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Mathematische Annalen

, Volume 373, Issue 1–2, pp 597–623 | Cite as

Construction of Nikulin configurations on some Kummer surfaces and applications

  • Xavier RoulleauEmail author
  • Alessandra Sarti
Article
  • 65 Downloads

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.

Mathematics Subject Classification

Primary: 14J28 Secondary: 14J50 14J29 14J10 

Notes

Acknowledgements

The authors thank the anonymous referee for useful remarks improving the exposition of the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aix-Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373MarseilleFrance
  2. 2.Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de PoitiersChasseneuilFrance

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