Rank two vector bundles over regular elliptic surfaces

Invent. math. 96,283 332(1989) Inventiones mathematicae �9 Springer-Verlag 1989

Rank two vector bundles over regular elliptic surfaces

Rober t F r i edman* Department of Mathematics, Columbia University, New York, NY 10027. USA

Introduction

Since the appearance of Schwarzenberger 's pioneering works over twenty-five years ago, rank two ho lomorph ic vector bundles over algebraic surfaces have been extensively studied. After Mumford and Takemoto in t roduced the not ion of stability, modul i spaces of stable bundles were constructed by Gieseker and Maruyama . Moreover , work of Maruyama , Taubes, and Gieseker shows that these spaces are not in general empty. Fo r instance, Gieseker has shown the following [9].

(0.1) Theorem. Let S be an algebraic surface and L an ample line bundle on S. Then for all c > 2 pg(S) + 4, there exists an L-stable bundle V over S with c t ( V) = 0 and c2 (V) = c.

Beyond the fact that the modul i spaces are nonempty , little is k n o w n of a qualitative nature about them. One basic result is due to Donaldson . To explain it, we introduce the following terminology.

For S an algebraic surface and L an ample line bundle on S, let 9)ls,L(c ) denote the modul i space of L-stable rank two bundles V over S with c~ ( V ) = 0 and c2(V)= c; it is a quasiprojective variety.

(0.2) Definition. A componen t M of 9J~s,L(c ) is good if there exists a point of M corresponding to a bundle V over S such that H2(S; a d V ) = 0 , where ad V is defined by the exact sequence

T r O - * a d V - - * E n d V , (SJs ~ 0.

Of course, if H2(S; a d V ) = O for one V e M , the same will be true for an open dense subset of bundles of M. The meaning of (0.2) is as follows. By

* Research partially supported by NSF grants DMS-85-03743 and DMS-87-03569 and the Alfred P. Sloan Foundation

284 R. Friedman

deformation theory, if 9J~s.L(c ) is nonempty, the dimension of every component is at least 4c-3z((gs). To say that a component M of ~lS.L(C) is good means precisely that d i m M = 4 c - 3 z ( ( g s ) and that M is reduced at a generic point (in its natural scheme structure).

We may now state Donaldson's theorem as follows:

(0.3) Theorem [7]. There exists a constant Co=Co(S, L) such that, if c>Co, then every component of TJ~S,L (C ) is good.

Given the lack of other general results on 9J~s,L(C ), one can focus instead on particular classes of surfaces. The case of ]p2 was studied by Barth, Hulek, and Maruyama. Hoppe-Spindler and Brosius considered ruled surfaces. Mukai has recently considered certain spaces of bundles over K3 or abelian surfaces (not necessarily of rank two, but with certain restrictions on c~ and c2). The next step in the Kodaira classification of surfaces is then elliptic surfaces. The first results in this direction were obtained by Donaldson in [6], for c = 1 and pg(S)=0, for special elliptic surfaces S. These methods were generalized in [8, 18, 16, 17].

The goal of this paper is to study L-stable rank two bundles V over general regular elliptic surfaces S, where L is a suitable ample line bundle (more precisely defined in w 2). In general, however, L depends on the choice of a second Chern class c. Since S is regular, the base of the elliptic fibration is necessarily ~?~. Let n: S--*~ '1 be the elliptic fibration and let /t be the number of multiple fibers of n. We shall also assume that S is generic, in a sense to be more precisely described in w Let c be a positive integer, let L c be suitable with respect to c, and let 9J~(C)=~IJ~S.Lc(C ) be as above. Note that the line bundle with respect to which we compute stability depends on c (although it is still the case that, if c' < c, then L c is c'-suitabte as well). Then

(0.4) Theorem. (i) With 9J~(c) as above, if c>=2(pg(S)+ 1), then there exist good components of ~JJ~(c).

(ii) I f c>max(2(1 + p g ) - 1, 2pg+2#), then every component of 9J~(c) is good.

A rather precise description of a Zariski open subset of the good components is given in w 7. They are all fibrations, with general fiber an abelian variety (in fact, a hyperelliptic Jacobian) over a base U whose dimension is specified by pg and c. If there are no multiple fibers, then U is just an open subset of a projective space. From this, one can show:

(0.5) Theorem. With S and Lc as above, if there are no multiple fibers, then, for all c->_ 2 (1 + pg), 9)1 (c) is irreducible.

If there are multiple fibers, then 9J~(c) is in general reducible for all large c. However, the components of 9J~(c), at least birationally, are covers of the corresponding moduli space for the Jacobian elliptic surface of S. This result parallels the work of Bloch-Kas-Lieberman on zero-cycles on elliptic surfaces [3].

A major motivation for this work has been the recent application by Donald- son of Yang-Mills theory to four-manifold topology [7]. Using the identification of stable bundles with solutions to the anti-self-dual Yang-Mills equations, it

Rank two vector bundles over regular elliptic surfaces 285

is possible to analyze Donaldson's polynomial invariants for simply connected elliptic surfaces. This analysis, and its consequences for the differential topology of elliptic surfaces, are joint work with J. Morgan and will appear elsewhere. It is worth noting, however, that the necessary calculations do not actually involve the knowledge of the component structure of the moduli space, so that w 3, w 6, and most of w 1 are not necessary for the topological applications.

We give a brief description of the contents of this paper. In w 1, we collect some of the technical preliminaries. After a discussion of elliptic surfaces, the main part of w 1 concerns certain torsion sheaves on algebraic surfaces which appear naturally as the cokernels of morphisms between two rank two vector bundles. In this context, these sheaves were first studied by Brosius I-4]. Our main result (1.14) is an upper bound for the number of local moduli of such sheaves. The results here seem quite related to Donaldson's study of holomorphic families of 2 • 2 matrices (Appendix III of [7]). In fact, it is possible to give a somewhat different proof of Theorem (0.3) than that in [7], which is based on (1.14) and methods similar to the first part of w 5 and (6.7), (6.9) of the present paper.

The analysis of stable bundles begins in w 2. First, we define suitability of line bundles along the lines of [6, 8]; this essentially amounts to taking an arbitrary ample line bundle and then adding to it a sufficiently large multiple of the fiber. The next step is to analyze the restriction VII t, where f = n - ~ ( t ) is a general fiber of n, using Atiyah's classification of vector bundles over elliptic curves. If VII ~ is decomposable, then we may write V i = ( g y , ( d t ) O 6 ) i , ( - - d r ) , where dr is a divisor on the elliptic curve f . Stability easily forces d e g d r = 0 for generic t. Thus, the line bundles _+d, of degree 0 on f fit together to give a bisection (7 of the Jacobian surface associated to S, invariant under _1 . There are two cases: either (T" splits into a union of two sections, or possibly a single section counted with multiplicity two (Case (A)), or (7 is irreducible (Case (B)).

Case (A) is studied in w 3. In this more elementary case, the bundle V may be described as an extension

(0.6) 0 ~ Cgs(D ) ~ V ~ (~s ( - D) I z ~ O,

where Iz is the ideal sheaf of a 0-dimensional subscheme of S and D is a divisor on S such that D.f=O. For small values of c, i.e. c < 2 ( l + p , ( S ) ) , all bundles satisfy Case (A). However, we show that, for large values of c, the set of bundles satisfying Case (A) forms a nowhere dense subvariety of 9J~(c) (although we do not explicitly characterize which bundles of the form (0.6) are actually stable).

The remainder of the paper is devoted to Case (B). Let C be the normalization of the bisection (7 of the Jacobian surface associated to S, so that C--* ~,1 is a two-sheeted cover. Let T--* C be the pullback of S ~ ~'~, T the minimal resolution of T and ~: T ~ S the corresponding map (generically 2 1). Essentially by construction, we have a canonically given extension

0 -~ (9~(D) --+ ~* V ~ (99( -- D) Iz -~ O,

where D is a divisor on T with D. n ' f = 0.

286 R. Friedman

In w the geometry of the double cover is studied and c2(V) is related to local invariants of C. In w 5, we compare V to the rank two sheaf ~. (9~(D). Roughly speaking, up to a twist by a line bundle on S, they differ by a torsion sheaf as described in w 1. The arguments here are inspired by the work of Brosius on ruled surfaces [4]. Finally, in w 6, we show that generically on the moduli space V= ~. (9~(D) (up to a twist). The proofs, while not difficult, are somewhat tedious owing to the number of special cases that must be dealt with: (7 may be singular, its singularities may not be in general position with respect to singular or multiple fibers . . . . . The main problem is to keep track of all the notation. In w 7, there is a brief discussion of the number of components and Kodaira dimension. These matters will be worked out more fully elsewhere.

The results in this paper may be generalized in several directions. It is not difficult to extend many of the results to the case where S is not necessarily regular. Similarly, the assumption that S is algebraic is made primarily for conve- nience. In this connection, Braam-Hurtubise have independently studied bundles over Hopf surfaces with c2 = 1. Finally, much of the discussion in w 2, w 4, w 5 extends to surfaces over an arbitrary algebraically closed field of characteristic different from 2. It would be interesting to know if Donaldson's theorem (0.3) holds in this generality.

Acknowledgements. It is a pleasure to thank Hyman Bass, Simon Donaldson, and John Morgan for helpful discussions.

Conventions. All schemes are defined over the field II; of complex numbers. We shall always identify a vector bundle (covariantly) with its sheaf of sections. If D is a divisor on a variety X, then I DI denotes the complete linear system associated to D; for line bundles L, ILl is similarly defined. When X is clear from the context, HI(X, D) and hi(X, D) are abbreviated to Hi(D), hi(D) and similarly for L. If F is a coherent sheaf supported on a 0-dimensional subscheme, l(F) is the length of F. For F=(gz , we let l(Z)=l((gz)=dimeH~ For a surface S and a point xeS, if C1 and C2 are two curves on S meeting properly at x, we set C l . xCz= loca l intersection number of C1, C 2 at x=l((gs, x/(fl,f2)), where f~ is a local equation for Ci at x. An elliptic surface S is always assumed to be relatively minimal (no exceptional curves lie in fibers of the elliptic fibration). For V a vector bundle on S, we shall use the shorthand e l ( V ) = 0 to mean det V= (9 s.

w 1. Preliminaries

(a) Geometry of regular elliptic surfaces

Let r~: S ~ P ~ be a regular (algebraic)elliptic surface. Associated to S is the Jacobian fibration, which we shall denote by j: J ( S ) ~ 1. It is again regular [5]. We let Zc_J(S) be the natural section. Let p~, ..., p, be the multiplicities of the multiple fibers of g. By results of Kodai ra [15], modulo the analytic Tate-Shafarevitch group H 1 (R J. (9,~s)/R).Z), all (possibly nonalgebraic) elliptic surfaces with associated Jacobian fibration J ( S ) are determined by the choice

R a n k two vector bundles over regular el l ipt ic surfaces 287

of pl-torsion points in p fibers of j of type I o or I,. Using the arguments of [8], I, w 3, it is not difficult to show that the moduli space of all such elliptic surfaces, possibly nonalgebraic, with prescribed multiplicities Pl , --- , Pu for the multiple fibers, is irreducible, although we shall not need this result. By a result of Dolgachev [-5], p. 128, given J (S) , a regular elliptic surface with a section, a choice of xl . . . . . xu~lP 1 corresponding to fibers of type I,, and positive integers p~ . . . . . pu, we can find an algebraic elliptic surface S with exactly p multiple fibers at the xl of multiplicity Pl, whose associated Jacobian fibration is J (S) . In particular, there is a finite cover of lP 1, over which the pullbacks of the two fibrations g: S ~ IP 1 and j: J(S)---, IW become birational.

For simplicity, we shall always make the following assumption on J ( S ) and on S:

(1.1) Assumption. All fibers o f j are irreducible, with at worst ordinary double points as singularities. Moreover on S, all of the multiple fibers have smooth reductions.

Using the natural section s of J ( S ) as origin, we have an involution J ( S ) ~ J (S) , defined by - 1 on each fiber. The quotient is then a smooth rational ruled surface ~ over ~,1 (in particular, no exceptional curves lie in fibers):

F 1

(1.2) Proposition. (1) r/(27)=~r is the negative section of IF,. (2) I f pg=pg(J(S)) , then we have n=2(1 +pg). (3) The branch divisor of r I on IF, is of the form B + a , where B~[3(cr+nl)[

is a smooth divisor with B c~ rr = O, and I is a f iber of the ruling on IF,. (4) Conversely, if n = 2 k is even and >0, if B is as in (3) and such that

the projection B-~ IP ~ has suitably generic branching behavior, the double cover of IF,, branched along B + o , is a smooth elliptic surface with a section satisfying (1.1), of geometric genus k - i .

Proof. With r/: J (S)-*IF, as above, a=r/(Z) is a section of the given ruling on IF,. By the canonical bundle formula [1],

K s ~s) = (Pg -- 1 ) f,

where f is a general fiber ofj. Thus, as 27 is a section of j, and so ~ 1 , _2=272 +X,.Ky(s)=~_,2+(pg--1), so that 272=-(1+pg) . As r/*o-=2Z', we have o 2 = 2(27) 2= - 2 ( 1 +pg)<0. Hence, a is the negative section of IF, and n = 2(1 +pg). This proves (1) and (2).

To see (3), let Bo~_9r be the branch divisor of q on J(S) . Since the involution on d;(S) is defined by - 1 in the fibers, B0=27+B', where B' is the trisection of j given by the 2-torsion points and B'm 27=0. Similarly, r/(B')= B

288 R. Friedman

is a trisection of the ruling on ~:, with B n ~ = 0. Thus B s[3 (a + n l)]. This proves (3), and (4) is a s t ra ight forward compu ta t i on of the canonical bundle of a double cover. [ ]

(b) Results of Iarrobino on O-cycles

Let S be an algebraic surface and c a positive integer.

(1.3) Notation. We denote by HilbC(S) the Hilber t scheme of 0-cycles of length c o n S .

Thus, Hilb~(S) is smoo th and is a canonical resolut ion of singularities of SymC(S). Let z: Hilb~(S) ~ SyruP(S) be the natura l map. We shall use the following nota t ion in this paper : (1.4) Notation. If Z~HilbC(S), then we can write s u p p Z = { x l . . . . . x,}, where the xi are distinct points of S. Thus, as schemes,

Z=[I zx~, i

where Z~ = t h e restr ict ion of Z to xi in the obvious sense. Also set ~ ( Z ) = 4t= supp Z.

With this said, I a r rob ino ' s result on 0-cycles on a surface is as follows:

(1.5) Theorem ( Ia r rob ino [14]). I f Z~HilbC(S) and 4~(Z)=r , then dim z - 1 (z (Z)) = c - r. [ ]

(c) Calculation of some Ext groups

We begin with a s i tuat ion tha t essentially dates back to Schwarzenberger ' s pape r [20] (cf. also [10]). Let V be a r ank two vector bundle on a surface, and (gs(F) a subline bundle, i.e. a line bundle on S which is also a subsheaf of V. If the cokernel V/(Ps(F ) is torsion-free (which may always be a r ranged by adding some- thing effective to F), then there exists a 0-dimensional local complete intersection subscheme Z, with ideal sheaf I z, and an exact sequence

(1.6) 0 -* (~s(F) -~ V-~ (gs(G) I z ~ O,

where (gs(G) is again a line bundle on S. To invert this procedure , we seek locally free extensions of (Os(G) lz by (~s(F)

as in (1.6). The set of all extensions is classified by Ext 1 ((gs(G)I z, (~s(F)), and we have an exact sequence

(1.7) 0 ~ H 1 ( (~s(F - - G)) --~ E x t I ((gs(G) I z , (~s(F))

-~ H ~ ((gz) ~


where we have identified H o m ( C s ( G ) l z , C s ( F ) ) ~ - C s ( F - G ) and (noncanonically) E x t 1 ((~s(G) lz, (gs(F))~ Ex t 1 (Iz, Cs)~ (9z. A locally free extension exists ,~ there is an everywhere generating section of H~ in kerO [10]. To analyze this condition further, one uses Serre duality. The dual sequence to (1.7) is

(1.8) O ~ H a ( C s ( K s - - F + G ) ) + - - H ' ( C s ( K s - F + G ) I z ) , - - H ~ ,--- H ~ (Cs(Ks- - F + G))

which is simply the long exact cohomology sequence associated to

0 ~ C s ( K s - F + G) Iz ~ Cs (Ks - - F + G) ~ C z ~ O.

