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Equations and Tropicalization of Enriques Surfaces

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Combinatorial Algebraic Geometry

Part of the book series: Fields Institute Communications ((FIC,volume 80))

Abstract

In this article, we explicitly compute equations of an Enriques surface via the involution on a K3 surface. We also discuss its tropicalization and compute the tropical homology, thus recovering a special case of the result of [18], and establish a connection between the dimension of the tropical homology groups and the Hodge numbers of the corresponding algebraic Enriques surface.

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Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute for Research in Mathematical Sciences. We thank Kristin Shaw for many helpful conversations and for suggesting Example 4.3. We thank Christian Liedtke for many useful remarks and suggesting Proposition 3.1. We thank Julie Rana for discussions and providing the sources for the introduction, and we thank Walter Gubler, Joseph Rabinoff and Annette Werner for sharing their insights. We also thank Bernd Sturmfels and the anonymous referees for providing many interesting suggestions and giving deep feedback. The first author was supported by the Fields Institute for Research in Mathematical Sciences; the second author was supported by the Fields Institute for Research in Mathematical Sciences, by the Clay Mathematics Institute, and by NSA award H98230-16-1-0016; and the third author was supported by the Polish National Science Center, project 2014/13/N/ST1/02640.

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Bolognese, B., Harris, C., Jelisiejew, J. (2017). Equations and Tropicalization of Enriques Surfaces. In: Smith, G., Sturmfels, B. (eds) Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7486-3_9

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