Automorphisms of Hilbert scheme of points of surfaces

If X is a K3 surface, then the Hilbert scheme of n points on it, denoted X[n], is a hyper-Kahler manifold. Every automorphism of the K3 surface induces an automorphism of X[n]. However, there are more automorphism on X[n] and the theory of automorphisms of X[n] is a rich and beautiful subject in its own right. In the talk we study what happens if we drop the condition on X to be a K3 surface. Do we still get interesting automorphisms of X[n] that do not come from the underlying surface? We will discuss some positive and some negative results to this question.

Joint work with Pieter Belmans, Chiara Camere and Jorgen Rennemo.