Here, the duality between the H~ in (1.7) and the H~ in (1.8) is given more intrinsically via Grothendieck residue. A simple argument then yields:

(1.9) Proposition [10]. Suppose that Z = { x l . . . . , xc} consists o f c distinct (reduced) points. Then a locally free extension as in (1.6) exists r162 every curve in I K s - G + FI which contains all but one of the xi contains all o f Z. []

(Note however that the condition (1.15) of [10] does not given the correct generalization of (1.9), even for reduced subschemes.)

We shall also consider extensions of the form

o ~ w ~ v ~ Q ~ O ,

where Q = l z / C s ( - E ) for Z some O-dimensional local complete intersection subscheme of S with ideal sheaf lz and E a possibly non-reduced divisor on S such that there is an inclusion I E = C s ( - - E ) ~ _ I z . The sheaves Q will be studied more systematically in the next section, where they will naturally appear as cokernels of maps between two rank two vector bundles. We shall use:

(1.10) Proposition ([4], w 3). With Q as above, suppose that W is a locally free sheaf o f rank r. Then there is a natural exact sequence

0 ~ W Q [(gs(E)/Cs] --* ExtX~=(Q, W) -~ C~z ~ O.

Proof From the exact sequence

O ~ C s ( - E ) ~ I z ~ Q ~ 0 ,

we obtain

, Horn(Q, W) , Hom(Iz , W) > Horn((~s(--E), W)

0 W W | Cs(E)

,Ex t l (Q , W) >Ext l ( Iz , W) ,0,

Uz

and collecting terms gives (1.10). []

290

(d) Moduli of the sheaves Q

R. F r i e d m a n

(o Let W , V be a homomorphism of locally free Cs-modules W, V, both of

rank 2. We wish to analyze coker (p locally. Let R ~ ~ [Ix, y]] be the completion

of the local ring of S at p. Thus locally tp corresponds to a map R 2 ~0 , R 2,

which in turn is given by a matrix M = ( J e- hi)(~p(1, 0)=(f, g), tp(0, 1)=(h~, h2)). h2

Suppose that g.c.d. (f, g, h a, h2)= 1, i.e., that we cannot extract a common factor from the coefficients of M. It is then easy to see that, possibly after a change of coordinates, we may assume that g.c.d. ( f g)= 1, i.e. that l z = ( f , g)R is the ideal corresponding to a 0-dimensional 1.c.i.Z. Further let det M = f h 2 - g h~ = t, say, and let IE = t R be the ideal corresponding to the divisor E. Then the natural map R 2 ~ R given by (a, b)~--~-g a +fb gives an isomorphism

Q = coker q0 ~- Iz/I~.

From this, it is clear that Q is determined up to isomorphism by Z, although Z is not in general determined by Q. Our goal is to analyze the number of local moduli for Q; to do so, we introduce the following invariant of Q:

(1.11) Definition. With notation as above, the Fitting scheme Z(Q) is the 0- dimensional subscheme of R defined by the ideal lz~2)=(f, g, hi, ha)R. (Note that Iz~) is the first Fitting ideal of Q, hence is canonically defined.)

Roughly speaking, the main result will assert that the flat deformations of Q as an R/tR module depend on at most l(Z(Q)) moduli. To make this more precise, we introduce the following notation. Let ~ denote the category of Artin local lI;-algebras, and define two functors ~ ~ Sets as follows:

(1.12) Definition. (i) DefEQ is the functor of deformations of Q as an R/tR- module, i.e. given A ~ , DefEQ(A) is the set of isomorphism classes of (R/tR) | modules 2, flat over A, together with an isomorphism ~/m.~ ~-Q (where m is the maximal ideal of A).

(ii) For Z a 0-dimensional 1.c.i. subscheme of R of length c with 1~___ I z, Hilb~(R) is the functor of flat deformations of Z preserving the condition IE ~-Iz. More precisely, given AE~, _H~(R)(A) is the set of isomorphism classes of R| Cz, flat over A, which are quotients of R | by l~r, such that I~ ~ I~ | and I~ induces lz on the closed fiber.

(1.13) Proposition. Suppose that Z is as in (1.12) (ii) and that Q=Iz/IE. (i) Def~ Q has a hull, and Hilb~(R) is prorepresentable.

(ii) There is a natural morphism of functors p: Hilb~(R)~ Def~ Q. (iii) The morphism p is formally smooth.

Proof. (i) is a standard application of Schlessinger's work [19] (cf. (3.6) especially). In particular, the Zariski tangent space to Def~Q is easily seen to be EXt1R/tR(Q, Q).


To prove (ii), given an ideal I~ of R | A containing IE | A and such that ( ~ = ( R | is flat over A, define ~ as the kernel of the natural map (R/t R) | A ~ (9 ~. From the exact sequence

0 ~ ~ -~(R/t R)| A ~ C.~ ~0 ,

it follows that Y is fiat over A, and this defines the morphism p. To prove (iii), we must show that, given a surjection B ~ A, B, Ae~, the induced map

Hilb~ (R)(B) --~ Hilb~ (R)(A) • Defe Q (B) D e.__f E Q(A)

is surjective. This

Let U and V respectively, and then:

(1.14) Theorem.

is a straightforward calculation, left to the reader. []

be the formal schemes prorepresenting Hilb)(R) and DefEQ again denote by p: U ~ V the induced map. Our result is

With Z and Q as above, the dimension of V is at most l(Z(Q)).

Proof Since dim V= dim U-dimension of a fiber of p, it clearly suffices to show that dim U < l(Z) and that the dimension of a fiber of p is > l(Z)-l(Z(Q)).

To prove the first inequality dim U<l(Z), note that Hilb~(R) is naturally a subfunctor of the Hilbert functor of length c 0-dimensional subschemes of R, which is prorepresented by the completion of HilbC(S) at Z. Moreover, using the map z: HilbC(S)~ SymC(S) of (b), we see that U is a subvariety of the completion of z-l(SymC(Ered)), using the natural inclusion SymC(Ered)~Sym~(S). An easy application of Iarrobino's result (1.5) shows that dimz-l(Sym~(Ered))<C =/(Z), and hence dim U < l(Z) as well.

To prove the second inequality, we use the well-known fact that the map ( t l , t2 ) ~ R/(f+ t~, g + t2) identifies a formal neighborhood of 0~ Cz @ Cz with a formal neighborhood of Z in Hilb~(S). Consider replacing the ideal (f, g) by the ideal (f+~h~, g+~h2). This just corresponds to an elementary operation on the matrix M and does not change detM, the condition 1E~_I z or the isomorphism class of Q. Moreover, the induced deformation of the scheme Z is trivial if and only if eh~ and eh2 are in lz. Choosing ~t~(gz/Anne~((h 1, h2)) thus gives a smooth subvariety of the fiber of p of dimension l(Cz/Anne~, ((h~, h2))).

Hence dim(fiber of p )> l (Cz)- l (Ann~((h l , h2))). One easily checks that

Anne~((hl, h2))= Home~.(Cz/(hl, h2) (9z, Cz)= Hom~z(Czto ), d)z).

But, by Gorenstein duality [2] (or via the Grothendieck residue pairing), we have l(Hom~z(((gz~q), Cz)) = l((gz~o )) = l(Z(Q)). Thus I(Annez((hl, h2))= I(Z(Q)) and dim(fiber ofp)>l(Z)- l (Z(Q)) . This proves (1.14). []

(l.15) Remark. In general, I(EXtlR#R(Q, Q))=21(Z(Q)). Thus, the dimension of the Zariski tangent space to DefEQ is never the actual dimension of Defv~Q, unless l(Z(Q))=O. One easily checks that this last condition is equivalent to Q ~-R/t R. Correspondingly, the locus in HilbCS corresponding to Hilb~(R) is typically singular (and even non-reduced) unless l(Z(Q))= O.

292 R. Friedman

The global corollary of (1.14) which we shall need is the following. Let E be a not necessarily reduced divisor on the complete surface S and Q an Ce-module which is locally of the form Iz/I~, Z a 0-dimensional 1.c.i. subscheme. Then we may define the 0-dimensional scheme Z(Q) as before.

(1.16) Proposition. The local deformations of the sheaf Q, as an (9~ module, may be parametrized by an analytic space of dimension at most h I (CE) + l(Z(Q)).

Proof. We have an obvious map from the global deformation functor of Q to the local one, concentrated at the support of Z. The fiber is prorepresented by the formal completion of H 1 ((ge), corresponding to the straightforward result that any two sheaves of (ge-modules, locally isomorphic to each other and of the form Iz/I~, differ by a twist of Hi(C*). Thus, (1.16) is an easy corollary of the local estimate (1.14). []

We shall also need the following local result:

(1.17) Lemma. Let R, M, Q=Iz/Ie , and Z(Q) be as above. Suppose that det M =x", where x is part of a local coordinate system, and that m is the smallest positive integer such that x"~(f , g) = Iz. Then I(Z(Q)) < l(Z).

, r o , notation h2] x m = f h z - g h l . Since Iz~Iz(e), l(Z(Q))

<l(Z), with equality if and only if h~, h2~I z. Hence, if I(Z(Q))=I(Z), xmEI~. But then

d d~ ( x " ) = r e x ' - ~ e l z ,

contradicting the choice of m. Thus, l(Z(Q))< l(Z). []

The author does not know if the analogous statement holds for characteristic p > 0 .

(e) Linear systems with large contact

In this section, we consider the following situation: S is a smooth complete surface and A, B are smooth curves on S. In general, we would expect A and B to meet transversally at A-B points. The point of the next result is to give an upper bound on the dimension of the subset of I A [ x I B I where the intersection is non-transverse. Both the statement and the method of proof are essentially just an explicit version of a very special case of a result due to Zariski [-21].

To explain the result, recall that if C is a smooth curve on S, the adjunction formula Kc = Ks+ Clc identifies [C[ with a subseries of [Kc | Ks~[c[. If C is singular, certain sections of C, those satisfying the adjoint conditions, wilI give a subseries of [K e | a* K s 1 I, where a: C ~ S is the normalization map. Zariski's result then asserts that, given an equisingular family U of curves C on S, the projective tangent space of U at a general point u~ U, viewed as a linear subspace

R a n k t w o vec tor bund le s over r egu la r elliptic surfaces 293

of the projective space I C], consists of curves satisfying a strong form of the adjoint conditions.

Let us apply this to C=A+B~_S. Let U be the subvariety consisting of A+B, A, B smooth, such that there exist points pt, . . . ,pkeAC~B where the local intersection number Ap, B = ~ i > 1. We may choose local coordinates x, y around pi so that the equation for A + B is given by f ( x , y ) = x Z - y Z ~ = O . Suppose that we choose a generic curve ~0(t) in U. Lifting q~(t) to the affine space H~ Cs(C)), we may suppose given a family of local equations st of the form

s~ = (x + t ~ (x , y) + t 2 ( . . . ) )2 _ (y + t ~ ( x , y) + t ~ ( . . . ) )~ , .

Thus

d St 1 dt t=o =2x~(x ' y ) - 2 ~ i y 2 ' ' - rl(x' Y)"

Zariskrs key observation is then that the meromorphic form O= dst dx dt t=o fy

satisfies the adjoint conditions. More precisely, restricting the form (2 to either smooth branch x = + y~' gives the holomorphic 1-forms

co+ = [ - ~(y' , , y ) + ~ iy " , - ' q(y ' , , y)] dy .

Moreover, using the coordinate y on either branch to compare co +, we have

(1.18). The tangents dst define co+eH~ K A | 11A), dt t=o

co_EH~174 such that co+-co_ is divisible by y~,-i at each point

Pl.

In other words, not only do the tangents satisfy the adjoint conditions, but they also satisfy cq - 1 additional linear conditions at each pl.

We shall need a slight generalization of (1.18), which is given as follows. Suppose that locally on the universal family of curves in the linear series [B[ we are given sections pl(t) . . . . . pj(t). Let cq . . . . . c~j, (~j+l . . . . . 0~ k be integers > 1 and let U consist of curves A, B, such that Ap,(t)B,=e i, 1 <__i<=j, and such that there exist points p j +1 . . . . . pk ~ A c~ B t with A b, Bt = ~i, J + 1 <_iN k. (Thus, we pre- scribe the first j intersection points on each member of B, at least locally.) Using the previous notation, if the x and y coordinates of p~(t) are x+t~ (x , y) + .... y + t I/(x, y) + .... respectively, then

(~(x, y), ~(x, y ) )= (x ' (0 ) , y ' ( 0 ) ) = ' ' dt t=0 '

dpi or, more intrinsically, (3, q) points in the tangent direction given by ~ ) - . Thus, we have,

294 R. Friedman

(1.19) Under the above assumptions, the relation

(4 ix, y), ~ ix, y)) = (x'i0), y' (0))

defines an additional linear relation on the pair (co+, co_)for each i= 1 . . . . , j.

Of course, in any given situation, we must check if the linear conditions (1.18) and i1.19) are independent.

w 2. Rough classification of stable bundles

Let g: S ~ IP 1 denote a regular algebraic elliptic surface whose associated Jaco- bian fibration satisfies (1.1). In particular, all fibers of rc are irreducible (but possibly multiple). Let f denote a generic fiber of re.

Fix a positive integer c. We begin by defining a class of polarizations on S for which we shall be able to analyze stability.

(2.1) Definition. A polarization L on S is c-suitable (simply denoted suitable when c is clear from the context), if for all HeP ic S with H 2 ~ - - c and H - f + 0, we have sign ( H . f ) = sign(H- L).

(2.2) Remark. 1) If co is the (1, 1)-form associated to a K/ihler metric, we may define suitability analogously for co.

2) Since f is in the closure of the ample cone of S, a general result about hyperbolic forms (cf. [8], II, (1.1)) implies that, for all H e P i c S with H2>O, we have either H e N . f or s ign(H-f )= sign(H. L). Hence, in the definition (2.1), it suffices to consider only those H with H 2 <0.

3) Suitability has the following geometric meaning. Let p = r k Pic(S). Inter- section pairing gives a quadratic form on P i c S | of signature (1, p - l ) . Let l-I + (S) denote the component of {x epic S @ N: x 2 = 1} which contains real multi- ples of polarizations. Thus, N+(S) is a hyperbolic space of dimension p - 1 . By [8], II, (1.6), the set ~ = { H e P i c S : - - c < H 2 < 0 } defines a locally finite set of walls of N+(S). After rescaling L by t e n +, we obtain a point xelH+(S), and f naturally defines a point of the ideal boundary of ~I+(S). The definition amounts to saying that x and f are not separated by any wall in ~#/;.

The following lemma should be compared with [8], III, (1.9).

(2.3) Lemma. For every positive integer c, c-suitable polarizations L exist. More precisely, i f Lo is an arbitrary polarization on S, then, for all t > t o = � 8 9 L t = L o + t f is a c-suitable polarization.

Proo f First note that, by the Nakai-Moishezon criterion, Lt is a polarization for all t >0. We must therefore show that, if - c < H 2 < 0 and if ( H . f ) > 0 , then (Lo + t f ) . H >O for all t > t o. Let n = ( H . f ) > O . Then

(n L 0 -- (Lo "f) H) - f= 0.

By the Hodge index theorem, as f 2 = 0,

(n L o - (L 0 .f) H) 2 = n 2 (Lo) 2 - 2 n (Lo . f ) (L 0 �9 H) + (L 0 .f)2 H z < 0.


Thus,

2 n (L 0 . f ) (L 0 �9 H) >-_ n 2 (Lo) 2 + (L 0 .f)2 H 2 > n 2 (Lo)2 _ c (L 0 .f)2 > _ c (Lo .f)2.

c (L o.f), and therefore It follows that (Lo' H) > - 2 n

(L o + t f ) . H) = (L o �9 H) + t ( f . H) >= (L o . H) + ~ (Lo "f)

c > --~nn ( L ~ (L~ "f) =-c2 (L~ "f)\(n-- 1~> O ' n ] =

i . e . L , . H > 0 . []

We turn now to vector bundles. Let V be a rank two vector bundle on S with c l ( V ) = 0 . We shall begin by analyzing V "bira t ional ly" , i.e. by studying its restriction to the generic fiber of re. More precisely, let k(lP 1) denote the function field of lP ~ and k(lP 1) be the algebraic closure of k ( ~ l ) . We set q = S pec k (~,1), 0 = Spec ~ , and define

S~=S x~,t/, S~=S x~,, 0 .

