Constructions of Kummer structures on generalized Kummer surfaces

Abstract

We study generalized Kummer surfaces ${\mathrm{Km}}_3(A)$, 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 nine disjoint configurations of two smooth rational curves C, $C^{\prime }$ with $C{C}^{\prime }=1$. This $9{\mathbf{A}}_2$-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 Shioda: Suppose that ${\mathrm{Km}}_3(A)$ is isomorphic to ${\mathrm{Km}}_3(B)$. Does that imply that A and B are isomorphic? We answer by the negative in general with two methods: a link between that problem and Fourier–Mukai partners of A, and construction of $9{\mathbf{A}}_2$-configurations on ${\mathrm{Km}}_3(A)$ that cannot be exchanged under the automorphism group.