Papers

Good models of Hilbert schemes of points over semi stable degenerations (2024). In this paper, we explore different possible choices of expanded degenerations and define appropriate stability conditions in order to construct good degenerations of Hilbert schemes of points over semistable degenerations of surfaces, given as proper Deligne-Mumford stacks. These stacks provide explicit examples of constructions arising from the work of Maulik and Ranganathan. This paper builds upon and generalises previous work in which we constructed a special example of such a stack. We also explain how these methods apply to constructing minimal models of type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.
Expansions for Hilbert schemes of points on semi stable degenerations (2023). Here we give a first construction of a good degeneration of Hilbert schemes of points on a semistable family of surfaces, and discuss stability conditions coming from GIT and from Li-Wu stability. The proper Deligne-Mumford stack we construct provides a geometrically meaningful example of a construction arising from the logarithmic Hilbert scheme constructions of Maulik and Ranganathan. It is special among all possible models because the necessary stability condition here is much simpler than what we expect in general.

Thesis

Expanded degenerations for Hilbert schemes of points

Abstract: The aim of this thesis is to extend the expanded degeneration construction of Li and Wu to obtain good degenerations of Hilbert schemes of points on semistable families of surfaces, as well as to discuss alternative stability conditions and parallels to the GIT construction of Gulbrandsen, Halle and Hulek and logarithmic Hilbert scheme constructions of Maulik and Ranganathan. We construct some good degenerations of Hilbert schemes of points as proper Deligne- Mumford stacks and show that these provide geometrically meaningful examples of constructions arising from the work of Maulik and Ranganathan.