The geometry of degenerations of Hilbert schemes of points

Abstract

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