Good models of Hilbert schemes of points over semistable degenerations

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 hyperk\"ahler varieties, namely Hilbert schemes of points on K3 surfaces.Comment: 40 pages, 6 Figures. Comments welcom

Expansions for Hilbert schemes of points on semistable degenerations

The aim of this paper 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 a good degeneration of Hilbert schemes of points as a proper Deligne-Mumford stack and show that it provides a geometrically meaningful example of a construction arising from the work of Maulik and Ranganathan.Comment: 44 pages, 7 figure