[Page 1] Zero-dimensional Schemes. on Abelian Surfaces. Antony Maciocia

[Page 1] Zero-dimensional Schemes on Abelian Surfaces Antony Maciocia Abstract. The moduli spaces of semistable torsion-free sheaves with c 1 = 0 and c 2 =?2 and?3 over a principally polarised complex torus are described explicitly in terms of zero-dimensional subschemes of the torus. The boundary structures are computed in detail. The rst moduli space is a compactied family of Jacobians and the second is a Hilbert scheme. 1991 Mathematics Subject Classication: 14J60, 14C17, 14C05, 14H40, 53C07. 0. Introduction In this paper we shall show how detailed information about zero-dimensional subschemes of a principally polarised complex Abelian variety (T; L) can be used to give us information about the moduli space of stable bundles. The information we are looking for is existence and connectedness of these moduli spaces. We shall show how the moduli spaces are related to Hilbert spaces of zero-dimensional subschemes of our torus (and its dual) using a combination of Serre's method of constructing vector bundles and Mukai's Fourier transform for tori. Properties of the vector bundles representing points of the moduli spaces can be related to geometrical properties of certain zero-dimensional schemes. The most important properties is whether various zero-dimensional schemes and their subschemes lie on certain divisors. These type of problems have been called `interpolation problems' by Geramita (see his article in this volume). We will consider the interpolation problem from an intrinsic viewpoint and call on results from [8] which dealt with many of these questions. However, the problem can also be viewed extrinsically by embedding the torus in some projective space. The most natural one would be CP 8 given by the very ample linear system jl 3 j. One could also gain some information from the singular Kummer surface in CP 3. Stability in this context means either -stability of Mumford-Takemoto or G- stability of Gieseker. To dene these notions we require a polarized variety (X; `) of The author is grateful to the organisers of the Ravello conference on Zero-dimensional subschemes for their support. He would also like to thank the Seggie-Brown trust for support while this work was carried out.

2 Antony Maciocia dimension n (or a complex manifold with a chosen Kahler form!). Let us denote the Chern characters of sheaves on X by (n + 1)-tuples: (r; c 1 ; 1 2 c2 1? c 2 ; : : :). Denition 0.1. We say that a torsion-free sheaf E is -stable (respectively, - semistable) with respect to ` if for all subsheaves F such that E=F is torsion-free we have (F ) < (E) (resp. (E) (F )), where (E) is the slope of E dened by (E) = c 1 (E) `n?1 =r(e): Denition 0.2. We say that E is G-stable (resp. G-semistable) with respect to ` if for all subsheaves F with E=F torsion-free, we have P m (F ) < P m (E) (resp. P m (F ) P m (E)) for all m suciently large, where P m (E) = (E L m )=r(e) for some representative line bundle L of `. There is a chain of implications: -stable ) G-stable ) G-semistable ) -semistable. Further properties of stabilities can be found in [7, Chapter 5] or [15, II.1]. The following is well known: Theorem 0.3. (Gieseker/Maruyama) [10,11], [4]. Let E be a G-stable vector bundle. Then the component of E in the moduli space of G-stable vector bundles of a given Chern character is a quasi-projective variety and its closure M(E) in the space of G-semistable torsion-free sheaves is projective. We shall denote the closure of the -stable moduli space of vector bundles in the moduli space of S-equivalence classes of G-semistable torsion-free sheaves with Chern character (2; 0;?k) by M k. We shall restrict our attention to M 2 and M 3. Our principal results are: Theorem 0.4. The moduli space M 2 is naturally bred over a CP 3 -bundle over the dual torus ^T with bres given by certain compactications of Jacobians of genus 5 curves. and Theorem 0.5. The moduli space M 3 is isomorphic to Hilb 6 ^T T. Similar calculations for the case M (2;L1;?1) were carried out in [1]. It is also well known that the open subspace of -stable bundles coincides with gauge equivalence classes of Hermitian-Einstein connections on the underlying smooth bundle [17], [2], [3]. In complex dimension 2 these connections are called instantons. We shall be interested in the case when (X; `) is a principally polarized Abelian surface (T; L), where we represent ` by symmetric line bundle L, (L) = 1 and 0 2 D L, the divisor corresponding to a non-trivial section of L. Taubes ([16]) and Gieseker ([5]) have solved the general existence question for instantons or -stable

