Constructions of Kummer structures on generalized Kummer surfaces

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.