Birational Geometry of Singular Moduli Spaces of O’grady Type

Following Bayer and Macr̀ı, 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-Macr̀ı map from the space of Bridgeland stability conditions Stab(X) to the cone of movable divisors on M to relate wall-crossing in Stab(X) to birational transformations of M . We give a complete classification of walls in 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.