Stable Vector Bundles on Algebraic Surfaces 1. Introduction

Stable vector bundles on algebraic surfaces

WeiPing Li and Zhenb o Qin

ABSTRACT We prove an existence result for stable vector bundles with arbitrary

rank on an algebraic surface and determine the birational structure of certain mo duli

space of stable bundles on a rational ruled surface

Intro duction

Let M r c c b e the mo duli space of Lstable in the sense of MumfordTakemoto

L 1 2

rankr vector bundles with Chern classes c and c on an algebraic surface X The

1 2

nonemptiness of M c has b een studied by Taub es Gieseker Artamkin

L 2

Friedman Jun Li etc The generic smo othness of M c c has b een proved

L 1 2

by Donaldson Friedman and Zuo For an arbitrary r and c Maruyama

proved that for any integer s there exists an integer c with c s such that

2 2

M r c c is nonempty however no explicit formula for the lower b ound of c was

L 1 2 2

given Using deformation theory on torsionfree sheaves Artamkin showed that if

c r max p then the mo duli space M r c is nonempty and contains a

2 g L 2

vector bundle V with h X adV where adV is the tracefree subvector bundle

of E ndV Based on certain degeneration theory Gieseker and J Li announced

the generic smo othness of the mo duli space M r c c

L 1 2

In the rst part of this pap er we determine the nonemptiness of M r c c in

L 1 2

the most general form and show that at least one of the comp onents of mo duli space

is generically smo oth Using an explicit construction we show the following

Theorem For any ample divisor L on X there exists a constant depending

only on X r c and L such that for any c there exists an Lstable rankr bund le

1 2

V with Chern classes c and c Moreover h X adV

1 2

This is proved in section Our starting p oint is the classical CayleyBacharach

prop erty A wellknown result see p in says that there exists a rank bundle

given by an extension of O L I by O L if and only if the cycle Z satises the

X Z X

CayleyBacharach prop erty with resp ect to the complete linear system jL L K j

that is any curve in jL L K j containing all but one p oints in Z must contain

Mathematics Subject Classication Primary D J Secondary J

Key words and phrases Mo duli space stable vector bundle ruled surface

the remaining p oint It follows that to construct a rankr bundle V as an extension of

(r 1)

O L I

X i Z

i=1

by O L we need only to make sure that Z satises the CayleyBacharach prop erty

X i

with resp ect to jL L K j for each i Now let L b e an ample divisor and

i X

normalize c such that r L c L Let L c r L and L L Our

1 1 1 i

main argument is that if the length of Z is suciently large and if Z is generic in the

i i

(Z )

Hilb ert scheme H il b X for each i then the vector bundle V is Lstable and

h X adV

Similar construction for stable rank bundles is wellknown

We notice that there have b een extensive studies for stable rank bundles on

P and on a ruled surface and for stable bundles with

arbitrary rank on P In the rest of this pap er we study the structure

of M r c c for a suitable ample divisor L on a ruled surface X In section we

L 1 2

prove that M r c c is empty if c f is not divisible by r and that M r tf c

L 1 2 1 L 2

is nonempty if r t and c r moreover we show that the restriction of

any bundle in M r tf c to the generic b er of the ruling must b e trivial

L 2

In section we assume that X is a rational ruled surface and verify that a generic

bundle V in M r tf c sits in an exact sequence of the form

L 2

M M

O n f V O

X i i f

i=1 i=1

where ff f g are distinct b ers with b eing the natural emb edding f X

1 c i i

and the integer n is dened inductively by The idea is a natural generalization

of those in Since the restriction of V to the generic b er is trivial V is a

rankr bundle on P thus we can construct r exact sequences

O n f V V

X i i1

where i r V V and V is a torsionfree ranki sheaf By estimating

r i

the numb ers of mo duli of V and V we conclude that for a generic V the sheaves

V V are all lo cally free and V O c n f I where Z consists of c

2 r 1 X 2 1 Z 2

p oints lying on distinct b ers Then the exact sequence follows

In section based on we dene a rational map from M r tf c to P

L 2

and show that the b er is unirational We thus obtain our second main result

Theorem Let X be a rational ruled surface Assume that the moduli space

M r tf c is nonempty where r r t and L satises the condition

L 2

Then M r tf c is irreducible and unirational

L 2

One consequence of Theorem is that the mo duli space M r c on P

L 2

which is known to b e irreducible is unirational In fact we shall show that any

irreducible comp onent of a nonempty mo duli space on a rational surface is unirational