Thus, S, is a curve of genus one over the nonalgebraically closed field k(lp1), and S~ is an elliptic curve defined by extending the field of definition to k(~)). Let V~, V, denote the induced rank two vector bundles over the curves S, and S~, respectively. F rom Atiyah's classification of vector bundles over an elliptic curve (all we need is what is done in e.g. [-12], p. 377), we obtain

(2.4) Lemma. With S~, V~ as above, either (i) Vo is indecomposable, and there is an exact sequence

o -~ Q ( F ) --, v~ ~ Q , ( F ) ~ o,

where (gs,(F) is a line bundle of order 2 in Pic S,, or (ii) V~ is decomposable and there is an isomorphism (defined over k ( ~ ) )

V ~ ( O s , ( r ) | 6~s,(--F)

for a line bundle (Os,(F)ePicS~; moreover, the line bundle 6's,(F) is unique up to sign. []

(2.5) Lemma. 1) In case (i) of (2.4), the bundle (Os~(F) and the map (Psq(F)~ Vo are rational over k(lPl).

2) In case (ii), /f the bundle (gs,(F) is induced by a bundle (gs,(F) which is rational over k(P1), then V, is k(p1)-isomorphic to (gs,(F) �9 (gs,(--F).

3) Finally, in case (ii), if 6)s,(F) is not rational over k(lpt), then G a l ( k ~ / k ( l P 1 ) ) permutes (~s,(F) and (gs , ( -F) . Moreover, there is a 2-1 map g: C-- ,~" , such that (gs,(F) is rational over k(C), and if q o = S p e c k ( C ) and S,o, 1,1,o are defined as beJore, then there is a k(C)-isomorphism V,o_~(Cs,o(F)| (gs,o ( - F ) .

296 R. Friedman

Proof First suppose that we are in case (i) of (2.4). The map O s , ( F ) ~ Vn is unique rood scalars, and hence the subbundle (gs~(F) of V~ is invariant under Gal (k(~/k( lP l ) ) . By standard descent theory [11], (Ps,(F) is rational over k(~ 't) and similarly for the map @s,(F)~ Vo. This proves 1). To prove 2), note by assumption that Horn (Os,(F), V,) has a section over k ( ~ , and hence over k(lP1). This proves (2). To see (3), assume that we are in case (ii) of (2.4) with F not rational over k(lP~). Thus, the bundle F cannot be invariant under the Galois action. Since the splitting of V~ is unique up to permuting the factors, this forces F + - -F; moreover, the Galois action must permute the factors + F. The remainder of the proof is similar to the proof of (1) and (2). []

By passing from the generic situation to S and introducing a suitable polarization, we can state the main result of this section:

(2.6) Theorem. Let L be a c-suitable polarization on S and V an L-stable rank two vector bundle over S with c l (V)=0 and c2(V)=c. Then V satisfies either Case ( A ) or Case ( B ) below:

Case (A). There exists a line bundle Os(D ) on S with D . f = 0 and a codimension 2 1.ci. subscheme Z on S, such that V is given as an extension

0 ~ Os(D ) ~ V ~ O s ( - D) I z ~ 0.

Case (B). There exist (i) a smooth curve C and a 2-1 map g: C ~ ]p1; (ii) a divisor D on ~F= minimal resolution of T= normalization of S x a,, C, such

that D . f ' = O, where f ' is the generic fiber of ~F~ C, and such that the restriction of O~(D) to the generic fiber is not rational over k(lPa);

(iii) a codimension two l.ci. subscheme Z ~_ ~F; (iv) an exact sequence

0 --* Of(D)--* ~* V ~ O r Iz ~ 0 ,

where ~: ~I'~ S is the natural map (generically 2-1). Moreover, if V is a rank two bundle over S with c l (V )=0 and c2(V)=c which

satisfies (i)-(iv) of Case (B) , then V is automatically L-stable for every c-suitable L.

Proof We begin by establishing two lemmas.

(2.7) Lemma. Let L be a c-suitable polarization on S and V an L-stable bundle with ca(V)=0, c2(V)=c.

(i) I f D is a divisor on S such that D . f > O and there exists an injective map qg: Os(D)~ V, then D.f=O.

(ii) I f in addition to (i), r factors through the inclusion (Ps(D)~(_gs(D+E), where E is effective, then E is supported in the fibers of It.

Proof Given q9 as above, there exists an effective E' such that ~0 factors through the inclusion Os(D)~_Os(D+E') and V/Os(D+E' ) has torsion-free cokernel. Thus, we may write

0 ~ Os(D+E') ~ V ~ O s ( - D - E ' ) Iw ~ O,


and hence - (D + E') 2 + I(W) = c2 (V) = c. It follows that (D + E') 2 > -- c. Since V is L-stable, we have L.(D+E')<O. By the definition (2.1) of c-suitability, f . (D + E') < 0 as well. Since f . D > 0 by hypothesis, and f . E' > 0 since E' is effective and f moves in a base-point-free pencil, we must have f . D = f . E' = 0. This proves (i). Moreover , since f . E ' = 0 , E' must be suppor ted in the fibers of n. Since, clearly, in the no ta t ion of (ii), Supp E ~ Supp E', the same mus t be true for E. [ ]

(2.8) Lemma. With L, V and c as in (2.6), suppose that n , V4:0. Then either V~ is indecomposable or V~ ~ (gs,(F) @ (g s, (--F) (i.e. V, is decomposable over k (~ 1) ).

Proof. In any case, n . V is a vector bundle over P~. If n . V+O, then there is a line subbundle ( 9~ , ( - a ) of n . V. F r o m the natural nonzero m a p n* n . V ~ V, there is an injection C s ( - a f ) ~ V , where a > 0 by stability. Applying (2.7) to this injection, we find that there is an effective divisor D' suppor ted in fibers of n and a 0-dimensional subscheme Z, such that the following is exact:

O--.(gs(--a f + D ' )~ V~(gs(a f - -D') l z ~ O .

Restrict ing to q, we get an exact sequence

O --, #s, -* V, ~ (f s --, O.

Thus, V~ is decomposab le ~ the extension class is zero over 0 "*~ the extension class is 0 over i;. Hence, either V~ is indecomposab le or V, = (gs, | (gs,, proving (2.8) (with f - 0 ) . [ ]

Proof of Theorem (2.6). We shall in fact p rove that if V~ satisfies (i) of (2.4) or if V,-~ r174 C s , ( - F) (i.e. if V, is decomposab le over k (IP 1)), then V satisfies Case (A) of (2.6), and that in the remaining case (V 0 is decomposab le but V, is not), V satisfies Case (B).

First assume that V, ~ (gs,(F) | (gs, ( - F). Possibly after replacing F by - -F , we m a y assume that degs , (F )>0 .

The inclusion Cs,(F) ~ V, cor responds to a m e r o m o r p h i c m a p ~0o: Cs(Fo) ~ V, where F o is some divisor on S inducing (~)s,(F) on S,; moreover , the only possible poles of q)o are a long fibers of n. Hence, after replacing F o by F o + b f, for an appropr ia t e b, we get a m o r p h i s m of sheaves ~01: Cs(Fo)-~ V. There is an effective divisor D' such that q~ extends to a m a p q): Cs(Fo+D')~ V and coker q) is tors ion free. As Fo.f=degs,(F)>O, we must have (Fo+D').f=O by (2.7). Set D=Fo+D' , we have shown that V satisfies Case (A) with D as above. The case where V~ is indecomposab le is similar, and shall be left to the reader.

Finally, we consider the case where V~ is decomposab le but V~ is not. Let g: C---~ ~? 1 be the 2-1 cover of IP l defined by the i so t ropy g roup of Gal(k(~-~)) acting on {d~'s~(F), (gs~(-F) } (nota t ion as in (2.5) 3)). Set T= normal iza t ion of S x ~ , C and let 7"=the min imal resolut ion of T, with ~7: 7 " ~ S and p: 7 " ~ C the natura l maps.

C ,p1 . g

298 R. Friedman

As in the previous case, we have an exact sequence

0 --. Ct(D) ~ ~* V ~ C t ( - D) Iz --* O,

where we may assume that D.f'__>0 f o r f ' the generic fiber ofp. By (2.8), n . V-0 , and hence p . (~*V)=0. By Riemann-Roch on f ' , this forces D.f'<=O. Hence D .f ' =0.

We must still prove the last statement of (2.5). Let V satisfy (i~(iv) of Case (B). Via (2.4) and (2.5), we see that (ii) of (B) implies that V n is decomposable and that V, is not. Suppose that Cs(F) is a subline bundle on V. We must show that b F<O. We may assume that V/Cs(F) is torsion free. Hence, there is an exact sequence

O ~ C s ( F ) ~ V~( f s ( - -F) Iz ~O.

Restricting this sequence to S,, we obtain

o ~ os~(v ) -~ v. --, ~ s . ( - F ) ~ o.

We have Extl(S,; Cs,(--F), Cs~(F))_~HI(S,; Cs,(2F)), which is naturally dual to H~ Cs,(-2F)). Thus, if degs,(F)= F.f>O, the extension splits, contrary to our assumptions on V,, and if degs,(F)=0, the extension does not split over k(~? 1) only if it does not split over ~ . This would imply that V~ is not decomposable, again contradicting our hypothesis. Hence, F.f<O. Since c2(V) = c = - F 2 + I ( Z ) , F2>=-c. As L is c-suitable, we must have L.F<O, because F . f < 0 . Thus, Vis L-stable. []

(2.9) Remark. An application of the Riemann-Roch theorem to V as in [6], (3.19) or [8], Part Two (2.6) shows that, if c<2 ( l+p g ) , then V satisfies Case (A). More precisely, there is an exact sequence

0 ~ Cs(D - Ks) ~ V-~ (gs(K s - D) Iz ~ O,

where D is an effective divisor supported in the fibers of n. However, as we shall see, for all c >= 2(1 + p,), there exist bundles V satisfying Case (B) with c2(V) = c, and, for c sufficiently large, the set of such bundles is dense in the moduli space.

w 3. Analysis of Case (A)

Our goal in this section will be to show that, if cz(V) is sufficiently large, then there are comparatively few bundles V satisfying Case (A) of (2.6). We begin with a reformulation of this case.

(3.1) Assumption. Let L be a c-suitable bundle on S and V an L-stable bundle with c l ( V) =0 and c2(V)=c. Throughout this section we assume that there exists a divisor D on S with D . f = 0 and a 0-dimensional 1.c.i. subscheme Z, such that V fits into an exact sequence

O ~ C s ( D ) ~ V ~ C s ( - D ) I z ~ O .


(3.2) Lemma. With V, D as above, we have D2<0. Moreover, D 2 = 0 ~ D is a rational multiple o f f modulo torsion ~ for all k > 1, hZ(kD)+O ~ there exists a k>= I such that hZ(kD)+O.

Proof By assumption, D. f=O, and so D 2<0, by the Hodge index theorem, with equality holding if and only if D is a rational multiple o f f (modulo torsion). To prove the remaining statements, note first that by the stability of V, L. D < 0. Hence, for all k > l , k D is not effective. Thus, h~ for all k > l . By the Riemann-Roch formula,

h 2 (kD) = h ~ (kD) + Z(Cs) + � 8 9 Ks)

= h I (kD) + (1 +pg)+ � 89

First suppose that D is a rational multiple of f (modulo torsion). Since Ks is a rational multiple of f up to torsion by the canonical bundle formula, D 2 = K s . D = O . Thus, h Z ( k D ) > l for all k > l . Conversely, if h Z ( k D ) > l for some k > l , then - k D + K s is effective, say C ~ [ - k D + K s [ . By hypothesis, C . f = - - k ( D . f ) + ( K s . f ) = O . Thus, C is supported in the fibers of n, and hence is a rational multiple of f As C = - - k D + K s , D is a rational multiple of f as well, modulo torsion. []

Thus, according to (3.2), we may divide up the V satisfying Assumption (3.1) into two sets, those for which D 2 < 0 and those for which D 2 =0. We shall deal with these cases separately, beginning with the first (and easier) case.

(3.3) Lemma. With L, V,, c as in (3.1) suppose that D2<0. Then there exists a O-dimensional 1.c.i. subscheme Z ' and a divisor D' with (D' . f )=0 , (D') 2 <0, and

- (2 c)(L.f) < (L. D') < O,

such that V is given as an extension

0 --* (gs(D') ~ V--+ C s ( - D') Iz, ~ O.

Proof For all k > 0, we have the natural exact sequence

0 ~ H~ + k f ) ) --~ H o m ( C s ( - D - k f ) , V)

H~ f ) Iz) ~ H 1 (Cs(2D + k f)) .

Note that ( 2 D + k f ) . f = O . If 2 D + k f were effective, it would thus have to be supported in fibers of g, and hence, modulo torsion, a rational multiple of f But this contradicts ( 2 D + k f ) Z = 4 D 2 + 4 k ( D . f ) = 4 D 2 < O . Hence, h~ + k f ) = 0. A similar argument shows that h 2 (2 D + k f ) = h ~ ( - 2 D - k f + Ks) = O. By Riemann-Roch, then, h l ( 2 D + k f ) = - 2 D Z - - ( l + p g ) . From the exact sequence

0 ~ H ~ Iz) --* H ~ ~ H~

we obtain dim H~ f ) Iz)) >= k + 1 - I(Z).

300 R. Friedman

As H ~ (Cs(2 D + k f)) = 0, we have

dim H o m ( C s ( - D - - k f ) , V)> k + 1 - I ( Z ) - - ( - 2D2-(1 + pg))

=k + 2 + pg--I(Z)+ 2D 2.

Since c2 (V)= c = - D 2 + I(Z), we have D 2 > - c , and

dim H o m ( ( ~ s ( - D - k f ) , g)> k + 2 + pg--( -D2 + l(Z)) + D 2

>=k+2+pg-2C.

Thus, if k=2c, we have dim H o m ( C s ( - D - ( 2 c ) f V)>0. Choose an injective map C s ( - D - ( 2 c ) f ) ~ K It vanishes a long a divisor E which is effective or 0. Thus, if we set D ' = - D - - ( 2 c) f+ E, we get an exact sequence

0 ~ Cs(D' ) --* V ~ Cs( - D') I z, ~ O.

By (2.7), E is a Q-mult iple of f. Hence (D') 2 = ( - D -- (2 c ) f + E) 2 = (-- D) 2 = D 2 < 0, and (D ' ) . f= - (D .f) = 0. Moreover ,

(L D') = /~ ( - D -- (2 c) f + E) = (-- 2 c)(I~f) + (-- D. L) + (E-L).

As V is L-stable, ( D . L ) < 0 , and (E.L)>=O since E is effective or zero. Hence (L D ' ) > ( - 2 c)(bf), and, again by the L-stability of K (L. D ' ) < 0, as desired. [ ]

(3.4) Proposition. Let L, V, D, and c be as in (3.1). Then the set of all V for which D 2 < 0 may be parametrized by a scheme of finite type of dimension < 3 c - -2- -pg .

Proof. By (3.3), we may always assume in addi t ion that D satisfies

(*) ( - 2 c ) ( D f ) < ( ~ D ) < O , D2> --c.

The set {DePic(S): D satisfies (,)} is finite, by a s t raightforward a rgument left to the reader. We choose one such D. Let l=c+(D2), so that O<l<c, and let U be the open subset of Hi lbtS corresponding to l.c.i, subschemes of S. If Z corresponds to a point of U, then by (1.7) there is an exact sequence

0 --* H 1 (2D) ~ gx t 1 ((gs(-D) Iz, Cs(D)) --* H~ --, 0

(where we have used (3.2) to conclude that H2(2D)=0) . Hence,

dim Ext i ((gs ( _ D) Iz, (gs(D)) = l(Z) + h ~ (2 D)

= l + ( - 2OZ) - ( l + p,).

Let ~ _ U x S be the universal subscheme, and n~, n2 the projections of U x S to its first and second factors. By s tandard arguments involving relative Ext groups (e.g. [4] or [13]), there exist a ~"-bundle ~: P ~ U and a rank two torsion free sheaf ~U over P x S such that

(i) the set 7~-l(u)is canonically F(Extl((gs(--D)I~.,Cs(D)), where ~e =r t i -~(u)n ~ ;

R a n k two vector bundles over regular ell iptic surfaces 301

(ii) if c~ is a non-zero element of Extl ( (gs(--D)l~, , (gs(-D)) and ~ is its image in IP(Extl((gs(--D)I~,,(gs(D)), then the restriction of ~ to {~} x S_~ a - l ( u ) x S, via the identification in (i), is the r ank two tors ion free sheaf

which is the extension of ( g s ( - D ) I ~ , by (gs(D) cor responding to ~. There is thus an open subset of P (possibly empty) which paramet r izes all

L-stable locally free extensions as in (3.4). It will therefore suffice to es t imate the d imension of P. But

by (i)

d im P = dim U + r = 2 1 + r = 2 1 + ( l + ( - - 2 D 2 ) - - ( 1 +pg) - - 1,

= 3 1 - 3 D 2 --(1 +pg)-- 1 + D 2

= 3 c - - ( l + p g ) - - l + D 2 < 3 c - 2 - p g , as D 2 < 0 . [ ]

Fo r the remainder of this section, we shall assume that D: = 0 ; equivalently, by (3.2), D is a Q-mul t ip le of f m o d u l o torsion. Again by (3.2), hZ(2O)>O; equivalently, K s - 2 D is effective. Finally, we clearly have c 2 (V) = I(Z) = c.