Zero-dimensional Schemes on Abelian Surfaces 3 bundles which, in our case, provide examples of -stable bundles for c 1 (E) = 0 and (E)?4. It is easy to show (see below) that there are no -stable bundles with Chern character (r; 0;?1). It is also known that the moduli space of simple torsion-free sheaves Spl on an Abelian surface or K3 surface is smooth for all E (see [13]) and are symplectic manifolds in a natural way. This contains the moduli space of G-stable torsionfree sheaves. This implies, for example, that M 3 \ Spl is smooth, consistent with Theorem 0.5 above. We shall make the simplifying assumption that T is irreducible (cannot be written as the product of elliptic curves). The reducible case can be treated similarly but the statement of our results and their proofs are more complicated. Our assumption implies that the Neron-Severi group of T is generated by L. Our primary tool for studying bundles and sheaves on such surfaces is the Fourier-Mukai transform whose denition and basic properties we shall now outline. 1 The Fourier-Mukai Transform This construction is due to Mukai (see [12], [14]). Let ^T denote the dual torus of T. This is just Pic 0 T and we shall denote the correspondence by ^x 7! P^x. Let P denote the Poincare bundle on T ^T. Given a sheaf E over T then we dene R i F(E) = R i^ ( E P), where T? T ^T?! ^ ^T are the projection maps. Denition 1.1. Following Mukai, we say that E satises WIT n if R i F(E) = 0 unless i = n and write ^E = Rn F(E). WIT stands for \Weak Index Theorem". If E satises WIT n then the bres of ^E over ^x 2 ^T are given by H n (T; EP^x ). This lead us to introduce the following stronger condition on E. Denition 1.2. We say that E satises IT n if for all i 6= n and ^x 2 ^T we have H i (E P^x ) = 0. Examples. (1) Any line bundle with non-zero Euler character satises IT n for some n. An ample line bundle always satises IT 0. (2) P^x satises WIT 2 and ^P^x = C(?^x), the skyscraper sheaf at ^x. (3) If E is -stable torsion-free sheaf with c 1 (E) = 0 then E satises IT 1. Proof. (of (3)). Since stability is preserved by twisting by line bundles and dualising it suces, by Serre duality, to show that H 0 (E) = 0. If f : O! E is a non-trivial section of E then f injects as the kernel must be torsion-free. This contradicts -stability of E. ut

4 Antony Maciocia If E satises WIT n then the Chern character of ^E is given by ch(e)i = (?1) i+n ch(e) 2?i ; where we have abused notation by omitting the isomorphism H i (T) = Hi ( ^T) = H 2?i ( ^T). Although we think of ^E as the Fourier transform of E, we can generalise to the derived categories of complexes of coherent sheaves on T and ^T to obtain a functor RF : D(T)! D( ^T). It is remarkable that this obeys a Fourier Inversion Theorem: Theorem 1.3. [12, Thm.2.2] RF is an isomorphism of categories with inverse given by (?1T) R (^? P). In other words, ( ^E)^ = (?1 T) E for a sheaf E satisfying WIT. More generally, even if E does not satisfy WIT we obtain a spectral sequence whose E 2 term is R p FR q F(E) and which converges to E1 p+q = (?1T) E when p + q = 2 and 0 otherwise. 2 Stable sheaves with c 1 = 0 and >?4 If E is a -stable bundle with Chern character (r; 0;?1) then by example (3) above it satises IT 1 and hence ^E is a vector bundle with Chern character (1; 0;?r). This is impossible and so no such bundles exist. However, M 1 is not empty but consists of non-locally-free G-stable and G-semistable sheaves. Our aim will be to construct points in the interior of the moduli space of -stable bundles of Chern characters (2; 0;?2) and (2; 0;?3). We shall use the well established Serre/Schwarzenberger method of constructing bundles (see [6, pp720{731]). The boundary of the rst moduli space has been studied in detail in [9]. Notation. We use the notation L x to denote L P x and we drop the tensor product sign when this does not lead to confusion. We use^to denote objects associated with the dual torus. However, since ^T = T via the principal polarization, the use of formulae such as x + ^x is permisable and mean that x originated in the torus and ^x originated in the dual torus. Proposition 2.1. Suppose that E is -semistable torsion-free sheaf of Chern character (2; 0;?k), k = 2 or 3. Then, for some ^x, EL^x admits a section. Moreover, if k = 2 then ^x can be chosen from a translate of some divisor in j^l2 j and if k = 3 then ^x can be chosen from V 2 Hilb 6 ^T. Proof. Observe that (EL^x ) > 0 and hence H 2 (EL^x ) = 0 by -semistability. Suppose that H 0 (EL^x ) = 0 for all ^x so that EL satises IT 1. If (E) =?2 then (EL^x ) = 0 and so EL has zero Fourier transform which is impossible. If (E) =?3 then d EL is a line bundle with Chern character (1; 2`;?2), but

