Zero - dimensional Schemes on Abelian Surfaces

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 compactiied family of Jacobians and the second is a Hilbert scheme. 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 calledìnterpolation 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 deene 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 KK ahler form !). Let us denote the Chern characters of sheaves on X by (n + 1)-tuples: (r; c 1 ; 1 2 c 2 1 ? c 2 ; : : :). Deenition 0.1. We say that a torsion-free sheaf E is-stable (respectively,-semistable) with respect tò if for all subsheaves F such that E=F is torsion-free