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In this paper, we complete the determination of the index of factoriality of moduli spaces of semistable sheaves on an abelian or projective K3 surface S. If v is a Mukai vector and H a generic polarization, let Mv(S,H) be the moduli space of H-semistable sheaves on S with Mukai vector v. First, we describe in terms of v the pure weight-2 Hodge structure and the Beauville form on the second integral cohomology of the symplectic resolutions of M v(S,H) (when S is K3) and of the fiber Kv(S,H) of the Albanese map of Mv(S,H) (when S is abelian). Then, 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. If S is abelian, we show that Mv(S,H) and Kv(S,H) are 2-factorial

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