and determine the irreducibility and rationality in rank case Details will app ear

elsewhere

Acknowledgment The authors would like to thank Jun Li and Karien OGrady for

some valuable discussions They are very grateful to the referee for useful comments

and for p ointing out a mistake in the previous version The second author also would

like to thank the Institute for Advanced Study at Princeton for its hospitality and its

nancial supp ort through the NSF grant DMS

Notations and conventions

X stands for an algebraic surface over the complex numb er eld C The stability

of a vector bundle is in the sense of MumfordTakemoto Furthermore we make no

distinction b etween a vector bundle and its asso ciated lo cally free sheaf

K the canonical divisor of X

p h X O K the geometric genus of X

g X X

Z the length of the cycle Z on X

H il b X the Hilb ert scheme parametrizing all cycles of length on X

r an integer larger than one

V c V LrankV where L is an ample divisor on X and V is a torsionfree

L 1

sheaf on X

adV kerTr E ndV O Then E ndV adV O

X X

x the integer part of the numb er x

When X is a ruled surface we also x the following notations

a ruling from X to an algebraic curve C

f a b er to the ruling

a section to such that is the least

r ba where L a bf and a

df d where d is a divisor on C In this case d stands for degreed

P the generic b er of the ruling

Existence of stable bundles on algebraic surfaces

The CayleyBacharach prop erty

Fix divisors L L L and reduced cycles Z Z on the algebraic surface

1 r 1 1 r 1

T S

X such that Z Z for i j Put Z Z and

i j i

(r 1)

W O L I

X i Z

i=1

Let W b e the obvious quotient W O L I It is well known that there exists

i X i Z

an extension e in Ext O L I O L whose corresp onding exact sequence

i X i Z X

O L V O L I

X i X i Z

gives a bundle V if and only if Z satises the CayleyBacharach prop erty with resp ect

i i

to the complete linear system jL L K j ie if a curve D in jL L K j

i X i X

contains all but one p oint of Z then D contains the remaining p oint Note that

(r 1)

1 1

Ext W O L Ext O L I O L

X X i Z X

i=1

In the following we study the existence of a bundle V sitting in an extension

O L V W

Prop osition There exists an extension e Ext W O L whose corresponding

exact sequence gives a bund le V if and only if for each i r the cycle

Z satises the CayleyBacharach property with respect to jL L K j

i i X

Proof Put e e e where e Ext O L I O L Let V b e the

1 r 1 i X i Z X i

subsheaf O L I of V Then V is given by the extension e

X i Z i i

O L V O L I

X i X i Z

Note that V is lo cally free outside the cycle Z and sits in an exact sequence

V V W

i i

Since W is lo cally free at the p oints in Z we see that V is lo cally free at the p oints

i i

in Z if and only if V is lo cally free at the p oints in Z that is Z satises the Cayley