Zero-dimensional Schemes on Abelian Surfaces 5 (2`) 2 =2 = 4 6=?2: a contradiction. If k = 2 then det(el) ^L?2 and so R 1 F(EL) must have non-trivial bres over at least a divisor in a translate of j^l 2 j. On the other hand, if k = 3 then the singularity set of R 1 F(EL) is at least V 2 Hilb 6 ^T.ut Now consider Q = E=L ^x. If this had any torsion, say T, then c 1(T ) ` 0 and E would surject to Q=T which is torsion-free. Now, (Q) = 2 and so (Q=T ) 2. The kernel of E! Q=T has slope?(q=t ) and is torsion-free. This implies that it is L m P^y I W for some ^y 2 ^T,?1 m 0 and zero-dimensional scheme W. If we assume that E is G-semistable then either m =?1 and we have a sequence: 0?! L ^x?! E?! L ^d+^x I X?! 0; (2:2) where X 2 Hilb 2?(E) T and det E = P ^d, or we have I W P^y! E with jwj = 1 if k = 2 and jwj = 2 or 3 if k = 3. Denition 2.3. If k = 2 we dene X 2 M 2 to be the set of S-equivalence classes of E's for which there are no extensions of the form 2.2. Similarly, we dene X3 2 and X3 3 M 3 in the k = 3 case where the superscript refers to the length of W as given above. We shall refer to these elements of X as exceptional sheaves. Notice that they are never -stable nor vector bundles nor satisfy WIT 1. We shall still consider extensions of the form We are now led to the following question: 0?! L ^x?! E?! Q?! 0: (2:4) Which extensions of the form 2.2 and 2.4 give rise to -stable/g-stable vector bundles? Before answering this question we shall draw some useful corollaries from the above proposition. First, a denition: Denition 2.5. We say that a zero-dimensional subscheme X of T is collinear if, for some x 2 T, X D x, the translate of D L 2 jlj given by x. We shall denote the scheme of collinear subschemes of length l by Hilb l ct and its complement in Hilb l T by Hilb l nt. Proposition 2.6. If E is -semistable torsion-free with Chern character (2; 0;?k) for k = 2 or 3 then E satises (i) WIT 1 if and only if E is -stable, (ii) IT 1 if and only if X (exists and) is not collinear and E is -stable. Proof. Without loss of generality assume that ^x = 0 = ^d. From the remarks in denition 2.3 we may assume that E is not exceptional. Apply RF to 2.2. Then we see that R 0 F(E) = R0 F(LI X ) and 0! R 1 F(E)! R 1 F(LI X )! R 2 F(L )! R 2 F(E)! R 2 F(LI X )! 0:

6 Antony Maciocia But if we apply RF to the twisted structure sequence of I X : we obtain the long exact sequence 0?! LI X?! L?! O X?! 0 (2:7) 0! R 0 F(LI X )! R 0 FL! H X! R 1 F(LI X )! 0 (2:8) and R 2 F(LI X ) = 0. The Mukai spectral sequence implies that R 2 ^FR 1 ^F(LI X ) = 0 but H X (= ^O X ) satises WIT 2 and so the middle map is not zero. Since ^L and H X are both locally-free, the kernel of R 0 FL! H X must be zero. Hence, R 0 F(E) = 0. On the other hand, applying H (P^x?) to 2.2 we see that H 0 (EP^x ) = H 0 (L^x I X ) and there is a long exact sequence 0! H 1 (EP^x )! H 1 (L^x I X )! H 2 (L P^x )! H 2 (EP^x )! 0: If E is -stable then H 2 (EP^x ) = 0 for all ^x and so R 2 F(E) = 0 implying that E satises WIT 1. E satises IT 1 if also H 0 (EP^x ) = 0 but this happens if and only if X is not collinear (this proves part (ii)). Conversely, suppose that E is not -stable. Then we have a short exact sequence P^y I X 0! E! P?^y I X 00. But the structure sequence of I X 00 implies that H 2 (I X 00) 6= 0 and so H 2 (EP^y ) 6= 0 and hence E does not satisfy WIT 1. ut Observe that if X is collinear then there is some at line bundle P^y mapping to E. This contradicts G-semistability and so such extensions cannot be in the closure of the moduli space. It suces to consider X's which are non-collinear. 3 Incidence of zero-dimensional subschemes on divisors In this section we shall recall some information about the incidence of zerodimensional subschemes on divisors on T. This was computed in [8] and we shall summarise the results below. Consider rst LI X and notice that (LI X ) = 1?jXj. Proposition 3.1. [8, x4{x8]. For all X 6= ;, LI X satises WIT 1. (i) If jxj = 1 then R 1 F(LI X ) is a line bundle of zero degree supported on a translate of D L. (ii) If jxj = 2 then R 1 F(LI X ) = ^Lx I X 0, where X 0 2 Hilb 2 ^T. (iii) If jxj = 3 then R 1 F(LI X ) is a rank 2 torsion-free sheaf with singularity set equal to a single point or empty. Proposition 3.2. [8, x8]. Suppose jxj > 3 then L 2 I X satises WIT 1 if and only if X is not collinear. Suppose that X is not collinear. When jxj = 4 then R 1 F(L 2 I X ) is line bundle of degree 3 supported on a divisor in jl 2 P j, where = P x2x x. When jxj = 5 then R 1 F(L 2 I X ) = ^L2 P y I X 0 for some y 2 T and X 0 2 Hilb 5 ^T.

