La singularit\'{e} de O'Grady

TL;DR

This paper demonstrates that blowing up the singular locus of certain moduli spaces of sheaves on abelian or K3 surfaces yields symplectic resolutions, clarifying O'Grady's constructions for specific cases.

Contribution

It provides a direct geometric construction of symplectic resolutions for O'Grady's moduli spaces using blow-ups of their singular loci.

Findings

Blow-up of the singular locus yields a symplectic resolution.

Provides explicit descriptions for O'Grady's resolutions.

Connects moduli space singularities with symplectic geometry.

Abstract

Let M be the moduli space of semistable sheaves with Mukai vector 2v on an abelian or K3 surface where v is primitive such that <v,v>=2. We show that the blow-up of the reduced singular locus of M provides a symplectic resolution of singularities. This gives a direct description of O'Grady's resolutions of M\_{K3}(2,0,4) and M\_{Ab}(2,0,2).