The 2-Factoriality of the O'Grady Moduli Spaces

TL;DR

This paper proves that certain moduli spaces of sheaves on K3 surfaces, specifically $M_{10}$ and $M_6$, are 2-factorial varieties and establishes a Hodge isometry via the Donaldson morphism, enriching the understanding of their geometric structure.

Contribution

The work demonstrates the 2-factoriality of O'Grady's moduli spaces and constructs a Hodge isometry linking the Mukai lattice to the second cohomology of their symplectic resolutions.

Findings

$M_{10}$ is a 2-factorial variety.

A Hodge isometry is established via the Donaldson morphism.

Similar results are obtained for $M_{6}$.

Abstract

The aim of this work is to show that the moduli space $M_{10}$ introduced by O'Grady in \cite{OG1} is a $2-$factorial variety. Namely, $M_{10}$ is the moduli space of semistable sheaves with Mukai vector $v:=(2,0,-2)\in H^{ev}(X,\mathbb{Z})$ on a projective K3 surface $X$. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between $v^{\perp}$ (sublattice of the Mukai lattice of $X$) and its image in $H^{2} (\widetilde{M}_{10},\mathbb{Z})$, lattice with respect to the Beauville form of the $10-$dimensional irreducible symplectic manifold $\widetilde{M}_{10}$, obtained as symplectic resolution of $M_{10}$. Similar results are shown for the moduli space $M_{6}$ introduced by O'Grady in \cite{OG2}.