Zero-dimensional Schemes on Abelian Surfaces 7 (Recall that the linear system jl 2 j is base-point free and contains only D x + D?x as reducible divisors.) Denition 3.3. Given a zero dimensional scheme X we dene n (X)^x = H 0 (L n P^x I X ): Over an open subset of ^T, n (X)^x will patch together to form a vector bundle. In any case, we can view n (X) as a torsion-free sheaf over ^T. The singularity set of n (X) we shall denote by S n (X) (standing for `sauf'). This can be given a natural scheme structure (see [8, x3]). Theorem 3.4. (i) If X has length 3 and is not collinear then S 2 (X) = X as schemes. The isomorphism is given by (x; y; z) 7! (x + y; y + z; z + x). 2 (X) has generic rank 0 and jumps to 1 on S 2 (X). (ii) If X is collinear, X D x say, and length 3 then S 2 (X) = D x+, where = P x2x x. 2(X) has generic rank 0 and jumps to rank 1 on S 2 (X). (iii) If jxj = 4 and X is not collinear then S 2 (X) 2 jl 2 P j, where = P x2x x. If S 2 (X) is irreducible then X contains no collinear length 3 subschemes, 2 (X) is generically empty and jumps to a single point over S 2 (X). (iv) If S 2 (X) is reducible then either or (a) 2 (X) is generically empty and jumps to a single point over S 2 (X), in which case X contains precisely one length 3 collinear subscheme, (b) 2 (X) is generically empty and jumps to a single point over S 2 (X), except that it has rank one over over one of the intersection points of the two components of S 2 (X). In this case X contains precisely two length 3 collinear subschemes. The importance of these propositions is in determining when extensions of the form 2.2 give rise to vector bundles E. There is a condition on X and the linear system jl 2 P 2^x j = T^x jl 2 j called the Cayley-Bacharach condition which determines whether E is locally-free or not. A good exposition of this can be found in [6, chapter 5]. In our case the condition reads: The Cayley-Bacharach Condition. X satises this if and only if for all X 0 X with jx 0 j = jxj? 1 we have 2 (X) 2^x = 2 (X 0 ) 2^x : Observe that Ext 1 (L^x I X ; L P?^x ) = H1 (L 2 P 2^x I X ) by Serre duality. When jxj = 4 the dimension of this equals the rank of 2 (X) 2^x which is generically zero. Hence, we are only concerned with 2^x 2 S 2 (X). Then X will satisfy the

8 Antony Maciocia Cayley-Bacharach condition if 2^x 62 S S 2 (X 0 ), where the union is over length 3 subschemes X 0 of X. If S 2 (X) is irreducible then this union consists of the six points (counted with multiplicities) fp + q; q + r; r + s; p + r; q + s; p + sg, where X = fp; q; r; sg. If S 2 (X) is reducible equal to D u + D v, say, then we must remove the component(s) corresponding to S 2 (X 0 ), where X 0 D x. If X contains only one collinear subscheme of length 3 then we must choose 2^x from D v minus some points. If X contains two collinear subschemes of length 3 then the Cayley- Bacharch condition is only met at a single point (one of the two intersection points of D u and D v ). In particular, we have the following Proposition 3.5. For all X 2 Hilb 4 T there is some ^x and a non-trivial extension of the form 2.2 such that E is locally-free. (E) =?2 4 Details of the moduli spaces We shall rst consider the case when E has Euler character?2. It is also convenient to relax the condition that det E = O. Then 2.2 gives rise to a sequence 0?! L ^x?! E?! L^x+ ^d I Z?! 0; (4:1) where Z 2 Hilb 4 T. Observe that any such extension has E -semistable. Furthermore, E is G-semistable provided Z is not collinear. Furthermore, E fails to be G-stable if there is a map from P^y I x! E. This must inject into L^x+ ^d I Z and hence Z contains a collinear Y 2 Hilb 3 T. This, in turn, implies that S 2 (Z) is reducible. Observe also that the cohomology jumping divisor S(E) of EL is T?^x? ^d S 2(Z). We also know from [8, Thm 7.5] that L 2 I Z is a line sheaf of degree 3 on S 2 (Z) when Z is not collinear. Hence, if we apply RF(L?) to 4.1 we see that d EL is a line sheaf of degree 2 on S(E). If S(E) is reducible then we need to be more precise about the degree of such line sheaves (which need no longer be locallyfree). This is done by restricting the line sheaf to irreducible factors and writing the degree as a tuple of the restriction degrees. In our case, the restriction degree of L 2 I Z is (2; 1) over D u + D v unless there are 2 collinear length 3 subschemes of Z in which case the restriction type is (1; 1). It will be convenient to introduce the term extension type to denote the degrees of the kernels of the restriction maps plus 2. For example, the restriction type of L 2 I Z when Z contains one collinear Y is (2; 1) and the extension type is (1; 2). If Z contains 2 collinear Y 's then the restriction type is (1; 1) but the extension type is (2; 2). Notice that the sum of the components of the extension type is minimised when the line sheaf is locally-free. Notice that a + c = b + d depends only on the Euler character of the sheaf. Finally, let us introduce the notation Jac (a;b) (c;d) (D u + D v ) to denote the space of line sheaves up to isomorphism of restriction type (a; b) and extension type (c; d).

