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

TL;DR

This paper determines the factoriality index of moduli spaces of semistable sheaves on abelian and K3 surfaces, describing their cohomological structures and providing examples of factoriality properties.

Contribution

It completes the classification of factoriality properties of these moduli spaces and relates them to their cohomological and geometric structures.

Findings

K3 case: moduli spaces are either locally factorial or 2-factorial.

Abelian case: moduli spaces and fibers are 2-factorial.

Explicit examples illustrating factoriality cases.

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.