Albanese map, and if / is a general fiber
of тг with g(f) = g, then c2(X) > Bg - 2)\(E) = 0, with equality only if
all fibers have smooth reduction of genus g. We may assume that g > 1,
for otherwise X is either ruled over an elliptic base or is elliptic. If nF is a
multiple fiber, then 2g-2 = Kxf = n(Kx • F) = nBg - 2). Thus, n = 1

10. Classification of Algebraic Surfaces and of Stable Bundles 297
and 7Г is a smooth morphism. The main problem now is that, unlike the
previous cases, the elliptic fibration is not in general given by the Albanese
fibration, and we shall have to construct it by other means. There are two
approaches to finding the desired elliptic fibration, an analytic approach
and an algebraic approach. We shall sketch both.
Analytic Proof. We have found a smooth morphism тг: X —> E, whose
fibers are curves of genus g > 1. There is an associated period map: Let Sjg
be the Siegel upper half space of symmetric complex g x g matrices Z =
X+iY such that Y is positive definite. It is known that f)g is biholomorphic
to a bounded domain in С9(э+1)/2. Let Г9 be the group SpBg, Z). Then Tg
acts properly discontinuously on Sjg. To a smooth curve С of genus g, its
period matrix is well defined mod Г9, and thus defines a point of Tg\Sjg.
Given a smooth morphism X —* E, all of whose fibers have genus g, there is
an associated holomorphic map p: E —* Tg\Sjg, the period map. Moreover,
p is locally liftable in the following sense: if ¦ф: С —* E is the universal cover,
with covering group Л = Z©Z, then there is a holomorphic map p: C-»Sj
and a homomorphism p: Л —» Г9, such that p{z + A) = p(X) • p(z). Thus,
there is an induced map С/Л = E —> Tg\Sjg, which is just p. Moreover,
there is a rank g vector bundle V over Sjg, which is holomorphically trivial,
together with a natural Г9-action. Thus, Л acts on p*V, and the induced
bundle p* V/Л is equal to the rank g vector bundle BPn*ilx,E.
The special feature of our situation is the following: the universal cover
of E is С, and there are no nonconstant holomorphic functions from С into
a bounded domain in CN. Thus, p is constant. If Imp = {x}, it follows
that Imp is contained in the isotropy subgroup of x in Г9. But as Г9 acts
properly discontinuously, the isotropy group of a point is finite. Thus, there
is a subgroup A' of Л of finite index such that p(\') = {Id}. Let E' —> E
be the induced unramified cover and let тг': X' —» E' be the pulled-back
family. It follows that Л°(тг')»П^, ,Е, is the trivial rank g vector bundle.
At this point, there are several ways to proceed. One way is to check
that X' is a surface such that Kx> is nef, K\, = 0, and Kx> is not numer-
numerically trivial, since Kx> is pulled back from X by the unramified covering
map. But P9(X') Ф 0, since Kx> = Ux'/E' ® {ж'УКе' = ?lX'/E" since
E' is an elliptic curve and so Ke> is trivial. So we can apply the previous
classification results to conclude that X' is elliptic, with к(Х') = 1, and
that X is the quotient of X' by a group of holomorphic covering transfor-
transformations. It is easy to check that the elliptic fibration defined by \nKx> I
for n large is preserved by the covering transformations, and thus induces
an elliptic structure on X. Another way to proceed is to use the fact that
Д°(тг/)»П]|С,/В, is a trivial rank g vector bundle, so that H°(ulx,/B,) = C9.
There is thus a morphism X' —» P9 and its image is a curve С of genus
g, since С1(П^7В,J = Ы^х'/В' ® (*T#s')a = K%. = 0. It follows that
there is a morphism X' —* E у. С which is in fact an isomorphism, and
X is a quotient of E' x C. Keeping track of the possible automorphisms

298 10. Classification of Algebraic Surfaces and of Stable Bundles
shows that X is an elliptic surface over C/G for some subgroup G of the
group of automorphisms of C. Thus, X is again elliptic.
Algebraic Proof. We begin with a series of lemmas:
Lemma 27. Let D be a divisor numerically equivalent to nKx + F, where
F is effective and F2 = Kx ¦ F = 0. Suppose that either n > 2 or n = 1
and F is not the 0 divisor. Then h°(D) = hl(D).
Proof. By Lemma 23, we know that h?(D) = 0. By assumption, x(Ox) =
0 and D2 = D Kx = 0 as well. Thus, by Riemann-Roch, h°(D) - h1 (D) =
о. а
Lemma 28. For alln>2, there exists a divisor D numerically equivalent
to пКх such that h°(D) ф 0.
Proof. We shall show that there exists a line bundle A of degree 0 on E
such that h°(Ox(nKx) ® тг*А) ф 0. By Lemma 27, it suffices to find a A
such that ^(Ox^ifxJSw'A) ф 0. To do so, let us fit together all of the line
bundles Ox(nKx)®n*\- choose a point t€E, and let V be the line bundle
0Exe(A - \{t} x E)) on E x E. Thus, on each slice E x {«}, V\E x {«}
is essentially just Oe{s — t), and as 8 ranges over E these line bundles
range over all of the line bundles on E of degree 0. Let tti : X x E —> X
and 7Г2: X x E —* E be the first and second projections. There is also
the induced map p: X x E —* Еу. Е denned by (тг о щ,тг2). Consider the
line bundle С = irlOx(nKx) ®p*V on X x E. For each fixed s e E, the
restriction of С to X x {«} can be identified with Ox(nKx)<S>n*OE(s -1),
and as 8 runs over E the line bundles Oe(s — t) run over all of the line
bundles of degree 0 on E.
Given n > 2, we may assume that Pn{X) = 0, for otherwise we can take
A to be the trivial line bundle. In particular, by the semicontinuity theorem
Я°7Г2*? is a torsion free sheaf on E which is zero in a neighborhood of t,
and thus Д°7Г2»? = 0. By Lemma 23, H2(Ox(nKx) ® тг*А) = 0 for all A of
degree 0 and hence Д27г2*-С = 0. Finally, if A is trivial, then Hx{nKx) = 0
by Lemma 28, since hl{nKx) = Pn(X) = 0, and so В}ъ2*С is a torsion
sheaf. Moreover, if Hl{Ox{nKx) ® тг*А) = 0 for all A of degree 0, then
}C = 0. Thus, it suffices to show that Д'тгг*^ Ф 0.
By applying the Leray spectral sequence to тг2 and C, we see that
X(C) = ?(-l)»*«W(?; Д»1га.Г) = -h°(EtRln2.?),
since Д9тг2»? = 0, q ф 1, and R1^*^ is a skyscraper sheaf. Thus, it
suffices to show that x(?) Ф 0. But we can calculate x(?) by using the

