The geometry of degenerations of Hilbert schemes of points

Given a strict simple degeneration \(f \colon X\to C\) the first three authors previously constructed a degeneration \(I^n_{X/C} \to C\) 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 \(I^n_{X/C} \to C\) is a dlt model. This is even a good minimal dlt model if \(f \colon X \to C\) has this property. We compute the dual complex of the central fibre \((I^n_{X/C})_0\) and relate this to the essential skeleton of the generic fibre. For a type II degeneration of \(K3\) surfaces we show that the stack \({\mathcal I}^n_{X/C} \to C\) carries a nowhere degenerate relative logarithmic \(2\)-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.