Birational Geometry of Singular Moduli Spaces of O'Grady Type

Abstract

Following Bayer and Macrì, 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-Macrì 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 Néron-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.

Final version

References in corpus (4)

• Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces
• On some moduli of complexes on K3 surfaces
• On the log minimal model program for irreducible symplectic varieties
• La singularité de O'Grady

Cited by in corpus (8)

• Projectivity and Birational Geometry of Bridgeland Moduli spaces on an Enriques Surface
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• Formality conjecture for K3 surfaces
• Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces
• An elementary description of nef cone for irreducible holomorphic symplectic manifolds
• On two families of Enriques categories over K3 surfaces