The geometry of degenerations of Hilbert schemes of points

Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration In X/C → C of the relative degree n Hilbert scheme of 0-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of f is at most 2. In this case we show that In X/C → C is a dlt model. This is even a good minimal dlt model if f : X → C has this property. We compute the dual complex of the central fibre (In X/C)0 and relate this to the essential skeleton of the generic fibre. For a type II degeneration of K3 surfaces we show that the stack In X/C → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.