Zero-dimensional Schemes on Abelian Surfaces 9 We omit the extension type if the line sheaves are line bundles. It will also be convenient to refer to h (2;1) (1;2) (1;1) (2;2) h (a;b) (c;d) i as the type of a line sheaf. We already know from [8] that the possible types of T = (L 2 P^x+ ^d I Z)^ are i h i and. In the latter case we do not allow torsion-free line sheaves over D u + D v which are restrictions of L I (because their Fourier transforms are not torsion-free see [8, x9]). From this it is possible to determine the types of kernels of maps T! O x. These are A = (1; 1) (1; 1) ; B = (0; 2) ; C = (2; 0) (0; 1) ; D = (2; 1) (0; 0) (2; 2) and B 0 and C 0 formed by interchanging D u and D v. Notice that if we have a column 0; 2 in a type then there is an extension M! d EL! N where M and N are of the form (L^y I p )^, for some p 2 T. This implies that E is not G-stable. This happens in types B, C and D. If we allow S-equivalence to act then it forces types B, C, D to be equivalent. It also identies B with B 0, etc. Type B corresponds to locally-free d EL. Lemma 4.2. If E is exceptional then d EL is the restriction of some L I Q, Q 2 Hilb 2 ^T, and so d EL has type D. Proof. Using the notation of 2.4 we see that d QL is torsion-free on Du +D v. Then [8, Lemma 9.3] implies that d QL is the restriction of L I. Applying RF to (2.4L) we obtain the short exact sequence d EL! d QL! O^x. If the composite L I! dql! O^x is zero then L I surjects to d EL. The kernel is torsion-free with Chern character (1;?L; 2) which is impossible. Hence, we have L I Q! d EL as required. The last part of the lemma follows from the obvious fact that Q = D u \ D v. ut If we assemble the Jacobians together to form a bundle of Jacobians over the space of divisors in translates of the linear system jl 2 j, which we can abbreviate to Pc L2, we obtain the full moduli space: Theorem 4.3. The moduli space of G-semistable torsion-free sheaves of Chern character (2; 0;?2) over a principally polarised Abelian surface (T; L) is given by a bration M 2! Pc L2 whose bres are Jac(D 0 ) if D 0 2 Pc L2 is smooth and Jac (1;1) (D (1;1) u + D v ) [ B if D 0 = D u + D v, where elements of B can be represented by elements of Jac (0;0). Points in B correspond precisely to G-semistable, non-gstable, torsion-free sheaves and taken together they are isomorphic to Sym 2 (T ^T). (2;2) The non--stable part coincides with the non-locally-free part and is given by line sheaves over a divisor D 0 2 Pc L2 which are of the form L y I Q =L z for Q 2 Hilb 2 D 0 and z; y 2 T and so includes X 2. This part of the moduli space is isomorphic to a

10 Antony Maciocia CP 1 bundle over Sym 2 ^T Hilb 2 T which is collapsed at the G-semistable points. In particular, M 2 is both non-empty and connected. The Donaldson moduli space of instantons can be obtained from this by applying S-equivalence corresponding to -stability rather that G-stability. This has the eect of blowing-down the CP 1 -bration given in the theorem as well as the diagonal in Hilb 2 T and provides an explicit complex analytic structure on the compactied moduli space. (E) =?3 We shall now turn our attention to M 3. Proposition 4.4. Suppose that ch(e) = (2; 0;?3) and E is G-semistable and torsion-free. Then R 1 F(EL) is of the form L 2 P y I V for some V 2 Hilb 6 ^T if and only if the cohomology jumping set of EL is zero-dimensional. Proof. We express E as an extension of the form 2.2. The assumptions on E imply that X is not collinear since any section of L^y I X lifts to EP^y?^x which would contradict G-semistability. We have a sequence P?^x! EL! L 2 P^x I X with jxj = 5. Apply RF and propositions 2.6 and 3.2 to obtain 0! R 1 F(EL)! R 1 F(L 2 P^x I X )! O^x! 0: But we know that R 1 F(L 2 I X ) is isomorphic to ^L2 P y I X 0 for some X 0 2 Hilb 5 ^T if and only if the condition of the jumping set of EL (which equals the cohomology jumping set of L 2 P^x I X ) given in the proposition holds. ut We must, therefore split our moduli space into those sheaves whose jumping set is in Hilb 6 T (denoted M 3 ) and those whose jumping set is a translate of D L (denoted ~ M 3 ). Apply RF to L 2 P y I V! L 2 P^y! O V to obtain T y c L 2! H V! (?1T) EL. Then det(el) = O implies that det(h V ) = P v2v v = L?2 det T y c L 2 = P?2y. We use here the fact that (FP y )^= T y ^F for any sheaf F satisfying WIT. Hence, the condition ^d = 0 becomes X v2v v =?2y: (4:5) Then proposition 4.4 tells us that M 3 is isomorphic to the subset H of Hilb 6 ^T T given by (V; y) satisfying the condition that L 2 P y I V has torsion-free Fourier transform. The latter condition is equivalent to the geometric condition that V does not contain any collinear subschemes of length 5 and if we want det(e) = O then we must also impose condition 4.5. If E 2 M 3 is G-semistable but not -stable then there is a short exact sequence 0?! P^a I Y?! E?! P?^a I Y 0?! 0; (4:6)

