Structure of the Moduli Space of Stable Sheaves on Elliptic Fibrations D. Hern Andez Ruip Erez

We describe a method for characterizing certain connected components of the moduli space M(a; b) of torsion-free sheaves on a elliptic surface X with a section which are stable with respect to a polarization of type aH + bb, H being the section and the elliptic bre (b 0) and whose relative degree is zero. We construct all sheaves in these components as Fourier-Mukai transforms of the pure dimension one rank one semistable sheaves supported on those (possibly nonintegral) curves D in X which are at over B. This generalizes and completes earlier constructions of stable bundles due to Friedman, Morgan and Witten. We prove that there exists a universal spectral cover over a Hilbert scheme and that a union of connected components of M(a; b) is isomorphic via Fourier-Mukai to a relative compactiied Jacobian of pure dimension one stable rank one sheaves on the universal spectral cover. We also study the relative moduli scheme of sheaves whose restriction to each bre is torsion-free and semistable of rank n and degree zero for higher dimensional elliptic brations X ! B with a section. By means of the Fourier-Mukai transform we prove that this scheme is isomorphic to the relative symmetric n-product of the bration. 1. Introduction Recently there has been a growing interest in the moduli spaces of stable vector bundles on elliptic brations. Aside from their mathematical importance, these moduli spaces provide a geometric background to the study of some recent developments in string theory, notably in connection with the conjectural duality between F-theory and heterotic string theory ((12], 13], 3], 9]). In this paper we study such moduli spaces, dealing both with the case of relatively and absolutely stable sheaves.