Algebraic surfaces and holomorphic vector bundles r friedman

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Springer (continued after index)

of |nKx| when X is a minimal surface of general type The original moti-

vation 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

Tomés Gémez and Titus Teodorescu for comments on various manuscript

versions, and Dave Bayer for doing an excellent job with the figures

Invariants of a surface Divisors on a surface Adjunction and arithmetic genus The Riemann-Roch formula

‘Algebraic proof of the Hodge index theorem -Ample and nef divisors

‘Exercises

t

2 Coherent Sheaves

What is a coherent sheaf?

‘A rapid review of Chern classes for projective varieties Rank 2 bundles and sub-line bundles

Elementary modifications Singularities of coherent sheaves Torsion free and reflexive sheaves Double covers

Appendix: some commutative algebra Exercises

a hữu

Definition of Mumford-Takemoto stability

Examples for curves

Some examples of stable bundles on P?

5 Some Examples of Surfaces

Rational ruled surfaces

General ruled surfaces

Linear systems of cubics

An introduction to K3 surfaces

Exercises

6 Vector Bundles over Ruled Surfaces

Suitable ample divisors

Ruled surfaces

A brief introduction to local and global moduli

A Zariski open subset of the moduli space

Exercises

7 An Introduction to Elliptic Surfaces

Singular fibers

Singular fibers of elliptic fibrations

Invariants and the canonical bundle formula

Elliptic surfaces with a section and Weierstrass models

More general elliptic surfaces

The fundamental group

Exercises

8 Vector Bundles over Elliptic Surfaces

Stable bundles on singular curves

- Stable bundles of odd fiber degree over elliptic surfaces

A Zariski open subset of the moduli space

An overview of Donaldson invariants

The 2-dimensional invariant

113

113 12:

13%

131 14]

'9 Bogomolov’s Inequality and Applications

-” Statement of the theorem

“ (he theorems of Bombieri and Reider The proof of Bogomolov’s theorem

Symmetric powers of vector bundles on curves

:Outline of the classification of surfaces

Proof of Castelnuovo’s theorem

The Albanese map

Proofs of the classification theorems for surfaces

‘The Castelnuovo-deFranchis theorem Classification of threefolds Classification of vector bundles

study of algebraic surfaces is by now over one hundred years old

sof the fundamental results were established by the Italian school of

akc: geometry, for example Castelnuovo’s criterion for a surface to be

al (1895), the theorem of Enriques that a surface is rational or ruled

vonly if Ps or Ps is zero (1905), and in general the role of the canoni- Wal ‘divisor in the classification of surfaces This theory was reworked from

‘modern perspective of sheaves, cohomology, and characteristic classes

l kmries of papers by Kodaira (1960-1968) and by the Shafarevich sem-

(1961-1963) In particular, new ideas were developed to attack those

Be stine:in the classification theory which had proved resistant to the

tc techniques of the Italian school, for example the classification of ke'surfaces or the structure of the moduli space of K3 surfaces and

‘tionship with the period map Another deep result which seems to Abceasible to the classical methods is the Bogomolov-Miyaoka- Yau in- ity:c‡ < 3ca Moreover, the new methods could be extended to the

of compact complex surfaces (Kodaira) or algebraic surfaces in pos- Micharacteristic (Mumford and Bombieri-Mumford) Despite the great Weteas in understanding algebraic surfaces, many open questions remain

@texdmple, the fundamental problem of whether there exists a classifi- ion scheme of some sort for surfaces of general type seems to require & Himpletely new insight

: By contrast, the study of holomorphic vector bundles on algebraic sur- faces is much more recent, and effectively dates back to two papers by

‘Schwarzenberger (1961) For the case of algebraic curves, Grothendieck

(3086) showed that every holomorphic vector bundle over P’ is a direct

am of line bundles (a result known in a different language to Hilbert,

Plemelj and Birkhoff, and prior to them to Dedekind and Weber) Atiyah

(4987) classified all vector bundles over an elliptic curve and made some

1 remarks concerning vector bundles over curves of higher genus In: 1960, the picture changed radically when Mumford introduced the no-

‘ton of a stable or semistable vector bundle on an algebraic curve and used

| geometric invariant theory to construct moduli spaces for all semistable

2 Introduction

vector bundles over a given curve Soon thereafter Narasimhan and Se-

shadri (1965) 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-

uli space of curves Explicit geometric constructions for the moduli space

were given for genus 2 curves by Narasimhan and Ramanan (1969) and for

hyperelliptic curves in general by Desale and Ramanan (1976)

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 Ox(— >; 4:E;), where the E; 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 1,L, where 7: Y — X isa

smooth double cover of X and L is a line bundle on Y He then applied

this construction to construct bundles on P? which were not almost decom-

posable; these turn out to be exactly the stable bundles on P? He showed

further that, if V is a stable rank 2 vector bundle on P?, then the Chern

classes for V satisfy the basic inequality c,(V)? < 4e2(V)

In the years after Schwarzenberger’s papers, the study of bundles over

surfaces diverged into two streams In the first, there were various at-

tempts to generalize Mumford’s definition of stability to surfaces and

higher-dimensional varieties and to use this definition to construct mod-

uli spaces of vector bundles Takemoto (1972, 1973) gave the straightfor-

ward generalization to higher-dimensional (polarized) smooth projective

varieties that we have simply called stability here (this definition is also

called Mumford-Takemoto stability, u-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 (1977) introduced

the notion of stability now called Gieseker stability or Gieseker-Maruyama

stability Gieseker showed that the set of all Gieseker semistable torsion

ì sgial geometric meaning of Mumford stability is the Kobayashi-Hitchin

od ture, that every stable vector bundle has a Hermitian-Einstein con-

Men unique in an appropriate sense This result, the higher-dimensional

serie of the theorem of Narasimhan and Seshadri, was proved by Don- son (1985) for surfaces, by Uhlenbeck and Yau (1986) for general Kahler folds, and also by Donaldson (1987) in the case of a smooth projective

siety (The easier converse, that an irreducible Hermitian-Einstein con-

inn defines 8 holomorphic structure for which the bundle is stable, was

‘ablished previously by Kobayashi and Liibke.) The geometric meaning of aeker stability is more mysterious, although Leung (1993) has obtained

tte in this direction A related general result is Bogomolov’s inequality

/>+able vector bundles, which follows from the Donaldson-Uhlenbeck- Yau

‘sim as well as from various purely algebraic arguments (Bogomolov,

le (117]), we do not discuss monads in this book The case of ruled

ng has been analyzed by Hoppe and Spindler and also by Brosius amoto briefly treated the case of abelian surfaces, but the study of ior bundles (not necessarily of rank 2) over K3 and abelian surfaces Rally got off the ground with a series of papers by Mukai This was the state Behe art until about 1985, when Donaldson theory gave a powerful impetus

he study of rank 2 vector bundles over surfaces We shall describe some

ifthe developments arising after 1985 at the end of Chapter 10

4 There is perhaps a third stream which should be mentioned, that of the

en ative geometry of the moduli space By now these questions have

Cowell studied for bundles over curves (Verlinde formula, cohomology Hing of the moduli space), and in some sense Donaldson theory is simply a

‘Question about the enumerative geometry of the moduli space of bundles

‘gver a surface Deep structure theorems and conjectures in gauge theory, due to Kronheimer and Mrowka and Witten, suggest that there is a very

imple enumerative structure to this moduli space, but as yet there is no

‘way to see why this should be true purely within the context of algebraic

geometry

|The goal of this book is to provide a unified introduction to the study of

‘algebraic surfaces and of holomorphic vector bundles on them I have tried

40 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 proofs 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-

' ical 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-

tic 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-

molov’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 proofs 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 proofs may be better adapted to handling

the classification of symplectic 4-manifolds The point of view of Mori the-

ory and the classification results for threefolds are briefly described toward

lo the chapter Because of the way we alternate

¥ ond of it may be a little disorienting to try to read the book

anologically, and certainly the chapters on surfaces can be, for the most reat ly of the chapters on vector bundles On the other

Mi, 4 the later chapters on vector bundles bundles over ruled or elliptic surfaces heavily on the description of the corresponding surfaces in the chap-

k precede them

en nts of length and time dictated that many topics had to be

out For surfaces, I would have liked to devote more time to rational

4s minimally elliptic singularities and to the classification of surfaces of

I 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 fe a definitive state, it seems

ý us on many concrete examples

ins te here are many exercises at the end of each chapter, and they

integral part of the book In particular, many results are left to the

nea and they are frequently used in later chapters I hope that the asis on examples, both in the text and the exercises, will help to serve introduction to this rich and beautiful field of mathematics

1

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 P" for some N Thus, “surface” is short for smooth (connected) complex algebraic surface By Chow’s theorem, & surface is also described as the zero set in P’ of a finite number of homo- geneous 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 P’ Just as with curves, we could organize this study in order of increasing complexity In terms of the ex- trinsic (synthetic) geometry of a surface in P', we could for instance try

to study and eventually classify surfaces in P 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- cuss birational geometry Chapters 5 and 7 will concern some of the main

examples of surfaces: rational and ruled surfaces, K3 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

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-

ented 4-manifold, its main topological invariants are its fundamental group

™(X, *), the Betti numbers b,(X) = bạ_;(X), and the intersection pairing

on H2(X;Z) Here by Poincaré uality H2(X;Z) H^(X ;Z) and inter-

section pairing corresponds under this isomorphism to cup product from

H?(X;Z)@H’ ?(X;Z) to H *(X;Z) >Z by taking the canonical orientation

Over R, the intersection pairing is specified by b2(X) and by bf (X), the

number of positive entries along the diagonal when the form is diagonalized

over R We also let by (X) = be(X) — by (X) If X = P? or if x is one of

an unknown but finite number of surfaces of general type whose universal

cover is the unit ball in C?, then H?(X;R) & R If X does not belong

to this finite list of examples, then H2(X ;R) 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 H2(X;Z)

mod torsion is specified by its rank, signature, and type, i.e., whether or

not there exists an element a € H2(X;Z) with a? = 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 Il, we use the

Wu formula, which says that a? = q [Kx] mod 2 Here [Kx] denotes the

homology class associated to the canonical line bundle K x via c)(Kx) and

Poincaré duality Thus, again by Poincaré duality, there exists an a with

a? =1 mod 2 if and only if the image of [Kx] in B,(X;Z) = H2(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 p,(X) of X, defined

by

4(X) = dime H°(X; 9) = dime H"(X; Ox),

Po(X) = dime H°(X;0%) = dime H?(X; Ox),

Thus, ¢(X) is the number of independent holomorphic 1-forms on X and

P(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 dim H'(X; Ox) (That the two expressions for p,(X) are equal follows

from Serre duality which holds in general.) Additional invariants are given

by hh1(X) = dim H1(X; 0) and c(X)? = [Kx]? The relation of these

invariants to the topological ones is as follows:

b(X) = 2q(X),

1 Curves on a Surface 9 ba(X) = 2pg(X) + h**(X),

b2 (X) = 2pg(X) + 1

the first two equalities follow by Hodge theory an

form of the Hodge index theorem for a surface We also have the Euler

- characteristic x(X) = 1 — by(X) + b(X) — b3(X) +1 a

= 2 — 2by(X) + b2(X) = 2 — 4q + 2p9(X) + hh (X) and the holomorphic Euler characteristic

x(Ox) = h°(Ox) — h*(Ox) + h?(Ox) = 1— 9(X) + p(X)

There is also Noether’s formula (in some sense a special case of the Riemann-Roch theorem for surfaces) which says that

c1(X) + œ(X) = 12x(0x),

; in other words that [Kx]? + x(X) = 12(1 - q(X) + ?ạ( = _ X)) An easy manipulation of the formulas (Exercise 1) shows that Noether’s formula is equivalent to the Hirzebruch signature theorem

by (X) — by (X) = š(f1(X) — 2ca(X))

A md this there are the “higher” holomorphic invariants of X, the

plurbenera P,(X) = dim H°(X; K"), defined for n 3 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 cen § own via new invariants introduced by Seiberg and Witten that the p mee ere

are diffeomorphism invariants of X (see for example [16] and [41]) We s discuss some of these developments further in Chapter 10

Divisors on a surface

We recall that a (reduced irreducible) curve C on X is an irreducible holo” morphic subvariety of complex dimension 1 Thus, locally C is nu

88 {ƒ(2Z\, z2) = 0}, where ƒ is a holomorpbic function of 21, 22 : cour ›

C need not be a (holomorphic) submanifold of X; if it is we say t at ni s |

smooth curve A divisor D on X is a finite formal sum des mCi os tine irreducible curves C;, where the n; € Z The set of all divisors Pị mà

thus the free abelian group generated by the irreducible curves on 4 °

divisor D is effective if the n; > 0 for all i An effective divisor z iw i

also be called a curve We write D > 0 if D is effective and Dy 7 2

D, ~ D2 > 0 If f; is a local equation for the curve Cis then Dis ee ly

described by the meromorpbhic function [[, ƒ;”?, which is in pet ho -

phic if and only if D is effective Conversely, a meromorphic func ion on

X has an associated divisor (f), which is the curve of zeros of f minus

curve of poles of f 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 (9) + DIU > 0, where DIU 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 Dz are linearly equivalent (which we

shall write as D, = Dz) if and only if Ox(D) & Ox(D2) if and only if

D,— D2 = (f) fora globally defined meromorphic function f 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

8 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 = © 'x(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

shail 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 : X 4 Y is a surjective map, then pullback 2*

induces a homomorphism Div Y — Div X 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 Div X — H?(X;Z) given by

D +> e(Ox(D)), followed by Poincaré 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 SPO), oO 7,

by taking c, to be the coboundary map H}(OX) = PicX — H?(X;Z) In

any Case, we shall denote by [D] 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 Div X — H2(X;Z) We begin by defining a

local intersection number for two curves C,C2 which have no component

in common

Definition 1 Let C,,C> be two curves with no component in common

and let x € X Define C, -, C2 = dime Ox,2/(fi, fz), where f; is a local

equation for C; at zx

1 Curves on a Surface 11

Note that to say that C,, C2 have no component in common is exactly to

say that ƒ¡ and f2 are relatively prime in the ring O Xx 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, N Ca supported at

x Of course this is empty if ¢ ¢ C1 M Co, and in this case it is easy to see

that C, -, Co = 0 Note also:

Lemma 2 The curves C; and C2 meet transversally at the point x if and

only if C, “a Ca = 1

Proof By definition C; and C2 meet transversally at x if and only if (fi, fa) = mz, the maximal ideal of Ox at x, and since Oxc/(f; fa) is a

nonzero C-algebra and thus always has dimension at least 1, this condition

is equivalent to the condition that C1 -„CŒ¿ =1 D1

To define an intersection pairing for divisors, we proceed as follows: sup-

pose as before that C,,C2 have no component in common and define

Ci - Ca = À ` C¡ -„ C¿

zcX

By hypothesis this is a finite sum

Lemma 3 IfC\ is a smooth irreducible curve and C\ is not a component

of Ca, then C; - C2 = deg Ox(C2)|C\ (here deg is the degree of a complex

line bundle on the curve C)) In particular, in this case C, -C2 only depends

on the linear equivalence class of C3

Proof Take the exact sequence

0 — Ox(—C2) — Ox - Oc, —¬ 0

and tensor it with Oc, Let f; and fg be local equations for C, and C2, respectively, at z An easy calculation using the fact that f; and fo are relatively prime shows that the resulting sequence

0 Ox(-Cx)® 0c, + 0c, > QB Oxe/ (fr, fa) +0

Thus, Oc, (C2) has a section vanishing at exactly C; - C2 points, counted

with multiplicity, and so deg Oc, (C2) =C,-Ce2 O

Theorem 4 There is a unique symmetric bilinear pairing from Div X to

Z, denoted by (D,, D2) which factors through linear equivalence and has

the property that (C\, Ca) = C - Ca for C\ and C2 distinct smooth curves

meeting transversally

Proof Note the following standard lemma:

Lemma 5 Let L be a line 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 € Div X is linearly equivalent to a difference of two very ample

divisors O

To prove Theorem 4, we begin with the uniqueness Given D, and D,

in Div X write D; = H{ — Hj/ We may assume that all of the H/, H?

are distinct and smooth and meet transversally Thus, necessarily we must

have

(L1) (Di, Da) = Hy - Hy — Hy Hy — HY Hy + HY HY

To see the existence, note that the above formula can be written as

(1.2) (Dị, Dạ) = deg Ox(Dì)| Hộ — deg 0x (D,)| Hy

Fix Dg and choose smooth curves Hj, Hy meeting transversally with Dạ

linearly equivalent to Hi — HY For an arbitrary divisor D2, we can define

(D1, Dg) by formula (1.1) Using (1.2), we see that (D,, D2) only depends

on the linear equivalence class of D;, and in particular does not depend

on the choice of H{ and H{/ By symmetry the same is true for Da Thus,

(D1, Dz) is well defined by (1.2), and is clearly symmetric It follows from

(1.2) that (D1, D2) is bilinear, and we are done O

We shall usually denote (Di, Dg) by D,- Dz As a corollary of the above

proof, note that, if D2 is smooth, then D, - Dg = degOx(D})|Dg In fact,

@ similar formula is true for an irreducible curve D2, noting that we can

define the degree of a line bundle LZ on an irreducible curve C in several

equivalent ways:

(i) As the degree of the pullback of L to the normalization C of C;

(ii) By writing L = Oc(o, nipi), where the p; are points in the smooth

part of C’, and taking deg L = )), ni;

(iii) Via the exponential sheaf sequence

0¬2Z— Ơc > 04-0 and the fact that for an irreducible curve we have H?(C; Z) % Z

The uniqueness part of the proof of Theorem 4 shows that D, - Dz =

[D1] - [Da], where [D,] is the homology class asociated to D; and we use

intersection product in homology Finally, two remarks that we shall often

we are the following: if H is ample and D is effective and nonzero, then

# D > 0 Moreover, if Ơ¡ and Ca are distinct irreducible curves, then

5 -Ca >0, and C, - C2 = 0 if and only if C; and C2 are disjoint

In the rest of this chapter we shall use intersection theory to analyze

urves and linear systems on X

Adjunction and arithmetic genus

hippose that C is a smooth curve on X Then C-C = degOx(C)IC eral results on smooth divisors (see for example Hartshorne [61, p I82]), Ox(C)IC = Nc;x is the normal bundle to C in X For an effective

livisor C, not necessarily smooth or reduced, we shall sometimes define

ye normal bundle of C' in X to be simply Ox(C)|C We shall also usually

Which gives det(Tx|C) = (Kx|C)~* = Kg! @Ox(C)I|C Thus, if 9(C) = 9

Š the genus of C, then

4) 2g — 2= deg Kc = (Kx + C) -

fore we use the same symbol Kx to denote the canonical line bundle and the canonical divisor (class)

\i For C' any nonzero effective divisor on X,, we can still define the dualizing

sheaf wo by the same formula as (1.3)

The significance of wo is that it is the unique line bundle on the (possibly

1 ) scheme C’ for which Serre duality holds: there is a trace map rita wc) — C such that, for every line bundle L on C, the induced map

ba perfect pairing (Serre [136| for the case where Œ is reduced and Barth- Peters-Van de Ven [7] in general) In particular H'(C;wc) is dual to H°(C; Oc) Warning: If C is reduced and connected, H°(C;Oc) = C and the trace map is an isomorphism In general, however, H°(C; Oc) may

be larger than C (Exercise 3) and thus H'(C;wc) may have dimension larger than 1 also

’ For a general nonzero effective divisor C' we define the arithmetic genus

of C by the same formula as before

(1.6) 2p„(C) - 3= (Kx +) -Ơ

Ng

In case C is reduced and irreducible, 2p,(C)—2 is therefore equal to deg we

In general an application of the Riemann-Roch theorem on X (see Exercise

8) shows that

Thus, if h°(C;Oc) = 1, for example, if C is reduced and connected, then

Pa(C) = h}(Oc) = h°(C; wo) As a result we have:

Corollary 6 If C is an irreducible curve on X, then Pa(C) > 0 Thus,

(Kx +C)-C > —2 Moreover, (Kx +C)-C = —2 if and only if p,(C) = 0,

and otherwise (Kx +C)-C>0 O

In fact, we can say more than the statement that (Kx +C)-C > 0 if

Pa(C) > 1:

Proposition 7 If C is an irreducible curve on X with pa(C) > 1, then

wo has no base locus O

For a proof of Proposition 7, see, for example, Catanese [17]

In general it will be useful to have various ways to calculate p,(C) We

begin with the case where C is reduced and irreducible In this case the

normalization C is a smooth connected curve, and has a well-defined genus

g(C) Let v: C — C be the normalization map Now consider the exact

sequence

(1.8) 0 ¬ Oc > % 0g — 1„Oz/Øc — 0

For x € C, we define the local genus drop at x to be the nonnegative integer

6z = dime [v.0g/Oc], - Thus, 5, = 0 if and only if z is a smooth point of C For example, if z is an

ordinary double point of C, then 6, = 1 Likewise, if x is a cusp point, so

that, locally analytically near z, C is described by the equation y? = 23, C

is a smooth curve with coordinate t, and v(t) = (t?, 3) in local coordinates,

we again have 5, = 1 We leave it as an exercise to show that conversely, if

6; = 1, then z is either an ordinary double point of C or a cusp The genus

drop of the curve C is the nonnegative integer 6 = )>,-¢ 52 Note that we

can still define the local invariant 6,, and hence 6, if C is only assumed

reduced, but not necessarily irreducible

Lemma 8 If C is reduced and irreducible, then p,(C) = g(C) + 6 Thus,

Pa(C) = 0 if and only if C is a smooth rational curve More generally,

let C be a reduced but not necessarily irreducible curve on X » and let

Ci, ,;Cy be the connected components of Ở, with 9(C;) = g; Then

Pa(C) = 091+ 6+1—n

Proof We shall just check the first statement, as the proof of the second

is similar Since C’ is connected, H°(Oc) = H°(O@) and the long exact

cohomology sequence of the sheaf sequence (1.8) gives

0— H°(v.06/Oc) > H'(Oc) > H`(0z) — 0

A dimension count gives h}(Oc) = h}(0g) +6 =9(C) +6 O

We will return to the study of 6 in Chapter 3 We now briefly discuss the nonreduced case The main tool for studying C in this case is the following

exact sequence: suppose that C and D are two nonzero effective divisors, not necessarily reduced or irreducible Here we will allow C and D to have components in common Then, by Exercise 9, there is an exact sequence

(1.9) 0> Øp(~C) — Øc+p — Øc — 0

Thus, (1.9) allows us to work out x(Oc+p) from the knowledge of x(Op(-C)) and x(Oc), and would allow us to work out h°(Oc+p) and

thus h'(Oc+p) if we could work out the coboundary maps in the associated

long exact sequence (which is usually impossible in general) An important

application of (1.9) is to work out Onc,, where Co is a reduced and irre-

ducible curve In this case, for n > 2, the exact sequence of (1.9) applied

toC = (n—1)Co and D = Cy becomes

(1.10) 0 > Oc,(—(n — 1)Co) > Oncy + O(n—1)G — 9

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- lowing formula

x(X;Ox(D)) = 3D: (D— Kx) +x(Ox)

Proof The formula is trivially valid for D = 0 Next, for D = C asmooth

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)IC)

On the other hand, 9(C) = (C? + Kx -C)/2 +1 and degOx(C)|C = C?

Thus, an application of the Riemann-Roch theorem for curves gives

x(Ox(C)|C) = Œ ~ (“#f+1)tt- Kx C,

verifying the formula in this case In the general case, write D = C, — C2

where C, and C2 are smooth, which is possible by Lemma 5, and use the

exact sequence

0 Ox(C; — C2) + Ox(Ci) ¬ Øx(Ci)|Ca — 0

Thus, x(Ox(C:~—C2)) = x(Ox(C1))—x(Ox (Ci)|C2) By the previous case

x(Ox(C,)) = 3(C?-Ci-Kx)+x(Ox), and adjunction and Riemann-Roch

Note that as a consequence of the Riemann-Roch formula, we recover

the Wu formula for divisors: Kx -D = D? mod 2

Closely tied in with the Riemann-Roch formula is Serre duality :

Theorem 10 If D isa divisor on X, then the vector space H'(X;Ox(D))

is naturally dual to H?-*(X;Ox(Kx —D)) O

Notice that the Riemann-Roch formula is indeed invariant under the

substitution Dre 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 H*(X;Ox(—D)) = 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 D? < 0, and if D? = 0, then D- E = 0 for all

divisors E

Proof We begin with the following lemma:

1 Curves on a Surface 17

Lemma 12 Let H be an ample divisor on X, and let D be a divisor such

that D? > 0 and D- H > 0 Then for all n >> 0, the divisor nD is nonzero

and effective

Proof Applying the Riemann-Roch formula and Serre duality to nD gives

4 A°(Ox(nD)) + h°(Ox(Kx — nD)) > x(Ox(nD)) = $D?n? + O(n)

Thus, for all n > 0 either nD or Kx — nD is effective However, H-(Kx — nD) < 0 ifn > (H-Kx)/(H-D), so that Kx —nD cannot be effective as goon as 7 is sufficiently large Thus, nD is effective for all n >> 0, and it is honzero since (nD)? = n?D?>0 O

‘ Returning to the proof of the Hodge index theorem, let D be a divisor such that D- H = 0 Suppose first that D? > 0 By Serre’s criterion (see,

for example, [61, II §7]), mH + D is ample for all m >> 0 Now D? > 0

‘and D- (mH + D) = D? > 0, so applying Lemma 12 to the divisor D

with H replaced by mH + D, it follows that nD is effective and nonzero

for all n > 0 But then (nD) - H > 0, and so D- H # 0, contradicting our

‘fypothesis on D

- Thus, we must have D? < 0 Suppose now that D? = 0 If there exists

a divisor E with D- E # 0, after replacing E by —E we may assume that

‘b E > 0 Next replace E by the divisor E’ = (H?)E — (H - E)H Then

Ề⁄ˆ.H =0and D- E' = (H2)(D - E) > 0 If we now set D' = rwD + E', hen D - H = 0 and (D’)? = 2m(D - E') + (E’)? For m > 0, D’ is thus a

divisor satisfying (D’)? > 0, D' - H = 0, contradicting the first part of the

proof It follows that either D? < 0 or D? = 0 and D- E = 0 for all divisors

5 é, ‘as claimed O

‘Definition 13 A divisor D is numerically equivalent to0 if D-E = 0 for all

divisors E Two divisors D, and D2 are numerically equivalent if D; — D2 is

‘numerically equivalent to 0 Note that linear equivalence implies numerical equivalence We let Num X be the quotient of Div X by the equivalence

elation of numerical equivalence

= Let H2(X;Z) = H?(X;Z) modulo torsion Then H?(X;Z) is a subgroup

of H?(X;C) According to Hodge theory, H?(X;C) ~ H?°(X)@H!!(X)@

H®?(X), where H?9(X) = H9(X;0%,) and HP.4(X) is the complex con- jugate of H%-?(X) Thus, the subspaces H2°(X) @ H®?(X) and H*!(X) are invariant under complex conjugation and are therefore the complexifica-

tions of subspaces of H?(X;R), which we denote by (H?°(X) @ H92(X))g

and H'!(X)p, respectively If D is a divisor, then the image of its associ-

ated cohomology class in H?(X;C) lies in H'!(X) as well as in the image

of H^(X;Z), which we can write as H?(X;Z)MH*(X) The usual Hodge

index theorem then says that the intersection pairing is positive definite

on (H?°(X) @ H°?(X)), and negative definite on the complement of an

ample divisor in H!:1(X) Combining the algebraic Hodge index theorem

above with the usual Hodge index theorem, we find:

Lemma 14 The natural map from NumX to H2(X;Z)n H}!(X) €

H?(X;C) is an isomorphism

Proof There is the natural map from Div X to H?(X;Z)NH""(X), which

is surjective by the Lefschetz theorem on (1, 1)-classes Clearly, if D is not

numerically equivalent to zero, then the image of D in H?(X;Z)NH'}(X)

is nonzero Conversely, suppose that the image of D in H*(X;Z)NH'1(X)

is nonzero Let H be an ample divisor on X If [D]-H #0, then D is not

numerically trivial If [D]- H =0, then by the usual Hodge index theorem

[D]? < 0, and so D is again not numerically trivial Thus, the kernel of

the map from Div X to H?(X;Z) 9 H'1(X) is exactly the subgroup of

numerically trivial divisors O

As a corollary, we have:

Corollary 15 The group Num X is a finitely generated torsion free

abelian group, of rank at most h}"(X) O

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 H? > 0, not necessarily

ample Then the intersection pairing on

H+={DeNumX: 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 H? > 0 and

H-C > 0 for all irreducible curves C on X O

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 ample

divisor, then H is ample O

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 @ R = R°’ has an intersection form q, the natural axtension of intersection pairing on Num X There is a real basis €1, ,€,

of Num X @ R such that

Lemma 19 With notation as above:

(il)Given € € P,, suppose that 1 lies in the closure P of P Then €-n = 0

a if and only if 7 = 0 Otherwise, 7 lies in the closure of P, if and only

-_ jf€ -r > 0

qh

Proof Part (i) is an easy consequence of the Cauchy-Schwarz inequality and is left to the reader For (ii), first suppose that €- = 0 Since n € P,

ˆ > 0 Thus, € and 7 span a positive semidefinite subspace of Num X @R,

which can have rank at most 1, and so 7 = 0 Now consider the function

P -+ {+1} defined by n + sign(€ - 7), which is well defined by the first part of the proof since €- 4 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 O

Fix an ample divisor H We can choose our basis above so that H € P, Since H - H' > 0 for every ample divisor H', H' € P, for every ample H’

We can take the convex hull of the ample divisors in Num X @ R It forms

a cone in P, C Num X @R, the ample cone A(X) If H € A(X)NNumX, then H is ample by the Nakai-Moishezon criterion, since H? > 0 and

H -C > 0 for every irreducible curve C

Lemma 20 A(X) is open

Proof Let H be an ample line bundle For an integral basis d,, ,d, of Num X, let D; be a divisor corresponding to d; By Serre’s criterion for

ampleness, there is a positive integer N; such that N;H + D; is ample for

every i Thus, A(X) contains the convex hull of the points H + (1/N;)d; 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 >, Ax Hx, where A; € R*+ and H, is ample, and

for every k there is an open neighborhood U; of A;,H;, contained in A(X)

It follows that A(X) contains >>, U;, which contains an open set around

nh:

Lemma 21 Let

A'(X) = {z € Py: 2-C >0 for all irreducible curves C};

A(X) = {z € P, : z - C >0 for all irreducible curves C},

where P is the closure of P, Then A(X) C A(X) C A(X) and A(X) is

the closure of A(X) Moreover, A(X) is-the interior of A'(X) and of A(X)

Proof We have already seen that A(X) is an open convex cone Clearly,

A(X) © A'(X) C A(X) and all three sets are convex Moreover, A(X) i is

closed and thus contains the closure of A(X) Conversely, let 1 € A(X)

Let ị, , dạ 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 + 3”, Í;¿ :

0 < t; < 1/n} is open in Num X OR, and thus contains a rational point An

Clearly, h? > 0 and hy, - C > 0 for every irreducible curve C, so that some

integral multiple of h, is ample by the Nakai-Moishezon criterion Thus,

hn € A(X), and clearly limp—.oo An = 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 R” 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) O

Note that, in the definition of A’(X) or A(X), it is enough to consider

only those irreducible curves C with C? < 0 Indeed, if C? > 0, then as

C-H > 0 for every ample divisor H, C € P, and thus A- C > 0 for every

A € P, Despite the fact that A(X) is described by a countable number

of inequalities defined by integral elements of Num X @ 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-C > 0 for all irreducible curves

C A divisor D is big if D? > 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 D? > 0 In fact this last condition is redundant:

Lemma 23 Suppose that D is a nef divisor Then D? > 0

Proof Fix an ample divisor H If D? < 0, there exists a to > 0 such that 'D + toH)? = 0 and (D + tH)? > 0 for t > to By the Nakai-Moishezon xiterion, D+tH is then ample for t > to,t € Q For such ¿, some multiple

if D + tH is then effective, so that D-(D+tH) = D? +t(D-H) > 0 iikewise, D - H > 0 By continuity, D? + t9(D - H) > 0, and so

43! A gtandard example of a nef and big divisor on X is obtained by taking a jivisor D such that the morphism defined by the complete linear system |D| nas no base points and is generically finite onto its image More generally,

by | analogy with ampleness we make the following:

away

finition 24 A divisor D is eventually base point free if for all n > 0,

he linear systen |nD| has no base points

khô

Suppose that D is eventually base point free Let @„p be the morphism defined by |nD| for n such that |nDj is base point free, and let Xp be the image of X under y,p We assume that /@„p Ìs generically finite, or iquivalently that D is big From the embedding Xo C PN, there is an ample line bundle Lp on Xo, the restriction of Opy(1), which pulls back

to Ox(nD) The morphism y,p has a Stein factorization X > X¬ Xo;

where X is normal and X — Xj is finite Let L be the pullback of Ly to X

Since X — Xp is finite, L is ample, and it pulls back to Ox (nD) If: X >

X is the natural map, then since X is normal (or by the construction

of the Stein factorization) ™4Ox = Ox Thus, 7,0x(nD) = 1.7 mL =

Ù@z,Ox = L It follows that H°(X;Ox(nD)) = H°(X; L) and similarly for H°(X;Ox(mnD)) Hence, for all k >> 0 and divisible by n, the image of

X under the morphism (xp defined by |k D| is the normal projective surface

X Moreover, if C is an irreducible curve on X, then yp(C) is a single

point if and only if C-D = 0 It follows that X is the uniquely specified normal surface obtained by contracting all such curves C By the Hodge

index theorem, the curves C for which C - D = 0 span a negative definite sublattice of Num X Moreover, they must be independent in Num X @ Q:

22 1 Curves on a Surface

Lemma 25 Let C,, ,C, be distinct curves spanning a negative defi-

nite sublattice of Num X Then the classes of the C; are independent in

Num X @Q

Proof If not, there exists a relation }>,n,C; = 0 with the n; € Z, not

all 0, and after deleting some of the C; we can assume that none of the

n; is 0 Since the C, are effective, not all of the n; are positive (otherwise

intersect with an ample divisor) Collecting the negative terms, and possibly

relabeling, there is a relation of the form

which is only possible if (}>j_, niC:)? = 0 Since the lattice spanned by the

C; is negative definite, it follows that }7j_, niC; is numerically equivalent

to zero, which is impossible since the n, are all positive O

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 H#(X;Ox(—D)) = 0 fori = 0,1 Dually H'(X;Ox(Kx + D)) =0

2 Let X be a smooth surface in P* of degree d Show using standard

facts about the cohomology of P* 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

the fact that q(X) = 0, determine c?(X) and p(X) Apply Noether’s

formula to find b2(X) and thus h!!(X) (Of course, you could also,

with somewhat more effort, find h!:!(X) directly and then find b2(X).)

When is the intersection form on X of Type I?

Let C be a smooth rational curve on X For n > 0, find dim H°(nC;Onc) and pa(nC) in terms of C? What can you say if instead g(C) > 0?

Let C be a reduced irreducible curve and z a point of C Show that

6, = 1 if and only if z is an ordinary double point or a cusp (One direction has already be done.)

Calculate 5, for a singularity of the form y? = z?*+! and also for a

singularity of the form y* = x?* What is the local picture of these

singularities?

Let Ài, ,Àna be distinct complex numbers and consider the local

singularity C given by

IIe — À¡z) =0

Thus, C is a union of n distinct lines meeting at the origin Compute 5p

- directly from the definition in this case (Note: C' is not obtained from

T

its normalization by identifying the common point on the n branches

if n > 2.) We will see an easier way to compute dp for this example in

Let C be an irreducible curve whose singular locus is a single ordinary double point z, and let p,q € C be the preimage under the normal-

ization map v of x Show that v*wc = Kg ® Og(p + q) Which local

sections of this line bundle are the pullbacks of sections of wc? Analyze

a cusp singularity similarly

Use the Riemann-Roch theorem applied to x(Ox(—C)) to show that, for every effective nonzero divisor C, pa(C) = 1—x(C; Oc) Also show that, for two such effective divisors C and D,

pa(Π+ D) = pa(C) + pa(D) + CC - D- 1

Verify that, in the notation of (1.9), the natural map Oc+p — Oc is

surjective (this is obvious) and that its kernel is Øp(—C)

Let D, and D2 be two divisors with D? > 0 Show that

(D1)?(D2)? < (Di - Da)?

and analyze the case where equality holds (‘This is an easy consequence

of the Hodge index theorem which applies to all nondegenerate sym- metric bilinear forms whose intersection matrix has just one positive eigenvalue.)

Show directly that, if H, is ample and Hz is eventually base point free, then H;+Hp is ample Using the Nakai-Moishezon criterion, show that

if H, is ample and Hz is nef, then H; + Hg is ample

12

13

Let 7: X — Y be a surjective morphism of surfaces Given a divisor D

on X, there is a divisor 7, D on Y defined as follows: For C irreducible,

1„Œ = 0 if x(C) is a point, and otherwise 7,C = dx(C), where ở is the degree of the map from C to 1(C) For general D we extend 1, by linearity Prove the projection formula *D - E = D - x„E Conclude that 7* induces a map NumY — Num X, also denoted 7*, and that

nm, induces a map Num X — NumY which we continue to denote by

Te

Let H be a nef and big divisor on the surface X Suppose that H = A+ B, where A and B are effective Show that A- B > 0, with

equality holding if and only if one of A or B is 0 We say that H is

numerically connected (Let E = aH — A with a = (A- H)/H? Since

0< A-H < (A+ B)-H = H?,0 < a < 1 By the Hodge index

theorem, E? < 0 Now estimate A- B = (aH — E) - ((1—a)H + E),

and analyze the case of equality.)

2

Coherent Sheaves

What is a coherent sheaf?

Let X be ascheme (or analytic space) with x € X and let R = Ox,z be the ring at z For our purposes X will usually be regular, and we could as

Ì work i in the analytic category, so that the reader can for the moment

R = C{z, ,%n} to be the ring of convergent power series at the

ae if so desired There are two paradigms for what a coherent sheaf F

on X should look like:

đ) -A locally free sheaf, locally modeled on the free module RỲ;

(i) An ideal sheaf, locally modeled on an ideal I ¢ R

e general torsion free coherent sheaf, roughly speaking, is a blend of these

wo models, and torsion, as we shall see, essentially corresponds to some

‘torsion free sheaf supported on a proper subvariety of X Another way to

‘think 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 or WY, 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 ox =

O* 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 F locally has a presentation

26 2 Coherent Sheaves

The local model of such a sheaf is then a finite R-module M The Noether-

ian properties of R are reflected in the statement (not trivial in the analytic

case) that the kernel of a map of Ox-modules OF — O¥ is also 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 F corresponds to an element m of the

R-module M which is annihilated by some f € R Thus, s is (at least near

x € X) supported on the subvariety {f = 0} Conversely, if s is a section

with support on a proper subvariety V C X, let f #0 € R be an element

corresponding to the ideal of V, and let V’ = {f = 0} > 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 — Ms = M@ Ry, where Ry

_ is the localization of R at f, and then it follows from the definition that

there exists an n > 1 such that f"m = 0 Thus, m is a torsion section,

annihilated by some power of f

In particular (using finite generation properties associated to coherence)

a torsion coherent sheaf F 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 Z be a coherent sheaf on X, corresponding to the R-module N Then

as R is an integral domain it has a field of fractions K, and the rank of

N is the dimension of the K-vector space N @p K Let r be the rank, so

that N @p K = K* Choose a basis 11, ,v, of N @n K After clearing

denominators we may assume that v; is the image of n; € N Thus, there is

a well-defined map from the free module R" to N which makes the following

diagram commute:

I |

R ——> R@rK=K', and so R" — N is injective If T is the cokernel, using the fact that

R’'@rK>~N@RK—-T@RK —Ũ

is still exact and that the first map is an isomorphism we see that T@pK =

0, and thus T is a torsion sheaf It follows by Example 1 that there is an

open subset U = Spec Ry of X such that F|U is free Its rank is then the

generic rank of F Pursuing this idea further, it is not hard to show that we

may stratify X by locally closed subvarieties S,, such that the restriction

of F to each Sq is free

Coherent but non-locally free sheaves play two different roles in the study

of the moduli of vector bundles:

1! Asa way to dismantle a vector bundle V into canonically defined pieces;

2 ‘As the necessary ingredient for compactifying ip " moduli spaces of vector +- bundle

Ù 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 & variety On the other hand, 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

6 in special position One goal of trying to break up a vector bundle irito more canonical pieces will be to make explicit the link between the élassification 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-

tions, and double covers The chapter ends with some commutative algebra

‘Ip Chapter 4, we will introduce the notion of stability and derive some of

, ite basic properties Chapters 6 and 8 describe vector bundles over ruled -and elliptic surfaces Finally, in Chapter 9, we return to the general theory

“and 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:

(2.1) ci(V) = ci(det V) € H?(X;Z)

if X is defined over C In general we could use an algebraic substitute for H?(X;Z) such as Pic X in the algebraic case (in which case we take c:(V)

to be actually equal to det V), or Num X To define c;(V) in general, we

first define it for a direct sum of line bundles V = 11 @ -® L, In this case, fotmally

(2.2) 1+ a(V) + -+ cn(V) = (1 + e1(L1)) tee (1 + e(L1));

the actual formula is obtained by equating the terms that lie in H ?(X;Z)

A similar formula holds if V, instead of being a direct sum of line bundles, has a filtration by subbundles V; such that V; /V._\ = L¡ is a line bundle

28 2 Coherent Sheaves

To define c;(V) for a general V, one shows that there exists a variety

Y and a morphism 7: Y — X for which 1*: H'(X;Z) > H'(Y;Z) is

injective, for all i, and such that 7*V has a filtration as described above

We have then defined c;(1*V), and one shows that these classes are in the

image of z* for all i Then we can set c;(V) to be the unique class such

that z°c;(V) = cœ¿(z*V) For our purposes, the Chern classes c;(V) will

lie in H?(X;Z) However, for a smooth projective variety X, there exist

algebraic analogues of H*(X;Z), namely the higher Chow groups A‘(X)

consisting of algebraic cycles of codimension i 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, A\(X) = Pic X But

for i > 1, the groups A(X) are poorly understood and can be quite large

In any case, one can define Chern classes c;(V) € A‘(X), which are a

much finer invariant of V For X defined over C, there is is a natural

homomorphism A‘(X) + H?*(X;Z) which sends a codimension i cycle to

its fundamental class, and the image of c;(V) under this homomorphism

is the usual Chern class Unless otherwise specified, we will always take

c(V) € H**(X;Z)

The Chern classes so constructed are functorial: c;(f*V) = f*c:(V), and

satisfy the Whitney product formula: for an exact sequence

The idea behind the construction of the Chern classes is called the splitting

principle and has the following useful extension: every “aniversal” formula

for Chern classes which holds for direct sums of line bundles holds in gen-

eral For example, for the dual bundle VY of a vector bundle V we have

the formula

where given a class œ = 3), o¡ € @ H?!(X;Z), œY is by definition the class

>, (-1)'a; Similar formulas can be given for SymẺ V, A* V, Hom(V, W),

V @W, For example, if L is a line bundle and V is a bundle of rank

r, then

œ(Y @ 7) = œ(V)+rei(L),

2.6

( ) ca(V ® L) = c2(V) + (r — 1)e(V) ci (L) + (Jaw

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

2 Coherent Sheaves

sheaf F on a smooth quasiprojective variety admits a finite resolution

by locally free sheaves: there exists an exact sequence of sheaves

0 >£*¬ ¬£?°¬7Z—0, where the £ are locally free We can then define the total Chern class of

‘tis 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, ie., the Tor sheaves, and we refer to Fulton [45] or Borel-Serre [12] for more details One impor- tant example which we shall use often is the following: let Z be a reduced frreducible subvariety of X of codimension r, and let j: Z > X be the -fnclusion map Then, as a consequence of the Grothendieck-Riemann-Roch

; teorem, c¡(j.Øz) = 0 for ï < 7 and

where [Z| € H?"(X; Z) is the cycle defined by Z A similar result holds for

a'vector bundle V of rank non Z: ¢(jx.V) = 0 fori <r and

œ(j.V) = (1) — 1)mỊZ!