10. Classification of Algebraic Surfaces and of Stable Bundles 299
Riemann-Roch theorem on the threefold X x E:
X(C) = f ch(?) • Todd(X x E)
JXxE
= f
JXx
where we have used the multiplicativity of the Todd genus to conclude that
Todd(X x E) = ttJ Todd(A"Or? Todd(S).
Since TE is trivial, Todd(?) = 1, and since c2(X) = 0, Todd(X) = 1 -
Kx/2. Now С = пЦпКх) ®р*0нхя(Д - ({t} x E)). Moreover, K\ = 0
and (Д - ({t} x Я)J = Д2 - 2Д • ({*} x E) = -2 because Д2 = 0 (apply
adjunction to Д с Е x 2?). Thus,
Г3 = Зтг*(пКх) -P*(A - {*} x ^J = -6(n/fx • ir*(t))
= -6(nA"x • /) = -6nBg - 2) = -12n(g - 1).
Likewise,
C2 ¦ jt'Kx = -2(Kx • ir*(t)) = -2{KX ¦ f) = -4@ - 1),
and putting this together we find that
X(C) = -2n(g - 1) + (g - 1) = -Bn - l)(g - 1).
It follows that the length of B}it2*C is Bn— l)(g-1) Ф 0, and so there exists
°
( )(
a A on E of degree 0 such that h°(Ox(nKx) ® тг*А) ^ 0, as desired. D
By Lemma 28, there exists an effective divisor D numerically equivalent
to nKx ¦ Suppose that D = ?V OiCj where the d are irreducible curves and
ai > 0. As in the proof of Lemma 25, since Kx is nef, the connected compo-
components of D are all numerically equivalent up to a positive rational number,
and the irreducible curves in a connected component span a negative semi-
definite lattice. Thus, either Cf < 0 or Ci is a connected component of
the support of D and Cf = 0. If Cf < 0, then d is smooth rational and
thus lies in a fiber of the morphism тг: X —» E. Using the first part of the
proof, all of the fibers of тг are irreducible curves of arithmetic genus g, so
this case is impossible. Thus, Q is a connected component of the fiber for
every i and Cf = 0. Hence Cj is an irreducible curve of arithmetic genus 1,
numerically equivalent to a positive rational multiple of Kx- (In fact, Ci
must be smooth, since otherwise its normalization would be rational, and
thus Ci would be contained in a fiber of the Albanese map, which as we
tiave seen is impossible.)
Consider the set of all irreducible curves D on X such that D has
irithmetic genus 1 and D2 = 0, or equivalently such that D ¦ Kx = 0.

300 10. Classification of Algebraic Surfaces and of Stable Bundles
Such a curve must be numerically equivalent to a positive rational mul-
multiple of Kx- Moreover, by adjunction Ox(Kx + D)\D = OD. Hence
Ox{nKx + nD)\D = Od for all positive integers n. Suppose that Dx
and D2 are two curves on X of arithmetic genus 1 with D? = 0, i = 1,2.
Then both D\ and D2 are numerically equivalent to a rational multiple of
Kx and in particular D\- D2 — 0. Thus, either D\ and D2 are disjoint or
Z?i = D2. Suppose that we can find D\ ф D2. Consider the exact sequence
0 -> OxBKx +Di+ D2) -* Ox{2Kx + 2DX + 2D2) -* ODl Ф ODt -» 0.
Since H2BKX +?>i + D2) = 0, it follows that the map H1(OxBKx +
2DX + 2D2)) -* Hl{ODl) 0 Hl{ODi) = С 0 С is surjective. Moreover,
h\OxBKx + 2DX + 2D2)) = h°(OxBKx + 2?>i + 2D2)), and so \2KX +
2Di + 2D2\ contains a pencil. Arguing as in the proof of Lemma 24, it
follows that the moving part of the pencil defines a morphism X —* В
whose general fiber is an elliptic curve, and thus that X is elliptic.
Thus, we are reduced to showing that there exist two disjoint curves
D\,D2onX with pa(Di) = 1 and Df = 0, i - 1,2. We know that one such
curve Di exists. For n > 2, consider the linear system \nKx + nDi\. By
adjunction Kx ® Ox(D\)\Di is the trivial line bundle, and thus the same
is true for Ox(nKx +nDi)\Di. Consider the exact sequence
0 - Ox(nKx + (n- 1)D,) - Ox(nKx + nDi) -» ODl -» 0.
Since #2(n#x + (n-l)?>i) = 0 for all n > 2 by Lemma 23 and Hl(ODl) =
C, hl(nKx + nDi) > 1. By Lemma 27, \nKx + nDi\ Ф 0. We have seen
that, if С is an irreducible component of an element of \nKx +nDi\, then
С is a curve of arithmetic genus 1 with С2 = С ¦ Kx =¦ С • D\ = 0. If
there is no such component С which is disjoint from D\, then, for every
n, nKx + nDi = mD\ for some positive integer m. Applying the same
argument to n +1, (n 4- l)Kx + (n + 1)D\ is linearly equivalent to m'D\
for some integer m', and thus Kx is linearly equivalent to (m' - m - 1)D\.
Since Kx ¦ H > 0 for every ample H, we must have m' — m — 1 > 0. In this
case Kx is effective, and so pg(X) ф 0. But we have already dealt with the
case pg (X) > 0 just after the proof of Lemma 24. Thus, in case pg (X) = 0,
we have found two disjoint curves as desired. D
We have showed that, if Kx is nef, Kx = 0, pg{X) = 0, and q(X) = 1,X
is elliptic and then all fibers of the elliptic fibration are smooth or multiple
curves with smooth reduction. Such surfaces do exist and are given as
follows: by the canonical bundle formula,
Kx = ir*(L ® Kc) ® Ox
where degL = 0 and the mi are the multiple fibers of multiplicity пц. It
follows that pg(X) = ft°(C; L ® Kc). Thus, if g(C) > 1, then pg(X) > 1.
If g(C) = 1, then as L has order 4 or 6 we see that Р\(Х) от Рц{Х) is

10. Classification of Algebraic Surfaces and of Stable Bundles 301
nonzero. However, if X has a multiple fiber, then P2(X) Ф 0 in this case. If
X does not have a multiple fiber, then Kx is numerically equivalent to 0. If
g(C) = 0, then L is necessarily trivial. To find an example of such surfaces,
let С be a curve of genus g > 2 such that there exists an automorphism
(p of order d with C/((p) = P1 (for example, if d = 2, this just says that
С is a hyperelliptic curve). Let E be an elliptic curve and ? a point of
order d on E. Then Z/dZ acts diagonally on С x Е, where the action on
the first factor is via (p and on the second by addition by ?. The quotient
(C x E)/(Z/dZ) = X fibers over С/{ф) = P1, and the general fiber is a
smooth elliptic curve. However, there are multiple fibers at the fixed points
of (p, with multiplicity equal to the ramification index. The quotient surface
X also fibers over E/(?) = E', where E' is an elliptic curve, and we leave it
as an exercise to show that the morphism X —> E' is the Albanese fibration,
with all fibers isomorphic to C.
To conclude the proof of Theorem 21, we must check that either P4(X) or
Рб(Х) is nonzero in case pg(X) = 0 and q(X) = 1. By the above remarks,
we can assume that X is an elliptic surface over P1 whose associated line
bundle L is trivial. Thus, Kx is numerically equivalent to rf, where
By assumption г > 0. The case г = 0 has been discussed in Exercise 7 in
Chapter 7, and corresponds to multiplicities
B,2,2,2), B,4,4), B,3,6), C,3,3).
In all cases P*{X) or Pe(X) is 1. Again by Exercise 7 in Chapter 7, there
are at least three multiple fibers, and the following cases are excluded:
B,2,m),m > 2; B,3,4), B,3,5). Now suppose that there are г multiple
fibers Fu with multiplicity ггц > 2. Thus, Kx = (r - 2)/ - ?<#> with
г > 3. If г > 4, then 2KX = 2(r - 2)/ - 2?\ Ft. Since г > 4, 2(r - 2) > r,
and so 2(r - 2) = г + s with s > 0. Thus,
2KX =af + ?(/ - 2Ft) = sf
t
which is effective as т4 > 2. If г = 3, the possibilities are 3 < mi < тг <
m3! mj = 2, 4 < тг < гпз or mi = 2, тг = 3, тз > 6. In the first case
= 3/ - 3Fi - 3F2 - З^з = (mi - 3)Ft + (m2 - 3)F2 + (m3 - 3)F3,
which is effective since m^ > 3. In the second case,
AKX = 4/ - 4Fi - 4F2 - 4F3 = 2/ - 2/ + (тг - 4)F2 + (m3 - 4)F3
= (m2 - 4)F2 + (m3 - 4)F3,
which is effective since rrc2 > 4 and m3 > 4. In the last case,
6KX = 6/ - 6Fi - 6F2 - 6F3 = 6/ - 3/ - 2/ - 6F3 = (m3 - 6)F3,

