Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces

Abstract

In this paper we study factoriality properties of some moduli spaces of semistable sheaves on abelian or projective K3 surface S. If v = 2w is a Mukai vector, w is primitive, w 2 = 2 and H is a generic polarization, let Mv(S,H) be the moduli space of H semistable sheaves on S with Mukai vector v. If S is abelian, we show that the fiber Kv(S,H) of the Albanese map of Mv(S,H) is 2 factorial. If S is K3, we show that Mv(S,H) is either locally factorial or 2 factorial, and we give an example of both cases.