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

Abstract

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.

arXiv comment14 pages; added calculation of the weight-two Hodge structures of the symplectic resolutions, bibliographical references corrected

References in corpus (2)

• Sur le produit de variétés localement factorielles ou Q-factorielles
• Albanese map of moduli of stable sheaves on abelian surfaces

Cited by in corpus (2)

• The Hodge diamond of O'Grady's 6-dimensional example
• On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces