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Stable Vector Bundles on Algebraic Surfaces 1. Introduction

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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 1 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 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 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 X that is any curve in jL L K j containing all but one p oints in Z must contain X 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) M O L I X i Z i 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 2 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 ) i Hilb ert scheme H il b X for each i then the vector bundle V is Lstable and 2 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 2 P and on a ruled surface and for stable bundles with 2 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 2 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 c r 2 M M O n f V O X i i f i 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 2 and the integer n is dened inductively by The idea is a natural generalization i of those in Since the restriction of V to the generic b er is trivial V is a 1 rankr bundle on P thus we can construct r exact sequences O n f V V X i i1 i 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 i i 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 c 2 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 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 X 0 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 2 a section to such that is the least 2 e r ba where L a bf and a L df d where d is a divisor on C In this case d stands for degreed 1 P the generic b er of the ruling K 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) M W O L I X i Z i i=1 Let W b e the obvious quotient W O L I It is well known that there exists i X i Z i 1 an extension e in Ext O L I O L whose corresp onding exact sequence i X i Z X i O L V O L I X i X i Z i 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 i (r 1) M 1 1 Ext W O L Ext O L I O L X X i Z X i i=1 In the following we study the existence of a bundle V sitting in an extension O L V W X 1 Prop osition There exists an extension e Ext W O L whose corresponding X 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 1 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 i 1 subsheaf O L I of V Then V is given by the extension e X i Z i i i O L V O L I X i X i Z i 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 0 Corollary If h X O L L K I for every i and for every X i X Z fxg i x Z then there exists a bund le V sitting in the exact sequence i 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 2 Thus by tensoring some line bundle to V we may assume that r L c V L 1 2 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 0 ii If p then h X O K I g X X Z Proof This is straightforward 0 Lemma Let maxp h X O r L c K Then a generic cycle Z g X 1 X +1 in the Hilbert scheme H il b X satises the CayleyBacharach property with respect 0 0 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 0 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 +1 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 +1 +1 Z f Z x V X j x Z g +1 +1 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 +1 Z V X V +1 +1 +1 1 Then
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