Construction of Nikulin configurations on some Kummer surfaces and applications

Abstract

A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $$\mathcal {C}$$ , then X is a Kummer surface $$X=\mathrm{Km}(B)$$ where B is an Abelian surface determined by $$\mathcal {C}$$ . Let B be a generic Abelian surface having a polarization M with $$M^{2}=k(k+1)$$ (for $$k>0$$ an integer) and let $$X=\mathrm{Km}(B)$$ be the associated Kummer surface. To the natural Nikulin configuration $$\mathcal {C}$$ on $$X=\mathrm{Km}(B)$$ , we associate another Nikulin configuration $$\mathcal {C}'$$ ; we denote by $$B'$$ the Abelian surface associated to $$\mathcal {C}'$$ , so that we have also $$X=\mathrm{Km}(B')$$ . For $$k\ge 2$$ we prove that B and $$B'$$ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface X that occurs naturally from our situation. Associated to the two Nikulin configurations $$\mathcal {C},$$ $$\mathcal {C}'$$ , there exists a natural bi-double cover $$S\rightarrow X$$ , which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $$k=2$$ is a Schoen surface.