(3.5) Notation. Let F//, i = 1 . . . . , #, denote the mult iple fibers of ~, and suppose that the multiplicity of F~ is p~. There are unique integers a > 0 , r~ such that 0 < r~ < pi, and with

/t

K s - 2D=aJ'+ ~ riF i. 1

F r o m the canonical bundle formula, we thus obtain

(3.6) - 2 D = ( a - p g + 1) f + ~(r i - -Pi+ 1)F/

= ( a - p g - # + 1)f+~,(r i+ 1)F i.

The following l emma is an analogue of [8] Par t Two, (3.7).

(3.7) L e m m a . Suppose that V is as in (3.1) and that D2=O. Then there exists an exact sequence

0 ~ (gs(D') ~ V ~ (gs(-D') I Z, --* O,

where again (O') 2= 0, and such that (i) for all effective nonzero divisors E supported in the fibers of 7z, we have

H ~ (C s (-- 2 D' -- E) lz,) = 0. (ii) h ~ ') I z , )< 1.

Proof Begin with V and D as in (3.1). Suppose that there exists an effective divisor E, suppor ted in the fibers of ~, and such that H~ Write E = b f + ~ s i F ~ , where b, s i > 0 and at least one of b, s~ is strictly positive. It follows that H ~ where E o = f or F~. F r o m the exact sequence

H~ (@s) ~ H~ ((-geo) ~ H1 (Cs ( - Eo)) ~ H1 (@s) = O,

302 R. F r i e d m a n

we have H a (Cs(-- Eo)) = 0. Using the exact sequence

H o m (Cs (D + Eo), V) ~ n ~ (Cs(-- 2 D -- Eo) Iz) ~ H 1 ((gs (_ Eo)) = O,

we see tha t there is a nonzero m a p Cs(D + Eo)--* V. After enlarging D + E 0 to a divisor Do, we m a y assume that V/Cs(Do)) has torsion-free cokernel. By (2.7) (ii), Do is again a ra t ional mult iple of f, so that (D0)2--0. In addit ion, L.Do > L- (D + Eo) > L. D. We have thus writ ten V as an extension

0 ~ Cs(Do) -* V~ C s (-- Do) lzo -* O,

with L.Do>L.D. IfH~ Izo):#O for some E as in (i), we m a y repeat this process. At each stage, b D strictly increases, and L.D < 0 by stability, so this process must terminate . We thus obta in a D' and Z ' as in (i). To see (ii) note that the moving pa r t of the linear system I C s ( - 2 D ' ) I z , I is clearly of the form n f n>=O. F o r this n, we have h~ But if n > 0 , then h~ contradic t ing our choice of D'. Hence either IC s ( - - 2D ' ) Iz.I is emp ty or h~ ') lz , )= 1. [ ]

We need some no ta t ion to measure the n u m b e r of condi t ions tha t Z imposes on - 2 D and on K s - 2 D.

(3.8) Notat ion. Let the 0-dimensional 1.c.i. subscheme Z be given. 1) Let xeE, where E is a nonmult ip le fiber of 7r, and set

w ~ = m i n { n : Cs( -n E)~-lzx} (notat ion as in (1.4))

2) IfF~ is a mult iple fiber and xeF~, set

w ~ = m i n { n : Cs((-pin--(ri+ 1)) F/)-~Iz~.}, if ri+ 1 <Pi;

= m i n { n : Cs(--pinF~)~_Izx} if r~+ 1 =p~.

3) Set w(E)=max{w~: xeSuppZc~E}, where E is the reduct ion of a fiber. 4) Set b = #{ i : S u p p Z n F ~ 4 : 0 , r ~ = p ~ - 1}. 5) Similarly define, for F~ a mult iple fiber and xeF~,

v ~ = m i n { m : Cs((--plm--ri)Fi)~_Izx}

and let vx=w~ if x lies in a nonmul t ip le fiber. Define v(E) as in 3). It is easy to see that v(E)= w(E) if E is nonmul t ip le and tha t

(i) If r i + 1 < pi, then w(F~) <_ v(Fi); (ii) I f r i + 1 =Pl , then w(Fi)<v(Fi)+ 1.

The following l e m m a is a simple consequence of (3.5), (3.6) and (3.8):

(3.9) L e m m a . 1) h~ a--Pg--l.t+b+ 2 - ~ w ( E ) } . Hence, if E

h~ 2D) Iz) < 1, then ~ w(E)>=a-p~-# + b + 1. E


2) I f h~ then h~ E

-- ~ w ( E) + b + 1. Hence, if h ~ ( (g s( K s-- 2 D ) I z) Oe O and D satisfies the conclusions E

of (3.7), then h~ I f moreover h~ = p , + g , then h~ 2D)Iz)= 1.

Proof. We have - 2 D = ( a - p g - # + b + l ) f + y' ( r i+l)Fi , and hence either r , < p ~ - 2

a+b+l<pg+l~ or the base locus o f - 2 D is ~ ( r i+l)F/ . Then 1) of(3.9) r , < p~ - 2

follows from the definition (3.8) of w(E), and the first two statements of 2) are similar. Finally, if h~ ~, then b = 0 and ~ w ( E ) = a

E - p ~ - p + b + l . Thus, we have a - p ~ - # + b + 2 - ~ w ( E ) = l . []

Next, keeping Z fixed we estimate the number of moduli in the extension class.

(3.10) Lemma. We have

dim Ext 1 ((gs(- D) lz, (gs(D)) = l(Z)-- (1 + p~) + h ~ ((9 s (K s - 2 D) Iz).

Proof. We have an exact sequence

0 ~ H 1 ((gs(D)) ~ Ext 1 ( (gs(- D) Iz, (gs(D)) ~ H ~ (Ext ~ ((gs(- D) Iz, (gs(D))

H2 ((gs(2 D)) --* ExtZ ( (gs(-- D) I z, (gs(D)) ~ 0.

By Serre duality, dim Ext 2 ((9 s ( - D) I z, @s (D)) = h ~ (6) s (K s -- 2 D) Iz). Hence

dim Ext ' ((9 s ( - D) lz, (gs (O)) = h i (2 D) - h2 (2 D) + l(Z) + h ~ ((9 s (K s - 2 D) Iz).

Since h~ hl(2D)--hZ(2D)=--(l+pg), by Riemann-Roch. This gives (3.10). []

The remaining lemma we shall use generalizes a necessary condition on a reduced collection of points for there to be a locally free extension as in (3.1).

(3.11) Lemma. Suppose that V is as in (3.1) and that h~ Let f be a nonmultiple fiber, and suppose that, for all xe (SuppZ)c~f Zx is a reduced point. Then ~ {x : (Supp Z) c~.[+ @} > 2.

Proof. This argument is similar to the proof of (I.9). Suppose that ( suppZ)c~f consists of a single reduced point {x}. By (3.5) and the hypothesis h~ --2D)lz)4=0, there exists a section of Ks--2D which does not vanish at x. Thus, the induced map /4~176 generates over x; by Serre duality, as in [10], the image of Exta((gs(-D)Iz,(gs(D)) in H~ ( - D) I z, Os (D)) ~ I4 o ((gz) cannot generate the stalk of (gz over x. This is a contradiction. []

We can now state the main result concerning the case D2= 0.

3 0 4 R. F r i e d m a n

(3.12) Theorem. The set of L-stable bundles V, given by an extension (3.1) with c2(V)=c and D2=0 may be parametrized by a scheme of dimension <max(5c + / ~ -2 , 3 c--(1 + pg)).

Proof. We may write V as an extension as in (3.1) with D2--0 and such that D satisfies the conclusions of (3.7). In particular, we may assume that h~ --2D)lz)<=l.

In the notation of (3.5) and (3.6) for D, we have

a<=~ w(E)+ pg + !a<=c + pg + #, E

by (3.9). This inequality, together with / ~ ( - D ) > 0 , gives only finitely many choices for the divisor D. We fix such a choice of D. For such a choice, consider the set of 1.c.i. subschemes ZeHilbC(S) with H~ 1 and such that Z satisfies (3.11) if h ~ (C s (Ks -- 2 D) Iz) �9 O.

By standard arguments, there exists a locally closed subset U of Hilbe(S), corresponding to 1.c.i. Z, and a stratification u = L I u~, such that, for ~e u the subscheme of S corresponding to u e U~, dim Ext I (Cs ( -D) I ~ , Cs (D)) is constant on U~, and such that locally free extensions (3.1) exist on U~. Arguing as in (3.4), there is a IP" bundle P~ over U~, with fiber ~'(Extl(Cs(--D)I~u, Cs(D)) over u, and a universal extension ~/F over (an open subset of P=) x S. We may further assume that the functions h~ h~ of u are constant on U~. Thus,

dim P~ = dim Ext i (C s ( _ D) I~ru, Cs (D)) - 1 + dim U~

= I (Z) - (1 +pg)+ h~ I~u) - 1 +d im U~, by (3.10).

Case (i). h ~ (C s (Ks-- 2 D) I~r~) = 0. Then

dim P~ = c - (1 + p g ) - 1 +d im U~< 3 c - ( 1 + p g ) - 1.

Case (ii). h ~ (Cs(Ks -- 2 D) I~. ) =t= O. We use the following notation (cf. (1.3)). If z: Hilb~(S) ~ Sym~(S) is the natural

d d"

map, write z (Z) = ~ n~ x~ + 2 m~ y~, where y~ ~ U Fk, the union of the multiple fibers, 1 1 k

and xieS--({,_ ) Fk). Suppose that El, ..., Ek are the fibers containing the xi's, k k k

and that =~({x~ . . . . . xd}c~Ej)=m~>O. Hence d=~mj . Clearly, dimz(U~)__<y. d' 1 1

( l+mi)+~(1)=k+d+d' . Hence, by Iarrobino's result (1.5), dimz-lz(U~)<=c 1

- (d+d ' )+k +d+d'=c +k.

C Claim. k < ~.

k

Proof of the claim. Set c(Ei)= ~ l(Zx). Clearly ~c(Ei)<c. Thus, it suffices XjEE i 1

to show that c(E~)>2. But this is clear if E~ contains a nonreduced point of


Z. Otherwise, all points of Ei ~ Supp Z are reduced. By (3.11), by the hypothesis of Case (ii), there must be at least two of them. Hence c(Ei)>2 in this case as well.

Returning to Case (ii), we thus have dim z-1 z(U~)< ~c. By (3.10) and (3.9.2), we have

z . .

3c 1 dim P~<-2--+ c - ( +p~)+h~

5c 5 <__T-O +p,)+p,+~- t =~ c + ~ - 2;

moreover, if equality holds, h~ 2D) l:zu) = 1. To complete the proof of (3.12), note that we have constructed bundles

~//~ over (open subset of P~) x S, and hence families of bundles over S. Restricting to further open subsets (possibly empty), we may assume that all of the bundles constructed in this way are L-stable. We have d i m ~ < 3 c - ( 1 +pg) in Case (i) and d i m P ~ < 5 c + # - 2 in Case (ii). If equality holds in this last case, then, if V is a corresponding bundle written as an extension (3.1), h~ 1. From the sequence

0 ~ C)s ~ Hom((f)s(D), V) ~ 6Os( - 2D) lz ~ O,

we thus obtain dim Hom(Cs(D), V) = 2, i.e., V can be written in the form (3.1) (with possibly variable Z) in oo 1 ways. If V is in addition stable, this shows that the map (open subset of P~)~ (moduli space) has one-dimensional fibers, so that, after taking an appropriate locally closed subscheme of P~, we get strict inequality in the estimate. This completes the proof of (3.12). []

(3.13) Corollary. I f c > max(2(1 +p~)- 1, 2pg+ then the generic L-stable bundle V over S with cl(V)=O, c2(V)=c, is of Type (B).

Proof By deformation theory, the dimension of the moduli space of such bundles, at every point, is =>4c-3(1+p~) . Thus, (3.13) follows from (3.4) and (3.12), by solving the inequality

max(Sc+p-3 ,3c-( l+pg)- -1)<=4c-3( l+pg) . []

(3.14) Remark. By reversing the above methods, one can show that the methods of this section are essentially sharp.

w 4. Analysis of Case (B): Chcrn class calculations

For the remainder of this paper, we shall assume that V is a rank two bundle on S satisfying Case (B) of (2.6):

(4.1) Assumption. Let V be a rank two bundle over S with cl (V)=0 and cz(V) = c. We assume:


(i) There exists a smoo th curve C and a 2-1 m a p g: C ~ ~ 1 . Set T = n o r m a l i - zat ion of S x r l C and T its min imal resolution, and let ~: T ~ S be the induced map. Let p: T--, C be the elliptic f ibration and f the generic fiber of p.

(ii) There exists a divisor D on T such that D . f = 0 and such that the restric- t ion of D to f is not ra t ional over k(~'~).

(iii) There exists a codimens ion two 1.c.i. subscheme Z of T and an exact sequence

0 ~ (9~.(D) ~ ~* V ~ (9~.(- D) I z --, O.

In this section, we shall relate the second Chern class of V to the geomet ry of the double cover C of ~ and the divisor D. In the next section, we shall s tudy the s tructure of V more carefully. Finally, in w 6, we shall es t imate the d imension of var ious s t ra ta of the modul i space and character ize those s t ra ta of max ima l dimension.

F r o m (4.1) (iii) and the fact tha t deg ~ = 2, we have

(4.2) L e m m a . 2 c z ( V ) = 2 c = - D 2+l (Z ) . []

(4.3) Next , we fix some notat ion. Suppose that g: C - ~ I W is b ranched over xa . . . . . x a, where n - l ( x i ) is a singular, non-mul t ip le fiber, l < i < a , and also over x , + l . . . . . x b, where n-~(x~) is either smoo th or a mult iple fiber of odd multiplicity, a + 1 < i_< b. The remaining b ranch points of g cor respond to mult i- ple fibers of even multiplici ty; the m a p 7"~ S is 6tale over these. If T ~ S is b ranched over a s ingular fiber, then T has an ord inary double point over the singular point of the s ingular fiber (a local equa t ion for T is x y = t 2 at such a point). Let 0: S ~ S be the b lowup of S at the singular points of n-l(x~), 1 < i < a, and v: T ~ S the induced finite map. We have a diagram"

p

)

g C ,IW.

Let cl be the except ional curve on S lying over the singular fiber n - l ( x i ) , and f/ the p roper t ransform of n - t ( x i ) on S. Thus, c/2= - 1 , f / 2 = - 4 , f i . c i = 2 and

- l (xi) =fl + 2 ci as divisors. Let v - 1 (fi) = ei, v - 1 (ci) = di. Then e 2 = d 2 = - 2, e i ' d i = 2, and el+ di is the fiber of p over g - l ( x i ) . As divisors, v* f i= 2 el and v* ci = dv Finally, set ~'= 4/* V, so tha t v* V= g* V.

(4.4). We shall also have to consider the Jacob ian f ibrat ion j: J ( S ) - - + ~ n (cf. w and the cor responding f ibrat ion J ( T ) of p: T--+C. Recall that J ( S ) is i somorphic to S on the c o m p l e m e n t of the mult iple fibers. Thus, j - ~ (x~) is again


a singular fiber o f j for i<~a. Let ~0: J ( S ) ~ J ( S ) be the b lowup of ~r at the singular points ofj-~(xi), l <i<_a. Finally, let k = ( 1 +pg). Using the no ta t ion of (1.2), we have a d i ag ram

j ( ~ ) s ,S(s)

C

r/ ' ~ 2 k

) l P 1 ,

Let d'i be the curve on ~r cor responding to di on T; i.e., d'i=f l(c'i), where

c'i is the except ional curve on ~r (S) m a p p i n g down to j-l(xi)~_ J (S).

(4.5). Referring to (2.6), we see that the choice of divisor +_D on T gives two sections of ~ ( 7 ~) over the generic point of C, and hence a bisection o f j over the generic point of IF' 1 which is invar iant under _+ 1. Thus, by (1.2), we obta in a section of N2k over the generic point of P~. This section extends uniquely to a smoo th section A of ~c:72k--~ ]p l . Set rl*A = (7. By assumpt ion , C is irreducible and birat ional to C. Thus, in the no ta t ion of (1.2), A 4= a (where a is the negat ive section of ~zZk ). Hence, if l is the class of a fiber in IF2k, there exists an r > 0 such that A e l a + ( 2 k + r ) l l . Conversely, the choice of a smoo th A ~ l a + ( 2 k + r ) l l for which rI*A=C is irreducible determines g: C-- . IP l, T, and a divisor D on the generic fiber of T--* C, well-defined up to _+ 1.

