(Seminar SAG) On the exceptional locus of O’Grady’s nonsymplectic resolutions

In this talk, we focus on some singular moduli spaces of sheaves on a K3 surface. More precisely, for any integer n > 1, we consider the moduli space M(n) associated with the Mukai vector 2(1,0,1-n). Looking for new deformation classes of hyper-Kähler manifolds, O’Grady constructed an explicit resolution of every M(n). O’Grady’s resolution is crepant and does give a hyper-Kähler manifold only if n=2. If n>2, it turns out that no crepant resolution exists for M(n), but one may still look for a categorical crepant resolution.
We will report on the preliminary step in this direction, which consists in a geometric analysis of O’Grady’s resolution and of its exceptional locus.