i i i i

Bacharach prop erty with resp ect to jL L K j Hence our result follows

i X

Corollary If h X O L L K I for every i and for every

X i X

Z fxg

x Z then there exists a bund le V sitting in the exact sequence

Construction of a rankr bundle V

Let L b e a very ample divisor on X and let V b e a rankr bundle Note that

c V O nL c V nr L

1 X 1

Thus by tensoring some line bundle to V we may assume that r L c V L

Without loss of generality from now on we x a divisor c with r L c L

1 1

We start with three lemmas In these lemmas we prove certain prop erties satised

by a generic cycle in the Hilb ert scheme H il b X when is suciently large

Lemma Let Z be a generic cycle Z in the Hilbert scheme H il b X

0 0

i If h X O r L c K then h X O r L c K I

X 1 X X 1 X Z

ii If p then h X O K I

g X X Z

Proof This is straightforward

Lemma Let maxp h X O r L c K Then a generic cycle Z

g X 1 X

in the Hilbert scheme H il b X satises the CayleyBacharach property with respect

to jr L c K j moreover h X O K I

1 X X X Z

Proof In view of Lemma ii we need only to prove the rst statement Dene an

op en dense subset U of H il b X such that if Z U then Z is reduced and

h X O r L K c I

X X 1 Z

By Lemma i this can b e done Dene V to b e the op en subset of H il b X

consisting of reduced cycles Hence U is an op en dense subset of V Dene Z to

b e the universal family in V X

Z f Z x V X j x Z g

Then there is a surjective morphism Z V given by Z x Z x Hence

+1 1 +1

Z U is a prop er closed subset of Z Dene the natural pro jection

Z V X V

+1 +1

+1 1

Then is a at surjection and Z U is a prop er closed subset of V So

+1 1 1 1

we can cho ose an element Z V Z U Hence Z U

this means that for any p oint x in Z Z x U that is we have

h X O r L K c I for any x Z

X X 1 Z x

So Z satises CayleyBacharach prop erty with resp ect to jr L K c j

X 1

The ab ove two lemmas will b e used to construct a rankr bundle while the following

lemma will b e used to show the Lstability of that bundle

2 2

Lemma There exists a reduced cycle Z of length Z r L such

0 2

that if h X O F I then we have F L r L

X Z

Proof Cho ose r distinct smo oth curves L L in the complete linear

2(r 1)

system jLj Cho ose a set Z of r L many distinct p oints in the op en subset

L L

i j

j =i

2(r 1)

of L Let Z Z Supp ose that h X O F I Then F is

i X Z

i=1

eective If F contains all the curves L as its irreducible comp onents then

F L r L

If F do esnt have L as its irreducible comp onent for some i then F L Z and

i i

r L F L F L Z

Z such Now for i r we can cho ose a reduced cycle Z Z

i i

that Z is chosen as in Lemma and Z is chosen as in Lemma moreover we

i i

r 1

may assume that Z Z are disjoint Put Z Z and

1 r 1 i

i=1

(r 1)

W O L I

X Z

i=1

for any x Z Since h X O r L K c I

X X 1

x Z

00 0

h X O r L K c I

X X 1

x Z Z

i i

Z Hence Z satises the CayleyBacharach prop erty with for any x Z Z

i i

resp ect to jr L K c j By Corollary there is a bundle V sitting in an extension

X 1

O c r L V W

X 1

Note that c V c and that since Z is nonempty the extension is nontrivial

1 1

LStability of the vector bundle V

In the following we show the Lstability of the bundle V constructed ab ove

Lemma The rankr bund le V in is Lstable

Proof Let U b e a prop er subvector bundle of V such that the quotient V U is torsion

free Let U b e the image of U in W and let U b e the kernel of the surjection

2 1

U U Then we have a commutative diagram of morphisms

O c r L V W

X 1

U U U

1 2

Case a U Then c U c r L E for some eective divisor

1 1 1 1 1

(r 1)

E From U W we have U W O L thus

1 2 X

r 1

r (r 1) r

2 2

O L O r L U

X X 2

where r is the rank of U Thus c U r L E for some eective divisor E and

2 2 1 2 2 2 2

c U c r r L E E

1 1 2 1 2

It follows that c U L c r r L L Therefore

1 1 2

c U L c r r L L c L

1 1 2 1

U V

L L

r r r

2 2

Case b U Then U W thus we see that

r r

U W O r L I

X Z

i2 i

r denotes the rank of U and runs over the set of r choices from r letters where

It follows that for some and for some i h X O r L c U I In

X 1 Z

r L c U I In view of Lemma we have particular h X O

1 X

2 2

r L c U L r L r L

So c U L r L r c Lr and U V

1 1 L L

Thus in b oth cases U V Therefore V is Lstable

L L

In the next lemma we are going to prove that h X adV that is the

irreducible comp onent of M r c c containing V is generically smo oth equivalently

L 1 2

this means that the versal deformation space of V is smo oth

Lemma Let V be the rankr bund le in If r L K L then

i HomW V O K

X X

ii h X adV

Proof i Let HomW V O K Then induces a map from W to

X X

V O K such that we have commutative diagram of maps

X X

(r 1)