From this it is easy to deduce that ci(j.Oz) = 0 for i<r and c,(j.0z) =

(cứ —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

coniponents of Z are Z¡, , Z4 and the length of Oz along Z; is Trị, then

: 1=): m2 For example, suppose that X is a smooth projective sur- face and that Z is a 0-dimensional subscheme of X supported at pi,.- - » Ph - Let j: Z — X be the inclusion Then Ox,,/1z,p, Ì8 8 finite-dimensional C-vector space Define £(Zp,) to be its dimension and let €(Z) = >>; £(2p.)- Then

c(j.Oz) = —(Z) € H*(X;Z) = Z

Applying the Whitney product formula to the exact sequence

0 — 1z > Ox > j20z > 9,

we see that co(Iz) = €(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

30 2 Coherent Sheaves

For an effective divisor D, the formula for ci(j+Op) is easy to verify:

using the exact sequence

0 — Ox(-D) — Ox > j.0n — 0, and the Whitney product formula, we see that

Lemma 1 Let L be a line bundle on the effective divisor D C X, and let

3: D — X be the inclusion Then

Cy (j.L) = [D],

œ(.L) = [DỊ — j.e((1)

Proof We may suppose that L = Op(V2—V;) is the line bundle associated

to a difference of two effective Cartier divisors Vi, V2 on D Begin with the

exact sequence

0— 3„Op(—VWi) — 7„Öp — kì,Öy, — 0,

where k;, is the inclusion of V; in X For simplicity of notation we shall

often drop the j, or k,, 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 (2.10) of c(j.Op) gives

(1+ei(Øp(—Wì)) + ca(Øp(—V\)) + - - -) =

= (1+[PJ + [Đ + -)(1~ [W] + -)””

Equating terms gives

e\(Op(-V)) = [DỊ, ca(Øp(—V.)) = [DỊ? + [Vị]

A similar calculation with the exact sequence

0 > jsOp(—-Vi) > jeOd(V2 — Vi) > kav, (V2 — Vi) 7 0

finishes the argument O

We shall also need the Riemann-Roch theorem for vector bundles on

curves and surfaces (see Exercise 17):

Theorem 2

(i) Let V be a vector bundle of rank r on the smooth curve C of genus g,

and let deg det V = d Then

x(C;V) = d+r(1- 9)

(ii) Let V bea vector bundle of rank r on the smooth surface X Then x(X;V) = a0) (10) = Kx) — œ(V)+rx(Øx)- n 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 8 smooth projective variety X The simplest method is to take a direct sum of two line bundles: V = L1® Lạ Of course,

we don’t expect to obtain especially interesting bundles in this way One

“way 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

‘EX ts a curve, then all rank 2 bundles can in fact be obtained in this way However, for a surface X, most interesting bundles ‘do not have such description

To classify such bundles V, we recall that all such extensions are classified

the group Ext! (Lo, Li) = H*(X; (L2)~'®L1) (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 ——¬ by —— V — l2 — > 0

I1 ]

0 —— Eị —— V' —— Lạ ——' Ô

A weak isomorphism of extensions is similary defined, except that we do

not require that the maps L; — L, 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 [; 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 scalars are endomorphisms

of V = V’, the diagonal subgroup of C* x C* acts trivially, and the weak

equivalence classes of extensions are the quotient of Ext (L2, lì) 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; @ Le

or are parametrized by a projective space P(Ext' (Lo, L)))

32 2 Coherent Sheaves

For example, if X = P, L; is necessarily Op2(a;) and

Ext} (Le, L1) = H'(P?; Op2(ai — az)) = 0

Thus, we cannot obtain any interesting bundles on the very simple surface

P? 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 H is a sufficiently ample line bundle Thus, there is an inclusion H—! >

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,, be the local ring of X at 2; it is a UFD since X is regular

Let y: L + V be a sub-line bundle After choosing local trivializations y

corresponds to an inclusion R — R@ R Thus, ¢ is (locally) determined by

¿(1) = (ƒ, g) € R@ R Now either f and g are relatively prime elements of

R or they are not relatively prime If they are relatively prime, consider the

map : R@ R — R given by (a, b) = ag— bf Clearly, Imp = (f,g)R=I

is the ideal generated by f 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

is exact

Proof If (a,b) € Ker, then ag = bf Since f and g are relatively prime,

fla and g|b Thus, a = fh and b = gh’ On the other hand, fgh = fgh’, so

that h = h’ Thus, (a,b) = @(h) cCImụ LÌ

It is also possible that f and g are not relatively prime Let t = ged(f,g),

where ¢ is not a unit in R, and write f = tf', g = tg’ Note that (R@®

R)/Imy has torsion, since (f',9') ¢ Imy but t(f',g’) € Imy There is

an induced map y’: R + R@ R defined by ¢'(1) = (f', 9’), and ¢ is the

composition of this map with multiplication by t Alternatively, y extends

to a map from (1/t)R to R @ R Globally, t defines an effective divisor D

on X and we can summarize these calculations in a coordinate free way as

't Lete: L— be a sub-line bundle Then there exists a unique effective

0 divisor D on X, possibly 0, such that the map ý factors through the

inclusion L + L @Ox(D) and such that V/(L® Ox(D)) is torsion

free

2 + +

(it) In the above situation, if V/L is torsion free, i.e., if D = 0, then there

` existg a local complete intersection codimension two subscheme Z of

ib -X and an exact sequence

ng n i if X=Ci = d V is an extension of

"Tn icular, if X = is a curve, then Z @ an line ales in Case (ii) Thus, since every rank 2 vector bundle has a sub- : Ife bundle, every rank 2 bundle over a curve can be written as an extension

ef line bundles Let us give a simple application of this, by classifying all

rank 2 vector bundles over a curve of genus 0 or 1 (The case of genus

@ special case of a theorem of Grothendieck [56], and the case of genus

due to Atiyah [4].)

Theo , rem 6

1): Let C =P" and let V be a rank 2 vector bundle over P! Then V = Op: (a) @ Op: (b) for integers a,b, unique up to order

ii) roa Che a smooth curve of genus 1 and let V be a rank 2 bundle on

» @, Then exactly one of the following holds:

a) V is a direct sum of line bundles;

a V is of the form € @ L, where L ‘is a line bundle on ơC and € is the (unique) extension of Oc by Oc which does not split into the direct

nts sum Oc @ Oc;

s{e) V is of the form 7p @ L, where L is a line bundle on C, p€ C, and

Zp is the unique nonsplit extension of the form

0 —= Øc — Fp > Oc(p) — 0

_ Proof (i) Let V be a rank 2 vector bundle on P' Since

deg det(V @ Op: (a)) = deg det(V) + 2a,

we may assume that degdet(V) = 0 or —1 Let us first assume that deg det(V) = 0, ie., detV = Op Then by the Riemann-Roch theorem applied to vector bundles on P!, we have x(V) = 2 Hence h (V) 2 2 Choose a 2-dimensional subspace of H°(V) and consider the associated

map 02, — V If this map is injective, then its determinant det Om =

Op: — det V = Op: is nonzero Hence the determinant map is an iso- morphism It follows that V ~ Oj:, which is certainly a direct sum of

line bundles Otherwise, the image of the map Of: — V is a line bundle L

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 Y Op: (k)

for k > 1 The map L —› V factors through a map L @ Op: (@) + V, where

£ > 0 and where the quotient is torsion free and thus a line bundle So in

this case we can write

0 — Opi (a) + V — Øp (~a) ¬ 0,

with a= k+£> 1 But the extensions of Op: (—a) by Op: (a) are classified

by H}(Op: (2a)), which is zero since a 2 0 Thus, the extension splits:

V & Op: (a) © Op: (—a)

In case degci(V) = ~1, then again by the Riemann-Roch theorem for

vector bundles, x(V) > 1 So there is a nonzero map Op: — V, which as

above factors through a map Op:(a) — V with a > 0 and the quotient is a

line bundle Thus, there is an exact sequence

0 Opi(a) + V > Opi(—a—1) 50

As H}(Op: (2a — 1)) = 0 for all a 2 0, this extension must again split and

V = Opi (a) ® Op: (—a — 1) We leave the uniqueness as an exercise

Next we prove (ii) Let C 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 deg det V = 0 or 1 First assume that deg det V = 0 Note the follow-

ing:

Claim Suppose that deg det V = 0 and that there is a nonzero map Ly >

V with deg Lo > 0 Then either V splits or V is of the form E€ @ 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 Tạ — V, so that

there is an exact sequence

0 ¬ LÙị >V ¬ Lạ —¬0,

with deg L, > 0 and deg Ly = — deg L, As we have seen, such extensions

are Classified by H1((L2)—! @ L;), which is Serre dual to H°(L2 @(L1)7})

If deg L, > 0, then deg Lo @ (Lì)~! = -2degL¡ < 0 Thus, H°(L¿ @

(L,)~1) = 0 and the extension splits, ie, V = L, @ Lo If deg L; = 0, then

either Lz @ (Li)? is nontrivial, in which case H' °(L2 @ (L1)~1) = 0 again,

or hạ @ (J1! = Oc, ie., Tị = hạ In this last case, dim H°(Oc) = 1,

and there is correspondingly a nonsplit extension € of Oc by Oc, unique

up to weak isomorphism Clearly, in this case, V = £ @ Thị O

Returning to the proof of (ii), with deg det V = 0, suppose that h°(V) #

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

at h°(V) = 0 Choose a point p € C, and consider the

oo Be ie Ve0c (p) By lạ Riemamn-Roch theorem, x(Ve0c(@)) =2

we chs h°(V @ Oc(p)) = 2 Choose a 2-dimensional subspace of H (Ve

Đi (p)) and consider the map O02, + V @ Oc(p) If the image of this map

4 line bundle, then as in the proof of (1) there is a line subbundle Lp o veOc0) with deg bọ > 2 Thus, V has a subbundle Tị =hạ@® Oc(-P)

‘with deg ZL; > 1 So we can apply the claim (and in fact V splits) n

the remaining case the induced map y: 02, + Ve Oc(p) is an ine usion

¥ comparing determinants, since we have det OZ = Oc and deg cà

Oc (p)] = 2, the determinant det ip must vanish at some point z € St us there is a nonzero v in the fiber C@C of OF, at with #z() = 0 Bu sine the induced map from H°(02,) to the fiber of OG over z is an isomorp ism, there exists a section s € H°(O2,) whose restriction to the fiber over x i

v Hence the induced map y|Oc - s vanishes at x Thus, the induc map

0 —» V @Oc(p) vanishes at x to order at least 1, and perhaps e ow ere

So there is a subbundle of V @ Oc(p) of the form Oc(d), where is an effective divisor on C containing x in its support Thus, V contains re

‘subbundle Oc(d) @ Oc(—p), which has degree > 0 Again by the claim,

oan we set 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

Oc — V If this map vanishes at some point, then we have an exac

Thus, V = L¡ @ Lạ The remaining case is where Oc is a subbundle of V

In this case we have an exact sequence

0¬ Øc ¬ V ¬ Ừc(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) O

Suppose that V satisfies (ii)(c) above Then it is a straightfor- and cxereio (Bxeriae 2) to show that V = V @ F for every line an PonC with F®? = Og 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 7p @ L = F, Otherwise, the descriptions of V in the three cases a

(ii) above are unique, up to permuting the factors of a direct sum of li bundles

36 2 Coherent Sheaves

We return to the general problem of understanding rank 2 vector bun-

dies The above discussion suggests that we should reverse the analysis of

Proposition 5 and try to construct vector bundles as extensions

0¬E—¬V—L'@1Tl;—0, where L and L’ are line bundles on X and Z is a local complete intersection

codimension 2 subscheme Thus, we must analyze Ext'(L’ @ Iz, L) (It

follows from (ii) of Lemma 7 below that the nontrivial weak isomorphism

classes of extensions will be parametrized by PExt!(1 @ 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

Ey" = H?(X; Ext’(L' @ Iz, L) => Ext?*4(L' @ Iz,L)

This spectral sequence is really just a long exact sequence, because of the

following:

Lemma 7 Let R be a regular local ring and let I = (f,g)R be an ideal

of R generated by two relatively prime elements Then:

(i) Homa(I, R) © R and the isomorphism is induced by the natural re-

striction map Homr(R, R) > Homn(1, R);

(ii) Homp(I, I) © R and again the isomorphism is induced by the natural

restriction map Homr(R, R) + Homa(I, R);

(iii) Exth(1, R) © Ext?,(R/I, R) & R/T;

(iv) ExtD(1, R) = 0 for k 3> 2

Proof Apply the functor Homa(-, R) to the resolution (2.11) of I We

obtain the short complex

0 — Exth(I,R)— R-% ROR Homa(I, R)

Here the transpose of y is the map (a,b) +> af + bg By the arguments

used to prove the exactness of (2.11), it follows that Homa(I, R) = R and

that Ext}(I,R) = R/I A check left to the reader shows that the restriction

map Homa(R, R) + Homa(I, R) in fact is an isomorphism From the exact

sequence

0¬Tï—¬ R— I/R—¬0,

we see that there is an inclusion

Homa(I, I) C Homn(1, R) > Homn(R, R) = R

On the other hand, R C Homg(I,Ï) by multiplication, and thus

Homr(J,I) = R acting by multiplication The higher Ext groups are zero

since J has a short free resolution Finally, the isomorphism Ext}.(1, R)&=

oN ‘ust as in Claim 4, the lemma is again a spec

ụ no bí Bxt (I, R) for an ideal I generated by a regular sequence

‘Jn the global case, Hom(L' @ Iz,L) =(L')"' @ L and Ext (L'S 2: )

da bundle supported on Z Tracing vhrovet tự TY sứ or

T i ify this line bundle with det(iz/1Z isnot to ae ie cow i ion, Iz/I2 is a locally free rank 2 flere, si i al complete intersection, Iz /1z

sheaf sae s its ite dual dual is by definition the norm: iti al bundle of Z in X This

rx iti l sở with the usual definition in case ition i Z is smooth, and is the oth,

eeneralization 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 seq’

