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On the Shafarevich conjecture for Enriques surfaces

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Enriques surfaces are minimal surfaces of Kodaira dimension 0 with \(b_{2}=10\). If we work with a field of characteristic away from 2, Enriques surfaces admit double covers which are K3 surfaces. In this paper, we prove the Shafarevich conjecture for Enriques surfaces by reducing the problem to the case of K3 surfaces. In our formulation of the Shafarevich conjecture, we use the notion “admitting a cohomological good K3 cover”, which includes not only good reduction but also flower pot reduction.

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Acknowledgements

The author is deeply grateful to his advisor Naoki Imai for his deep encouragement and helpful advice. The author also thanks Tetsushi Ito for the proof of Proposition 2.5, Yohsuke Matsuzawa for helpful suggestions, and Yuya Matsumoto for pointing out my misunderstandings about cohomologies of K3 double covers (Remark 3.3). Moreover, the author thanks the referee for many helpful comments.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Teppei Takamatsu

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Takamatsu, T. On the Shafarevich conjecture for Enriques surfaces. Math. Z. 298, 489–495 (2021). https://doi.org/10.1007/s00209-020-02623-4

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