W W O L

y 0

V O K

X X

To show that it suces to show that H X V O K L

X X

Since c L and K L r L c r L K L Thus

1 X 1 X

H X O c r L K

X 1 X

By our choice of the cycles Z H X O K I Thus

X X

H X W O K L

X X

Now tensoring by O K L and taking cohomology we see that

X X

H X V O K L

X X

ii We follow the argument as in the pro of of Lemma in By the Serre

2 0

duality we have H X adV H X adV O K Let

X X

0 0

H X adV O K H X E ndV O K

X X X X

Then we obtain a map from V to V O K Consider the diagram

X X

O c r L V W

X 1

O c r L K V O K W O K

X 1 X X X X X

Thus By our choice of the cycles Z H X O r L c K I

X 1 X

HomO c r L W O K

X 1 X X

so Applying HomO c r L to we obtain

X 1

H X O K HomO c r L V O K

X X X 1 X X

HomO c r L W O K

X 1 X X

It follows that there exists H X O K such that

X X

where Id is the identity morphism in EndV Thus Id Applying

V V

Hom V O K to we get an exact sequence

X X

HomWV O K H X E ndV O K

X X X X

HomO c r L V O K

X 1 X X

From i we conclude that Id Since Tr Hence

h X adV

Finally we state and prove the main result in this section

Theorem For any ample divisor L on X there exists a constant depending

only on X r c and L such that for any c there exists an Lstable rankr bund le

1 2

V with Chern classes c and c Moreover h X adV

1 2

Proof We may rescale the ample divisor L such that L is very ample and that

2 2

r L K L Note that c W r L and c W Z r r L

X 1 2

From the exact sequence we see that c V c and

1 1

c V Z r c L r r L

2 1

By the construction of the cycle Z we get

(r 1)

Z Z Z

i i

i=1

0 2 2

r maxp h X O r L c K r L

g X 1 X

Let b e the integer

0 2 2

r maxp h X O r L c K r L

g X 1 X

r c L r r L

Then dep ends only on X r c and L By Lemma for any c there exists

1 2

an Lstable rankr bundle V with Chern classes c and c

1 2

2 2

Moreover since r L K L h X adV by Lemma ii

Remark In Artamkin showed that M r c is nonempty whenever

L 2

c r max p

2 g

in particular when we only consider the case of c the lower b ound of the integer

c do es not dep end on the ample divisor L By contrast the constant in Theorem

dep ends on L In fact if we want a universal lower b ound of c for all c this

2 1

b ound must dep end on the ample divisor L We shall see this fact from Theorem

in the next section that on a ruled surface there exists a divisor c such that for any

integer c we can nd an ample divisor L with M r c c b eing empty

2 L 1 2

Restriction of a stable bundle on a ruled surface to the generic b er

From now on we study stable bundles on a ruled surface X Our rst goal in this

section is to show that if c f r and if r then M r c c is empty

1 L L 1 2

Theorem Let c f r Then there exists a constant r depending only

1 0

on X r c and c such that M r c c is empty whenever r r

1 2 L 1 2 L 0

Proof Assume that V M r c c Let c a bf then a r For any

L 1 2 1

divisor k on C we see that c V O kf a r b r kf and that

1 X

r r

kf c V O kf c r a bf kf

2 X 2

By the RiemannRo ch formula we conclude the following

ea a

V O kf a k a b g c

X C 2

Let k g b c a ea Then V O kf Thus

C 2 X

h X V O kf where i or On the other hand put

k r b r O er k r b

r maxfe e g

r a r a

Then r is a numb er dep ending only on X r c and c If h X V O kf

0 1 2 X

then there exists an injective map O kf V By stability of V we see that

kf L a bf Lr By direct calculations we get

k r b

r e

r a

but this contradicts with the choice of the numb ers r and r

0 L

2 0

If h X V O kf then h X V O K kf Hence

X X X

there is a nonzero map V O K kf which can b e extended to

X X

V O K kf O E I

X X X Z

for some eective divisor E By the stability of V we must have

c V Lr K L kf L E L K L kf L

1 X X

By a straightforward calculation we obtain that

r O er k r b

r e

r a

again this contradicts with our choices of r and r

0 L

Therefore if r r the mo duli space M r c c is empty

L 0 L 1 2

Remark Theorem only says that for a xed c with c f r and for

1 1

a xed c the mo duli space M r c c is empty for some sp ecial ample divisor L

2 L 1 2

eg when r r For other ample divisor L M r c c can b e nonempty see

L 0 L 1 2

when r we will discuss this issue in other places

In view of Theorem our next goal is to study the mo duli space M r tf c

L 2

where r t Let V M r tf c where L is of the form r f with

L 2 L

r maxfe O r g jc j jej r g jc jg

L X C 2 C 2

We want to show that the restriction of the stable bundle V to the generic b er is