Ve 0— H1((1)~!& L) ¬ Ext'(L' @ lz, L) —

—¬ H9(Ert1(U' @ Iz,L)) ¬ H°(()ˆ} ® 1)

TY enced to decide when an extension V, which a priors 18 Just 8

‘coherent sheaf, is in fact locally free IfV corresponds toe te ED) Close

t € Ext}(L’ @ Iz, L), we have the image of € in H°(Ext!(L' @Iz,L)),

there is the following theorem of Serre:

IÈbeorem 8 The extension corresponding to € is locally free if and only

if the section € generates the sheaf Ext)(L’ @ Iz,L), ie., the natur p

and the image of Id € Hompr(R, R) in Ext}(I, R) corresponds to the value

of the extension class in the stalk TP x Moreover, Bate (a D is ae et shơws that Extt(M, R) % ExtR(1, R) = 0 Thus, Extp(M, oS

> 1 if and only if Ext},(M, R) =0 if and only i

Tecondlade, wwe use the following theorem of Serre whose proof is deferred

to Theorem 17 below:

38 2 Coherent Sheaves

Theorem 9 Let R be a regular Noetherian local ring and let M be a finite

R-module Then M is free if and only if Ext‘,(M, R) = 0 for alli > 1

Corollary 10 Suppose in the above situation that X is a surface and that

H?((L')—' @ L) = 0 Then there exist locally free extensions V of L' @ Iz

byL O

Example On P?, take L = L’ = Ops Then H?(Op2) = 0, and so there

exist locally free extensions V on P? of the form

0- Op —- V1, > 0

We leave it as an exercise to show that, if Z 4 @, 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

H}((L')"} @ L) ——> Ext}(L' @ Iz, L) ———» H®(Ext(L' @ Iz, L))

Ext(U,L) ——— Ext!(L’@Iz,L) ——> Ext?(Oz, L),

where the bottom row is the exact sequence obtained by applying the long

exact Ext sequence to the exact sequence

0-L'@lz>L' 407-0

Proof The spectral sequence for Ext gives an isomorphism

Ext?(Oz, L) & H°(Ext?(Oz, L)) ¥ H°(Ext!(L' @ Iz, L))

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 O

Thus, it will suffice to analyze the image of Ext!(L’ @ Iz,L) in

Ext?(Oz, L), noting that the isomorphism

H® (Ext?(Oz, L)) ¥ H°(Ezt'(L' @ Iz, L))

has the property that a section of Ext!(L' @ Iz, L) is a generating section

if and only if the corresponding section of Ext?(Oz, L) is generating Now

we can apply Serre duality on the surface X to the Ext groups above, in

2 Coherent Sheaves

m= form which says that, for a coherent sheaf F on X,

Ext'(F, Kx) is dual 'eo H?-'(X:7')- Thus, the sequence

Ne pow suppose that Z = {pi,. sPn} consists of distinc D

“hus Oz = @, Op, and the duality pairing between Ext (Oz, y) enc

: , Jocal i it is induced by a direct sum

eee ata On Le On § Ces sy a ‘identi Oz,L) C" an

sat t each p:, we may identify Ext*(Oz,

5 ne 5 5 Thus the pairing between Ext?(Oz, L) and H® (Oz) i of the

° ben c ”) = » =>; aids where A, is a nonzero complex number , We have the following result:

Thee free extension of L' @ Iz by L exists!

Tem mm tion Aer! @ Kx which vanishes at all

but one of the pi vanishes at the remaining point as well

: Ha › f Let s be a section of Ext?(Oz, L) Then s is the image ofa see

g.Ext'(L’ @ Iz,L) if and only if O(s) = 0, where @ is the conne ting

Ề, iomorphism in the Ext long exact sequence Now 0(s) ; i xo) a0

F(Ø(2),e) = 0 for alle € H°(L'® L-! @ Kx) if and only if (s,

on e€ H9(L/ @ L~! @ Kx), where ô(e) is the image of e in

8 88 shove with s; z 0 for all i which is liftable toe et CVAY

HT nh le ho n nh te ° for ence of the folowing linear

The proposition is now an immediate Co

algebra lemma:

Lemma 13 Let (,) be the bilinear form on C” given by

(x, y) = > NLiYis

40 2 Coherent Sheaves

where 4; # 0 Let W be a vector subspace of C" Then there exists an

8 = (81, ,5n) € W+ with s; # 0 for alli if and only if W does not

contain 6; = (0, ,1, ,0) for any z

Proof If W contains 6; for some i, then (s,6;) = 48; Thus, if s ¢ W+,

then s; = 0 Conversely, if every s ¢ W*+ satisfies 5; = 0 for some i, then

W+ C UL, Hi, where H; is the hyperplane {s € C” : 5; = 0} As C is

infinite, there exists an i such that W+ C Hj But then W D H} = Cé,

and so 6; € W for somei O

Definition 14 We say that Z = {pi, ,pn} has the Cayley-Bacharach

property relative to the linear system L' @ L~! @ Kx if every section of

L' @ L~! @ Kx which vanishes at all but one of the p; 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-

quence of Lemma 17 in Chapter 5) Thus, nine such points have the

Cayley-Bacharach property relative to pz(3) Taking L = Op2(—3) and

L’ = Op:(3), so that L’ @ L~! @ Kx = Op:(3), 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 — Opa(—3) + V — Opa(3) @ 1z — 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 {p1, ,pn}

in general position on X, then there will be no section of L’ @ L~! @ Kx

which vanishes at all but one of the p; Thus, vacuously Z has the Cayley-

Bacharach property relative to L'@ L~1@K,x In this way we can construct

rank 2 vector bundles V on X with det V = L @ L’ such that co(V) is

arbitrarily large

More generally, a set of n points {pi, ,pn} in general position on X

will impose independent conditions on sections of L’ @ L~! @ Kx For such

points, we can count how many vector bundles we can construct by Theo-

rem 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 | DỊ

2 Coherent Sheaves

Elementary modifications

hat D is an effective divisor

son 15 Suppose that X is regular and t

te by j: D ¬ X the inclusion Let V be a rank 2 bundle on

vn * L a line bundle on D, and suppose that we are given a surjection

re joL Define W as the kernel of vn we“ the given surjection V ¬ j„L; thua

here jg an exact sequence

ty Site

is say that W is a elementary modification of V

Lemma 16 An elementary modification is locally free Its Chern classes are given by

| œ(W) = œ¡(V) — [DÌ,

co(W) = c2(V) — e1(V) - [D] + joes (Z)

oe Locall n X, with R = Ox,z, D is defined by i = i by an equation ý and !

root V- L is given by a surjection ø: R ® H — R/tR Since

Ro R is free, @ lifts toa map yp: ROR R which reduces mod £ to Ø-

Since y: ROR — R is surjective mod ý, it is surjective by Nekayen femme Since R is free, the surjection ¢ is split The kernel of ¢ is a ¥: " gummand of R@R, and since R is local it is a free rank 1 R-module and so

‘fsomorphic to R This gives a possibly different direct Tr Cooney W

9 i i he form (x,y) + y (mod ¢) Thus, 1o

Re ee ROtR + ei hic to R@tR & R@ R and so W is is locally free The statement locally free

‘Poout the Chern classes of W follows from the Whitney product formula

Given an elementary modification 0 — W — V — j„L — 0, the surjec-

tion V — j,L gives an exact sequence

0-L'-ViID—L-), where L’ is a line bundle on D, defined as the kernel of the map vịp >, L

Thus, there is an induced surjection W — 3„L Since detW = detV

@

Ox(—D), it follows that

det(W|D) = det(V|D) @ Ox(-D) = L @ L'@ Ox(—D)

42 2 Coherent Sheaves

Thus, Ker(W|D — L’) is the line bundle L @ Ox(—D) We leave it as an

exercise to show that if U is given by the elementary modification

0¬U¬W ¬7,L' ¬0, then % V @x(—D) In other words, repeating an elementary modifi-

cation in the obvious way essentially brings us back to where we started

Elementary modifications are the first case in understanding the struc-

ture 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-

lows: given a rank 2 bundle W on X and a line bundle L on the divisor

Dc X, we can try to construct extensions V of j,L by W Such extensions

are classified by the Ext group Ext)(j,L,W), and an easy exercise gives

Ext}(j.L, W) © H°(D; (W @ Ox(D))|D @ L~’)

We could also see this by noting that if W is an elementary modification

of V, then V, or equivalently V’, is also obtained as an elementary modi-

fication

0- VY = WY = j,(L~!) @ Ox(D) > 0,

and these are classified by

Hom(WY, j.(L7') @ Ox(D)) = H°(D;(W @ Ox(D))|D @ 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 R-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 regular local Noetherian ring and let M be a

finite R-module Then M is free if and only if Ext,(M, R) = 0 for alli > 1

if and only if Ext,(M, N) =0 for all i > 1 and all R-modules N

Proof Clearly, if M is free, then Ext,(M,N) = 0 for all i > 1 and

all R-modules N and, in particular, Ext,(M, R) = 0 for all i > 1 Con-

versely, suppose that Ext},(M,R) = 0 for all i > 1 First we claim that

Ext},(M,N) = 0 for all finite R-modules N Indeed choose a surjection

R™ — N — 0 with kernel N, Thus, there is an exact sequence

(M Mi) = Ext3,(M, Na) Eventually we obtain Exta(M Nao

Ha (M, Na), where d= dim F By the Hilbert syzygy

We claim that this exact sequence splits, i.e., that there is a map M — R®

ing identity map M — M If so, then M is a summanc i d of a free

ane ‘and thus it is projective Since R is local, this will imply

that

H we

TM Homn(M, R") —= Homp(M, M) — Exth(M, N)

Since Ext},(M, N) = 0, we can indeed lift Id € Homr(M, M) to an element

# Homg(M, tr), as desired L1

‘Gi -module M, recall that the support Supp M is defined to be

pe sper: M, # 0} = V(Annpr(M)), where, for a an ideal of Xà

là the closed subset of Spec R of prime ideals containing 4, corrosp

nding

to the support of the subscheme R/a In particular, Supp is a closed

aubecheme of Spec R, and is the smallest closed subscheme S se sheaf M on Spec R corresponding to M is zero on Spec R — 5

Torsion free and reflexive sheaves

Our goal in this section will be to describe two classes of sheaves wih rather mild singularities As in the previous section, we shall only consider

44 2 Coherent Sheaves

the local case where R = Ox,,, or more generally R is a regular local

Noetherian ring, and M is a finite R-module

Definition 19 Let Mors be the torsion submodule of M and define the

rank of M to be the dimension of the K-vector space M @p K, where K is

the fraction field of R M is torsion free if Miors = 0 A coherent sheaf F

on a regular scheme X is torsion free if and only if the R-modules F, are

torsion free for every x € X

Proposition 20 The following are equivalent:

(i) Mtors =0

(ii) There exists an inclusion M C R" for some n

(iii) There exists an inclusion M C R" where n = rank M

Proof Clearly, (iii) implies (ii) implies (i) To see that (i) implies (iii),

note that Mtors = 0 is equivalent to the natural map M > M®p K = K"

being injective, where n = rank M Let e1, ,€,, be a K-basis for M@pK

and let m, ,m, generate M over R Write m; = 3» rije;, where the

rij € K If r € R is a common denominator for the rij, Le rrụạ € H for

Thus, all torsion free modules are given as submodules of R” for some n

The standard example is the case of an ideal I, i.e., a submodule of R In

general, ideals I will not be free Indeed, since rank J = 1, J is free if and

Only if it is projective (recall that is local) if and only if it is principal

Corollary 21 Meors = Ker(M — MVYY)

Proof Clearly, Miors © Ker(M ++ MY”) To see the opposite contain-

ment, we may as well replace M by M/Mtors and show that in this case

(M torsion free) M injects into MY’ Using Lemma 20, choose an inclusion

M C R” for some n, and consider the following diagram:

the last section of this chapter (see also [117])

fe If M is torsion free, then codim S(M) > 2 In particular,

# example, i = i i ed by two relatively prime

2 Ro le, if M = (f,g)F is an ideal generat y ty

dyaments of R, neither one of which is a unit, then it is easy to see that

'g(M) is the scheme Spec R/(f,9)R which has codimension exactly 2

SỨ ig an obvious restatement of Proposition 22 for coherent sheaves:

% X be a regular scheme and let F be a torsion free sheaf on X Then

de exists a closed subscheme Y of X of codimension at least 2 such that

#|X - Y is locally free

‘a ini Xã 23 A finite R-module M is reflexive if the na i i ive i tural map M — ;

My so comorpbistn, Hence a reflexive module is torsion free Reflexive

yes on a regular scheme are similarly defined

Me We have the following results concerning reflexive modules, whose proofs

ie given at the end of the chapter (see also {117])

“Proposition 24 The following are equivalent:

Py There exists a finite R-module N such that MENY omV(a)>2

a M is torsion free, and, for all ideals a of R such that codim / (a) 22,

At the natural map H°(Spec R, M) — H°(SpecR — V(a),M) is an iso-

¬ morphism, where M is the coherent sheaf on Spec R naturally associ- ated to M

pha

Proposition 25 If M is reflexive, then codimS(M) 2 3 Hence if

dim R < 2, every reflexive R-module is free

We leave to the reader the formulation of the above results for a reflexive regular scheme

_— Corollary 21 for a regular local ring of dimension 2, we see inet every torsion free R-module M canonically sits inside a free rans it module, namely MY’ Conversely, we can obtain every torsion ree x module as follows: starting with R”, choose a submodule M such that ih has finite length Then M is a torsion free R-module with M oR 5 pe simplest example of this construction is @®?_¡ lz, C R", where a the

ideal of a 0-dimensional subscheme of Spec R Of course, not every M is

this form

Our final result about reflexive modules is the following

46 2 Coherent Sheaves

Lemma 26 A rank 1 reflexive R-module M is free

Proof Let M be a rank 1 reflexive R-module and set X = Spec R and

Y = S(M) C X If M is the sheaf on X induced by M, then M|X — Y

is a locally free Ox—y module of rank 1 and is thus a line bundle Since

codimY > 2 by Proposition 25, Pic(X — Y) = Pic R = 0, for example,

by (61, 11.6.5 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 O

Thus, to find examples of reflexive modules which are not free, we must

consider R-modules of rank at least 2, where dimR > 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

