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Construction of Nikulin configurations on some Kummer surfaces and applications

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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.

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Acknowledgements

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

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Authors and Affiliations

  1. Aix-Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France

    Xavier Roulleau

  2. Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de Poitiers, Boulevard Marie et Pierre Curie, Téléport 2, Futuroscope, 86962, Chasseneuil, France

    Alessandra Sarti

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  1. Xavier Roulleau
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Correspondence to Xavier Roulleau.

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Communicated by Vasudevan Srinivas.

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Roulleau, X., Sarti, A. Construction of Nikulin configurations on some Kummer surfaces and applications. Math. Ann. 373, 597–623 (2019). https://doi.org/10.1007/s00208-018-1717-5

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  • DOI: https://doi.org/10.1007/s00208-018-1717-5

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