Equations and Tropicalization of Enriques Surfaces

  • Barbara Bolognese
  • Corey Harris
  • Joachim Jelisiejew
Part of the Fields Institute Communications book series (FIC, volume 80)


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.

MSC 2010 codes:

14T05 14J28 14N10 



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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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Barbara Bolognese
    • 1
  • Corey Harris
    • 2
  • Joachim Jelisiejew
    • 3
  1. 1.School of Mathematics and Statistics, University of SheffieldSheffieldUK
  2. 2.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.Institute of Mathematics, University of WarsawWarszawaPoland

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