(4.6). By (1.2), t/ is b ranched over a smoo th curve of the form or+B, where B is a smoo th m e m b e r of [ 3 ( a + 2 k / ) [ , disjoint f rom o. In the no ta t ion of (4.5), we have

A.(~+ B)=(~ +(2k +r)l).(4~ + 6 k l )=6k +4r.

Generically, then, C ~_ C ~ IP ~ is b ranched at 6 k + 4 r points. However , in special cases, either the intersection of A and a + B fails to be t ransverse or (7 passes th rough the singular points of some singular fibers. We shall s tudy these cases separately.

(4.7) Non-transverse intersections. Let p cA n (or + B), let x ~ P~ be the point lying under p, and let ~x=Ap(c r+B) be the local intersection n u m b e r at p. Thus, over p, the m a p t/: C ~ A looks like xZ=y ~x.

(i) The case c~ x = 2 fix + 1 is odd. Then C has a cusp at p, but is analyt ical ly irreducible. If C = normal iza t ion of C and q is the unique point of C such tha t g(q)=x, we have l((gC,q/(ge, p)=flx, i.e. the local genus d rop is fix. The pull back of C to ~r splits into two sections C~, C2 which meet over q, with local intersection n u m b e r ~x = (2 fl~ + 1) (Fig. 1).

308 R. Friedman

ip 1

J (S)

(i) odd case

Cz

J (Y)

P J IS}

ipl x ql q2

(ii) even case

Fig. 1

J (Y)

(ii) The case ~x = 2 fix is even. Then C has two local branches at p, meeting tangentially. Let C be the normalization of C and ql, q2 the two points of C mapping to p. We have I((~c,ql ~) (gC, qJ(gC, p)=fix" The pull back of C to J(7") splits into sections C1, C2 which meet over ql and over q2; over each point the local intersection number is fix (Fig. 1).

(4.8). C passes through the singular point of a singular fiber. Since E is a bisection of j, it must be smooth at such an intersection point and meet the two local branches transversally. Note that g: C ~N1 is automatically branched at such

a point. The proper transform C' of C on J (S) is then r where

the c;, are the exceptional curves on J (S ) coming from blowing up points through which C passes. Locally on J ( T ) , C' splits into C~ and C2, two sections which do not meet on d~, in the notation of (4.4). The situation is summarized in Fig. 2.

(4.9). In the notation of (4.4), the pullback f * C' of C' to j (7") splits up into C 1 Jr- C2, where C~, C 2 a r e sections of the elliptic fibration J(7") -~ C. If t: J ( T ) ~ J ( T ) is the involution corresponding to the double cover, then l ( C 0 = C2.

Let Z=rl*a be the 0-section of J ( S ) and identify S with ~o'2; on J(S). Then f * S = S ' is the natural section on J(7"), and C I - Z ' has degree 0 on each irreducible singular fiber of J (T ) . For the reducible fibers, C i - S ' has degree 0 on both components if C does not pass through the double point of the


C' ~ d~ C' C'i

fi~ , ei~

,

fiber = f'k + 2C~ ]E' J (S] J ($1 J (~1

Fig. 2

corresponding fiber on J(S). If C does pass through the double point, then C i - Z' has degree - 1 on e~ and + 1 on d~, (Fig. 2).

We now relate the divisor D on ? ' to the divisor C i - X ' on J (T) . Let U = 7" - (multiple fibers)~ J (T) - (the corresponding smooth fibers). For a topological space X, set Hi(x)~-HI(X; 7l) modulo torsion. We have a natural map

Hz(U)_.+ H2(~ ) P_o. , H Z ( T ) ~ ( ~ . ) ,

qJt

where the map in the middle is just Poincar6 duality. Similarly, there is a natural map ~p~: HE(U)-- ,Hz(J(~) ) .

The following lemma is an application of the Gysin sequence. Its proof is left to the reader.

(4.10) Lemma. (i) The image of tpt is contained in {~e/~2(~): ~. i f ] =0} (where [ f ] is the class of a general fiber), and the quotient {~e/~2 (~): ~. I f ] = 0}/lm tpl is generated by a rational multiple of I-f]. Moreover,

Im q~= = { a ~ 2 ( j (~) ) : a. [ f ] =0}.

(ii) Im ~p ~ ~ Im q~2 ~ A, say. (iii) Via (i), Im q)i has a natural quadratic form, and the isomorphism in (ii)

preserves this form. []

Via (4.10) (i) we may view D and C 1 - Z ' as defining elements of A + Q [ f ] , which we denote by [D] and [C1-27 ]. Similarly, from the inclusion A ~_ n 2 ( f (~)), we have classes IdOl, [ f ]m A.

(4.11) Lemma. Possibly after replacing C~ by C2, there exist rational numbers hi, 1 <_ i <_ a and m such that

[D ] = [ c , - z ' ] + ~] n, [d',] + m [ f ] .

310 R. Friedman

Proof There exists a smooth multisection C of 7" such that the map C-~ C is unramified at points which are the images in C of singular fibers. Let 7"= normalization of Tx c C ~ J (T) x c C, and let p 1 : ~--* T,, P2 : T--* J (T) be the natural maps. We have 0 =pi -a(U)=p2~ (U)_ T. Since Pi is proper, there is an inclusion Hz (U) ~ H 2 (0 ) and a commutative diagram

H2(U) r ,tq2(T)

N2(0) ~ ,~q2(%,

and similarly for ~o z. By construction, and possibly after switching C1 and C2, the line bundles

p*(9~.(D) and p*(9~ ,{ t ) (Ci -Z ' ) induce the same divisor over the generic fiber of 7"~ C. Hence, they differ by some combination of fiber components. But the only reducible fibers are the preimages of the fibers d'i+e'~. If we write p21 ' _ J_ - ( d i ) - ~ , d i - p l ~(di), we must therefore have [p* D] = [p* (C1-- Z')]

J

+ ~, au[d{] +mo [ f ] , where au, m o eQ. As clearly [p* D]. [d/] = [D]. [di] is independent of.l, and similarly for [p* ( C 1 - 2;')]-IdOl, we can write

~, a u [d/] + mo I f ] = ~ n, [P2 d,] + m [p* f ] ,

where n~, mell~. Restricting to U, we obtain (4.11). []

We have the following natural conditions on n~:

(4.12) Lemma. (i) ( D . d i ) = e i - 2 n i , where ei=O if C1 does not pass through d'i and el = 1 if C 1 does pass through d'i (cf. Fig. 2).

(ii) (D. d,) < O.

Proof (i) By (4.11),

(D. d~) = (C, - Z'). d'~ + n,(d'~, d'~) = ~ , - 2 n,,

since U. d'i = 0. (ii) We have

v* ~'b, = v* (~%) = v* ((~c, | (%)= (~d, | (~d,.

On the other hand, we have an inclusion (9~.(0)~ v* V,, which is of maximal rank except at a finite number of points and hence generically along di. Thus, there is an injective map (9~, (D) ~ (9~, �9 (rid,, and hence (D- dl) < 0. []

(4.13) Lemma. nieTZ for all i.

Proof By (4.12), 2n~e7l for all i. Suppose that not all n~e7Z,; we will produce a contradiction. From (4.11), using the inclusion A~_H2(j(T)) , we have ~, ni [d'i] + m~ I f ] an integral class in H2( j (T ) ) , for some ml eQ.


Case (i). The denominator of m is odd. Then after multiplying by twice the denominator and subtracting appropriate integral cohomology classes, there is a nonempty set I such that ~ d l is divisible by 2 in H z ( J ( T ) ; ~ ) . We can

i E l

then form a double cover of J ( T ) branched along ~d'i, say ~, corresponding i e l

to a choice of L e P i c J ( T ) with L| Note that, i f f is a generic fiber, i e l

then (9:t(L) is a line bundle of order 2 in Pie f,. If (5:,(L) is nontrivial, then there is a nontrivial 2-torsion point of the generic fiber of J ( T ) rational over k(C). Since [k(C): k(~X)]=2, and the nonzero 2-torsion points of the generic fiber of J ( S ) are not rational over k(~'~), this is impossible. Thus, 6~:,(L! is trivial for general t. It follows that T is a smooth elliptic surface; near d~ it is locally a base change of order 2. However, in T, the inverse image of d'~ is an exceptional curve, and after contracting it the fiber is an irreducible curve. This is absurd.

Case (ii). The denominator of m is even. After multiplying by the odd part, as before, we may assume that the denominator of m is 2 N, a > 1. If N > I , then [ f ] is divisible by 2 in H 2 ( ~ ( ' F ) ; Z), which is a contradiction. If N = 1, then there is a nonempty subset I as before with ~ d'~+f divisible by 2 in

i ~ I

Pic~(T) . Choosing i~I, write f=e'i+d'i. Then e'i+ ~ d~ is also divisible by j 6 l , j 4 i

2, and the method of Case (i) handles this case also. []

We now proceed to the calculation of D 2. By (4.11), using C~.d'~=e, i ( = 0 or 1) and Z'. d'~ = 0, we have

a

D 2 = ( C 1 - U ) 2 + 2 f n , ( C ~ - U ) . d ; - 2 ~ n { 1 1

a

= (Ca - ~,)2 + ~ ( 2 ni ~ i - 2 n~). 1

Expanding out, we have

(C1 - U ) 2 = (CO 2 - 2(C1.2; ' ) + (Z') 2.

We ca lcu la te these terms separate ly , beg inn ing w i t h (Z') 2. As Z' = f * ~ = � 8 9 q* and o 2 = - 2 k, we have (2;') 2 = - 2 k.

(4.14) Claim. C I ' U = r .

Proof As I(Z ' )=Z' and z(CO=C2, we have

2(C1 "2;') =(C1 + C2).U = ( / * C').X,'=2(C'.r,)=2(C.2;),

since 2; is disjoint from the exceptional curves in the blowup J(S)--* J (S) . But

2(C.X)=C.(2Z)=~I*A.~I*a=2(A.a)=2r. []

312 R. Friedman

(4.15). To calculate C 2, we u s e I ( C 1 ) = C 2 again. Thus,

(C~ + CE) 2 = C~ + 2(C~. C2) + C~ = 2(C1) 2 + 2(Ct- C2).

On the other hand,

(C1 + C2) 2 = ( f * C') 2 = 2(C') 2 = 2((/)* ( 7 - ~ C'k) 2 = 2 ((7)2 _ 2 e, k

where e is the number of singular fibers such that (7 passes through the singular points (e = ~ ~i)- Moreover ,

i

Hence

((7) 2 = (q* A) 2 = 2 A 2 = 2(a + (2 k + r) 1) 2 = 2(2 k + 2 r).

(Ca) 2 = 4 k + 4 r - e - ( C 1.C2).

(4.16). Lastly, to calculate (C 1 "C2), we use (4.6), (4.8), and (4.9):

(Cl "C2)=6 k + 4 r - e + fl~v,

where fle~=Z{�89 ~x is even} =Z{f lx : ~ = 2fix}, in the nota t ion of (4.7). Thus, x x

( C 1 ) 2 = - 2 k - f l ~ and ( C 1 - S ' ) 2 = - 4 k - 2 r - f l ~ .

(4.17) Proposition. With the notations of this section, a

(i) D Z = - 4 k - - 2 r + ~ ( 2 n i e i - - 2 n 2 ) - - f l e v ; 1

a (~ t(Z). (ii) c 2 ( V ) = 2 k + r + ~ ( n ~ - n i e ~ ) + + 2 '

1

a

(iii) 4 c 2 ( V ) - 3 ( i +pg(S) )=5k+4r+21(Z)+2f lev+~(4n{-4nie~) . [] 1

(4.18) Corollary. I f V satisfies (4.1), then c2(V) > 2(1 + pg(S)).

Proof As l+pg(S)=k, it suffices to show that all of the terms on the right hand side of (4.17) are nonnegative. This is clear except possibly for the terms nZi--niei=ni(ni-ei). However , as ni~Z (4.13) and ei=O or 1, ni(ni-ei)>O always. [ ]

(4.19) Remark. In fact, as ~i-2ni<O and n~TZ, we have either e~=0 and n~>0 or e i = l and n~>l . Thus, in bo th cases, n~ and n~-e~ are nonnegative, and ni(n i - ei)= 0 only when either ~i = ni = 0 or e i = n i = 1.


w 5. Case (B): Descent conditions

If V is a rank two bundle on S satisfying (4.1), then v* V is an extension of C f ( - - D ) I z by Cf(D), for appropriate D and Z, and is thus given by a class in Ext~((9~-(-D)Iz, (9f(D)). Conversely, given such an extension V', when is it of the form v* P for an appropriate V? In principle, the answer is given by descent theory: if t: T--, T is the natural involution, there should exist an

isomorphism l* V' , V', restricting to the identity over the branch locus of

v. Since this condition seems a little difficult to apply in our situation, we shall proceed somewhat differently. We shall try to build V on S by finding standard pieces and then realizing P as an extension of these. We retain the notations and assumptions of w 4.

By construction, t*D and - -D have the same restriction to the generic fiber of p: T ~ C. Thus l* D = - - D - R, where R is numerically equivalent to a ~- l inear combination of a general fiber f and the di (see (4.3) for the notation). Moreover, we have

(5.1) - R = D + t* D = v * v , D,

where v, : Pic ~ Pic S is the natural map on divisors. Referring to (4.3), we assume that g ' C ~ 1 is branched over xl . . . . . x,

where 7r- t (xi) is a singular, nonmultiple fiber, 1 < i < a, and also over x, + 1 . . . . . Xb, where fl = l-7t-l(Xl)]red is a smooth fiber or a multiple fiber of odd multiplicity (and possibly over some points of ~'1 corresponding to multiple fibers of even multiplicity, which will not contribute to the branch locus of v). Setting f~ = proper transform of ~ - 1 (xi) on S and e i = v - 1 ( f / ) for all i < b, we have:

b

(5.2) Lemma. The branch locus of v in S is B = ~ f i . The divisor B is linearly b 1

equivalent to 2 G. We have v* B = 2 F, where F = ~, ei, and v* G is linearly equivalent

to F. [] x

The sheaf v, (9~.(D) is a rank two bundle on S, isomorphic to V on the generic fiber. By a standard argument (see [12], p. 306, Ex. 2.6)

(5.3) de tv , C~.(D)=v,D--G and v*de tv , C f ( D ) = - R - - F .

From the inclusion Cf(D) ~ v* V,, we have an injection

v, r ~ v, v* P= P | v, (~- = P | ((~t | ( ~ - ( - G)).

We may project v, Cf(D) into either factor of the direct sum.

(5.4) Lemma. The natural map v, C~.(D) ~ V/(-- G) is injective.

Proof. The direct sum decomposition v, v* ~'= V �9 ~'( - G) is obtained by taking the (+1) and ( - 1 ) eigenspaces of the t-action on v,v*V. But v,C~.(D) is a module over v, C~-, which contains ( - 1 ) -eigenvectors for the t-action. Thus, v, (9~.(D) does not lie in either eigenspace. Hence, the induced map v, (9t(D)

314 R. Friedman

--* V(-G) is nonzero. To see that it is injective, restrict to the generic fiber, using the fact that D is not rational over k(lP1). It follows easily that the restriction of v, (9~,(D) to the generic fiber cannot have a quotient line bundle of degree < 0 and that the restriction of V(-G) to the generic fiber cannot have a subline bundle of degree > 0 (cf. (2.5)(3)). Thus v, C~(D)~ V(-G) is injective over the generic fiber, and hence everywhere. []

(5.5). Using v* G = F, we have by (5.4) an injection

v, Ct(D + v* G) = v, (9t(D + F) --* [I.

Set W=v, (9~(D + F) and define the torsion sheaf Q by the exact sequence

o ~ w ~ P--.~2 ~ o .

By (5.3), de tW=v,D-G+2G=v,D+G. Since det ~ ' - ~ * d e t V=Cs, v,D+G = - - E for some effective divisor E, and the scheme theoretic support of Q is E. Moreover, - R + F = v* v, D + v* G = - v* E = -- E', where E' is again effective. As (v, D) -f-- 0, E and E' are orthogonal to the class of a general fiber, and hence are supported in fibers of ~ and p respectively. We also have:

(5.5.1) t(D) = - D-- E ' - F.