trivial To start with we prove the following technical lemma

Lemma Let U be a ranks bund le with an injection U V

i For any divisor d with d r g jc j h X U O df

C 2 X

ii If c U af with a r sg jc j and c U c then U sits in

1 C 2 2 2

U U O nf I

1 X Z

where U is a ranks bund le with an injection U V moreover c U

1 1 1 1

a nf with a n r s g jc j and c U c

C 2 2 1 2

2 0

Proof i By the Serre duality h X U O df h X U O K df If

X X X

h X U O K df

X X

then we have O df K U V by the stability of V we obtain that

X X

tf L

df K L

On the other hand we have df K L d e O r in view of

X X L

the assumption but this is a contradiction

ii By the RiemannRo ch formula one checks that

U O kf s k s O a c U s k s O a c

X X 2 X 2

Let k g c as Then U O kf Since

C 2 X

c r sg jc j

2 C 2

k g r g jc j

C C 2

h X U O kf by i thus there is an exact sequence

U U O kf E I

1 X Z

where E is eective and Z is a cycle Since U U is torsionfree U is a bundle Let

1 1

E f Then moreover when e and e when e

We claim that otherwise then

c U L a k f L

1 1

r e a k

r e jej a k

But

a k r sg jc j g c as

C 2 C 2

r sg jc j g jc j

C 2 C 2

r s g jc j

C 2

r g jc j

C 2

So c U L r jej r g jc j by our assumption ab out r but this

1 1 L C 2 L

contradicts with the stability of V Therefore E is supp orted in the b ers of the

ruling and U sits in the desired exact sequence moreover c U c U c Note

2 1 2 2

that c U a nf and that a n a k r s g jc j By the

1 1 C 2

stability of V a ns tr Thus a n

Theorem Let V M r tf c where r t and L satises Then

L 2

O V j

Proof By Lemma ii and by induction on the rank of subbundles of V we conclude

that there exists a ag of subbundles of V V V V V V such that

1 2 r 1 r

rankV i c V c c V b f with b r g jc j for i r and

i 2 i 2 1 i i i C 2

O V V O b b f I where Z is a cycle Hence V j

i i1 X i1 i Z i

Next we prove the following simple observation

Lemma If the moduli space M r tf c is nonempty then it is smooth with

L 2

2 2

dimension r c r g in particular c g r r

2 C 2 C

Proof Since L satises K L By a wellknown result of Maruyama

M r tf c is smo oth with the exp ected dimension r c r g

L 2 2 C

We notice that the ample divisor L in Theorem dep ends on the integer c

that is the condition However in our existence result Theorem the

integer c has to b e bigger than some constant dep ending on L Thus Theorem

can not apply to the present situation to guarantee the nonemptiness of the mo duli

space M r tf c The following result deals with this problem

L 2

Prop osition Let r r t and L r f with r jej r

L L

If c r then the moduli space M r tf c is nonempty

2 L 2

We omit the pro of since it is a slight mo dication of the pro of of Theorem

replacing the L in W by f It seems to us that a stronger result should hold that is

if c r t then M r tf c is nonempty see Theorem iii

2 L 2

Generic bundles in M r tf c on a rational ruled surface

L 2

From now on X will b e a rational ruled surface In this section we will study the

structure of a generic bundle in M r tf c where L satises and r t

L 2

Exact sequences asso ciated to a bundle V in M r tf c

L 2

In this subsection we will construct r exact sequences for each vector bundle in

the mo duli space M r tf c We b egin with two lemmas

L 2

Then O Lemma Let U be a ranki bund le with c U af and U j

i U is a ranki bund le on P

ii deg c U a c U

1 2

Proof i Note that U is always torsionfree Thus U is a vector bundle Since

the rank of U is equal to i is equal to O U j

O R U is a torsion sheaf supp orted in some p oints thus ii Since U j

deg c R U By the GrothendieckRiemannRo ch formula see p in

ch U chR U chU tdT i a c U pt

where T is the relative tangent bundle tdT e f and pt stands for

the class determined by a p oint Therefore

deg c U deg c R U a c U a c U

1 1 2 2

If O Lemma Let U be a ranki bund le with c U af and U j

1 1 1

U O n O n O n

1 ij

P P P

where j i and n n n then

1 ij

i in i j c U a

0 1

ii in h P U O n c U a i

iii the bund le U sits in an exact sequence of the form

O nf U W

(i1)

