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 hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.