To further analyze the relationship between W and V,, we use:

(5.6) Lemma. There is a commutative diagram with exact rows and columns:

0 0

c; ~(D) C~(D)

1 0 ) v * V

' ( 9 r ' (9t( iD)Iz

0 0

,v*Q ,0

/ ,v*Q ,0.

Proof The canonical surjection v*v, Ct(D+F)~(97:(D+F) gives an exact sequence (where the kernel is calculated by comparing determinants)

O--,(9~(-D-R)---r v* W--*Ct(D + F)~O.


Since v* Wis invar iant under t* and t * ( D + F ) = - - D - R + F = - D - E ' , we have an exact sequence

0 ~ (9 ~.(D) --, v* W-~ (9~.( -- D -- E') ~ 0.

C o m p o s i n g the injection (5)~(D) -~ v* W with the injection v* W-~ v* ~ we obta in a nonzero m a p (_9~-(D) ~ v* E There is an exact sequence

0 -* H~ (g0 j --* Hom((fi~(D), v* V) ~ H~ ( ( 9 ~ ( - 2 D ) lz).

By assumpt ion, D is not equivalent to - D on the generic fiber of p. Thus, H~ - 2 D ) l z ) = 0, and the m a p ((~-(D)~ v* V is unique m o d scalars. The rest of (5.6) now follows f rom the flatness of v and by a d i ag ram chase. [ ]

(5.7) Corollary. v*Q~-(Iz/6~(-E'))| []

As a result of (5.7), there is an inclusion Cg~(-E')_~Iz. We may explain the intuitive significance of Q as follows: for a given irreducible fiber f of 7i,

W[ , - - : -~ V I, unless either ~/[, = (9i(d) (9 (5',(-d), deg d > 0, in which case supp Z

meets v - l ( f ) , or VI, is a nontr ivial extension of ~9,(d) by Cg/(d), where 2 d - 0 in P i c f Thus, it is reasonable to expect that E' is the " smal les t " divisor such that (9~-(- E') _~ Iz, at least away from the fibers f of p such that (~,(D) = Cg,(t* D), i.e. for which (~,(2D + U +F)= (9,. Moreover , at fibers which satisfy this condition, we would like some control on the size of E'. This is the point of the next series of lemmas.

(5.8) Lemma. Let f be an irreducible nonmultiple fiber of p, and set p ( f ) = x . Suppose that the inclusion (9~.(--E')~_I z factors through the natural map (9~.(-E')~_(f~(--E' +r f)~_Iz. Then r + o r d l F <=l(Rl p , (5~:(2D+ E' + F)j. (Here o r d / F = 1 / f f _ ~ s u p p F and 0 otherwise.)

Proof Via the inclusion v* W ~ v* ~" and (5.6), we have

Hom(( f f (D) , v* W) ~ , Hom(g~.(D), v* V ) ~ C ,

i.e., there is a unique h o m o m o r p h i s m (Ot(D)~ v* ~" (mod scalars), and it is induced f rom the cor responding h o m o m o r p h i s m (fi~.(D)~v*W. Apply ing t* to this h o m o m o r p h i s m and using the t- invariance of v* ~" and v* W, the same statement is true with (gf (D) replaced by (g f ( - D - - E ' - F ) . F r o m the natura l m a p

H o m ( ( 0 ~ ( - D - E ' - - F ) , v* V) ~ H o m ( ( g f ( - D - E ' - F), (9~.(-D) Iz),

we thus obta in a section ~eH~ Moreover , ~ is the image, under the m a p H~ --* H ~ ((fi~.(E'+ F) lz), of the element of

Hom((9~.(- D-- E ' - F), (O~.(-- D-- E'))~- H~

obta ined by applying the same cons t ruc t ion to v* W instead of v* V..

316 R. Friedman

We may view z as a meromorph ic function on T with polar set contained in E'+F. Since z is in the image of the natural map H~176 + F)Iz), the polar set of ~ is in fact conta ined in F.

Claim. Viewed as a meromorph ic function on T,, z is in fact a nonvanishing constant.

We defer the p roof of the claim to the end of the p roof of (5.8). Assuming the claim, suppose that we have a sequence of inclusions Ct(--E')c_(9t(--E' +rf)C_Iz. Then there is a nonzero global section zo~H~162 We may again view ~o as a meromorph ic function on T,, and choosing Zo in the image of H ~ ((9 ~) in H ~ ((9 ~,(E'--r f )Iz) , we m a y assume again that Z o is a nonvanishing constant .

Choose a small disk A a round x = p ( f ) , and let t be a coordinate on A, vanishing at x. Set Y=p-I(A). Then H~176 It follows that the map (gy(--D--E'--F) ~ v* ~'lr is unique up to multiplying by an element of H~ as H~ Cy (2 D + k f ) )= 0 for all k. On the other hand, we have an exact sequence

H~ D - E'-- F), v* V'lr))--* H~ (gy(E' + F) Iz)

~ H ~ R ~ p, Ct(2D+ E' + F)).

Given the section z o ~ H ~ ((9f(E'--r f )Iz) , we may restrict to Y and then multiply by t -1 . . . . . t - r (resp. t -( '+1) if f c _ S u p p F ) , to obtain sections of H~ Cy(E' + F ) I z ) . By the choice of z and Zo, none of these is of the form f-z , where

f~H~ If we had r>= I +h~ R~ p, Cf(2D+ E' + F))= I +I((R 1p, (97r(2D +E'+F)x) (resp. r > l ( R l p , Cr(2D+E'+F)x)) then a nonzero combina t ion of these would lift to H~ This is a contradiction. [ ]

We must still prove the claim used in (5.8). The p roof follows immediately from the following more general result.

(5.9) Lemma. Let L be a line bundle on (i) There is an exact sequence

O~ t* L | v*v,L-. ' .L~O.

(ii) The map L | (9 ~ ( - -F )~ L which is induced by the composition of the natural maps

L @ (9~(-- F) ~ t* v* v, L ~- v* v, L--* L

is the natural map arising from the inclusion (9~(- F)c_ 0~.

Proof. The existence of the exact sequence (i) follows from the calculation v* det v, L = L | t* L | (g f ( - -F) as in (5.5). To prove (ii), let U be an open subset of S over which v is 6tale and such that v- l (U) is disconnected. Then v*v, LJv-,<u)~-(L| and the natural map (t*L| G(L | is an i somorphism over v-l(U). It follows that the map L | (_g t ( - F ) ~ L in (ii) can only vanish along F, proving (ii). [ ]

Rank two vector bundles over regular elliptic surfaces 3 1 7

The next two l emmas are the analogues of (5.8) for mult iple and reducible fibers. The proofs are entirely analogous to the p roof of (5.8), and are left to the reader. First, however, we introduce some nota t ion which we will use for the rest of the paper.

(5.10) Definition. Let H be the reduct ion of an irreducible fiber of p or of n. Define m (H) = the multiplicity of H = min {m ~ 77 + : (9 n (m H) ~- Cn}. (Thus, m (H) is just the multiplicity in the usual sense of elliptic fibrations.)

(5.11) Lemma . Let H be the reduction of an irreducible fiber of p, and let x=p(H)~C. I f the inclusion Ct(--E')~_I z factors through the natural map C ~ ( - E')~_ C~(-- E' + r H), then

(r + o rdn F)/m (H) < l(R 1 p , C ~ (2 D + E' + F)x).

(5.12) Lemma . Let di + ei be a reducible fiber of p. Suppose that H = W l dl + w2 el, w~, wz>=O, and that the inclusion C~(-E')~_I z .factors through the natural map C~,(-- E')~_ C~,(-- E' + H). Then, setting x = p(di)= p(ei), we have

min(wt, w 2 + l ) < l ( R l p , C~r(2D+E'+F)x). []

In order to apply the above lemmas, we need to know l ( R l p , Cr + F)~):

(5.13) Lemma . Let f = p - ~ ( x ) be a fiber of p (possibly reducible or multiple), and let y=g(x)E~ '1. Then

l(R 1 p , Cf(2D+ E' + F)x)=[II'

O~y/2,

if % "is odd, unless f is reducible and satisfies (4.8); if % = 1, f is reducible and satisfies (4.8); if % is even,

where % is the integer defined in (4.7).

Proof First we claim that we m a y as well assume that 7 " - ~ ( ~ ) . Since the s ta tement is local a round a ne ighborhood o f f , this is clear i f f is not a mult iple fiber. N o w suppose that f is a mult iple fiber, and let a~77. We have an exact sequence

O~C~,(2D+ E' + F +(a--1) f ) ~ C r + E' + F + a f ) ~ C I ( 6 ) ~ O ,

where degy 6- -0 . Apply ing R i p , and using R i e m a n n - R o c h on f and the fact that

R~ p, C~(2D+ E' + F +a f ) = R 2 p , C~-(2D+ E' + F +a f)=O,

we see that l(R 1 p , Cf(2D + E' + F + a f)~) is independent of the choice of aeTZ. Let ~ be the elliptic surface with a section obta ined by base extension of 7"~ C const ructed in (4.1 1), and let Pl: T ~ T, P2: T ~ J ( T ) be the na tura l maps. Then, in the no ta t ion of w p* (9~r(D) and p*C~(~)(C~-Z') induce the same divisor on the generic fiber of T. Thus, after adjust ing by a mult iple o f f , we can assume

318 R. Friedman

that p~(9~ . (2D+E '+F +a f ) is pulled back from a divisor on ~(7"). Let a: --} 7"xcC be the normalizat ion map, ~: ~ C and tT: 7 " X c C ~ C the natural fibrations, and q: 7"xcC-- ,T the projection. By the arguments above, I (R I~ ,p '~C~ . (2D+E'+F+af )x ) is independent of a. Now there is an exact sequence

(,) O--}(9t• ~ a . C~.--rE ~ O ,

where g is the direct image of a vector bundle on f of degree 0. Let L be the line bundle of fiber degree 0 on 7" corresponding to the divisor 2D + E ' + F. Tensoring the exact sequence (*) by q*L and applying Riemann-Roch on f as in the first part of the proof gives

l(R ~ fi, (Pt(p* L)x) = I(R 1 p , 60~. • cc(q * L)x).

But now, by applying the flat base change theorem [12] III (9.3) to q and to P2: T ~ ( 7 " ) simultaneously, it is easy to see that we may assume that 7"= J (i3.

By (5.5.1), 2 D + E ' + F = D - - t ( D ) . Using freely the notation of w on J(7"), D = C1 - S' + ~, ni dl + m f Hence D -- t (D) = C1 - C2. Moreover, the two sections CI, C2 of J(7") satisfy by (4.7)

C~ .xC2=]0,

[ a / 2 ,

So we must calculate R l(p ') , (9r C2)x in these cases, where p': ~ ( 7 " ) ~ C is the elliptic fibration. By the formal functions theorem,

if ~y is odd, unless f is reducible and satisfies (4.8) if ay = 1, f is reducible and satisfies (4.8) if ~y is even.

a I ( p ' ) , (fij(~-)(C 1 - - C 2 ) x =!im H~ - - C2)),

There is an exact sequence

0 ~ (Qf "-+ ( Q n f ( C 1 - C 2 ) ---} (~(n -- 1) f ( C 1 - - C2) -~ 0,

where we have used (_9I(C1-C2)=(9y. A nonvanishing section of (91n_I)y(C1 --C2) is equivalent to a trivialization of (9.,t~)(C1- C2) to order n - 1. Moreover, the coboundary map from H~ to H'((9y) is the obstruction to extending such a trivialization to order n. From this, a straightforward argument (left to the reader) shows that dirnr ~ H ~ ((9,I(C ~ - C2)) = C~. x C2.

The formula for CI'xC2 then gives (5.13). [ ]

Putting (5.8)-(5.13) together, we obtain a direct description of V, provided that the double cover C ~ '1 (or equivalently the section A e r a + ( 2 k + r ) l l ) is suitably generic, and that Z = 0:


(5.14) Theorem. Let V be a rank two bundle on S satisfying (4.1). With notation as in w 4 and above, suppose that

(i) Z=0; (ii) C_~C is smooth (i.e. ~y--0 or 1 for all yelP1);

(iii) g: C ~ I P 1 is not branched over any point corresponding to a multiple fiber;

(iv) Either every reducible fiber satisfies (4.8) or, if di + ei is a reducible fiber not satisfying (4.8), then Ordd, E' =0. Then E '=0 . Hence V - v , (5~(D + F) and V=qJ, V..

Proof. By assumption (i), l z = ( ~ . Thus, the map ~ ( - - E ' ) ~ _ I z factors through (9~( -E '+rH) , where r=ordnE' , for every divisor H. The assumptions (ii)-(iv) and (5.8), (5.11), and (5.12) now imply that o r d n E ' = 0 for all divisors H. Thus, E '=0 , and hence Q = 0 by (5.7). Thus, ~ / = W = v . ( ~ ( D + F ) . Since V=~*V, V = ~ ' , ~. [ ]

If E' 4:0, then ~" is still determined by W, Q, and the extension class in Ext I (Q, W). In the next section, we shall calculate dim Ext ~ (Q, w). In principle, then, we could describe all bundles V (and hence V) by giving a more precise description of which sheaves Q can arise as in (5.5) and which extension classes correspond to locally free extensions. However, we shall not need such detailed information.

w 6. Case (B): Number of moduli

Let V be a rank two bundle over S with c2(V)=c and c l (V)=0, satisfying Case (B), i.e. Assumption (4.1). In this section, we will consider generic S (where generic means outside of a countable union of proper subvarieties of the parame- ter space). Effectively, this means that B and the multiple fibers are chosen generically. Our main goal will be to prove most of:

(6.1) Theorem. Let S be generic, and let V be as above. 1) For all c > 2 ( l +pg(S)), the set of V as above may be parametrized by a

nonempty scheme M of dimension 4 c - 3 (1 + pg). 2) The set of V of the .form v. COT(D+F) is Zariski open and dense in M,

and is scheme-theoretically reduced.

We shall prove that M is nonempty and reduced at points corresponding to V as in 2) in the next section. The rest of the argument consists in estimating various strata of M. Here is the outline of the argument. The bundle V determines the section A of FZk, the double covers C-~lP 1 and T ~ S , and the divisor D on T modulo the involution t. From this, we obtain canonically the subsheaf W of V and the quotient Q. Reversing this procedure, the choice of a section A of IF2k determines the double covers C--* ~,a and T--* S and the divisor D on the generic fiber of T. From this, there are q(7")=g(C) continuous parameters involved in choosing D. Having chosen D, we determine W. The remaining choices are 1) the moduli of the torsion sheaf Q and 2) the extension class in Ext 1 (Q, W) corresponding to V,, modulo the action of Aut Q on Extl(Q, W).

320 R. Friedman

(It turns out that Aut W-112", and we gain nothing by factoring in this action.) We shall tally up the number of moduli for various combinatorial strata. The final count will show that only if E ' = 0 do we get a stratum of dimension 4 c - 3 ( 1 + pg); all other strata have strictly smaller dimension. To carefully construct the strata as schemes of the appropriate dimensions, one would have to stratify into spaces in which various Ext groups have constant dimension and various families of sheaves are flat over S. We leave the details of these standard constructions to the reader.

(6.2) The number of moduli of the pair (C, D).

Let A ~ l a + ( 2 k + r ) l ] be a smooth section, and let n: C - - * C = q - I ( A ) be the normalization map. Thus ~0: C ~ A ~ - F 1 is 2 - 1 , so that C has a natural g~z (and is a hyperelliptic curve if g(C)>2). The linear system ]a+(2k+r) l l is a projective space of dimension 2 k + 2 r + 1. For a general A, C = C has genus 3 k + 2 r - 1 > 2, and the total number of moduli involved in choosing C and D is 5 k + 4 r = d i m l a + ( 2 k + r ) l l + g ( C ) . For singular C, however, g ( C ) = 3 k + 2 r - l - X p f l p , peC, where tip is the local genus drop described in (4.7). We shall use the following estimate for the dimension of various strata of l a+(2k+r ) l l and hence for the number of moduli of pairs (C, D):

(6.3) Proposition. Let S be a generic elliptic surface. Let ~1 . . . . , c~ k be integers >= 1. Let U be the subvariety of l a + (2 k + r) ll consisting of smooth sections A such that there exist points P l . . . . . pk e A C~ ( B + a) with A . m ( B + a ) = c q. Suppose moreover that a of the points P l . . . . . Pk correspond to singular nonmultiple fibers of S (i.e. to branch points of the map B ~ ~1) and I% of the points Pi correspond to multiple fibers of S. Then

d i m U < 2 k + 2 r + l - e i - 1 - a - g o .