1 1

O W j where W is a torsionfree ranki sheaf with W j

P P

K K

Proof i Since n n n by Lemma ii we have

1 ij

n in i j c U a deg c U j n

k 2 1

k =1

0 1

ii Note that h P U O n j Therefore by i

0 1

in h P U O n c U a i j j c U a i

2 2

iii Since there is a natural injection U U we have

O nf U

We claim that the quotient W U O nf is torsion free otherwise we have

O nf O nf D U

X X

is equal to O D is supp orted where D is some nontrivial eective divisor since U j

in the b ers of put D df where d applying to we obtain

1 1

O n O n d U

P P

but this is imp ossible in view of the assumption

Thus we have the exact sequence Since W is torsionfree W is lo cally free

outside p ossibly nitely many p oints Restricting to P we see that

1 1 1

W j W j O O

P P P

K K K

(i1)

1 1

O W j Since c W a nf we conclude that W j

1 1

P P

K K

Prop osition Let V M r tf c where L satises the condition and

L 2

r t Then there exist r exact sequences

V O n f V

i1 X i

where i r V V and V is a torsionfree ranki sheaf such that

r i

1 1 1

O n O i V n O n with n n

i i1 iij i ik

P P P

i i

1 1

O V j ii V j

P P

K K

t n iii in i j c V

k i i 2

k =i+1

0 1

t n i O n c V iv in h P V

k i 2 i

i i

k =i+1

Now the exact sequences and the O Proof By Theorem V j

prop erties i and ii follow from induction and Lemma iii Note that

n f t n f c V c V c V

i+1 k 1 i 1 1

i+1 i

k =i+1

Therefore the prop erties iii and iv follow from Lemma i and ii

The numb er of mo duli of V and V

In this subsection we estimate the numb er of mo duli of V and V These estima

tions will b e used in the next subsection to study generic bundles in the mo duli space

M r tf c where L satises the condition and r t To b egin with we

L 2

collect some prop erties satised by the sheaf V

Lemma i For each i there exists a canonical exact sequence

Q V V

i i

where Q is a torsion sheaf supported on nitely many points in X

ii dim HomV O n f dim AutV

i X i+1

i+1

iii Ext V O n f

i X i+1

iv V O n f c V t n i n i

i X i+1 2 i k i+1

k =i+1

Proof i This is a standard fact The torsion sheaf Q is supp orted on those p oints

