Degenerations of Hilbert schemes of points

Abstract.

Let $C$ be a smooth curve and $ f  :  X  \to  C $ a simple degeneration with smooth fibers away from a point $0 \in C$. I will present a "good" compactification of the relative Hilbert scheme of points associated to the smooth locus of $f$. This compactification is obtained by GIT-methods, and gives a new, alternative, approach to earlier constructions of J. Li and B. Wu.

I will focus on the case where $f$ is a degeneration of surfaces, and explain how this set-up can be used to study degenerations of irreducible holomorphic symplectic varieties. This is joint work with M. Gulbrandsen and K. Hulek.