In particular, U = 0 if the right hand side is negative.

Proof Apply the analysis of (1.18) and (1.19) to Ac~(B+a). Thus, let U o be the set of pairs (A, B+a) satisfying the hypotheses of (6.3). Then an element of the affine tangent space to U o, which has dimension dim Uo + 1, maps to the subspace of

HO(A, KAQK~2~)GHO(B, KB | 1 KFzK) 0 H~ K~ @ Kn:~)

of sections satisfying the appropriate linear conditions (1.18) and (1.19) with a one-dimensional kernel corresponding to A + B + m Note that deg~(K~ | -1 HO(a, -1 KF2, , )=a . a<0 , so K~| KF~)=0. Also,

so that

degA(Ka | KF21k) = A. A = 2 k + 2 r,

H ~ (A, KA | K~2~) ~ H ~ (N1, (gF, (2 k + 2 r))

R a n k t w o v e c t o r b u n d l e s o v e r r e g u l a r e l l ipt ic su r faces 321

has dimension 2 k + 2 r + 1, and finally - 1 K~ | KF~k = KB | 0B(A), where A is an effective divisor on B of degree 6 k + 6.

N o w suppose that Uo maps onto an open subset of [BI. In particular, dim Uo = dim I B I + dim of a generic fiber = dim H ~ (B, K~ @ Kn;21k) + dim of a generic fiber. Fixing an element ~o of H~ K~| K~21k) plus the zero element

k

in H~ K~ @ K~I) determines ~(~i - 1) + a linear condit ions on 1 k

H~ KAQK~21)=H~ 1, (_gP,(2k+2r)). If ~ ( ~ - l ) + a < 2 k + 2 r + l , these 1

condit ions are linearly independent; this follows from considering the exact sequence

0 --* C~n,, (2 k + 2 r) | (~F1 ( - ~ m, p,) -~ (9 F, (2 k + 2 r) --* (9~,/(9 F, ( - ~ m~ p~) ~ 0

and the vanishing of Ht(6"~F,(2k+2r)| ) if ~ m ~ < 2 k + 2 r + l .

Thus, the dimension of a general fiber is _ - < 2 k + 2 r + 1 - (c~i- 1 ) - a . If ~(c~ i 1

- 1) + a > 2 k + 2 r + 1, the above a rgument shows that there is at most one section ~k in H~ K A - 1 | KF2~) which satisfies the compatibi l i ty condit ions imposed by ~0. Fixing ~, we then impose at least one nontrivial condi t ion on H ~ (B, KB - 1 | K~:z~), so that dim Uo < dim ] B] - 1 and Uo cannot map onto an open subset of IB]. Doing this for all choices of the a~ and r, we obtain a countable union of proper subvarieties of ]B]. Finally, since every curve A + B + a has only finitely many singularities, for each choice of componen t of a generic fiber of Uo ~ ] B ] and each choice of the ~i's and r, we can choose multiple fibers to impose nontrivial condit ions on A (i.e. to obtain a proper subvariety of a generic fiber of Uo --* I B ]). Thus we obtain the final estimate

d i m U _ - < 2 k + 2 r + l - c~/-1 - a - g o . [ ] \ l

Having chosen C, we obtain T and a divisor on the generic fiber of 7"~ C. Next we claim:

(6.4) Lemma. Let Dt and D2~Pic(7" ) satisfy v. D1 =v . D2 and suppose that D 1 and D2 have the same restriction to the generic fiber of T.. Then DI - D 2 e P i c ~ ( T ) = line bundles on 7" numerically equivalent to O.

Proof. By hypothesis, p. (9 f (D1-D2) is a line bundle ~ on C. Thus we have a natural injective map p* ~ --~ (9~-(Dt --D2). Since both bundles have fiber degree 0, this map can only vanish along fiber components . Letting - denote numerical equivalence, we must therefore have/)1 - D 2 + r f, r ell~, where f denotes a generic fiber of p. F r o m v. D~ = v. D2, we obtain r = 0. [ ]

We have an exact sequence

0 ~ Pic~ ~ Pic*(T) ~ H Z(T; 7/)tor~ ~ 0.

In the next section, we shall show that p*: P i c ~ Pic~ is an isomorphism, so that dim Pic:(T) = g(C). Thus:

322 R. Friedman

(6.5) Corollary. Let U be as in (6.3) and let D be a divisor on T which is specified on the generic fiber of p by the choice of C. Then the number of moduli of the pair (C, D), C~U, is at most

k k l i] 5 k + 4 r - ~ , ( e i - 1 ) - ~ f l l - a - I ~ o , where fli= ~ ~ .

1 1

[]

(6.6) The number of moduli for Q

The number of moduli for Q may be analyzed in two steps. First, the number of moduli for S u p p Q = E is clearly at most the number of general fibers in Ered, i.e. those fibers which are smooth and non-multiple and do not correspond to singular points of C-call this number 7. Secondly, fixing the (possibly nonreduced) scheme structure on E, we can ask for all possible flat deformations of a given sheaf Q with support E.

(6.7) Proposition. The number of moduli of the pair (E, Q) is < h 1 (Ce) + II(Z).

Proof The number of moduli for E is clearly ~. Next, for a fixed E, let Z(Q) denote the Fitting scheme of Q as in (1.12). By (1.16), the number of moduli for Q with support E is < h I (Ce) + l(Z(Q)). On the other hand, by straightforward calculation we obtain

v* Z(Q)= Z(v* Q)

and thus t(Z(Q))=�89 From the description of v*Q~-(Iz/C~. ( - E')) | C T ( - O), we have Iz-~ Izt~.e) and thus l(Z(v* Q)) < l(Z). Moreover, an application of (5.8), (5.13) and (1.17) shows that we actually have l (Z(v*Q))<l(Z) -27 , i.e. at a general fiber f in suppE, there exists a point x ~ S u p p Z n [ e i t h e r component of v - l ( f ) ] and a coordinate t at x such that E = { t " = 0 } and m is the smallest integer such that t '~Izx. Thus, by (1.17), l(Z(v* Q)x) < l(Zx).

Thus, the total number of moduli for (E, Q) is at most

~,+�89 []

(6.8) The number of moduli for the extension

We begin with a simple lemma on the size of Ext '(Q, W).

(6.9) Lemma. dim Ext '(Q, W ) - d i m Ext2(Q, W)=l(Z)- �89 2.

Proof Since Q is a torsion sheaf, Horn(Q, W)=0. As Q has a short free resolution (5.5), Ext2(Q,W)=O. Thus, by the Ext spectral sequence, Ext~(Q,W) = H i - l ( E x t 1 (Q, W)). Next we use:

(6.10) Lemma. I f ~ is a sheaf on S and f# a sheaf on T, there is a natural isomorphism

v, Ext~(v* ~ , fr ~ , Extq~(~ , v, ~).

Proof This follows by taking a resolution of .~- and using the flatness of v. []


Returning to the p roof of (6.9), we have

E x t I (Q, W ) = E x t 1 (Q, v , ( g t ( - D - E')) ~ v , E x t I (v* Q, ~ ( - D - E')).

By (1.10), there is an exact sequence

0 --+ ~ . / ( g t ( - E') --* E x t 1 (v* Q, ( 9 ~ ( - D - E')) ~ ~ z ~ O.

Combining this with the above remarks, we obtain

dim Ext I (Q, W ) - dim Ext 2 (Q, W)

=Z(g r E x t ' ( Q , W)) = z(T, E x t ' ( v * Q, ( 5 ' ~ ( - D - E ' ) )

= z ( r ~ / c ~ ( - E')) + Z ((~'z) = Z(C%) - Z (~'~( - E ' ) )+ l(Z).

By Riemann-Roch ,

Z ((or ( - E')) - Z ((9 t) = �89 ((E') 2 + (E'. Kt)).

Since E' is supported on fibers and K t is numerically a rational multiple of the fiber, (E'. K t ) = 0 . Thus, we have

dim Ext ' (Q , W ) - d i m ExtZ(Q, W ) = - I ( E ' ) 2 + I ( Z ) . []

(6.11) Lemlna. The group Ext2(Q, W ) is dual to the kernel o f the natural map H ~ ((9~(E')/(9~ | K~-) ~ H ~ ((~z).

Proof. The surjection C9~ ~ (9 z induces an exact sequence

0 -~ [ I z / (~ . ( - E')] | C97.( -- D) ~ (~t( -- D)/((,'~.(-- D -- E') ~ (9 z ~ O,

where the left-hand term is v* Q. F r o m the p roof of (6.10), we have

Ext2(Q, W) = H 1 (Ex t 1 (Q, w ) ) = H 1 (v , E x t I (v* Q, (o~( -- D - E'))

= H~ (Ex t ~ (v* Q, C g f ( - D - - E ' ) )= ExtZ(v * Q, 6 J f ( - O - E')).

Using the above exact sequence, ExtZ(v * Q, ( 9 f ( - D - E ' ) ) is the cokernel of the map

Ext 2 ((gz, (r - D - E')) ~ Ext 2 ((~/. ( - D)/(o t ( - D - E'), ( g t ( - D - E')).

Applying Serre duality, Ext 2 (Q, W) is dual to the kernel of the map

H~ | Ks.)--* H~ []

We next analyze the kernel of this map.

(6.12) Lemma. dim Ext 2 (Q, W) = dim H ~ (6~ (E")/(~ ~ | K ~), where E" is the larg- est divisor such that there ex i s t s a sequence o f inclusions

(9~.(-- E ' ) ~ ( ~ ( - E' + E" )~_ I z .

324 R. Friedman

Proof Let f be a c o m p o n e n t of E'red which does not lie in a reducible fiber. Then (9r | K t ly has a filtration, namely Cr f ) /C~ @ K~, 0 < a <_ ord I E', whose successive quot ients are line bundles over f of degree 0. In part icular , given s~H~174 it has a well-defined fil tration level, say s~H~ If moreove r the image of s in H~ is zero, then s defines an inclusion (9~(af)~-Iz(9~.(E'). Hence we have (9~r(-E')~_(9~ ( - - E ' + a f ) ~ _ I z . Conversely, given any sequence of inclusions ( 9 ~ ( - E ' ) ~ ( 9 ~ ( - - E ' + a f ) ~ l z , the sections of (9 f (a f ) / (9~c|162176162 project to 0 in H~

N o w suppose that we are at a reducible fiber d + e and suppose tha t ordd E' = a, orde E ' = b. If for example a > b, then we can write

(9 r (E')/(9 ~ | K ~le + e = (9 ~ (ad + b e)/(9 ~ = (9_9 ~ (b (d + e) + (a - b). d)/(9 ~.

It follows easily tha t

H ~ ((9 ~ (ad + b e)/(9 ~.) ~- H ~ ((9 ~ (b (d + e))/(9 ~r)

under the natura l inclusion. Again, (9r162 is filtered by subsheaves whose successive quot ients are of degree 0, indeed trivial, on d + e. An a rgument similar to the case of an irreducible fiber will now handle this case also. [ ]

To use (6.12), we shall use the control over E" given by (5.8)-(5.13). First, however, we mus t in t roduce some notat ion. Recall the no ta t ion f rom (4.3): the m a p v is b ranched over f l . . . . . fa, fa+ 1 . . . . ,fb, where f l . . . . . f , are the p roper t ransforms in S of singular fibers and f , ~ ~ . . . . ,fb are irreducible (possibly mult i- ple) fibers in the branch locus. Also let fb+~ . . . . . f~ be mult iple fibers of even multipl ici ty p~ = 2 q~ such that v - ~ ( f ) = ei is irreducible. Thus, the multiplici ty m(ei)=qi. We m a y write

a b c d

(6.13) (i) E=Z(uici-}-vifi)+ Z r i f i+ ~ r i f i + ~ t j g J 1 a + l b + l 1

where the gj are possibly singular or mult iple fibers such that v-1 ( g j ) consists of two componen t s g) and g~. We may assume tha t u~, v~, r i> 0 and that t i > 0. Let us write

(6.13) o b c .

(ii) E ' = ~ ( u i d i + 2 v i e , ) + ~ 2 r i f i + Z r i f i + Z t j ( g ) +g2) 1 a + l b + l 1

(6.13) i t t t t t t t 1 t t 2 (iii) E " = Z ( u i di-t-v i el)q- r i f .q -~( ( t j , 1)g j +(tj,2)gj).

1 a + l 1

Thus, u'i'<ul, v~'<2vl, r'i'<2ri, a+l<_i<_b, r'i'<ri, b + l < i < _ c , and t~,i<tj, i = 1 , 2 .

Finally, as in (5.10) we let re( f ) denote the mult ipl ici ty of a mult iple fiber. Set xl = g(P(fi)), Yj = g(P(gj)). Then the following is a direct t ransla t ion of (5.8)- (5.13):

R a n k two vec to r bund les over r egu la r ell iptic surfaces 325

(6.14) Lemma. In the previous notation, we have (i) " i tj, i/m(gj) <=c~rj/2 , 1 <=j <=d, i = 1, 2;

(ii) ( r ' /+ 1)/m(f~) < ~ , , a + 1 <i<b; (iii) r'//m(f)<ct~,, b+ 1 <i<_c;

< c%, unless dl + ei satisfies (4.8), 1 < i < a (iv) min(u'/, VI{ + 1)

= 0 in case (4.8). [ ]

(6.15) Lemma. In the above notation, we have

v ' q + V r r ' / - - l l h~174 min(u'{, i , z_., + 1 a + l

d ,, 1 r t ' : - - 1 1 \ �9

Proof It clearly suffices to estimate separately

h~ ' di+v'i'ei)/(9f) and h~ | K~c),

for f an irreducible fiber of p. We shall only do the second calculation and leave the first to the reader.

Let m = m(f) . Locally near f, K t = (9 f ( (m-1 ) f ) . We have an exact sequence, f o r a ~ r

(,) 0 --* ( (gt ( (a- 1)f)/(gf) | K t -~ ((9f(a f)/C~r) | K~r --* C y((a + m - 1)f) --* 0.

Moreover , h~ - 1 ) f ) = 0 if a ~ 1 (modm) and = 1 if a = 1 (modm). Fo r r ~

1 < a < r, this gives a nonzero contr ibut ion exactly for a = 1, m + 1 . . . . . [ r ~ [ m k ,,~ i + 1. Thus

h ~ | Kt )=<[rml]+ 1. [ ]

(6.16) Then

Corollary. Let 6 be the number of nonmultiple fibers in the branch locus.

dim Ext2(Q, W ) < ~ m i n ( u ' [ , v'/)+ c % + ~ c % - 6 . [ ] 1 a + l i

To rood out the Ext 1 term by the act ion of Aut Q, we use

(6.17) Lemma. dim Aut Q < h ~ ((9~).

Proof Via multiplication, there is an injection (9 E ~ End Q, and hence (9* ___ Aut Q. As dim (9~ = h ~ ((gE), we get (6.17). [ ]

326 R. Friedman

We now begin to tally up the number of modul i of a given s t ra tum: Fix the integers e~ and the combinator ia l configurat ion above. Then

# moduli (C, D) + # moduli(E, Q) + dim Ext 1 (Q, W) - dim Aut Q

= < 5 k + 4 r - Z ( c t ~ - l ) - ~, ( a y ~ - l ) - Z f l ~ , i ~ y j * 0 i

- - ~ fly, - - a - - # o + 3 l ( Z ) + h ' ((gE) - h ~ ((PE) - �89 (E ' ) 2 J

+ ~ min(u'/, v'i')+ ( ~ , ) + ~ % - - 6 1 a + l 1

) 5 k + 4 r + 31(Z) + min(u'/, = v ' , ' ) - % + (

-- E flxi - - ]A1 -~- h l ((gE) - - h~ ((gE) - - �89 ( E ' ) 2 i

where p~ is the number of multiple fibers among the gj. Here, the fi~, correspond - - 1 to eyj even, f l y j -~ ayj, and we can ignore the ayj = 0.

To handle the reducible fibers, we use:

(6.18) Lemma. With hi, ei as in (4.12), we have a

(i) h 1 ((ge)- h ~ ((gE)- �89 2 = �89 ~ [(2 v , - ui) 2 + (2 v i - u,)] l

(ii) 2 vi - ur + (1 + ~i) = 2 ni and 2 v i - ul + 1 > O. (iii) 4 n 2 - 4 ni el >= �89 [(2 v i - ul) 2 + (2 vl - ui)], wi th equal i ty holding i f and only

i f ni = el, i.e. 2 v i - ui = e i - 1.

