Degenerations of Hilbert schemes of points

I will present a “good” compactification of the relative Hilbert scheme of points associated to the smooth locus of a (simple) degeneration $X\to C$ of varieties. This compactification is obtained by GIT-methods, and yields a new, refined, approach to earlier stacky constructions of J. Li and B. Wu. In particular, I will discuss various aspects of the geometry of this degeneration (e.g. its birational geometry and dual complex), when $X\to C$ is a family of surfaces. This is joint work with M. Gulbrandsen, K. Hulek and Z. Zhang.