Zero-dimensional Schemes on Abelian Surfaces 11 where jy j + jy 0 j = 3 and jy j > 1. This gives two possibilities. Either jy j = 2 or jy j = 3. Notice that E is forced to be G-stable. This suggests a further splitting of M into: M 0 3 -stable vector bundles jsing(e)j = 0 M 1 3 -stable non-vector bundles jsing(e)j = 1 M 2 3 G-stable sheaves with jy j = 2 jsing(e)j = 3 M 3 3 G-stable sheaves with jy j = 3 jsing(e)j = 3 Similarly for ~M we dene ~M i 3 by the same denition. From [14, Cor.4.5] we know that M 1 consists only of non-locally-free non-stable sheaves and hence it is impossible for E to lie in M 1 and hence the singularity set of E must have lengths 0, 1 or 3. This explains the last column in the denition above. Since S 2 (V ) is a translation of the singularity set of E for E 2 M we can deduce the immediate geometrical proposition: Proposition 4.7. There is no length 6 subscheme of T with js 2 (V )j = 2. In other words, if V lies on two divisors in Pc L2 then it must lie on a third. Lemma 4.8. If a torsion-free sheaf F has Chern character (2; 2L;?1) and satises WIT 1 then it must be G-stable. Proof. If L m P^z I W! F! L 2?m P^z I W 0 with m > 0 then we must have 2? m > 0 since F satises WIT 1. Hence, m = 1 and so F is -semistable. Since L satises IT 0 we cannot have jwj = 0 and if jwj = 1 then R 1 F(LI W ) is a torsion sheaf which cannot map to L 2 I V. Hence, jwj > 1 and so F is G-stable. ut In particular, R 1 F(L 2 I V ) is G-stable if V contains no collinear length 5 subschemes. Lemma 4.9. X 2 3 M 2 3 and X 3 3 6 M 3. Proof. Let E 2 X 2 3. Refering to 2.4 we have a sequence S! LQ! L^y I q, where S is a line bundle over a translate of D L of degree 0. Now, S satises WIT 1 and the transform of the above sequence is of the form L I! d QL! S 0 and hence d QL is torsion-free. This implies that EL satises WIT 1 and d EL d QL is also torsion-free and hence isomorphic to some L 2 P y I V. Hence, using lemma 4.9, E 2 M 2 3. On the other hand, if EL d = L2 P y I V then ^Q = L 2 P y I X 0 for X 0 2 Hilb 5 ^T. Hence, S must have degree 0 and so jwj = 2. ut In fact, the above proof shows that ~M 2 3 = ;. Theorem 4.10. (i) E 2 M 0 3 i V is generic with S 2 (V ) = ;.

12 Antony Maciocia (ii) E 2 M 1 i S 3 1(V ) has length 1. (iii) E 2 M 2 3 i V contains at least one collinear length 4 subscheme (but no collinear length 5 subscheme). i V contains 3 collinear length 4 subschemes (but no collinear (iii)' E 2 X3 2 length 5 subscheme). (iv) E 2 M 3 i V satises 3 3(V ) y = 3 (X 0 ) y for some y and H 1 (L 3 P y I V ) 6= 0 and such that V contains no collinear length 5 subschemes. The codimension of M n 3 in M 3 is n and the codimension of X3 2 is 6. Proof. (i) and (ii) are trivial. (iii) EL ts into extensions of the form LI Y! EL! LI p if and only if there is an extension LI Y 0! L 2 P y I V! (LI p )^. This happens if and only if V n Y 0, which has length 4, is collinear. (iii)' E 2 X 2 if and only if each 3 X0 contains a collinear length 4 subscheme. If there were four collinear subschemes then there would have to be a collinear length 5 subscheme in V. (iv) Applying RF to L I Y! EL! L gives the short exact sequence ^L P?! G! L 2 P y I V, where G is torsion-free, has Chern character (2; L;?1) and satises WIT 1. Hence, G is -stable. If G were not locally-free then there is some L I p! G and the induced map L I p! L 2 P y I V implies that L 2 I V does not have a torsionfree Fourier transform; a contradiction. Hence, G is locally-free. Conversely, suppose that V does satisfy Cayley-Bacharach with respect to L 3 P y for some y so that H 1 (L 3 P y I V ) 6= 0. Then the resulting extension gives rise to G and hence to L I Y! EL! L for suitable ;. To compute the dimensions of M n observe that 3 dimm0 = 14. Now 3 dimm2 = 3 2 + 4 + 4 + 2 = 12 given by 2 dimensional choice of D u, a 4 dimensional choice of Z D u, a 4 dimensional choice of 2 other points and then a 2 dimensional choice of y. If V contains 3 collinear length 4 subschemes then the freedom of choice is given by choosing 4 points on any translate of D L and the other two points are determined by the intersection of a pair of translates of D L each passing through 2 of these four points. Hence, dimx 2 = 2 + 4 + 2 = 8. 3 For M 1 we observe that Cayley-Bacharach fails for X when 2^x + ^d 3 lies on S 2 (Z) n S 2 (X) for some Z X. This is a codimension 1 condition. For M 3 we 3 require 2^x + ^d to lie on the intersection of S 2 (Z i ) for three Z i X. This imposes two conditions on ^x and a further condition on X giving codimension 3. ut We can likewise classify sheaves in M ~ 3. Points E 2 M ~ 3 n X 3 3 are characterised by the fact that if EL=P?^x = L 2 P^x+ ^d I X then X contains a collinear length 4 subscheme which we shall denote by Z. For the moment we shall assume that ^x = 0 = ^d. Then L 2 I X satises WIT 1 but its Fourier transform A, say, is not torsion-free. Applying RF to LI p! L 2 I X! T we obtain S! A! ^L y I q, where ch(s) = (0; L;?1). If the composite S! A! O 0 vanishes then S! d EL injects which implies LI p! EL injects contradicting the G-semistability of E. Hence,

