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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  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-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  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.

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