R-module M defined by the short exact sequence

0—¬ R— R3¬ M¬0, where the map # — RÖ sends 1 to (¿, ƒ,g) We leave it as an exercise to

show that M is reflexive If S = R/tR and ƒ, ø are the induced elements of

S, we may think of M as a typical 1-parameter deformation of the torsion

free S-module S @ J, where J = (f,9)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 Y be two smooth varieties and let f: X — Y be a finite

morphism of degree 2 If D is a divisor on X and Ox(D) is the associated

line bundle, then f,0 x (D) is a rank 2 vector bundle on Y 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 Y be a smooth variety and let B be a reduced effective divisor on

Y Suppose that Oy (B) is divisible by 2 in PicY and let L be a choice of

a square root, ie., L®? = Oy(B) Then we construct the double cover X

of Y branched along B associated to DL as a subvariety of the total space

of the bundle L:

X = {x: 2® = 5},

where s is any section of L®? corresponding to B It is easy to see that X

is smooth if and only if B is smooth In this case f*L = Ox(B), where

we have identified B with its inverse image on X Conversely, given f :

X — Y, we can recover B and L as follows Let 1 : X — X be the sheet

e involution Then B is the image under f of the fixed set of t

nd f.Ox = Oy @ L', where the direct sum decomposition corresponds

; taking 1 and —1 eigenspaces of v

° Let ne OT be the group of divisors on X and similarly for Div Y There

' the patural map f.: DivX — Div Y which is defined on a reduced

eweducible effective divisor D as follows: f.(D) = rf(D), where r is the

a of f|D: D — f(D) Hence r is either 1 or 2 In general we extend

“B: to all divisors by linearity (Of course, we can define f for any finite 'morphiam by the same procedure; see also Exercise 12 in Chapter 1.) It is

‘av to see that this f, is compatible with the map fa: H?(X) > H?(Y)

Veauced by Poincaré duality In addition, ƒ,ƒ” is multiplication by 2 in

'BRY and ƒ*/.(D) = D+ 4D) for all D € Div X We can similarly define

“don cycles of any degree It is easy to check directly that the projection ormula f.(D-f*E) = f.D-E holds if, for example, D and f*E are effective visors meeting transversally; the general case is done in [45] th

‘Note that f.(D) is a divisor whereas f.Ox(D) is 8 rank 2 bundle The

Slatlonship between the two is given by the following (cf also (61, IV, p

Bes of For D = 0, we have c1(f.Ox) = [det(Oy © L™)] = [L~*], which verifies the formula in this case since f.0 = 0 Next consider a general

divisor D = 55, n; Dj, where the D; are distinct reduced irreducible divisors

6a X The proof will proceed by induction on n = Yi Jm| Ifn=0, D=0

&éiid we have verified the proposition in this case Otherwise, choose some

in #0 If nj < 0, then we set D’ = D + D; Thus, D’ = TiniDi, where

ni =n, if i # j, and nj =n; +1, 8O that |nj| = |n;| — 1 Consider the

0 + Ox(D) — Ox(D’) > Op; (D’) > 0

Now f, is an exact functor since f is finite and therefore affine Thus, there

is an induced exact sequence

0¬ fr.Ox(D) — ƒ0x(D) = frOv,(D') — 0

The restriction of f to D; is a morphism of degree r = 1 or 2, and f.Op, is a vector bundle of rank r on ƒ(D;) By the comments after (2.8), œ(ƒ.Øp,) = r|[ƒ(D;)Ì = [f.D;] By the Whitney product for- mula we thus have c:(f,Ox(D)) = e1(feOx (D’)) - [f.D,] By induction,

48 2 Coherent Sheaves

œ(ƒ.Ox(DÐ')) = [ƒ.D"] — [L] Thus,

œ(ƒ.Øx(D)) = [f.D’] - (L] — [f.D;] = [#.DỊ - LH

This concludes the argument in case some ø; < 0, and a similar argument

handles the case where all ø; are >0 L)

A straightforward argument shows that the homomorphism from Pic X

to Pic Y in Proposition 27 is just the norm homomorphism

H!(X;Ø%) > H*(Y; OF)

To illustrate Proposition 27, let f: P! + P! be the degree 2 map given

by taking the double cover of P! branched at two points Then in the above

notation B consists of two points and L = Op: (1) Thus, c:(f.Op:(d)) =

d—1 We leave it as an exercise to show that, for d = 2k even, f,Op:(d) =

Op: (k) © Op: (k — 1), whereas for d = 2k + 1 odd, f.Op:(d) = Op: (k) @

Op: (k)

Our next result is a calculation of co(f.Ox(D)), originally proved by

Schwarzenberger via the Grothendieck-Riemann-Roch theorem

Proposition 28 With f: X — Y as above and D a divisor on X, the

following equality holds in H*(Y; Z([3)):

ca(ƒ.Ox(D)) = ‡(Œ.D)? — f.(D?) — f,D-L)

Proof To begin with, we note the following:

Lemma 29 There is a natural exact sequence

0+ Ox((D)) @ f*L —¬ ƒ*f,0x(D) > Ox(D) > 0

Proof The natural map ƒ*ƒ,Øx(D) — Ox(D) is surjective because f

is an affine map The kernel is thus the line bundle det f*f,Ox(D) @

Ox (—D) Using Proposition 27, det f* f.Ox(D) = ƒ*(0y(ƒ.D) 6 L~}) =

Øx(ƒ*ƒ.D) @ ƒ*L~1 = Ox(D + u(D)) @ ƒ*L~! Combining gives Lemma

29 O

Returning to the proof of Proposition 28, the Whitney product for-

mula applied to the exact sequence in Lemma 29 gives f*c2(f.Ox (D)) =

co(f*f.Ox(D)) = D-u(D) — D- f*L On the other hand, we have

god thus, in H*(X;@Z[B]), D-1(D) (with all coefficients in Z(3)) = 34°(f.D)? — 3D? — 3D) Thus,

-_ a(f,Ox(D)) = #.f*ea(/.Ox(Đ)) = fe(f* ƒ.Ox(D))

ch = f.[D-o(D) - D- ƒ*L]

: = (f.D)? — fa(D?) - (fx) -L,

‘where at the last stage we have used the projection formula and the fact

“that f.t(D)? = f,(D?) Dividing by 2 then gives the & formula in Proposition

‘93 O

by mple Let Q = P? x P! be a smooth quadric in P* and let 7: Q > Pp?

‘th 3 projection of Q onto a plane Then Q is a double cover of E2 via 7;

J nch locus is a smooth conic Hence, in the notation of this section,

sr (0) A basis for Pic Q is given by {fi, fo}, where the f, are lines

ay elonging to the two distinct rulings Thus, 7 (ƒ,) is a line in P° Let

2y be the rank 2 bundle 7,O(af: + bf2) on P’ Using Propositions 27

‘and 28, det Vay = Op2(a +b — 1) and c2(Vas) = 2((6” — 4) + (0? — b))

; le leave it as an exercise to determine when Vi, is @ direct sum of line

i ble cover However, the nonuniqueness of the construction makes this

‘vesult: mainly of theoretical interest

‘Proposition 30 Let Y be a smooth curve or surface and let V be a rank

\3 vector bundle on Y Then there exists a smooth double cover f: X > Y

“and a line bundle M on X such that V = f.M

De

‘Proof Let P = P(VY) + Y be the projectivized bundle associated to V

re our sign conventions are the opposite ones to those in, for example, [61] or EGA) Then for the tautological bundle Op(1) on P, we have (61, _ I, p 253, Ex 8] 7.Op(1) = V Let L bea sufficiently ample line bundle

on Y and let L' = Op(2) @ 1*L Then L’ is very ample on P (61, I, 7.10]

If X is the linear system corresponding to L’, then for X € X and £a

fiber of x, either 2c X or X meets (in exactly two points, counted with

multiplicity

‘Claim 31 There exists a smooth X € such that the induced map ƒ†:X — Y is finite and so is a double cover

50 2 Coherent Sheaves

Proof of the Claim Consider the incidence correspondence IC X x Y

given by

I = {(X,p): 07 '(p) c X}

and let 71, 72 be the projections of J to VY, Y, respectively Then 71 (J) is

the set of X which contain some fiber of 7, and we will be done by Bertini’s

theorem if we show that 7,(J) is a proper subset of 7 Let dim = N

Then we claim that dimI < N — 1 (it is here that we use the assumption

dimY < 2) To see this, let p € Y and let £ = 1~1(p) Then dim 73 ‘(p) =

dim{X € ¥: 2c X} = h°(P; L' @ I;) — 1, where I; is the ideal sheaf of 2

Consider the exact sequence of sheaves

0¬ ứ/@1,— Ư ¬ O,(2) ¬ 0

For L sufficiently ample on Y, for example, L = L2? where Øp(1) ® x* Lọ

is very ample on P, we claim that the map H°(P; L') + H°(é;O;(2)) is

surjective Indeed, since Op(1) @ x*Lp is very ample on P and restricts to

Oz(1) on 2, the map H°(P;Op(1) @ 1*Lo) — H°(€;O¢(1)) is surjective

But then L! = (Op(1) ® 1*Lo)®", so that H°(L’) contains the image of

Sym? H°(Op(1) @ m*Lo), and thus the image of H°(L’) in H°(é;O¢(2))

contains the image of Sym? H°(é;O;(1)) = H°(;O;(2))

As dim H®(é; O;(2)) = 3, it follows that dim '(p) = h°(P; L' @ Ie) =

(h®(1) — 1) - 3 = N — 3 Hence dimï = dimz; '(p) + dimY < N - 1

Thus, 7(Ï) must be a proper subset of 1’, proving the claim O

Returning to the proof of Proposition 30, choose X as in the claim It

suffices to prove that f,(Op(1)|X) = V Apply 7, to the exact sequence

0 Op(1) @ Op(—X) ¬ Øp(1) = Øp(1)|X — 0

We thus obtain a map V = z„Op(1) — T„(Øp(1)|X) = ƒ.(Øp(1)|X) To

see that this map is an isomorphism, note that for all € Y, Øp(1) @

Op(—X)|x~1(y) = Op:(—1) and that H°(Op:(—1)) = H'(Op:(-1)) = 0

Thus,

Rn (Op(1) @ Op(—X)) = Rit (Op(1) ® Op(—X)) = 0,

proving that the map V — f,Op(1)|X is an isomorphism O

Consider the example Y = P? Every smooth double cover 7: X — P? 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 Opa(1) But 7.7*Op2(n) = Opa(n) @ Opa (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 P? implies that there are many configurations of plane curves

a and Cz in special position, in the sense that C, is smooth and C2 is everywhere tangent to Ci

&

Appendix: some commutative algebra

Ou goal is to prove some of the more technical results used above We

‘begin by recalling the following definitions (cf [87] or [139]:

Definition 32 If R is a regular local ring and M is a finite R-module,

define the projective dimension proj dim M, by any of the following equiv-

alent ways:

1 the minimal length of a free resolution of M;

Š sup{k : there exists a finitely generated R-module N such that

“we set depth, M to be the maximal length of an M-sequence 21, ,T%

ch that z;¡ € a for all i Let us collect some salient facts about depth:

i ifdepth, M > k

' (iv) Ifa is an ideal of R with codima > k, then Hi(R) =0 for alli < k—1

Proof (i) This is [87, p.105]

(ii) This is the dimension formula of Auslander-Buchsbaum [87]

(iii) This is (61, p 217, Ex 3.4]

(iv) It suffices by (i) and (iii) to show that depth R, > k for all primes p€ V(a) Since R is regular, R, is regular, and thus

depth R, = dim R, = d— dim V(p) = codim Ví)

Since a has codimension at least k, codim V(p) > & and thus depth R, >

k oO

52 2 Coherent Sheaves

Lemma 34 For every R-module M,

dim Supp Ext(M, R) < d- i,

¡.e., codim Supp Ext’,(M, R) > i

Proof If p € Supp Ext’,(M, R), then the localization Ext,(M, R)p is not

0 As there is a natural isomorphism

Exti,(M, R)p & Extz, (Mp, Rp) Extz, (Mp, Rp) #0, and hence proj dim M, > i, where the proj dim is as

an R,-module As Ry is again a regular local ring of dimension d—dim V(p),

from

proj.dim M, + depth M, = d— dim Vip),

we obtain dimV(p) < d — 1 for all p € Supp Extj(M, R) Thus,

dim Supp Ext2(M, R) < d- ¡ ñ

Definition 35 M isa k“ syzygy module if there exists an exact sequence

0 M— R™ — Rm +++ > R™? > RB

Lemma 36 Let M be a kỲh syzygy module Then:

(i) proj.dimM <d—k

(ii) codim S(M) >k+1

(iii) For all ideals a of R, depth, M 2 k

Proof We claim that, for all 1 > 1, dim SuppExt(M, R) < d— k~— ¡ and

that, in particular, Supp Ext>(M, R) = Ú if ¿ > d—k Thus, by definition

dim S(M) < d—k—1 Moreover, we must have Ext,(M, R) = 0 fori > d—k,

so that proj.dimM <d—k as well We will prove the claim by induction

on k For k = 0, ie., no condition on M, this is just the syzygy theorem If

now M is a (k + 1)* syzygy module, then there exists an exact sequence

0¬ M—R"' ¬ M' ¬0,

where M’ is a k** syzygy module An argument with the Ext exact se-

quence shows that Ext,(M, R) = Extit!(M’, R) for all i > 1 Thus, by the

inductive hypothesis dim Supp Ext,(M, R) = dim Supp Extt'(M', R) <

d—k—(¡+1)=d—(k+1)T—¿ 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 R, is a flat R-module, M, is a k*" syzygy module

as well (for R,) Thus,

VM is a first syzygy module

“For all proper ideals a, depth, M 2 1

llary 38 If M is torsion free, then codim S(M) 2 2 In particular,

~, dimension 1, then every torsion free R-module is free

proof This is immediate from Lemma 36 0

& ‹

y

A) M is reflexive , tion 39 The following are equivalent:

brhere exists a finite R-module N such that M = NV

ii) "‘M is a second syzygy module

M) For all ideals a of R such that codim V(a) > 1, depth, M 2 1, and for

Sổ! all ideals a of R such that codim V(a) 2 2, depth, M > 2

M M is torsion free, and, for all ideals a of R such that codim V(a) > 2,

bi

% the natural map H°(Spec R, M) > H®(Spec R — V(a), M) is an iso-

morphism, where M is the coherent sheaf on Spec R naturally associ-

module is torsion free, H°(M) = 0 for all proper ideals a Thus, if M isa

second syzygy module we have an exact sequence

D—› M ¬ Rh' > M' >0

with M’ C R™ and hence torsion free Applying the long exact sequence

of local cohomology, we obtain

= H9(M’) > H}(M) + H3(R™)