302 10. Classification of Algebraic Surfaces and of Stable Bundles
which is again effective as m3 > 6. Thus, in all cases, either P*{X) от Рв(Х)
is nonzero. ?
The Castelnuovo-dePranchis theorem
We recall the statement of the Castelnuovo-deFranchis theorem:
Theorem 29. Let X be an algebraic surface with к(Х) > 0. Then c2(X) >
0, with equality holding if and only ifX is a minimal elliptic surface whose
only singular fibers are multiple fibers with smooth reduction or X is an
abelian surface.
Proof. We may clearly assume that X is minimal, since c2 can only in-
increase under blowing up. The theorem is clear if q(X) = 0, since then
c2(X) = 2 + ЬгРО > 0. Next suppose that the Albanese morphism тг has
image a curve C. Then g(C) = q > 1 and the general fiber of n has genus
g > 1 since X is not ruled. By Exercise 5 in Chapter 7,
c2(X) = x(X) > B - 2g)B - 2q) = Bg - 2)Bq - 2) > 0.
Moreover, equality can hold only if either g = 1 and all fibers of тг are
multiple fibers with smooth reduction (by Corollary 17 in Chapter 7) or
q = 1 and C2(X) = 0. In this case 4 = 4q = 2 + 2pg + ft1-1. Since ft1'1 f 0,
pg(X) = 0 and x(Ox) = 0. Thus, by Noether's formula if|=0. Since X
is minimal, X is elliptic by Theorem 26. Again by Corollary 17 in Chapter
7, the only singular fibers in the elliptic fibration are multiple fibers with
smooth reduction. Thus, we have proved the theorem in case the Albanese
image of X is a curve, and the above proof also works in case there is a
morphism from X to a curve С of genus at least 2.
Next suppose that there exist two linearly independent holomorphic 1-
forms i/j.ijjonX such that т/i Л щ = 0. We claim that in this case there
is a morphism from X to a smooth curve С of genus at least 2, and hence
that the theorem is true in this case as well. More precisely, we have:
Lemma 30. Suppose that X is a smooth surface and that т/i, щ are two
iinearly independent holomorphic 1-forms on X such that щАщ =0. Then
there exists a fibration тг: X —* С, where С is a smooth curve of genus
д > 2, and two holomorphic 1-forms V'ltVb on С such that щ = тг*^<,
i = 1,2.
Proof. Let C(X) be the function field of X, so that C(X) is a finitely
generated field extension on С of transcendence degree 2. Let zu z2 ? C(X)
be two algebraically independent meromorphic functions on X. Then the
module of Kahler differentials Пс(х)/с 'ш а vector space of dimension 2
over C(X), with basis dz\,dz2- Moreover, two elements 0:1,0:2 €

10. Classification of Algebraic Surfaces and of Stable Bundles 303
are linearly independent over C(X) if and only if Qi Л q2 Ф 0 in &ctx)/c-
Now the 1-forms щ induce elements of fic(X)/c> ^V restriction to the generic
point of X, which we will continue to denote by rji, and clearly т/i Л % = 0
in fic(x)/c- Thus, there exists / 6 C(X),f Ф 0 such that щ = frfc. A
holomorphic 1-form on X is closed, and so
0 = drji — df Л %•
Thus, щ = gdf for some g e C(X). Since % is closed as well,
0 = din = dg A df,
and thus df and dg are linearly dependent in ^.,Хус. It follows that
^c(/ s)/c nas dimension 1 over C(/, g), and thus that the transcendence
degree of C(f,g) over С is 1. The inclusion C(f,g) С С(Х) defines a ra-
rational map (p: X --+ D, where D is the smooth curve corresponding to
the function field С (/,</). Let X —> D be a blowup for which y> becomes a
morphism, and let тг: X —» С be the Stein factorization of X —» ?>. The
1-forms fgdf and gdf are meromorphic 1-forms on D which pull back to
meromorphic 1-forms ^i > Vb on C, and then further under тг* to the forms
Vi. %• Since T/i and % are holomorphic and linearly independent, it is easy
to see that Vi and $2 are two linearly independent holomorphic 1-forms on
C. But then necessarily g(C) > 2. In particular, every exceptional curve
on X from the resolution of indeterminacy must map to a point of C, and
so 7Г factors through a morphism тг: X —» С. By construction rji = n*tpi on
a dense open subset of X, and hence everywhere, as claimed. D
Thus, we may assume that, whenever щ and % are two linearly inde-
independent holomorphic 1-forms on X, т/i Л % ф 0. In this case, we claim
that pg > 2q - 3. To see this, let V = Я°(ПХ), with dimV = q and
let W = H°(Q2X) with dimW = pg. Inside Д2^. there is the cone С of
elements of the form v\ Л V2, where v\ and tvj are two linearly indepen-
independent vectors inside V, and clearly С is the affine cone over the Grass-
mannian GB,q) of 2-dimensional subspaces of V. Thus, С has dimen-
dimension dimGB,q) + 1 = 2(q - 2) + 1 = 2q - 3. There is the natural map
Д2 V —* W. Let К be its kernel and let / be the image. By hypothesis,
К П С = {0}, so that dim К + dim С < dam/\2V. On the other hand,
dim К + dim / = dim Д2 V. Hence
2
dim К + dim С < dim Д V = dim К + dim / < dim К + pg.
It follows that dim С = 2q - 3 < pg.
Thus, if X is any surface for which сг(Х) < О,
c2(X) = 2 - 4q + 2pg + ft1-1 > 2 - 4q + 2Bq - 3) + ft1'1 = ft1'1 - 4 > -3.
On the other hand, suppose that X is a surface such that сг(Х) < 0. Then
in particular q(X) > 0, so that H\(X; Z) has rank at least 1. It follows that,

304 10. Classification of Algebraic Surfaces and of Stable Bundles
for every n > 0, there exists a surjective homomorphism ir\{X) —» Z/nZ,
and thus an unramified cover X of X of degree n. An elementary Euler
characteristic argument shows that C2(X) = nc2(X) < -n. Applying the
above argument to X, we see that — n > сг(Х) > -3, which is absurd as
soon as n > 4. Hence, we must have сг(Х) > О.
Finally, we consider the case where ot(x) = 0. If K% — 0 as well, then
k(X) = 0 or 1, and by the classification of such surfaces either X is an
abelian surface or X is an elliptic surface with all fibers smooth or multiple
with smooth reduction. Otherwise, K\ > 0, and so by Noether's formula
1 + pg > q, or in other words pg > q. Next, we claim the following:
Lemma 31. Suppose that X is a smooth surface and that there does not
exist a fibration n: X —* C, where С is a smooth curve of genus д > 2.
Then h^iX) >2q-l.
Proof that Lemma 31 implies the Castelnuovo-dePranchis theo-
theorem. If there exists a fibration к: X —> С, where С is a smooth curve of
genus д > 2, then either X is elliptic or the fibers / of ж have genus at
least 2. We have already handled the elliptic case above, and the other case
follows from the inequality сг > Bg(f) — 2)B<j(C) - 2). So we can assume
that pg > 2q - 3 and that Л1-1 > 2q - 1. It follows that
+ ft1'1 > 2q - 5.
If q = 0 or q > 3, then automatically ог{Х) > 0. If q = 1, then pg > 1
and ft1-1 > 1. Thus, сг(Х) >2-4 + 2 + 1>1. Ifq = 2, then pg > 2 and
Л1'1 > 3, so that C2PO >2-8 + 4 + 3>l. In all cases, <b{X) > 0. D
Proof of Lemma 31. Let V\ be the complex vector space (^)
HXfi{X) and let V2 be the conjugate vector space Н°Л(Х). Wedge product
(tyb V2) *-* Vi A % induces a complex linear map Vi ® V2 —* W — Hl<l(X).
Suppose that there exist щ € V\,rfr 6 Vi such that щ Ф 0, i = 1,2, but
that щ A tJ = 0 as an element of Я1-1^) С H*(X;C), in other words
т/1 Л t}2 is exact. It follows that
liAtfeA т/1 Лт/2 = »?i Л % Л ^2 Л »h
is also exact. Set щ Л т/2 = u>, so that
Since a) is a holomorphic 2-form, шЛш = 0 only if u> = 0. In this case, by
Lemma 30, either there is a fibration -it: X —*C, where С is a smooth curve
of genus g > 2, or fJ is a multiple of т/i. In this case щ Ащ is exact. But since
т/1 is closed, locally we can write щ = df for some holomorphic function
/, and after replacing / by / + с for some constant с we can assume that
there is a nonempty open set where / = 21 is part of a coordinate system
on X. Thus, rii = dz\, and clearly rj\ А щ = dz\ A dz\ ф 0.

10. Classification of Algebraic Surfaces and of Stable Bundles 305
Thus, we can assume that, for all vi ф 0, v\ 6 V\, the induced map
Ц-tW defined by V2 *-* the image of v\ ® v2 in W is injective, and
symmetrically in V^- It is now an easy exercise in algebraic topology, left
to the reader, that in this case dim V\ + dim V2 — 1 < dim W. In our case,
since dim Vi = dim V2 = q and dim W = ft1-1, this gives ft1-1 > 1q - 1. D
Classification of threefolds
The classification of threefolds and higher-dimensional algebraic varieties
has been revolutionized by the deep ideas of Mori. We shall just outline
some of the known results concerning threefolds here, referring to [18] and
[75] for more details and references.
Following the order of the theorems in the first section of this chap-
chapter, let us first consider threefolds X with к(Х) = —oo. First, we need a
preliminary definition:
Definition 32. Let X be a variety of dimension n. We say that X is
unirationaJ if there exists a dominant morphism X' —* X, where X' is a
rational variety. Equivalently, X is unirationaJ if there exists a dominant
rational map P" - -+ X. We say that X is uniruled if there exists a dominant
rational map Р'хУ —+ X, where dim У = n — 1.
Unlike the case of curves or surfaces, a unirational threefold need not be
rational. It is easy to see that, if X is uniruled, then к(Х) = —oo. Miyaoka
has proved the converse to this statement in case dim X = 3:
Theorem 33. If X is a. smooth threefold with к(Х) = -oo, then X is
uniruled.
In a curious twist, the proof uses the strong form of Bogomolov's inequal-
inequality proven by Donaldson (Theorem 27 in Chapter 4): if V is a stable bundle
on a surface Y with Cx(V) = C2(V) = 0, then there is a fiat connection on
V and so V is associated to a unitary representation of the fundamental
group 7Г1(У,*).
We turn now to the analogue of Theorem 4, in other words to the question
of what keeps the canonical bundle of a threefold from being nef. The
answer is a theorem of Mori, based on his theory of extremal rays:
Theorem 34. Suppose that X is a smooth three/old such that Kx is not
nef. Then one of the following holds:
(i) — Kx is ample and PicX has rank 1;
(ii) There exists a morphism n: X —> Y, where Y is a surface and the
generic fiber of тг is a P1;

306 10. Classification of Algebraic Surfaces and of Stable Bundles
(Hi) There exists a morpbism n: X —* Y, where У is a curve and the
generic fiber of n is a del Pezzo surface;
(iv) There exists a birational morphism n: X —* Y, where У is a normal
projective variety, whose exceptional set is one of the following five
types:
(a)	Y is smooth and n is the blowup of a smooth curve on Y;
(b)	Y is smooth and ж is the blowup of a smooth point on Y;
(c)	Y is singular and n contracts a smooth divisor on X isomorphic to
P1 x P1, with normal bundle 7rJ0|.i(-l) ® n^Opi(-l);
(d)	Y is singular and n contracts a singular divisor D on X isomorphic
to a singular quadric in P3, with normal bundle the restriction to D
of 0рз(-1);
(e)	Y is singular and n contracts a smooth divisor on X isomorphic to
P2, with normal bundle 0pi(-2).
In this theorem, Cases (i), (ii), (iii) correspond to the rational and ruled
surfaces in Theorem 4. In Case (iv), (a) and (b) are manageable since
the threefold Y is again smooth. In the remaining cases, however, Y is
singular, and we cannot just keep going until Kx becomes nef. However, a
difficult theorem of Mori shows that there is a sequence of contractions and
somewhat more general birational operations beginning with X as above
such that, at the end, the result is a possibly singular threefold X' with
"mild" singularities such that either X' satifies an analogue of Cases (i),
(ii), (iii) above (and in fact X' is uniruled) or Kx> is nef. Here some care
must be taken in defining nef, since X' may be singular and Kx> is not
always a Cartier divisor. However, some multiple of the Weil divisor Kx>
will be Cartier, and then nef can be denned as in the smooth case. Such an
X' will be called minimal or a minimal model of X. The precise definition of
minimal involves understanding the allowable singularities on Xr, and will
not be given here. Unlike the surface case, there is not always a morphism
from a smooth X to a minimal model, which need not be unique.
Finally, we have the generalization of the abundance theorem to dimen-
dimension 3, proved by Kawamata:
Theorem 35. Suppose that X is & minimal (not necessarily smooth)
threefold such that Kx is nef. Then one of the following holds:
(i) k(X) = 0 and Kx is numerically equivalent to 0.
(ii) k(X) = 1, Kx is not numerically equivalent to 0, and K\ is numer-
numerically equivalent to 0. In this case, some multiple of Kx defines a
morpbism to a curve whose general fiber is a surface with к = 0.
(iii) k(X) = 2, K\ is not numerically equivalent to 0, and Kx is numer-
numerically equivalent to 0. In this case, some multiple of Kx defines a
morphism to a surface whose general fiber is an elliptic curve.
(iv) k(X) = 3. In this case, X is of general type and Kx > 0.

10. Classification of Algebraic Surfaces and of Stable Bundles 307
Classification of vector bundles
In this section, our goad is to survey some results concerning the general
structure of the moduli space of vector bundles on surfaces. Due to the
efforts of many mathematicians, substantial progress has been made. On
the other hand, the theory outlined here is not in any sense complete.
Throughout this section, X will denote a smooth algebraic surface and H
an ample divisor on X. With X and H understood, given w € H2(X; Z/2Z)
and p € H*(X;Z), we have defined the moduli space ffl(w,p). We will
discuss the overall dependence on H later, and will write VRh(w, p) when we
wish to make the dependence on H explicit. The first result is an existence
result:
Theorem 36. If p < 0, Wl(w,p) Ф 0.
General existence results of this type, for surfaces and higher dimensional
varieties, go back to Maruyama [85]. In the context of gauge theory, and
for general 4-manifolds and principal bundles with arbitrary (compact)
structure groups, results of this type were established by Taubes [145]. In
case w = 0, there are sharper results. For example, one has the following:
Theorem 37. For aUc > 2pg(X) + 2, there exists an H-stable rank 2
vector bundle V with det V = Ox and C2(V) = с
Theorem 37 was proved by Gieseker [49], under the slightly weaker as-
assumption that с > i(\pg(X)/2] + 1J, and in the form stated above in [37].
All known approaches are deformation-theoretic in nature. Taubes uses the
gluing theory of anti-self-dual connections (which is a C°° version of de-
deformation theory). Gieseker studies the deformation theory of bundles on
the singular scheme which is the union of X blown up at a certain number
of points together with the corresponding number of copies of P2, where
each copy of P2 is glued to the blowup of X by identifying a line in the
P2 to an exceptional divisor in the blowup of X. The approach in [37] is
to consider the problem of smoothing certain torsion free sheaves on X to
vector bundles.
All of the above constructions actually show that there exist good vector
bundles V with the given Chern numbers. If we relax the condition that
the vector bundles are good, can we significantly improve the bounds in
the theorem? Are the bounds of the theorem optimal for good bundles?
Qi Xia, in his Columbia thesis, showed that for simply connected elliptic
surfaces X with no multiple fibers, there are no good vector bundles V
with det V -Ox and c2(V) = с for с < 2pg{X) if pg{X) is even and for
с < 2pg(X) if pg(X) is odd. So the bounds given above are close to the
best possible in this case.

<,o\ otaDie Bundles
There are less explicit existence results in case detV is not trivial, as
well as in the case of higher rank [62], [82].
Once existence results have been established, we can go on to ask about
the overall structure of Ш1(ги,р), in both very general terms and in a re-
refined sense. For example, deformation theory tells us that, once Ш1(го,р) is
nonempty, it has dimension at least -p—3x(©x) at every point. One of the
most surprising results is that of Donaldson on the dimension of Wt(iu,p):
Theorem 38. There exists a constant С depending only on X, H, w such
that, ifp < C, then all components of Ш1(ги,р) are good. In particular,
p) = -p - 3x(<9x) at all points.
Donaldson's theorem is given in [26]. For a related approach to the proof,
see [37]. The general idea is the following: by deformation theory, we know
that Ш1(ги,р) has dimension at least -p - Zx{Ox) at every point. Donald-
Donaldson's method is to estimate the dimension of the space of vector bundles
which are not good, or in other words for which /i2(adV) Ф 0. The gen-
general goal is to show that the set of such bundles has dimension at most
— |p + O(y/\p\). By Serre duality, it is enough to estimate the dimension
of the space of V such that /i2(ad V ® Kx) ф 0. In other words, we must
consider the V such that there is a homomorphism (p: V —> V® Kx whose
trace is 0 (as a section of Kx)- There are two cases to consider. la the
first and elementary case, det ip = 0 as well, and thus ip has rank 1. Setting
Ker<p = Ox(D), there is an exact sequence
0 -> OX(D) -> V -> OX(A - D) ® lz -> 0,
and an inclusion Ох(Д — D)®IZ С V®Kx- Moreover, both V and V®Kx
are stable, so that n(D) < n(V) and n(D) > ц(А) - fi(Kx) - (*(V). The
argument now involves some straightforward manipulation with the Hodge
index theorem.
In case det ys is not 0, the argument is much more subtle, and involves
looking at the double cover f:Y-*X defined by the eigenvalues of \p. In
this case, we can compare V with the pushforward W = /»L of a certain
line bundle L on Y. It turns out that W is a rank 2 subsheaf of V with
torsion free quotient, and thus can be viewed as a generalized elementary
modification of V. The theory of such modifications can then be used to
bound the number of moduli of V for which /i2(ad V ® Kx) Ф 0.
The dimension of the moduli space is asymptotically much larger than
the dimension of its singular \ocias, and deformation theory shows that,
once the moduli space has the correct dimension it is a local complete in-
intersection. Thus, Donaldson's theorem has the following corollary, observed
by Jun Li [81] (and generalized by him to the Gieseker compactification):

10. Classification of Algebraic Surfaces and of Stable Bundles 309
Corollary 39. There exists a constant С depending only on X, Я, w such
that, ifp < C, then all components of f0t(w,p) are reduced, normal local
complete intersections of dimension -p - 3x(C?x) et *U points.
It is not known if there are further restrictions on the singularities of the
moduli space <M(w,p) for p < 0. It seems reasonable to expect that, in
general, the singularities can be quite complicated in high codimension.
In the above discussion, the canonical bundle plays no special role. Thus,
similar methods can prove the following generalization:
Theorem 40. Let M be a fixed line bundle on X. There exists a con-
constant Cm depending only on X,H,w such that, ifp < См, then for every
component Y of OTt(iu,p), there exists a vector bundle V 6 Y such that
h°(adV®M)=0.
Let С be a smooth curve on X. If we apply the above to M = OxiC) ®
Kx, it says that, as long as p is sufficiently negative, where the implied
constant only depends on X,H,C, the general bundle V with u^(V) =
w and Pi(adV) = p satisfies /i°(adV® OX(C) ® Kx) - 0. By Serre
duality, h?(adV® Ox(-C)) = 0. Now the group #2(adV® Ox(-C)) is
the cokernel of the natural map
which measures the differential of the restriction map from the deforma-
deformation theory of V on X to the deformation theory of V\C on C. Thus, if
#2(ad V®OX{-C)) = 0, then JJ^ad V) = 0, and the morphism of moduli
functors is s submersion in an appropriate sense. Roughly speaking, this
says that the deformations of V give a general deformation of V\C. Now if
g(C) > 2, the general deformation of a vector bundle on С is stable (with
similar results if g(C) = 0,1). Thus, we have:
Theorem 41. Let С be a smooth curve on X of genus at least 2. Then
for all p <g. 0 and for every component Y of DJl(w,p), the general bundle
V in Y satisfies: V\C is stable.
We see that the discussion of vector bundles over ruled or elliptic surfaces
is in a sense a special case of this result. In a sense, Theorem 41 is dual
to the restriction theorems of Mehta and Ramanathan and Bogomolov
(Corollary 3 in Chapter 9), which says that, fixing the stable bundle V,
the restriction of V to a smooth curve С е \kH\ is stable as soon as fc is
sufficiently large. In Theorem 41, we fix instead the curve С and conclude
that the restriction of the generic bundle V e SDT(u;,p) to С is stable as
soon as \p\ is sufficiently large.
Donaldson's proof of Theorem 38 does not give any effective information
on the size of the constant C. Recently, new ideas have been brought to

