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Supersingular K3 surfaces are unirational

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Abstract

We show that supersingular K3 surfaces in characteristic \(p\ge 5\) are related by purely inseparable isogenies. This implies that they are unirational, which proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct, we exhibit the moduli space of rigidified K3 crystals as an iterated \({{\mathbb P}}^1\)-bundle over \({{\mathbb F}}_{p^2}\). To complete the picture, we also establish Shioda–Inose type isogeny theorems for K3 surfaces with Picard rank \(\rho \ge 19\) in positive characteristic.

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Acknowledgments

It is a pleasure for me to thank Xi Chen, Igor Dolgachev, Gerard van der Geer, Brendan Hassett, Daniel Huybrechts, Toshiyuki Katsura, Frans Oort, Matthias Schütt, Tetsuji Shioda, Burt Totaro for discussions, comments, and pointing out inaccuracies. I especially thank Olivier Benoist and Max Lieblich for pointing out mistakes in earlier versions of Sect. 3, and for helping me to fix them. Finally, I thank the referee for careful proof-reading, for pointing out inaccuracies, and for helping me to improve the whole exposition. I gratefully acknowledge funding from DFG via Transregio SFB 45, as part of this article was written while staying at Bonn university.

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  1. TU München, Zentrum Mathematik, M11, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Christian Liedtke

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Liedtke, C. Supersingular K3 surfaces are unirational. Invent. math. 200, 979–1014 (2015). https://doi.org/10.1007/s00222-014-0547-7

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