Higher-dimensional O'Grady's resolutions and their exceptional locus

TL;DR

This paper investigates the explicit resolution of singularities of certain moduli spaces on K3 surfaces, describing the exceptional locus and exploring the potential for categorical crepant resolutions.

Contribution

It provides a detailed analysis of O'Grady's explicit resolution for higher-dimensional moduli spaces and describes their exceptional loci.

Findings

Explicit description of the exceptional locus in O'Grady's resolution

Insights into the structure of singularities in moduli spaces on K3 surfaces

Discussion on the possibility of categorical crepant resolutions

Abstract

For any $n \ge 3$, we consider the moduli space $M_n$ of semistable sheaves with Mukai vector $2(1,0,1-n)$ on a K3 surface. The moduli space $M_n$ is singular and lacks a crepant resolution, but might still admit a categorical crepant one. As a preliminary to exploring this possibility, we study the explicit resolution of singularities of $M_n$ constructed by O'Grady, and provide a global description of its exceptional locus.