where V is not lo cally free

ii Applying the functor HomV to the exact sequence we have

i+1

HomV O n f EndV HomV V

X i+1 i

i+1 i+1 i+1

where Id p for the identity endormorphism Id in EndV Thus

i+1 i

i+1

O n f dim HomV dim EndV dim AutV

X i+1

i+1 i+1 i+1

Similarly applying the functor Hom O n f to we obtain

X i+1

HomV O n f HomV O n f

i X i+1 X i+1

i+1

thus dim HomV O n f dim HomV O n f Hence

X i+1 i X i+1

i+1

dim HomV O n f dim AutV

i X i+1

i+1

1 1 1

we see that O and V j O iii Since O K n f j

i X X i+1

P P P

K K K

H X V O K n f By the Serre duality

i X X i+1

Ext V O n f H X V O K n f

i X i+1 i X X i+1

iv Recall that by denition F F dim Ext F F for two

1 2 1 2

i=0

sheaves F and F on X Let tdX b e the To dd class of X and let chF b e the

1 2

Chern character of a sheaf F Then we have the formula

F F chF chF tdX

1 2 1 2 4

2i i

where acts on H X Z by multiplication of Thus we obtain

K c V i n i V O n f c V

X 1 i i+1 i X i+1 2 i

Since c V t n f the conclusion follows immediately

1 i k

k =i+1

Next for convenience we intro duce some notations

Notation i Let h X Q for i r

i i

ii Let mo duli of V dim AutV for i r

i i i

iii Let mo duli of V dim AutV for i r

i i i

Now we estimate the numb er of mo duli of Q V and V

i i

Lemma i mo duli of Q dim AutQ l

i i i

ii i l

i i

iii V O n f h X V O n f

i1 i1 X i X i

i i

Proof i From we have an exact sequence

V O n f V

i X i+1

i+1

Applying Hom Q to we obtain

Q i l dim HomV Q dim HomV

i i i i

i+1

Applying Hom Q to we get

Q HomV Q HomQ Q HomV

i i i i i

1 1

Q Ext Q Q Ext V

i i i

It follows that

Q dim Ext Q Q dim HomQ Q dim HomV Q dim HomV

i i i i i i i

i i

i i i

Since mo duli of Q dim Ext Q Q and dim AutQ dim HomQ Q

i i i i i i

mo duli of Q dim AutQ l

i i i

ii From the exact sequence we see that

Q mo duli of Q dim HomV mo duli of V mo duli of V

i i i

i i

dim AutQ dim AutV

mo duli of Q dim AutQ

i i

Q dim HomV

i i

i l

Since dim AutV we obtain that i l

i i i

iii Similarly from the exact sequence we have

mo duli of V dim Ext V O n f mo duli of V

i1 i1 X i

dim AutV dim HomO n f V

i1 X i

dim Ext V O n f h X V O n f

i1 i1 X i X i

V O n f h X V O n f

i1 i1 X i X i

dim HomV O n f

i1 X i

where we have used Lemma iii in the last equality By Lemma ii

O n f V O n f h X V

X i i1 i1 X i

i i

r 1

c l i il Prop osition

2 k i1

i1 i

k =i

Proof By Lemma iv and Prop osition iv we have

V O n f c V t n i n i

i1 X i 2 i1 k i

k =i

t c V n in i

2 k i

k =i+1

0 1

O h P V c V c V n i

2 2 i

i i i

O n f i h X V c V

X i 2

i i

r 1

O n f i c l h X V

X i 2 k

k =i

Therefore by Lemma ii and iii we conclude that

O n f il V O n f h X V

X i i1 i1 X i

i i1 i

r 1

c l i il

2 k i1

k =i

Generic bundles in the mo duli space M r tf c

L 2

Our purp ose is to determine the structure of a generic bundle in M r tf c

L 2

Lemma Assume M r tf c is nonempty where r t and L satises

L 2

Then for a generic bund le V in M r tf c there are r exact sequences

L 2

O n f V V

X i i i1

for r i with the fol lowing properties

i V V V is a ranki bund le for i r and

r i

V O t n f I

1 X i Z

i=2

ii Z c and Z is supported in c distinct bers

1 2 1 2

c t n

2 i

c t

k =i+1

iii n and n for i r

r i

By Prop osition dim AutV mo duli of V Proof Note that

1 1 1

r r 1

X X

c l i il

2 k i1

1 r

i=2

k =i

r 1

r c r il

2 i

i=1

r 1

Since mo duli of V and l c we have

i 2

i=1

r 1

2 2

mo duli of V r c r il r c r

2 i 2

i=1

By Lemma since M r tf c is nonempty we always have

L 2

mo duli of V r c r

thus in particular all the inequalities in and Prop osition iii b ecome

equalities Hence for a generic bundle V in M r tf c we conclude that

L 2

a since is an equality l l so l c It follows that

2 r 1 1 2

V V are bundles and comes from Since V is of rank

2 r 1 1

V O t n f I

1 X i Z

i=2

for some cycle Z on X Thus Q O and Z c This proves i

1 1 Z 1 1 2

b since is an equality and Q O

1 Z

mo duli of Z mo duli of Q c

1 1 1 2

Thus for a generic bundle V Z is reduced and supp orted in c distinct b ers This

1 2

proves ii

c since Prop osition iii is an equality for i r we have

r r

X X

t n c t n i n i j c V

k 2 k i i 2

k =i+1 k =i+1

c t

note that i j i thus n and

i r

c t n

2 i

k =i+1

for i r This proves iii and completes the pro of

Prop osition Assume that M r tf c is nonempty where r t and L

L 2

satises Then a generic bund le V in M r tf c sits in an exact sequence

L 2

M M

O n f V O

X i i f

i=1 i=1

where the integer n is dened by induction as fol lows

c t n

2 i

c t

k =i+1

for i r with n n

r i

i r

and ff f g are distinct bers with being the natural embedding f X