Zero-dimensional Schemes on Abelian Surfaces 13 del ts into an extension of the form 0?! R?! d EL?! Ly I q?! 0; (4:11) where R is a degree?1 line bundle over some translate D^y of D ^L. Extensions 4.11 cannot split. On the other hand, if E 2 X 3 then we have a sequence P 3?^x! EL! Q for some torsion sheaf Q. We also have (4.6) with jy j = 3 which shows that EL satises WIT 1. Then L^a I Y! Q and Q satises WIT 1. Since EL d is not torsion-free we see that ^Q is not torsion-free. Applying RF to the short exact sequences gives ^L?1 x! K 0! EL d and K0! K! O^x, where K is locally-free and -stable of Chern character (2; l; 0). Let T denote the torsion subsheaf of ^Q. Then ^Q=T = ^Ly I p for some p as K maps to ^Q. This implies that d EL ts into an extension of the form 4.11 again. Now suppose that we are given a non-trivial extension R! A! L y I q. Then A satises WIT 1 and its Fourier transform ts into an exact sequence K! ^A! S, where ch(k) = (2; L; 0), K is locally-free (because R is IT 1 ) and S is a degree 0 line bundle over a translate of D L. If T is the torsion subsheaf of ^A then it must be a line bundle over supp(s) of degree less than 0. But S=T is supported in dimension 0 and K! ^A=T! S=T is short exact. This implies that ^A=T = K S=T which contradicts the fact that ^A=T is torsion-free. Hence, ^A is torsion-free. We know from lemma 4.8 that ^A must be G-stable and so lies in ~M 3. We also see easily that extensions 4.11 are uniquely determined by E. So we have proved: Theorem 4.12. There is an isomorphism between non-trivial extensions of sheaves of the form L y I q and degree?1 line bundles over translates of D ^L, and ~M 3. Using the fact that R can be written as L x I Y =O with Y supp(r) and the structure sequence L y I q! L y! O q we can compute Ext 1 (L y I q ; R) which is isomorphic to H 1 (R L y ) = C4 if y 62 supp(r) and isomorphic to coker(ext 1 (O q ; R)! H 1 (R L y )) Hom (R; O q ) = C 4 if y 2 supp(r). This provides us with an isomorphism between ~ M 3 and a CP 3 bration over ^T ^T ^T T. This latter factor consists of (^a 1 ; ^a 2 ; q; y) where (^a 1 ; ^a 2 ) 7! R given by R = T ^a 2 (ker(o(d ^L )! O^a1 )). This implies that dim M ~ 3 = 11. Theorem 4.13. X 3 3 = ~M 3 3 and ~ M 3 = ~M 1 3 [ ~M 3 3. Proof. Suppose E 2 ~M 3 nx 3 3 and we assume, as before, that ^x = 0 = ^d. Then there is a collinear length 3 subscheme of X and hence a sequence L I N! L 2 I X! Q where N 2 Hilb 2 T and Q is a line bundle over a translate of D L of degree 1. The composite EL! L 2 I X! Q has kernel K 0 which has Chern character (2; l;?1) and this factors through K = ker(d EL! L2 I X =L I p ) which is locally-free. This implies that K 0 is torsion-free and hence ~M 0 1 = ;.

