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

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.