From (iv) of Lemma 33, if codimV(a) > 2, then H}(R™) = 0 and hence

H}(M) = 0 The statement (iv) then follows from (iii) of Lemma 33

To see that (iv) ==> (v), note that depth, 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 depth, M > 2, and so H}(M) = 0 by (iii) of Lemma 33 again

Setting M = the sheaf on Spec FR naturally associated to M, we have a

long exact sequence

0 —= Hệ(M) —› H®(Spec R, M) — H*(Spec R — V(a), M) — HẠ(M)

Thus, H°(Spec R, M) H°(Spec R — V(a), M)

Finally, to see that (v) ==> (i), since M is torsion free we have M + MV

and codim $(M) > 2 Let M be the sheaf on Spec R naturally induced by

M and MVY that induced by MY’ We have a commutative diagram

H°(Spec R— S(M), ) ——— H°(Spec R— S(M), MỸ*)

H°(SpecR,M) —*°—¬ H°(SpeR,MVYV)

By hypothesis f is an isomorphism Since MY” is the dual of the R-module

MY, by using the implication (ii) => (iii) g is an isomorphism Thus, h is

an isomorphism, and so M = MYY, 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 R-module is a second syzygy module, the corollary

follows immediately from Lemma 36

Remark (1) The proof that (vi) implies (i) also shows that for a general

torsion free R-module M, MYY = H®(Spec R — S(M), M) For example, if

I is an ideal in R such that codim Spec R/I > 2, then IVY = R

(2) One can more generally show the following An R-module M is a k*®

syzygy module if and only if, for all 7 < k and for all ideals a of R with

codim V(a) > j, depth, M > j There is also an equivalent statement in

terms of local cohomology

Exercises

1 Let L, and Lz be line bundles on a scheme X Give an alternate

description of Ext'(Z2, L;) = H!((L2)~! ® L;) as follows If V sits in

the exact sequence

0-L1,-V—-L,-0, then the transition functions for V can be taken to be upper triangular, with the diagonal entries transition functions for Dạ and L2 How does the nonzero off-diagonal entry transform? Define in this way a section

of H!((L2)~! @ Ly), independent of choices, and then show that this

LG can be reversed Generalize this to vector bundles Wị, V2 of , arbitrary rank

; Prove the uniqueness assertions of the remark after the proof of The- orem 6: if F, is the nonsplit extension of Oc(p) by Oc, where C’ is

an elliptic curve, then F, ® L = F, if and only if L is a line bundle

of order 2 Conclude that Hom(F,,F,) = Oc @ Li © Lz © L3, where

the L,; are the nontrivial 2-torsion line bundles on C’ Moreover, for all

q€ C, there exists an L such that F, @ L = F, (here L is unique up v1, to a 2-torsion line bundle)

Let V be a vector bundle on P? which sits in an exact sequence

’ If Z #@, show that V is not a direct sum of line bundles

‘Let Z be a set of nine points in P? 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 — Op2(—3) — V — Opz(3) @ Iz — 0

#1 Show that in fact V is the trivial bundle

'§ Show, for a rank 2 vector bundle V, that VY = V @ det V-! Now 8Mppose that

0- W ¬VY_—¬j7,L':—>0

dc is an elementary modifcation of V, where j: D —› X is the inclu-

sion of a smooth divisor Show that VY is obtained as an elementary

modification of WY, and thus that V is an elementary modification of

„ Let 0 — W — V — j,L — 0 be an 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

0>U—W ¬+U”/¬Q0, then U is just V @ Ox(—D)

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 H and H' are line bundles, and let L = H'|D For the elemen-

tary modification W corresponding to the surjection V — j,D, show

that there is an exact sequence

0— H'(-D) > W ¬ H — 0

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

‘(Oc ® Oc) ® L, i-e., up to a twist by a line bundle, V is obtained

10

11

via elementary modifications from the trivial bundle (Starting with

an arbitrary exact sequence 0 —' H — V — H' — 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.) ‘

Let R be a regular local ring and F: R? — R? a homomorphism of

R-modules, corresponding to a 2 x 2 matrix Suppose that the matrix

is of the form ( h with gcd(ƒ,g) = 1 Show that R?/F(R?) =

Iz/Ig, where Iz = (ƒ, g)R and Ig = tR, where t = ƒhạ — ghì is the

determinant of the matrix

(Working a little harder, one can show that for every 2 x 2 matrix

whose entries are relatively prime, there exists a choice of bases for

the domain and range R?’s for which the first column has relatively

prime entries Note that if E is singular, for example nonreduced, then

the modules Iz/I¢ can be very complicated.)

Let R be a regular local ring of dimension at least 3, and choose a

regular sequence (t, f,g) Consider the R-module M defined by the

short exact sequence

0¬ R¬R°—M—0, where the map R — R° sends 1 to (t, f,g) Show that M is reflexive

(Show that there is an exact sequence

0— MỸ ¬ R3 —¬ I— 0,

where I = (t,f,g), and that Extp(J, R) = 0.) What is S(M) in this

case?

12 Let M be a complex manifold of dimension at least 2 and let Y C M

"be a closed analytic subspace of codimension at least 2 Suppose that Lis 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 = C?,

_ 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 refiexive 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 f: Pp! — P! be the degree 2 map given by taking the double cover

of P' branched at two points Show that, for d = 2k even, f.Op:(d) =

Op: (k) BOP (k—1), whereas for d = 2k +1 odd, f.Op:(d) = Op: (k)®

Op: (k) (It suffices by the projection formula to do the cases d = 0,1 The case d = 0 is done For d = 1, write f,Op:(1) = Opi(a) ® Op: (b) and calculate H®(Op:(1)) = H°(f,Op:(1)), and H°(f.Op: (1)@

Op: (—1)) = H°(Op:(1) @ ƒ„Om (—1)).)

{4 Let Q = P! x P! be a smooth quadric in P? and let 7: Q — P2 be the

‘double cover obtained by projecting Q onto a plane Let Va, be the

‘rank 2 bundle z„@q(øƒt + bƒ;) on P2, where ƒ and f2 are the two

rulings on Q, as in the example at the end of the last section Show

that, if a = b, then Vaa = 1(Og @ 1* Op2z(a)) = Op2(a) ® Op2(a — 1)

Tf la — b| = 1, say b = a +1, then show that Va,a41 = Op2(a) @ Op2(a)

In all other cases, show that V,, is not a direct sum of line bundles

15 Let X be a regular scheme Given a rank 2 bundle W on X and a line

&: bundle L on the divisor D C X, show that

Se Ext! (,L, W) & H°(D; (W @ Ox(D))|D ® L~)

18: “Let X be a regular scheme and Z a codimension 2 local complete inter- section subscheme of X Verify that there is a canonical isomorphism Ext!(Iz,Ox) & det Nz;x, where Nz/x is the dual of the locally free

rank 2 sheaf Iz/I3

17 Prove the Riemann-Roch theorem for vector bundles V of rank r on a

* gmooth curve or surface (Theorem 2) (For a curve C, show first that

there is some exact sequence

0>-LoV'—0, 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 E—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

0— V' W >Q—¬‹0,

18

where W is a vector bundle and Q has finite support Apply induction and use (2.8).)

Let f: X — Y be the double cover constructed by choosing a square

root L of the line bundle Oy(B), where Y and B and therefore X are

smooth Show that hi(X;Ox) = h'(Y; Oy )@hi(Y; L*) 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 Basa smooth divisor on X Conclude that f,.Kx = Ky @ (Ky @ L) and

thus that p,(X) = p(Y) + h°(Y; Ky @ L) Finally, if X and Y are surfaces, then K2, = 2K? + 4(Ky - L) + 2L? For example, if Y = P?

and B is a smooth curve of degree 2d, show that p,(X) = 0 if and only

if d = 1,2, that otherwise p,(Y) = (5 3 and that KỆ = 2(d—3)?

In this chapter, we describe the basic properties of surface birational ge-

ometry After reviewing the operation of blowing up a point on a surface

fy and the relationship between the invariants of the blown up surface X

Haine ithe invariants of X, we prove the Castelnuovo criterion for blowing

down a curve Using the Castelnuovo criterion, we show that every bira-

onal morphism between two smooth surfaces is a composition of blowups,

j discuss various notions of minimal models At the end of the chapter,

we discuss more general contractions to normal surfaces, with particular attention to rational singularities and rational double points

operation of blowing up goes by many names in the literature: o-

process, 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 F = p~'(p) be the exceptional divisor Thus, E=P' and deg Np, z = —1, 80 that E? = —1 and E- Kx = —1 by the adjunction _ formula We may describe X locally as follows: let U be a coordinate neigh-

| borhood of p, with coordinates x,y centered at p Let Ũ = p}`(U) Then

U is covered by two coordinate charts Ũ; and U2 Here Ũ has coordinates

z’,y’ and the map p is given by x = 2’, y= z'y' The chart U2 has coordi- nates x”, y” and in these coordinates p is given by z = zy", y=y" We glue U, — {y’ = 0} to U2 — {x” = 0} via

1 g! = „

y”= mhự

The exceptional divisor E is defined in Ún by z and in U2 by y” From this

it is easy to check that E % PÌ and that Np/x is identified with Op:(—1)

60 3 Birational Geometry

Thus, deg Nyx = —1 Also recall the universal property of X [61, p 164]:

if @: Y — X is a morphism such that gp”! (mz)Oy is the ideal of a Cartier

divisor on Y, then ¢ factors through X: there exists a unique morphism

2: Y +X such that yp = po

We will frequently use the following lemma:

Lemma 1 Let m, be the maximal ideal of p in X Then

Ox(az) = { PP ifa=—n<0,

POV Ox, ifa>0

Proof For a neighborhood U of p, let f € T'(p~'(U), Ox (aE)) Thus, f

defines a section of Oz over p~1(U) — E = U — {p} or in other words

a holomorphic function on U — {p} By Hartogs’ theorem f extends to

a holomorphic function on U It follows that there is a natural inclusion

j: psOxz(aE) — Ox for all a If a > 0, we can reverse this inclusion

too: if g € T(U,Ox), then p*g is a section of Ox over p~'(U) and thus a

section of O z(aE) over p~!(U), i.e., a section of p.Oz (aE) over U Clearly,

j(p* 9) = g and so p,Oz (aE) = Ox

For a = —n < 0, we must determine the image of j Equivalently, given a

function g holomorphic in U, it suffices to determine when p*g € Ox (aE)

Now we can write 9(z, y) = Scp m 9v (x,y), where g is homogeneous of de-

gree and gạ, # 0 Thus, m is the multiplicity multy g and g € my"; indeed

gem - met, Working, for example, in U, and using the coordinates

described above, we have

p°9(2',y') = g(a’, 2'y’) = (2')™(9m(1,y') + 29"),

where z’g’ vanishes along E and g,,(1,y’) does not vanish identically on

E Thus, we see that

p*g €T(p~}(U),Øz(—mE)) — T(ø~`(U),Øz(~(m + 1)E))

It follows that p*g € I'(p—?(U), Oz (—mE)) if and only if g € my*, which

was what we needed to complete the proof of Lemma 1 O

Definition 2 Let C be a nonzero effective divisor on X We define the

multiplicity multp C to be mult, g, where g is any local defining equation

for C 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 C’ 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 C’ M E can be identified with the projective

tangent cone to C' at p

’ We turn now to the structure of Pic X and Num X

hạ Proposition 3

i) pt: PicX — Pic X and ø*: Num X — NumX are injective

i) Num X = p* Num X © Z{E], and this direct sum is orthogonal with

¡› pespect to intersection pairing on X and X

ee

Proof The proof of (i) in the case of p*: Pic X — Pic X follows from the

projection formula p.p*L = L ® p.Ox = L The proof for p*: NumX —

Num X also follows from the projection formula and the definition of ps giTo see (1i), let Pic’ X be the set of L € PicX such that deg(L|E) = 0 Thus, Pic’ X is the kernel of the natural restriction map Pic X — Pic E = E;Note that deg@¿(aE)|E = ~a, so that, identifying a[E] € Z[E] with

9 (aE), we have a splitting Pic X = Pic’ X @Z[E] To prove (ii) it therefore fices to identify Pic’ X with p* Pic X Clearly, p* Pic X is contained in

XK Conversely, suppose that L € Pic’ X Consider L|X — E = X — {p}

ince Pic(X — {p}) = Pic X the line bundle L on X — {p} extends uniquely

» a line bundle on X (compare the proof of Lemma 26 in Chapter 2) Let

4 be the extension of L|X —E to a line bundle on X; in fact M = (ø„r)YY

n p*M|X — E % L|Ấ — E Thus, p*M @ L~' has a regular nowhere yanishing section over X — E = X — {p} Such a section extends to give Ì.meromorphic section of p*M @ L~' which can only have zeros or poles itdag EF Thus, p*M = L ® Ox(aE) for some integer a As both p*M and fhave trivial restriction to E, a = 0 This concludes the proof of (ii)

wTo see the direct sum part of (iii), it again suffices to show that every

De NumX with D- E = 0 is of the form p*D’ Representing D as the

pumerical equivalence class of a line bundle L with deg L|E = 0, this follows

from (ii) The fact that the direct sum is orthogonal is an easy consequence

a the projection formula, which implies that p*D - p*D' =D-p.p*D' =

D' D! and that p*D-E=D-p,.E=0 O

| The proof also shows that the projection of Num X onto the factor p* Num X ~ Num X is given by px Likewise, the projection PicX — p*PicX ~ PicX is given by L > (ø„L)YY In general we see that

p.(o*M ® Oz(aE)) is M if a > 0 and is M @ mỹ iŸ ø = —n < 0

' Here are some easy consequences of Proposition 3 (which could also be

checked directly, cf Exercise 1)

62 3 Birational Geometry

(iii) Ky = p*Kx + E

Proof Write p*C = C’ + mE, where C’ is the proper transform of C

Since E and p*C are orthogonal,

C! - E = (p*C — mE) - E = —m.E? = m

Next we note that (C’)? = C?—m? Finally, we can write Kz = p*Kx+aE

for some integer a By adjunction Kz - E + E2 = —2, and thus a= 1 LÌ

Corollary 5 c2(X) = c3(X) — 1 and p(X) = p,(X) More generally

P,(X) = Pa(X) for all n > 1

Proof The equality for c?(Ÿ) follows immediately from (iii) above To

see the other statement, note that

H°(X; KỆ") = H°(X,p` KỸ" @ Ox(nE))

= H%(X;p.|p' KỸ" ® Og(nE)Ì)