312 10. Classification of Algebraic Surfaces and of Stable Bundles
Some of the above questions seem to be deeply connected to the rela-
relationship between stable bundles and 0-cycles. Let A2(X) denote the Chow
group of 0-cycles on X modulo rational equivalence, and let Z(X) be
the subgroup of 0-cycles of degree 0. There is a natural homomorphism
from Z(X) to кЩХ). Mumford has shown that, if pg(X) > 0, then
Z(X) is quite large [108]; in fact, it behaves somewhat like an infinite-
dimensional algebraic variety. On the other hand, Bloch has conjectured
that, if pg(X) = 0, then the homomorphism from Z(X) to Alb(.Y) is an
isomorphism, and this conjecture has been verified in many cases. Fixing
a base point po € X, there is a function dJl(w,p) —> Z(X) defined by
V ^ c2(V) - с ¦ po,
where ca(V) is computed in the group A2(X) and с = degC2(V). The
induced function $Jl(w,p) —> А1Ъ(Х) is in fact a morphism. Fitting together
the above functions, there is an induced function
P<0
which can be shown to be surjective. What can we say about the fibers
of this map? To some extent, we ask if, at least for p<0, the moduli
space ffl(w,p) behaves more like X or more like the space of 0-cycles on X
modulo rational equivalence.
In all of the above discussion, we have tacitly assumed that the surface
X is minimal. Let us now discuss the effect of blowing up: suppose that
p: X —v X is the blowup of X at a point p, and that E is the exceptional
divisor. For an ample divisor H on X, we will choose the ample divisor
NH — E on X, where the exact value of TV will depend onj?. Let SDt(tw,p)
be the moduli space of Я-stable bundles V on X and let 3Jt(w,p) be the
corresponding moduli space for X, where we identify w € H2(X;Z/2Z)
with p*w € H2{X\7ij2Z). In this case, the map V ^ p*V defines a mor-
morphism SDt(«;,p) —> Ш1(ги,р). There is the following theorem, which can be
proved by using the fact that for p <K 0 the generic V" € Ш1(го, р) satisfies
V\E *O O
Theorem 48. The map p* embeds ffl(w,p) as a Zariski open subset of
ffl(w,p). Foip<?. 0, this subset is also dense, and thus Wl(w,p) andV3l(w,p)
are bir&tional.
Instead of considering vector bundles Von^ such that det V is the
pullback of a line bundle on X, we can look at general vector bundles
V". After twisting by Ox(aE) for the appropriate a e Z, we can assume
that det V = Ox(p*A) or that det V = Ox(p* A - E). We have already
dealt with the first case. In the second case det V • E = 1 and, for p <S 0,
the restriction of V to E is Ов Ф Oe{\)- In this case, Brussee [15] has
shown that V is obtained from a semistable vector bundle V on X by an

10. Classification of Algebraic Surfaces and of Stable Bundles 313
elementary modification of the form
0 -* V -* p*V -* j,OE -* 0,
where j: E —» X is the inclusion. (If det V = A and c2(V) = p, then by the
formulas for elementary modifications det V = Д — E and c2(V) = C2(V).)
Since Hom(p*V,jmOE) = Hom@? Ф Oe, Oe) has dimension 2, there is a
P1 of vector bundles V corresponding to V. Thus, we have the following:
Theorem 49. Let w' =jfw + e, where e is the mod 2 class of [E]. For all
p <?. 0, the moduli space ffl(w',p) is birationally a P'-bundfe over $Jl(w,p).
For more details on the relationship between vector bundles on X and on
X, see Chapter 9 and [38] and [15]. It would be nice to have more precise
results; for example, can we obtain moduli spaces on X from those on X
by a series of "flips," or in other words a reasonably explicit sequence of
blowups and blowdowns (at least when the moduli spaces for X are all
good)?
Beyond these results, are there more concrete instances where the geom-
geometry of X is reflected in the geometry of the moduli space SDt(tw,p)? This
question has been studied for genus 2 fibrations (in other words, surfaces X
such that there exists a fibration ж: X —> С whose general fiber has genus
2) by D. Gomprecht [52]. By using the description of the moduli space of
stable bundles over a genus 2 curve С given by Narasimhan and Ramanan
[111] and a really intricate analysis of the restriction picture discussed in
Chapters 6 and 8, he showed the following:
Theorem 50. Let n: X —» P1 be e genus 2 fibration such that all singular
fibers of ж are irreducible nodal curves. For an integer c, let H be a c-
suit&ble polarization on X. Then for с » 0, there is a Zariski open and
dense subset of the moduli space SDT(O, c) ofH-st&ble rank 2 vector bundles
V with det V = Ox and °2(V) — c which is a finite cover of a Zariski open
subset of a F°+1-bundle over p3c-x(Ox)-ij aD(j ^ne degree of the cover is
22c-x(Ox)-2
A genus 2 fibration X is a double cover of a rational ruled surface, in
other words there is a degree 2 morphism from X to a P1-bundle over P1.
Thus, the moduli space carries a vestige of the structure of the genus 2
fibration.
Exercises
1. Let Vi and Vi be two complex vector spaces, and let F: v\ ® Vi —* W
be a complex linear map such that, for all nonzero v\ G Vi, the map
t;2 i-+ F(y\ ® t^) is injective, and symmetrically for v2 € Vi- Show that
dim W > dimVi + dimVi — 1. (There is an induced continuous map

314 10. Classification of Algebraic Surfaces and of Stable Bundles
F: P(Vi) x P(V2) -> V(W). Calculate F* in cohomology.) Is the same
result true for real vector spaces?
2.	Suppose that X is a surface such that Kx is nef and big. Let X
be the surface obtained by contracting the finitely many — 2-curves
orthogonal to Kx- Show that Kx is the pullback of an ample divisor
on X, and use this to give another proof of the first part of Proposition
10 (for n large).
3.	Let ж: X —> С be an elliptic surface over the base curve C, with
multiple fibers of multiplicities mi,... ,тпг- Let L be the dual of the
line bundle ИхжшОх, with degL = d, and let g(C) = g. For n > 2 and
g > 2, show that
Pn(X) = Bg-2 + d)n + 52 [П{П^ 1}] + 1 ~ 9-
What are the formulas in the remaining cases? In particular, Pn is not
8 linear polynomial in n.
4. Let E be an elliptic curve and let G be a finite subgroup of the transla-
translation group of E. Let С be a curve on which G acts faithfully, such that
C/G ^ P1. Let G act diagonally oaCxB, and let X be the quotient
(C x E)/G. Show that there is a morphism px: X —> P1 whose general
fiber is E, and which has multiple fibers where G does not act freely.
Moreover, there is a smooth morphism pa: X —» E/G whose fibers are
all isomorphic to C. Finally, show that pa: X —» E/G is the Albanese
morphism. (Using the fact that X is an elliptic surface over P1, show
first that q(X) = 1. Thus, there is a finite morphism AlbX -¦ E/G.
Now use the fact that the fibers of X -» Alb X are connected.)

