Degenerations of generalized Kummer varieties

TL;DR

This paper introduces a method to explicitly construct degenerations of higher-dimensional generalized Kummer varieties starting from simple degenerations of abelian surfaces, and studies their geometric properties.

Contribution

It provides a new explicit degeneration construction for generalized Kummer varieties and analyzes the geometry and stratification of these degenerations.

Findings

For n=2, obtains a projective Kulikov model of Kummer surfaces.

For n=3, discovers new phenomena in the degeneration.

Shows the dual complex of the degeneration is PL-homeomorphic to a 2-simplex.

Abstract

We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an abelian scheme over $C \setminus 0$ and we can form the relative generalized Kummer variety $K^{n-1}_{\circ} = \mathrm{Kum}^{n-1}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. This is naturally a closed subscheme of the relative Hilbert scheme $\mathrm{Hilb}^{n}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. In previous work (joint with Gulbrandsen) we had constructed a compactification $I^n_{\mathcal{Y}/C}$ over $C$ of the latter scheme. The closure $K^{n-1}_{\mathcal{Y}/C}$ of $K^{n-1}_{\circ}$ inside $I^n_{\mathcal{Y}/C}$ yields a canonical way to degenerate the family of generalized Kummer varieties, and is the degeneration we propose. This paper contains a detailed study of the geometry of the scheme $K^{n-1}_{\mathcal{Y}/C}$ and its natural stratification. For $n=2$ we obtain a projective Kulikov model of Kummer surfaces, whereas already for $n=3$ new phenomena occur. We study in detail the dual complex of $K^{2}_{\mathcal{Y}/C}$ and show that this is PL-homeomorphic to the standard $2$-simplex.