Now by the projection formula, if n > 0,

p.|o` KỆ" @ Øz(nE)] = KỆ" @p.Ox(nE) = KỸ" @ Ox = KỸ",

by Lemma 1, since n > 1 Thus, H°(X;K$") = H°(X; K$"), and so

P,(X) = P(X) O

Thus, the invariants P, are unchanged after blowing up Such a state-

ment would fail if we had considered the equally “natural” bundle K;' =

det Tx, for example

Finally, we note that q(X) is also unchanged under blowing up:

Proposition 6 q(X) = 4(X)

Proof One proof uses the topological fact that H!(X;Z) = H'(X;Z);

indeed an easy Mayer-Vietoris argument shows that Z\ (X;*) % m(X;*)

via py A second proof uses the fact that Pic? X = Pic 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 Pic? X A third

proof uses the Leray spectral sequence for the map p, (or the easy special

case that is in [61]) as follows We shall show that R'p,O = 0 Then by

the Leray spectral sequence

H(X; Og) = H'(X; p.Ox) = H*(X;Ox) = 4(X)

To see that R'p,Oz = 0, we apply the formal functions theorem [61, p

To evaluate H'(nE; Onz), use the exact sequence

0 @Øg(—(n — 1)E) > Onze > O(n-1)E — 0

of (1.10) of Chapter 1 Since Og(—(n — 1)E) & Op:(n — 1), it follows

that

H!(Oz(—(n — 1)E)) = 0 for all n > 0 Thus, by induction, starting with

case 1 = 1 where we know that H!(E;Op) = 9(E) = 0, we see that H}(Onz) = 0 for every n > 1 This concludes the proof that R'p,Oz =0 and thus of Proposition 6 O

_ To conclude this section, we will say more about the arithmetic genus

of an irreducible curve Let C be a (reduced) irreducible curve on X and suppose that p € C Let m be the multiplicity of C 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)

- initely: 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 C which is smooth This is embed-

ded resolution for curves on a surface Keeping track of the total change in Pa(C), we arrive at the classical formula for ốp:

by = > Mq(™Mg — 1)

2 q—P where the q are the “infinitely near” points to p and are defined as follows: they include p, all the points on C’ 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 C 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 — 1)/2

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 7: Y — X be a birational morphism between two

smooth varieties Let y € Y and let x = m(y) Then either there is a

Zariski open subset U of X containing x and a morphism U — Y which is

an inverse to 7, or there exists a hypersurface V on Y containing y such

that the Zariski closure of (V) has codimension at least 2 on X

Proof There is an inclusion of local rings Ox,2 C Oy,y and the two rings

have the same field of fractions An easy argument shows that Ox,2 = Ory

if and only if 7 is invertible on some Zariski open subset of X containing z

Writing Oy., as the localization of C[t, ,tn]/Z for some ideal J, we can

write t; = fi/gi, where fi,9; € Ox,2- By standard commutative algebra

[87], Ox,2 is a UFD Thus, we may assume that fj and g; are relatively

prime in Ox,z If g; is a unit for all ¿ (i.e., gi(y) # 0), then Ox,2 = Ory

and the first conclusion of the theorem holds Otherwise, g; is not a unit

for some i, so that {7*g; = 0} defines a hypersurface V on Y containing y

Moreover, in Oy,, ™* fj = 7* gi ti, and so both 2* f; and 1*g; vanish on V

Thus, f; and g; vanish on 1(V), and so the closure of m(V) is contained in

{fi = 9: = 0} Since f, and g; were assumed relatively prime, {fi = 9i =

0} has codimension 2 in X, and so the second alternative of Theorem 8

holds O

We can now prove the following fundamental result, known as the Castel-

nuovo criterion:

Theorem 9 Let Y be a smooth surface, and let E be a curve on Y such

that E ~ P! and E? = —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 Y as the blowup of X

Step I To find X, choose a very ample divisor H on Y We may also assume

after replacing H by a multiple that H'(Y; Oy (H)) = 0 Let a = H-E>0

Consider the linear system corresponding to the divisor H + aE Note that

(H + aE) - E = 0 We claim that the linear system |H + aE| has no base

points and that the image of the corresponding morphism y: Y — P* is

a surface X such that y(E) is a single point p € X and y|Y — £ is an

jsomorphism from Y — E to X — {p} To see this, note that since a > 0,

| H+aE| contains the subseries |H| which, since H is very ample, separates ints and tangent directions on Y —E Next we claim that |H+aE| has no base points along E Since Oy (H+aE)|E = Or, it will suffice to show that the map H°(Oy(H+aE)) > H°(Øy(H+aE)|E) = H°(Oz) is surjective

‘The cokernel of this map is H!(Øy(H + (a — 1)E))

Claim For0<k<a+1, H'(Oy(H + kE)) = 0

Proof of the Claim Consider the exact sequence 0¬ Øy(H +(k— 1)E) > Øy(H + kE)¬ Øg(H + kE) — 0

By assumption H'!(Oy(H)) = 0 Moreover, H'\(E;Og(H + kE)) = H}(Op: (a — k)) and this group vanishes for k < a+ 1 Thus, by induc- ton on k we see that H*(Oy(H + kE)) =0far0<k<a+l D

Bass

Thus, |H +aE| has no base locus and so defines a morphism y: Y — PN for some N Since (H + aE)- E = 0, y(E) is a point p Moreover, given xpoint q € Y — E, since |H| has no base locus, there exist curves in

|H + oE| vanishing along E but not along q, by using the subseries |H| Thus, y separates y(E) = p from the points of Y — E as claimed The morphism defined by y maps Y onto a projective surface Xo Taking X

to be the normalization of Xo gives the candidate for the blowdown of Y This concludes the proof of Step I

Step II Let p = @(E) as above We must identify Y with the blowup

„ X — X of X at p First we claim that X is smooth at p It suffices

‘to show that the completion Ox,» of the local ring of X at p is a formal power series ring Since X is normal and ¢ is birational, Ox = y.Oy By

‘the formal functions theorem,

Oxp = lim H°(Y; Oy /ms Oy)

But the sheaf Oy/m?Oy is supported on E and is thus annihilated by

some power of Ig = Oy(—E) In particular there is a surjective map

Oy /Oy(—NE) — Oy/m"Oy and it will in fact suffice to show that lim H°(Y ; Oy / Oy(—nE)) is a formal power series ring Consider the exact

It follows that the map

(Oy /Oy(—(n + 1)E)) + H°(Oy/Oy(—nE))

is surjective for every n > 0 For n = 1, we have the inclusion

H°(Oy(—E)/Oy(-2B)) c H°(Oy/Oy(-28)),

As Oy (~E)/Oy(-28) = Op (1), dim H®(Oy (—E)/Oy(—2E)) = 2, and

we can choose 2{" 2”) a basis for H°(Oy(—E)/Oy(—2E)) For all n > 1,

we can choose 2h), 2h) € H°(Oy(-E)/Oy(—(n + 1)E)) mapping onto

2°") 2f"-) The natural map

Sym” H®(Oy(—E)/Oy (-2E)) + H°(Oy(—nE)/Oy(—(n + 1)E))

can be identified with the isomorphism Sym" H°(Op:(1)) + H°(Op:(n))

From this, an easy induction shows that the C-algebra map

C[z”, 2] — H°(Oy /Oy(-(n + 1)E))

is surjective for every n Taking z; = lim 2{”, it also follows that the

induced map

C21, 22]] — lim H°(Y; Oy /Oy(—nE))

is surjective and hence an isomorphism (since Ox must have Krull dimen-

sion 2) Thus, X is smooth at p

Let mp be the maximal ideal of p Clearly, (y'm,)Oy C Oy(—-E) = Ip

Moreover, the images of 2; and z2 generate Iz mod I2, since Op:(1) is

generated by its global sections, and thus 2 and 22 generate Iz It follows

that (p~'m,)Oy = Ig Thus, by the universal property of blowing up the

morphism factors through the blowup p: X — X: there is a morphism

@: Y — X such that @ = øo Ø Clearly, Ø is birational Let # be the

exceptional curve on X Since Y is Projective, ¢ is surjective, and so we

must have ¢(E) = E In particular, Ø() is a curve on X But by Theorem

8 above, if ¢ is not an isomorphism, then there must exist a curve C on

Y such that £(C) is a point As ? is an isomorphism on Y — E, the only

possibility for such a curve C is C = E, but we have seen that G(E) is

again a curve Thus, ¢ is an isomorphism, identifying Y with the blowup

of X atp O

Definition 10 A curve E on a smooth surface Y such that EF = P! and

E? = —1 is called an exceptional curve The smooth surface X obtained

from Y via Castelnuovo’s criterion is called the contraction of Y along E,

or the contraction of E We also say that X is obtained from Y by blowing

down E

Here are two other characterizations of exceptional curves:

emma 11 E is an exceptional curve if and only if E? = E- Ky = —1 if

ind only if E? <OQ and E- Ky <0

| roof is left as an exercise — Next we turn to the factorization of birational morphisms

[heorem 12 Let t: Y — X be a birational morphism Then 7 is a

wmposition of blowups

Proof Given x € X, suppose that 7° is not a 2

oz) = C = LU, Ci is a curve on Y, and by Zariski’s connectedness

he am [61, p 279|, Œ is connected We will find a component C; of Cơ suet

tC, “Ky < 0 and C? < 0; by Lemma 11, C; is therefore exceptional anc

ye can contract it by Castelnuovo’s criterion The result is a new surtee

i induced continuous map 7: Y + X It is easy

Oe ‘hism: supposing that U is an affine neighborhood of ® contained nh

te for some n, the coordinate functions on A” define functions on T w)

ind thus functions on m—!(U)—C; = x~'(U)— {p}, where p = T( cản se then extend to regular functions on #~'(U) by Hartoge theorem nd 9

tỉ hism We may reapply the argument to the morp x; ting that sinee C has only finitely many components this procedure will

wr find C;, we shall first prove the following two claims:

Claim 1 There exist positive integers r; with Ky = £*Kx +33; rC¡+D,

where D is an effective curve disjoint from C

2

Claim 2 There exists an i such that 2; r;(C;-C;) < 0 and C? < 0

Proof that the claims imply Theorem 12 We have C? < 0 and

C;: Ky = » r;(C; - C¡) < 0 Thus, C¡ is exceptional

Proof of Claim 1 Choose local coordinates 2, 2¿ at z € * Given $

i , and let w 1, W2 be local coordinates or sooo at nha ae ww ; we) and given the local generating section

dw, A dwe

Proof of Claim 2 Choose a very ample divisor H on X Thus, H on and m*H -C; = 0 for all j There exists a curve in |H| passing through z,

68 3 Birational Geometry

so that m*H = H"+Ð`„ s;C; with s; > 0 for all j, where H’ is a curve in Y

which does not contain any of the C; in its support Moreover, by Zariski’s

connectedness theorem 2*H is connected, and so H’ C,; # 0 for some i

For every j,

0=z7*H- C;= H' C; +> si(C; , C;) + 8/(Œ;)`

tj

Moreover, H’- C; > 0 and (C; -C;) > 0 if i # j, and by the connectedness

theorem at least one of these is > 0 Thus, s;C? < 0, and therefore we

must have C? < 0 for all j

Suppose now that » r;(C; - Œ;) > 0 for all i By the Hodge index

theorem, since }°,1;C; is orthogonal to x*H, which has positive square,

(0, 75C;)? < 0, and ()°,1r;C;)? = 0 if and only if 7, r;C; is numerically

trivial But

0> (Đ»ø)' = 3 ri r;(C, + Œ,) >0

so that (5>; r;C;)? = 0 It follows that 3¬; r;C; is numerically trivial But

H’.C;, > 0 for alli and H’-C; > 0 some i Since the r; are positive, this

implies that H” - (57, r;C;) > 0, and so 3”, r;C; is not numerically trivial,

a contradiction Thus, there exists an i such that 3”, r;(C; - Œ;) < 0, as

claimed O

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 + Y be a rational map from the smooth surface

X to a projective variety Y Then there exists a sequence of blowups X,

-»—+ Xo = X and a morphism f: X, — Y such that ƒ and f agree ona

Zariski open subset of X„

Proof Clearly, we may assume that Y = P” and that f(X) is nondegen-

erate, so that f corresponds to a linear system £ on X without fixed curves

We may further assume that N > 1 and that £ actually has base points If

p is a base point for £, let p: X; — X be the blowup of X at p and let E be

the exceptional curve If D € £ and D’ is the proper transform of D, then

p*D = D' + kE Let ko be the minimum possible value for k as D ranges

over the elements of £ Thus, p* D—koE is effective for all D € CL, and there

exists a Do € L such that p* Do — kp E = Dy is the proper transform of Do

Since p is a base point, ko > 1 Thus, (p* Do — ko E)2 = Dậ — kệ < Dậ Set

£ = {p*D - kọE : DE L} Thus, L’ consists of effective divisors and the

base points of £ are the base points of £ other than p together with some

possible base points along E The only possible fixed curve of CL’ would be

t, by the choice of ko, E is not a fixed curve of £L’ Hence £L’ has no

Ba res It follows that (D,)? > 0 for D; € L’ Clearly, Le L', and the rational map from X, to P% defined by CL’ agrees with f away from E

and the points of X; corresponding to points of indeterminacy of f other

than p If C’ has no base points, we are done Otherwise, continue this procedure If D;, denotes a typical element of the linear system at stage k,

then 0 < Dz < D?_, Thus, this procedure cannot continue indefinitely,

and eventually we reach a base point free linear series O

Corollary 14 Let f: X » Y be a birational map between two smooth surfaces Then there exists a smooth surface Z and morphisms 7 : Z¬Xx, 1: Z — Y, such that 7m, and m2 are sequences of blowups and f 07 = %2

in the sense of rational maps (i.e., where defined)

in homogeneous coordinates This map (called a Cremona transformation)

is defined as long as (20, 21,22) ¢ {(1,0,0), (0, 1,0), (0,0, 1)} We leave it

as an exercise to show that f becomes a morphism on the blowup of P* at these three points and that the birational morphism from the blowup to

‘the target P? consists exactly in contracting the proper transforms of the

‘three lines joining pairs of points in {(1,0,0), (0, 1,0), (0,0, 1)}

A base point p of a linear series £ is called a simple base point if the following holds: if p: X — X is the blowup of X at p, the linear series

L'={p*D-E:DeEL}

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 £ 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 £ = |D — p| of all curves in D passing through p We say that p is an assigned base point of CL Any other base points of £ (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 or are singular at p

(this corresponds to looking at |p*D — E — qj, where qé E) Likewise, the linear system of all curves in |D| containing p which are singular at p,