P r o o f (i) We have

--�89 = -- u i d i + 2 v i e i

a a

= -- �89 ~ ( - 2 u~ + 8 ul v i - 8 v]) = ~ ( 2 v i - ui) z. 1 1

Next, h t ((9~) - h ~ ((_gE) = - X ((9~). F r o m the exact sequence

and R i e m a n n - R o c h on S, we obtain -- Z((gE)= �89 2 + � 8 9 KS). Using (6.13)(i), together with c 2 = - 1 , f 2 = _ 4 and cl, f~ ~ ~,1, a calculation gives

a

�89 2 + �89 KS) = �89 ~, [(2 v i - ui) -- (2 v l - ui)23, !


so that combining we get

a

h a (CE) - h ~ ((9~) - �89 (E) 2 = �89 ~ ' [(2 v, -- ul) 2 + (2 v, -- u,)]. 1

(ii) Recall f rom (4.12) that ( D . d l ) = e i - - 2 n i < O . On the other hand, f rom 2 (D) = -- D -- E ' - - F and t (di) = di, we have

2 (D. dl) = (O + l (O)). d~ = - (E'. d i ) - (F . dl) = 2 u l - 4 vi - 2 < O.

Thus 2 n i= 2 v i - ui +(1 + ei) and u l - 2 v l - 1 <0 , i.e. 2 v i - ui + 1 > 1. (iii) We shall just write out the case e~=l , 2 v i - u ~ + 2 = 2 n ~ . Note that in

this case, as n~e2g, n~ > 1.

4 n{ = 4 n i g ' i - 3(2 V i - U l ) 2 - - 3(2 V i - Ui)

= 4 n2 - - 4 n i - - �89 ni-- 2)2 -- �89 ni- - 2 )= 2 n2 -- nl - 1 >0 ,

with equali ty if and only if n~= l. [ ]

(6.19) Theorem. The dimension o f all s t rata o f M is < 4 c - 3 (1 + pg), wi th equal i ty holding only f o r E' = O.

Proof. We must compa re the two numbers :

and (4.17):

a

5 k + 4 r + 3 l (Z) + ~, [(min (u'/, v'i') - ex,) + �89 v l - ui) z + (2 v i - ui))] a

+ Z Ctyj ~l~ 0 i

a

5 k + 4 r + 2 1 ( Z ) + ~ ( 4 n Z - 4 n i e , ) + 2fle~. 1

As min(u'i', v'i ')<min(u'i ' , v'i '+ 1)<c%, we see t e rm-by- t e rm that the n u m b e r of modul i is __<4c-3( l+pg) , with equali ty holding only if Z = 0 , f i x = 0 for all i, fly~=0 for all j, i.e. Cr or 1 at all points p. Since Z = 0 , E " = E ' . Moreove r u'i '=ui, v ' i '=2vi , and 2 v i - u i = e i - 1. Since c~x= I, we may assume min(ui, 2 vi) # 0 , hence min(u/, 2 v i ) = m i n ( u i , 2 v i + 1)= 1 and e i=0. Thus, u i = 1 and vi > 1, hence 2 vl - u~ > -- 1 and we get strict inequali ty unless u~ = v~ = 0.

The remaining p rob lem is c~x,= 1 for certain mult iple fibers. Changing notation, suppose that f t . . . . . f , , are mult iple fibers at b ranch points pi, and C = C (Cr or 1 for all p). We count the modul i of such a s t ra tum. By (6.3), the n u m b e r of modul i of ( C , D ) is 5 k + 4 r - m . By (6.14) and (6.15), given Q, d im Ext a(Q, W ) = m , and dim Aut Q=h~ But the n u m b e r of modul i of Q is at mos t h I (CE) -m. Indeed, we have

Q | 1 7 4 '~s0 and v * Q ~ - ( C E , ) |

328 R. Friedman

is fixed by D. Thus, the restriction of Q to the reduced fiber Jl is specified for each i, and this gives one condition on H ~ ((9*) for each fiber f . Hence, the total number of moduli for this stratum is at most

5 k + 4 r - m + m + hl (CE)- h~

=5k +4r -m- - (E )2=5k + 4 r - m < 5 k +4r,

unless m = 0 . Thus, for the strata of maximal dimension, e v = 0 or l for all p, all branch points are at nonmultiple fibers, and there are no reducible fibers. By (5.14), the corresponding bundles V are of the form V= V= v, (gT(D + F). []

w 7. Some global properties of the moduli space

Fix a generic S and c>2(1 +pg). Let L be a c-suitable line bundle and let ~Jl(c), as in (0.4), be the space of all L-stable rank two bundles V over S with c t (V)=0 and c2(V)=c. Let M be the subset of 9Jl(c) consisting of bundles V satisfying Case (B), By (6.1)(2), the subset Mo of M consisting of bundles V of the form v, (gr(D + F) = v, (9 r ( -D) , corresponding to C ~ C with C ~ p l unbranched at singular or multiple fibers of S, is open and dense in M. By (3,13), if in addition c>2pg+Zp, then Mo is dense in ~JJ~(c) as well. In this section, we shall give a more global description of Mo.

Let U~la+(2k+r)l[~--lP 2k+2"+~ be the open nonempty subset of smooth sections A of lF2k such that A meets a + B transversally, and such that A c~ (a + B) does not contain any points corresponding to singular or multiple fibers of S. Then there is a morphism M o-~ U, and, as we shall see below, the fiber over uE U is a finite union of copies of J(C), where A ___IF2k is the section corresponding to u~U and C ~ J ( S ) is its preimage in J (S) . Thus, all components of M o (and hence of TA(c) if c is sufficiently large) fiber over finite covers of U, and the fibers are hyperelliptic Jacobians. In particular, the Kodaira dimension of every component of M0 is < 2 k + 2 r + 1. (This statement can be shar- pened, but we shall not do so here.)

We give a more precise construction of Mo. Let ,~_~ U x 1F2~ be the universal family of sections of •2k and ~f - - ,Uxr162 be the pullback of ,~ under the double cover J (S)~IF2~. From the map J ( S ) ~ l P 1 we have cg__, U x p t ~ l p t . Set ~ - = ~ f x r , S. Note that there are maps J - ~ g - - * U, and that the fiber of J - ---, U over ueU is just C x~,, S, where C is as above. We have a diagram

P @ ~ U x S

(g ~ U x P 1

\ / U.


Similarly let J ( , Y - ) = ~ x F , ~ r ). The inclusion cg_~Ux~r defines a morph ism (of schemes over U) s: cg ~ r which is a section of the natural map J ( 3 - ) ~ . Moreover , J ( J - ) is an elliptic curve over the base scheme cog, and is naturally Pic~ The obstruct ion to the section s defining a relative divisor ~ on J - of fiber degree 0 lies in the Brauer g roup Br(k(C~)) (cf. [11]). By Tsen's theorem, B r ( C ) = 0 for C a curve over an algebraically closed field. Thus, there is a finite extension K of the function field k(U) of U such that over the pullback of ~ to K the section s does define a relative divisor. Let U ' ~ U be the normal cover corresponding to K, and let ~ ' , ~--' denote the pullbacks of cg, j - to U'. By construct ion there is a relative divisor 63 on J ' corresponding to s.

Fix ue U', and via the m a p U ' ~ U let A denote the corresponding section of ]a+(2k+r) I I . Similarly let C, T, etc. have the usual meaning given A. Let P ic (J - ' /U ' ) ~ U be the relative Picard scheme. Define Pic%Y-'/U') on each fiber by the rule

Pic%Y-'/U'), = {L~Pic~(U): v . (6~g, (~) l r@L = - G},

where G is as in (5.2). Let c~a be the Poincar6 line bundle over J ' xv , PicS(,Y-'/U'). Define a rank two bundle • over S x P ic ' (g - ' /U ' ) as follows:

"g" = q . (~* (9~, ( -- ~ ) | 5~ - '),

where ~1: r xv , Pic=(J-'/U') ~ J ' is the natural map, and

q: J ' x v, Pic=(Y'/U') ~ S x PicS( y ' / U ')

is the map naturally induced by

p x l d g-' xv , PicS(~ J xvPicS(~- ' /U ') , (U x S ) x v P i c % Y - ' / U ' ).

Of course, we are just gluing together the bundles v . ( g r ( - - D ) = v . ~ O r ( D + F ) , where v. D = G.

No te that there is an equivalence relation ~ on Pic%Y-'/U') which is compat i - ble under the projection to U' with the equivalence relation on U' obtained by identifying points with the same image in U, and is an isomorphism on the fibers. Let J / / b e the quotient of Pic%Y-'/U') by ~ .

(7.1) Lemma. There is a natural bijection Jig --* M o.

Proof By the universal proper ty of coarse moduli spaces, the bundle ~K" induces a morphism P i c = ( g / U ' ) - ~ M o . This map is easily seen to be surjective and to factor th rough ~ . To check that the induced map ,//g--* M o is injective, note that, for a given uc U corresponding to the section A, the divisor D is determined on the generic fiber of T up to _+ 1, and these choices are distinct (2 D is nontrivial on the generic fiber). However, the two possible choices correspond to the two different sections of ~r which arise from the two different maps C ~ J ( S ) ,


corresponding to the identity and the hyperelliptic involution respectively. The construction above singles out the section of J ( T ) corresponding to the identity. Hence the map ~ ' ~ Mo is injective. []

Now we check that the bijection above is an isomorphism of schemes.

(7.2) Lemma. Mo is smooth and scheme theoretically reduced of dimension 4 c - 3 (1 + pg), i.e. all components of Mo are good in the sense of (0.2).

Proof. Choose V = v , (gr(--D), corresponding to an arbitrary point of Mo. By deformation theory, we must show that H2(S; adV)=O where a d V is defined by the split exact sequence

Tr (7.2.1) O ~ a d V ~ E n d V ~(9s~O.

As ad V is self-dual, it suffices to show that H~ ad V | Ks)=0, or, using the fact that (7.2.1) is split, that h~ End V | Ks)=h~ But

H ~ (End V | Ks) ~- Horn (V, V | Ks) ~- Horn (V,, v. ((0 r ( -- D) | v* Ks))

=Hom(v* V, (gT(--D)|

Using the exact sequence

O--* CT(D)~ v* V--* (gT(- D)--*O

and the fact that - 2 D is nontrivial on the generic fiber of T ~ C, we obtain Horn(v* V, O r ( - D ) | Ks)~-H~ v* Ks). But

H~ v* Ks)= H ~ (S, K s | v,(9 T)= H~ (S, Ks) G H ~ (S, K s | (gs (-- G)).

Thus, it suffices to show that H~ K s | (gs( -G))=0. Let f be a generic fiber of S and F~ the multiple fibers, i= 1, ..., #, of multiplicity pi. Under the assumption that C is generic, we have G - (3 k + 2 r ) f Moreover, by the canonical bundle formula,

Ks = (gs((k - 2 ) / + ~ (Pi- 1) Fi). i

Hence Ks | (gs( - G) = (gs(( - 2 k - 2 r - 2 ) f + ~ ( P i - 1) F/), and this clearly has no

sections. [] i

(7.3) Corollary. The map JC/~ Mo is an isomorphism of smooth schemes. In particular, M o is smooth, nonempty, and of dimension 4 c - 3 (1 + pg) = 5 k + 4 r.

Proof The first statement follows from Zariski's Main Theorem (easy case), and the second is clear. []

To analyze M 0 further, we analyze the fibers of the map ~ ' ~ U. Thus fix u~ U, and let T, C, etc. be the corresponding objects. Set PicS(T)= {D~Pic T: v . D = - G and D=(g r (~ ) on the generic fiber of T--* C}. By (6.4), Pic~(T) is, up to a translation, just PicS(~;/U)c~ c~- l(u), the fiber of ~ over the point u.


(7.4) Lemma. (i) There is a natural exact sequence

0 ---, Pic ~ T--* Pic * T--* H 2 (T; 2g)t .... ~ 0.

(ii) Pic ~ T~ Pic ~ C via p*. (iii) Pics T is a principal homogeneous space under the subgroup of Pic ~ T which

maps to ker(HZ(T; 2g)t .... ~* ' H2(S; 2g)tor~.)"

Proof (i) is clear. To prove (ii), consider the diagram

0 ,H1(C; 2g) ,HI (C; (%) ,Pic~ C ,0

1 l 1 0 , H i ( T ; ~) , H i (T ; C~ r) ,P icOT _ ,0,

where the vertical maps are p*. It suffices to prove that H I ( C ; Z ) ~ H I ( T ; 2 g ) is an isomorphism, for then the same will be true for H ~ ((Cc)-~ H ~ ((c;r). Dually, it suffices to show that the map H~(T;2g)~H~(C;2g) is surjective with finite kernel. The surjectivity is clear, and the s tatement about the kernel follows easily from the discussion in [5] on H~ for elliptic surfaces.

To prove (iii), let Dt, D2ePic~ T. By (6.4), D ~ - D z e P i c ~ T and v , ( D ~ - D 2 ) = 0 . It follows that the image of D t - D 2 in H 2 (Y; 2~)t . . . . maps to 0 under v, . Converse- ly, the p roof that the subgroup of PicOT described in (iii) does in fact act on Pic~Tis straightforward, and left to the reader. [ ]

(7.5) Corollary. I f S has no multiple fibers, then ?Ol(c) is irreducible for all c >2(1 +p~).

Proof In this case, Mo is dense in ~l(c), so it suffices to verify that Mo is irreducible. As Mo fibers over U with irreducible fibers (by [5], H2(T; ~)t .... =0) this is clear. [ ]

(7.6) Remark. It is easy to see that 9X(c) is not in general irreducible if S has multiple fibers. For example, if S has a single multiple fiber of odd multiplici-

p + l ty p, then ~J~(c) has exactly ~ components for all c>2 (1 +p~). It would

be interesting to know if a similar result holds for the spaces ~JJ~s,t,(c), where L is a f ixed polarizat ion on S, and c is arbitrari ly large.

References

1. Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Berlin-Heidelberg-New York: Springer 1984

2. Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82, 8 28 (1963) 3. Bloch, S., Kas, A., Lieberman, D.: Zero cycles on surfaces with pg--0. Comp. Math. 33, 135 145

(1976) 4. Brosius, J.E.: Rank-2 vector bundles on a ruled surface I. Math. Ann. 265, 155 168 (1983) 5. Dolgachev, I.: Algebraic surfaces with q=pg=0. In: Algebraic Surfaces, proceedings of 1977

CIME, Cortona, Liguori Napoli, 1981, pp. 97 215

332 R. Friedman

6. Donaldson, S.: Irrationality and the h-cobordism conjecture. J. Differ. Geom. 26, 141 168 (1987) 7. Donaldson, S.: Polynomial invariants for smooth four-manifolds. Preprint 8. Friedman, R., Morgan, J.: On the diffeomorphism types of certain algebraic surfaces 1. J. Differ.

Geom. 27, 297-369; II. J. Differ. Geom. 27, 371-398 (1988) 9. Gieseker, D.: A construction of stable bundles on an algebraic surface. J. Differ. Geom. 27,

137 154 (1988) 10. Griffiths, P., Harris, J.: Residues and zero-cycles on algebraic varieties. Ann. Math. 108, 461-505

(1978) 11. Grothendieck, A.: Technique de descente et th6or~mes d'existence en g6om6trie alg~brique I.

Sem. Bourbaki 190, 12 (1959-1960) 12. Hartshorne, R.: Algebraic geometry. Berlin-Heidelberg-New York: Springer 1978 13. Hoppe, H.J., Spindler, H.: Modulrhume Stabiler 2-Biindel auf Regelflhchen. Math. Ann. 249,

127-140 (1980) 14. Iarrobino, A.: Punctual Hilbert schemes. Bull. Am. Math. Soc. 78, 819-823 (1972) 15. Kodaira, K.: On compact analytic surfaces II. Ann. Math. 77, 563-626 (1963) 16. Liibke, M., Okonek, C.: Stable bundles on regular elliptic surfaces. J. Reine Angew. Math. 378,

32-45 (1987) 17. Maier, F.: Tulane Thesis 18. Okonek, C., Van de Ven, A.: Stable bundles and differentiable structures on certain elliptic surfaces.

Invent. Math. 86, 357-370 (1986) 19. Schlessinger, M.: F unctors of Artin rings. Trans. Am. Math. Soc. 130, 208-222 (1968) 20. Schwarzenberger, R.L.E.: Vector bundles on algebraic surfaces. Proc. London Math. Soc. 11,

601-622 (1961) 21. Zariski, O.: Dimension-theoretic characterization of maximal irreducible algebraic systems of

plane nodal curves of a given order n and with a given number d of nodes. Am. J. Math. 104, 209-226 (1982)

Oblatum 5-IV-1988

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Rank two vector bundles over regular elliptic surfaces