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Equations and Tropicalization of Enriques Surfaces

  • Barbara Bolognese
  • Corey Harris
  • Joachim Jelisiejew
Chapter
Part of the Fields Institute Communications book series (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.

MSC 2010 codes:

14T05 14J28 14N10 

Notes

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.

References

  1. 1.
    Wolf P. Barth and Michael E. Larsen: On the homotopy groups of complex projective algebraic manifolds, Math. Scand. 30 (1972) 88–94.Google Scholar
  2. 2.
    Wolf P. Barth, Chris A.M. Peters, and Antonius Van de Ven: Compact complex surfaces, A Series of Modern Surveys in Mathematics 4, Springer-Verlag, Berlin, 2004.Google Scholar
  3. 3.
    Arnaud Beauville: Complex algebraic surfaces, Second edition, London Mathematical Society Student Texts 34. Cambridge University Press, Cambridge, 1996.Google Scholar
  4. 4.
    Mauro C. Beltrametti and Andrew J. Sommese: The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics 16, Walter de Gruyter & Co., Berlin, 1995.Google Scholar
  5. 5.
    Vladimir G. Berkovich: Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33. American Mathematical Society, Providence, RI, 1990.Google Scholar
  6. 6.
    Morgan Brown and Tyler Foster: Rational connectivity and analytic contractibility, arXiv:1406.7312 [math.AG].Google Scholar
  7. 7.
    Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, and Kristin Shaw: Brief introduction to tropical geometry, in Proceedings of the Gökova Geometry-Topology Conference 2014, 1–75, Gökova Geometry/Topology Conference (GGT), Gökova, 2015.Google Scholar
  8. 8.
    C. Herbert Clemens: Degeneration of Kähler manifolds. Duke Math. J. 44 (1977) 215–290.Google Scholar
  9. 9.
    François Cossec: Projective models of Enriques surfaces, Math. Ann. 265 (1983) 283–334.Google Scholar
  10. 10.
    François R. Cossec and Igor V. Dolgachev: Enriques Surfaces I, Progress in Mathematics 76, Birkhäuser Boston, Inc., Boston, MA, 1989.Google Scholar
  11. 11.
    Igor V. Dolgachev: A brief introduction to Enriques surfaces, in Development of moduli theory - Kyoto 2013, 1–32, Adv. Stud. Pure Math. 69, Math. Soc. Japan, Tokyo, 2016.Google Scholar
  12. 12.
    Federigo Enriques: Introduzione alla geometria sopra le superficie algebriche, Mem. Soc Ital. delle Scienze 10 (1896) 1–81.Google Scholar
  13. 13.
    Gino Fano: Nuovo ricerche sulle congruenze di retta del 3 ordine, Mem. Acad. Sci. Torino 50 (1901) 1–79, www.bdim.eu/item?id=GM_Fano_1901_1.Google Scholar
  14. 14.
    William Fulton: Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993.Google Scholar
  15. 15.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.Google Scholar
  16. 16.
    Walter Gubler, Joseph Rabinoff, and Annette Werner: Tropical skeletonsm arXiv:1508.01179 [math.AG].Google Scholar
  17. 17.
    Walter Gubler, Joseph Rabinoff, and Annette Werner: Skeletons and tropicalizations, Adv. Math. 294 (2016) 150–215.Google Scholar
  18. 18.
    Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov: Tropical homology, arXiv:1604.01838 [math.AG].Google Scholar
  19. 19.
    Philipp Jell, Kristin Shaw, and Jascha Smacka: Superforms, tropical cohomology and Poincaré duality, arXiv:1512.07409 [math.AG].Google Scholar
  20. 20.
    Lars Kastner, Kristin Shaw, and Anna-Lena Winz: Computing sheaf cohomology in polymake, in Combinatorial Algebraic Geometry, 369–385, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  21. 21.
    Christian Liedtke: Arithmetic moduli and lifting of Enriques surfaces, J. Reine Angew. Math. 706 (2015) 35–65.Google Scholar
  22. 22.
    Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.Google Scholar
  23. 23.
    David Mumford: Varieties defined by quadratic equations, in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), 29–100, Edizioni Cremonese, Rome, 1970.Google Scholar
  24. 24.
    Sam Payne: Analytification is the limit of all tropicalizations, Math. Res. Lett. 16 (2009) 543–556.Google Scholar
  25. 25.
    Theodor Reye: Die Geometrie Der Lage, volume 2. Hannover, Carl Rümpler, 1880, available at www.archive.org/details/geoderlagevon02reyerich.Google Scholar
  26. 26.
    Hal Schenck: Computational algebraic geometry, London Mathematical Society Student Texts 58, Cambridge University Press, Cambridge, 2003.Google Scholar
  27. 27.
    Jessica Sidman and Gregory G. Smith: Linear determinantal equations for all projective schemes, Algebra Number Theory 5 (2011) 1041–1061.Google Scholar
  28. 28.
    Bernd Sturmfels: Fitness, apprenticeship, and polynomials, in Combinatorial Algebraic Geometry, 1–19, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  29. 29.
    Ravi Vakil: The rising sea: Foundations of algebraic geometry, available at math.stanford.edu/~vakil/216blog/.Google Scholar
  30. 30.
    Alessandro Verra: The étale double covering of an Enriques surface, Rend. Sem. Mat. Univ. Politec. Torino 41 (1983) 131–167.Google Scholar
  31. 31.
    Magnus Dehli Vigeland: Topics in elementary tropical geometry. PhD thesis, Universitetet i Oslo, 2008, available at folk.uio.no/ranestad/mdvavhandling.pdf.

Copyright information

© 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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