## Abstract Cattaneo

**Beauville-type involutions on the Hilbert scheme of two points on a K3 surface**

It is well known that if S is a generic quartic in P^3, then the Hilbert scheme S^{[2]} admits a non-symplectic involution, which we call a Beauville involution. The aim of the talk is to study the automorphism group of the Hilbert scheme of two points on a generic projective K3 surface S, and in particular to show that this group can be either trivial or of order two. Our argument will highlight the relationship between the automorphism group and the ample cone of the Hilbert scheme. We will also give a characterization of the presence of the non-trivial automorphism in terms of the self-intersection of the ample divisor on S and in terms of the geometry of the Hilbert scheme. This is a joint work with S. Boissière, M. Nieper-Wisskirchen and A. Sarti.In the very last part, we will also sketch some ideas for giving an explicit geometric realization of these automorphisms (work in progress).