Birational Geometry of Singular Moduli Spaces of O'Grady Type

TL;DR

This paper explores the birational geometry of singular moduli spaces of sheaves on K3 surfaces, classifying walls in stability conditions and relating them to birational models and symplectic resolutions.

Contribution

It provides a complete classification of walls in Bridgeland stability space and links all birational models of the moduli space to Bridgeland semistable objects.

Findings

Classified all walls in the stability condition space.

Connected birational models to moduli spaces of semistable objects.

Showed symplectic resolution is deformation equivalent to O'Grady's 10-dimensional manifold.

Abstract

Following Bayer and Macr\`{i}, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macr\`{i} map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland semistable objects on $X$. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of $X$ and the N\'{e}ron-Severi lattice of $M$ which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of $M$ is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.