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Index
Abelian surface, see surface, abelian
abundance theorem, 279, 293, 306
adV, 99
adjoint bundle, see ad V
adjunction formula, 13
Albanese map, 118, 289
Albanese variety, 288
algebraic function field, 271, 275
allowable elementary modification,
see elementary modification,
allowable
almost decomposable, 2
ample cone, 19
Andreotti, A., 278
anti-self-dual connection, see
connection, anti-self-dual
arithmetic genus, see genus,
arithmetic
Artin's criterion, 75, 80
Atiyah's theorem on vector bundles,
1,33
Atiyah, M., 1
Atiyah-Singer index theorem, 207
Auslander-Buchsbaum dimension
formula, 51, 199
Barlow surface, see surface, Barlow
Barth, W., 3, 162, 310
base point
assigned, 69
infinitely near, 69
simple, 69
unassigned, 69
basic elliptic surface, see surface,
elliptic, basic
Bauer, S, 240
big divisor, see divisor, big
Bloch, S., 312
blowing down, 66
blowing up, 59
Bogomolov number, 274
Bogomolov's inequality, 3, 22, 107,
149, 163, 245-247, 305
Bogomolov, F., 3, 245
Bogomolov-Miyaoka-Yau inequality,
1,282
Bombieri's theorem, 79, 248
Bombieri, E., 1, 248
botany, 282
Broeius, J.E., 3, 150, 310
Brussee, R., 312
Canonical bundle formula, 176
canonical curve, 134, 139
Castelnuovo's criterion, 1, 64-67
Castelnuovo's theorem, 278
Castelnuovo-deFranchis theorem,
283, 302
Catanese, F., 14
Cayley-Bacharach property, 40, 126
chambers of type (w,p), 100
Chern class, 27-30
total, 28
Chevalley, C, 275

324 Index
Chow group, 28, 312
Chow's theorem, 7
class field theory, 271
class of type (w,p), 100
classification of threefolds, 305
Clifford's theorem, 163, 256
coarse moduli space, see moduli
space, coarse
compatible pair, 186
complex torus, see torus
computation sequence, 80, 168
connected
numerically, 24, 191
connection, 101
anti-self-dual, 105, 206
flat, 102
Hermitian, 102
Hermitian-Einstein, 3, 104, 245
integrable, 102
irreducible, 102
self-dual, 105
unitary, 102
connection matrix, 102
contraction, 66
Cremona transformation, 69
cubic surface, see surface, cubic
curvature, 102
cusp, 14, 23, 138, 173
Deformation theory, 155
degree
normalized, 85
of a torsion free sheaf, 198
del Pezzo surface, see surface, del
depth, 51
Desale, U.V., 2
descent theory, 145, 265
destabilizing, 86
Div X, 9
divisor, 9
big, 20, 72
effective, 9
nef, 20, 72
divisor class, 10
Dolgachev surface, see surface,
Dolgachev
Dolgachev, I., 189, 278
Donaldson invariant, 101, 197, 203,
205
Donaldson polynomial, 205
on P2, 206
Donaldson, S., 3, 208, 308
Donaldson-Uhlenbeck-Yau theorem,
106,245
double cover, 2, 46-50, 57
double point
ordinary, 14, 23, 133, 138, 173, 241
threefold, 210, 242
rational, 77-78, 81-83, 128, 129,
134, 138, 178, 182
dual complex, 75
dual graph, 75, 173
dualizing sheaf, 13
Dynkin diagram, 173
Effective, see divisor, effective
elementary modification, 41-42, 55
allowable, 148
elementary transformation, 114, 118
elimination of indeterminacy, 68
elliptic normal curve, 128
elliptic surface, see surface, elliptic
embedded resolution, 63
Enriques surface, see surface,
Enriques
Enriques' theorem, 1, 280
Enriques, F., 278
enumerative geometry, 2, 3
equivalence
linear, 10
numerical, 17
of families of vector bundles, 153
Euler characteristic, 9
holomorphic, 9
Euler exact sequence, 109
eventually base point free, 21, 72
exceptional curve, 66-67
exceptional divisor, 59
expected dimension, 158
extended root system, 173
extremal rays, 305
Factorization of birational
morphisms, 67
faithfully flat, 275
Fernandez del Busto, G., 245

Index 325
fibration, 167
fine moduli space, see moduli space,
fine
Fintushel, R., 240
fiat connection, see connection, flat
formal functions theorem, 62, 65, 74
Preedman, M., 278
Pulton, W., 4, 28
fundamental cycle, 79-82, 191
Galois cohomology, 184, 185
Galois morphism, 272
general type, see surface, general
type, 277
generic rank, 26
genus
arithmetic, 13
geometric, 8
genus drop, 14
local, 14
geography, 282
geometric genus, see genus, geometric
geometric invariant theory, 1, 154
Gieseker compactification, 206
Gieeeker stability, see stability,
Gieseker
Gieseker, D., 2, 154, 245, 257, 259,
307, 310
Godeaux surface, see surface,
Godeaux
Gottsche, L., 206
Gomez, Т., 311
Gomprecht, D., 313
good generic surface, see surface,
good generic
good vector bundle, see vector
bundle, good
Grauert's contraction theorem, 72
Grauert, H., 72
Griffiths, P., 4
Grothendieck's theorem on vector
bundles, 1, 33
Grothendieck, A., 1, 154
Grothendieck-Riemann-Roch
theorem, 4, 29, 48, 212,
213, 215
Harder-Narasimhan filtration, 97,
112
Harris, J., 4
Hartogs' theorem, 60, 67
Hartshorne, R., 3, 259
Hermitian-Einstein connection,
see connection, Hermitian-
Einstein
Hilbert scheme, 155, 164, 204
Hirzebruch signature theorem, 9, 22
Hodge index theorem, 9, 16-18, 23
Hodge theory, 8-9, 17
homological invariant, 186
Hoppe, H.-J., 3, 310
Horikawa, E., 282
Horikawa surface, see surface,
Horikawa
Hulek, K., 3, 162, 310
hyperelliptic surface, see surface,
hyperelliptic
Infinitely near, 63
intersection pairing, 8, 11
irregularity, 8
birational invariance of, 62, 70
Italian algebraic geometry, 1, 278
j-function, 185
Jacobian elliptic surface, see surface,
elliptic, Jacobian
Jordan-Holder filtration, 112, 154
КЗ surface, see surface, КЗ
Kahler identities, 105, 106
Kahler manifold, 101, 103
Kametani, Y., 218
Kawamata-Viehweg vanishing
theorem, 245
Klein, F., 83
Kobayashi, S., 3, 106
Kobayashi-Hitchin conjecture, 3
Kodaira classification, 7
Kodaira classification of surfaces, 278
Kodaira dimension, 7, 277
of the moduli space of bundles, 311
Kodaira vanishing theorem, 22, 247
Kodaira, K., 1, 70, 131, 175, 176,
186, 187, 189, 249
Kodaira-Spencer class, 156
Koszul complex, 32
Kotschick, D., 278

