Factoriality properties of moduli spaces of sheaves on abelian and K3 surfaces

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 $v=2w$ is a Mukai vector, $w$ is primitive, $w^2=2$ and $H$ is a generic polarization, let $M_v(S,H)$ be the moduli space of $H-$semistable sheaves on $S$ with Mukai vector $v$. First, we describe in terms of $v$ the pure weight-two 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 $K_v(S,H)$ of the Albanese map of $M_v(S,H)$ (when $S$ is abelian). Then, if $S$ is K3 we show that $M_v(S,H)$ is either locally factorial or $2-$factorial, and we give an example of both cases. If $S$ is abelian, we show that $M_v(S,H)$ and $K_v(S,H)$ are $2-$factorial.