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Kummer surfaces associated with group schemes

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Abstract

We introduce Kummer surfaces \(X={\text {Km}}(C\times C)\) with the group scheme \(G=\mu _2\) acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type \(A_1\), together with a rational double point of type \(D_4\). We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant \(\sigma \le 3\), and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces.

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Acknowledgements

We wish to thank the referee for valuable comments and pointing out inaccuracies. The research of the first author is partially supported by Grant-in-Aid for Scientific Research (S) No. 15H05738. The research of the second author is conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the DFG.

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Shigeyuki Kondō

  2. Mathematisches Institut, Heinrich-Heine-Universität, 40204, Düsseldorf, Germany

    Stefan Schröer

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  1. Shigeyuki Kondō
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  2. Stefan Schröer
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Correspondence to Stefan Schröer.

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Kondō, S., Schröer, S. Kummer surfaces associated with group schemes. manuscripta math. 166, 323–342 (2021). https://doi.org/10.1007/s00229-020-01257-4

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