1 c i i

Proof First of all we notice that if c t ar with r then

a if i r

a if i

In particular n n if i j By Lemma for a generic bundle V in M r tf c

i j L 2

we have r exact sequences Consider the rst two exact sequences

r 1

O n f V V

X r r 1

O n f V V

X r 1 r 1 r 2

Then the subsheaf p O n f of V sits in an exact sequence

X r 1

r 1

O n f p O n f O n f

X r X r 1 X r 1

r 1

Since n n Ext O n f O n f thus

r r 1 X r 1 X r

p O n f O n f

X r 1 X i

r 1

i=r 1

We check that V O n f V O n f V Thus V sits in

X i i1 X r 1 i2

i=r 1

O n f V V

X i r 2

i=r 1

By induction and the fact that HomO n f V H X O c f I we

X 1 1 X 2 Z

conclude that V sits in an exact sequence

O n f V V O n f

X i 1 X 1

i=1

Now the exact sequence follows from the observation that

V O n f I O c f O

1 X 1 Z X 2 i f

1 i

i=1

where f f are the c distinct b ers supp orting the cycle Z

1 c 2 1

Remark i By Theorem for any stable bundle V in M r tf c V is

L 2

a lo cally free rankr subsheaf of V with the quotient Q b eing supp orted on the b ers of

the ruling over which the restriction of V is nontrivial Another p ossible approach

to prove Prop osition is to study the exact sequence

V V Q

and to estimate the numb er of mo duli of these V s in terms of the data of Q and the

rankr bundle V on P In fact this approach has b een used very successfully by

Friedman to study stable rank bundles on an arbitrary ruled surface However

for r the diculty of this approach lies in the observation that the deformation of

Q is quite complicated

ii From the exact sequence we conclude that

V O n f

X i

i=1

for a generic bundle V in the mo duli space M r tf c

L 2

The mo duli space M r tf c on a rational ruled surface

L 2

In this section based on the results from the previous section we determine the bira

tional structure of the mo duli space M r tf c on a rational ruled surface where L

L 2

satises and r t First of all we intro duce the following notations

Notation i Let n f and b e as in Prop osition Put

i i i

r r

M M M

W O n W W O n f and Q O

0 i 0 X i i f

i=1 i=1 i=1

ii Let M b e the Zariski op en and dense subset in M r tf c parametrizing

L 2

all bundles sitting in exact sequences of the form

iii Let M U b e the morphism dened by

V f

i=1

c 1

where U is a Zariski op en and dense subset in Sym P P

Next we want to determine the b er u for u U We start with a lemma

Lemma i HomW V EndW

ii dim AutW r and dim AutQ c

iii dim Ext Q W r c

Proof ii and iii follow from and the denitions of W and Q In the following

we prove i Since W W EndW EndW Since Q is torsion and

0 0

0 1 0

H P Q H X Q

W W Thus Q must b e zero Applying to we have V

0 0 1

HomW V H X V W H P V W

0 1

EndW H P W W

0 0

EndW

Prop osition Let u U Then the ber u is birational to Ext Q W

modulo the c r dimensional group actions from AutW C and AutQ

EndW From the pro of of Lemma we Proof By Lemma i HomW V

see that generic extensions in Ext Q W must corresp ond to bundles in the Zariski

op en and dense subset M It follows that u is birational to Ext Q W mo dulo

the group actions from AutW C and AutQ By Lemma ii

dim AutW r and dim AutQ c

Therefore the group actions are c r dimensional

Now we prove the second main result in this pap er

Theorem Assume that the moduli space M r tf c is nonempty where r

L 2

r t and the ample divisor L satises the condition Then

i M r tf c is irreducible and unirational

L 2

ii a generic bund le V in M r tf c sits in an exact sequence

L 2

r c

M M

O n f V O

X i i f

i=1 i=1

where the integer n is dened by induction as fol lows

c t n

2 i

c t

k =i+1

for i r with n n

r i

i r

and ff f g are distinct bers with being the natural embedding f X

1 c i i

iii c t r

Proof i By Lemma iii the extension group Ext Q W has dimension r c

By Prop osition we have a rational map from the mo duli space M r tf c to

L 2

P such that a generic b er u is birational to

2r c

AutW C n C AutQ

Therefore M r tf c is irreducible and unirational

L 2

ii This is the same as Prop osition

iii Since O n f V and V is Lstable n f L tf Lr thus

X r r

n Since n c tr c tr we get c t r

r r 2 2 2

Remark In the Theorem of Artamkin showed that if c r then

M r c is nonempty and irreducible Therefore by Theorem iii we conclude

L 2

that M r c is nonempty if and only if c r

L 2 2

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Department of Mathematics Hong Kong University of Science and Technology Clear

Water Bay Kowlo on Hong Kong Email address mawpliuxmailusthk

Department of Mathematics Oklahoma State University Stillwater OK USA

Email address zqhardymathokstateedu