14 Antony Maciocia We also have part of a long exact sequence Ext 1 (Q; O)?! Ext 1 (L 2 I X ; O)?! Ext 1 (L I N ; O): Then EL 2 ~M 2 only if the representative of 3 K0 in the last Ext group vanishes. But this is impossible as K 0 K. Hence, ~M 2 3 = ;. We can repeat this with a length 3 subscheme Y instead of N to show that E 62 M ~ 3 3. This proves the theorem. ut We shall now relate M ~ 3 to the part of Hilb 6 ^T T which is not isomorphic to M 3. Consider an extension of the form 4.11 above and consider a suitable structure sequence for R: L?1 P x! P w I p! R. Without loss of generality assume that x = 0 = w. The Ext 1 (R; L?1 ) = C and this choice of extension gives a canonical map (up to a scalar) : Ext 1 (L y I q ; R)! Ext 2 (L y I q ; L?1 ) via the cup product. This map forms part of a long exact sequence. The previous terms are 0! Ext 1 (L y I q ; I p ) because H 1 (L 2 P y I q ) = 0. Suppose that 4.11 was the image of I p! G! L y I q. Then we obtain L?1! G! d EL. This implies that E 2 X 3 3. Conversely, suppose that E 2 X 3 3. Then we have L?1 P x! G! d EL and the induced map G! L y I q has kernel P w I p for some w and p. This implies that 4.11 is in the image of some element in Ext 1 (L y I q ; P w I p ). Hence, the kernels of as we vary x, w, p and q give rise precisely to X 3 3. Serre duality implies that ker() is isomorphic to the dual of the cokernel of : Hom (L?1 ; L y I q )! Ext 1 (R; L y I q ). Claim. Points of (coker( )) correspond to collinear elements V 2 Hilb 6 ^T. Proof. (of claim). Points of the image of correspond to injections I p! B where B ts in L y I q! B! R. If B has torsion then the torsion must take the form T! B! L and there must be an induced injection I p! L. Let the quotient be S. Applying RF we obtain ^L! R 0 F(S)! P p and R 1 F(S) = O 0. But Ext 1 (P p ; ^L) = 0 and so R 0 F(S) = ^LPp. The Mukai spectral sequence for S gives O! L O p! S and hence S has torsion over its support. But S = L y I q =L?1 which implies that S is torsion-free; a contradiction. Hence, B must be torsionfree of the form L 2 P y I V with V not collinear but containing a collinear length 5 subscheme. The converse argument also holds showing that points of the image of correspond to non-collinear V 's. This implies that the dual of the cokernel corresponds precisely to the collinear V 's as desired. ut This implies that there is a canonical isomorphism between the collinear part of Hilb 6 ^T T and X 3. This shows that dimx 3 3 3 = 8 and provides a canonical section of the CP 3 -bration ~M 3! ^T 3 T. We have also shown in the proof of the claim that the dual bration M ~ 3 is naturally isomorphic to the subvariety of Hilb 6 ^T T consisting of V 's with collinear length 5 subschemes. On the other hand, we know that M 3 and Hilb 6 ^T T are birationally isomorphic via M 3 and the there is a canonical isomorphism between two sections of the CP 3 -brations. This shows that the CP 3 -bration is holomorphically self-dual and proves:

Zero-dimensional Schemes on Abelian Surfaces 15 Theorem 4.14. There is a canonical isomorphism M 3 = Hilb 6 ^T T. In particular, M 3 is connected. In general, one might expect that M 2p+1 = Hilb 4p+2 ^T T References [1] Braam, P.J., Maciocia, A., Todorov, A., Instanton Moduli as a Novel Map from Tori to K3-Surfaces, Inv. Math. 108 (1992) 419{451. [2] Donaldson, S.K., Anti-self-dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles, Proc. Lond. Math. Soc. 50 (1985) 1{26. [3], Innite Determinants, Stable Bundles and Curvature, Duke Math. J. 54 (1987) 231{247. [4] Gieseker, D., On the moduli of vector bundles on an algebraic surface, Ann. Math. 106 (1977) 45{60. [5], A Construction of Stable Bundles on an Algebraic Surface, J. Di. Geom. 27 (1988) 137{154. [6] Griths, P., Harris, J., Principles of Algebraic Geometry, Wiley (1978). [7] Kobayashi, S., Dierential geometry of complex vector bundles, Princeton University Press (1987). [8] Maciocia, A., Incidence of points on divisors in a principally polarised abelian surface, Edinburgh University Preprint (1992). [9], Classifying sheaves in the Maruyama boundary for surfaces with trivial canonical bundle, Edinburgh University Preprint (1992). [10] Maruyama, M., Moduli of stable sheaves, I, J. Math. Kyoto Univ. 17 (1977) 91{126. [11], Moduli of stable sheaves, II, J. Math. Kyoto Univ. 18( 1978) 557{614. [12] Mukai, S., Duality between D(X) and D( ^X) with its application to Picard sheaves, Nagoya Math. J. 81 (1981) 153{175. [13], Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Inv. Math 77 (1984) 101-116. [14], Fourier Functor and its Application to the Moduli of bundles on an Abelian Variety, Adv. in Pure Math 10 (1987) 515{550. [15] Okonek, Ch., Schneider, M., Spindler, H., Vector Bundles on Complex Projective Spaces, Birkhauser (1980). [16] Taubes, C.H., Self-dual connections on 4-manifolds with indenite intersection matrix, J. Di. Geom. 19 (1984) 517{560. [17] Uhlenbeck, K.K., Yau, S-T., On the Existence of Hermitian-Yang-Mills Connections in Stable Vector Bundles, Comm. Pure & Appl. Math. Supp. 39 (1986) 257{293.