Given a strict simple degeneration f : X → C the first three authors previously constructed a degeneration IX/Cn → 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 IX/Cn → 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 (IX/Cn )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 IX/Cn → C carries a nowhere degenerate relative logarithmic 2-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.