Constructions of Kummer structures on generalized Kummer surfaces

TL;DR

This paper investigates generalized Kummer surfaces derived from abelian surfaces with order 3 automorphisms, exploring their configurations and the conditions under which these surfaces determine the original abelian surfaces.

Contribution

It demonstrates that isomorphic generalized Kummer surfaces do not necessarily imply the original abelian surfaces are isomorphic, using methods involving Fourier-Mukai partners and automorphism group actions.

Findings

Counterexamples to the reconstruction problem for generalized Kummer surfaces.

Identification of unique configurations that are not exchangeable under automorphisms.

Abstract

We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint configurations of two smooth rational curves $C,C'$ with $CC'=1$. This $9{\bf A}_{2}$-configuration plays a role similar to the Nikulin configuration of $16$ disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of T. Shioda: suppose that Km$_{3}(A)$ is isomorphic to Km$_{3}(B)$, does that imply that $A$ and $B$ are isomorphic? We answer by the negative in general, by two methods: by a link between that problem and Fourier-Mukai partners of $A$, and by construction of $9{\bf A}_{2}$-configurations on Km$_{3}(A)$ which cannot be exchanged under the automorphism group.