326 Index
Kronheimer. P., 3
Kummer surface, see surface,
Kummer
Lang, S., 275
leading coefficient of the Donaldson
polynomial, 240
Lefschetz theorem
on A, l)-clas8es, 18
on hyperplane sections, 22
Leray spectral sequence, 4
Leung, N., 3
Li, J., 207, 308, 310, 311
logarithmic transformation, 187
Lubke, M., 3, 106
Liiroth problem, 287
/j-map, 205
Af-sequence, 51
Maruyama, M., 3, 154, 158, 162, 307,
310
Matsumura, H., 4
Mayer, A., 135
Mehta, V.B., 246
Mehta-Ramanathan restriction
theorem, 246, 309
minima) model, 70-71
of a threefold, 306
relatively, 171
strong, 71-72
minimal surface, see surface, minimal
Miyaoka, Y., 245
moduli functor, 153
moduli space, 153
coarse, 154
fine, 155
Moishezon, В., 175, 189, 249
monads, 3
monoidal transformation, 59
Morgan, J.W., 207, 240
Mori theory, 4, 277, 279
Mori, S., 4, 279, 305
Mrowka, Т., 3
Mukai, S., 3, 310
multiple fiber, 169
multiplicity, 60
of a multiple fiber, 169
Mumford's vanishing theorem, 22,
247
Mumford, D., 1, 21, 22, 72, 78, 154,
246, 312
Nagata, M., 91
Nakai-Moishezon criterion, 18-21, 23
Narasimhan, M.S., 2, 313
Narasimhan-Seshadri theorem, 2, 3,
105
nef
relatively, 172
nef divisor, see divisor, nef
Noether's formula, 9, 22
Noether's inequality, 281
normalized degree
see degee, normalized, 85
Num X, 17
numerically connected, see
connected, numerically
O'Grady, K., 240, 310-311
Ogg, A., 194
Okonek, C, 278
orbifold, 188
orbifold fundamental group, 188
ordinary double point, see double
point, ordinary
oriented wall, 100
Period map, 133, 297
Picard number, 18
Pidstrigach, V., 278
plurigenera, 9
birational invariance of, 62, 70
Poincare duality, 8
Poincare line bundle, 263
principal homogeneous space, 183
projection formula, 24
projective dimension, 51
projective tangent cone, 60
projectively normal, 128
proper transform, 60
Qin, Z., 163, 278
Quot scheme, 155
Ramanan, S., 2, 313
Ramanathan, A., 246
Ramanujam's lemma, 191

Index 327
rational double point, see double
point, rational
rational equivalence, 28
rational normal curve, 116
rational ruled surface, see surface,
rational ruled
rational surface singularity, 74-78,
80-82, 138
reflexive, 45, 53-54, 56
Reider's theorem, 249, 274
Reider, I., 248
relative duality, 177
relative Picard variety, 263
relatively minimal model, see
minimal model, relatively
representable functor, 153
Riemann-Roch theorem, 4
for a curve, 4, 30, 57
for a surface, 4, 9, 15-16, 23, 30,
57
ruled surface, see surface, ruled
a-process, 59
S-equivalence, 154
Sato, Y., 218
Schwarzenberger, R.L.E., 1, 2, 46,
48,49
Segre, C, 91
Segre-Nagata theorem, 91, 123
Seiberg, N., 9
Seiberg-Witten invariants, 9, 278
self-dual connection, see connection,
self-dual
semistable vector bundle, see vector
bundle, semistable
Serre duality, 16
Serre, J.-P., 37
Seshadri, C.S., 2
Shafarevich, I.R., 1, 194
simple, 2, 88
Simpson, C, 154
singular support, 43
slant product, 205
Spindler, H., 3, 310
splitting principle, 28
stability
Gieseker, 2, 3, 96-97, 154
Gieseker-Maruyama, 2
Mumford-Tlakemoto, 2, 86
slope, 2
stable vector bundle, see vector
bundle, stable
standard quadratic transformation,
59
Stein factorization, 21
Stein neighborhood, 74
Stern, R., 240
strictly semistable vector bundle,
see vector bundle, strictly
semistable
strong minimal model, see minimal
model, strong
sub-line bundle, 32
suitable ample divisor, 142
support, 43
surface
abelian, 132, 137, 293
Barlow, 278
cubic, 129
del Pezzo, 128-130, 306
Dolgachev, 278
elliptic, 1, 132, 138, 281
basic, 186
Jacobian, 184-186
properly, 281
Enriques, 192, 278, 281, 293
general type, 1, 7
Godeaux, 278
good generic, 130, 138
Horikawa, 282
hyperelliptic, 192, 281, 293
КЗ, 3, 132-137, 139, 192, 274, 281,
293
Kummer, 133
minimal, 70
ruled, 117, 137
geometrically, 117
rational, 113-117
syzygy module, 52
Tp,4,r graph, 81
Tp,,,r lattice, 81
Takemoto, F., 2, 3
tangent bundle to P2, 109
Tate-Shafarevich group, 185
analytic, 187
Taubes, C.H., 307

328 Index
termination of adjunction, 284
theta divisor, 240
Torelli theorem
for КЗ surfaces, 133
torsion, 26
torsion free, 44, 53
torsion free sheaf on a singular curve,
198
torus, 132, 192
Tsen's theorem, 275
twisted G-representation, 265
standard, 265
type of a quadratic form, 8
Uhlenbeck compactiflcation, 207
Uhlenbeck, K., 3
Uhlenbeck-Yau theorem, 3
unirational, 160, 305
uniruled, 305
unitary connection, see connection,
unitary
universal property of blowing up, 60,
66
Xia, Qi, 307
Vaccaro, G., 278
Van de Ven, A., 278
Van der Waerden, В., 64
vector bundle
flat, 245
good, 157, 308
on P1, see Grothendieck's theorem
on vector bundles
on a hyperelliptic curve, 2
on an elliptic curve, see Atiyah's
theorem on vector bundles
regular
on an elliptic curve, 227
semistable, 1, 86
stable, 1, 86
on P2, 2, 91-96, 109-111, 310
on an abelian surface, 310
on a blowup, 165, 260-262, 312
on a curve, 1
on an elliptic surface, 311
on a genus 2 fibration, 313
on а КЗ surface, 310
on a ruled surface, 310
on a singular curve, 199
strictly semistable, 86
unstable, 86
Verlinde formula, 3
Veronese surface, 113, 117
Walls of type (w,p), 100
weak isomorphism, 31
Weierstrass equation, 179
Weierstrass model, 179
Whitney product formula, 28
Witten, E., 3, 9
Wu formula, 8, 16, 189
Yau, S.-T., 3
Zariski tangent space, 156
Zariski's connectedness theorem, 67
Zariski's Main Theorem, 64

Universitext	(continued)
Rotman: Galois Theory
Rnbel/Colliander: Entire and Meromorphic Functions
Sagan: Space-Filling Curves
Samelson: Notes on Lie Algebras
Schiff: Normal Families
Shapiro: Composition Operators and Classical Function Theory
Simonnet: Measures and Probability
Smith: Power Series From a Computational Point of View
Smoryrtski: Self-Reference and Modal Logic
Stlllwell: Geometry of Surfaces
Stroock: An Introduction to the Theory of Large Deviations
Sunder: An Invitation to von Neumann Algebras
Tondeur: Foliations on Riemannian Manifolds
Zong: Strange Phenomena in Convex and Discrete Geometry