Theta Characteristics of Tropical K4-Curves

  • Melody ChanEmail author
  • Pakawut Jiradilok
Part of the Fields Institute Communications book series (FIC, volume 80)


A K4-curve is a smooth proper curve X of genus 3 over a field with valuation whose Berkovich skeleton Γ is a complete graph on four vertices. The curve X has 28 effective theta characteristics—the 28 bitangents to a canonical embedding—while Γ has exactly seven effective tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a K4-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.

MSC 2010 codes:

14T05 (primary) 14C20 14H45 14H50 (secondary) 



We are grateful to M. Baker, Y. Len, R. Morrison, N. Pflueger, and Q. Ren for generously sharing their ideas on tropical plane quartics developed in the paper [3]. Thanks also to M. Baker, Y. Len, and B. Sturmfels for helpful comments on an earlier version of this paper, and J. Rabinoff for helpful references. We heartily thank M. Manjunath, M. Panizzut, and two anonymous referees for extensive and insightful comments on a previous version of this paper. We also thank W. Stein and SageMathCloud for providing computational resources. MC was supported by NSF DMS-1204278. PJ was supported by the Harvard College Research Program during the summer of 2014.


  1. 1.
    Matthew Baker: Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008) 613–653.Google Scholar
  2. 2.
    Matthew Baker and David Jensen: Degeneration of linear series from the tropical point of view and applications, in Nonarchimedean and Tropical Geometry, 365–433, Simons Symposia, Springer International Publishing, 2016.Google Scholar
  3. 3.
    Matthew Baker, Yoav Len, Ralph Morrison, Nathan Pflueger, and Qingchun Ren: Bitangents of tropical plane quartic curves, Math. Z. 282 (2016) 1017–1031.Google Scholar
  4. 4.
    Matthew Baker and Serguei Norine: Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math. 215 (2007) 766–788.Google Scholar
  5. 5.
    Matthew Baker, Sam Payne, and Joseph Rabinoff: On the structure of non-archimedean analytic curves, in Tropical and non-Archimedean geometry, 93–121, Contemp. Math. 605, Centre Rech. Math. Proc., American Mathematical Society, Providence, RI, 2013.Google Scholar
  6. 6.
    Matt Baker and Joseph Rabinoff: The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves, Int. Math. Res. Not. IMRN (2015) 7436–7472.Google Scholar
  7. 7.
    Vladimir Berkovich: Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, American Mathematical Society, Providence, RI, 1990.Google Scholar
  8. 8.
    Vladimir Berkovich: Smooth p-adic analytic spaces are locally contractible, Invent. Math. 137 (1999) 1–84.Google Scholar
  9. 9.
    Barbara Bolognese, Madeline Brandt, and Lynn Chua: From curves to tropical Jacobians and back, in Combinatorial Algebraic Geometry, 21–45, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  10. 10.
    Siegfried Bosch and Werner Lütkebohmert: Stable reduction and uniformization of abelian varieties I, Math. Ann. 270 (1985) 349–379.Google Scholar
  11. 11.
    Sarah Brodsky, Michael Joswig, Ralph Morrison, and Bernd Sturmfels: Moduli of tropical plane curves, Res. Math. Sci. 2 (2015), Art. 4, 31pp.Google Scholar
  12. 12.
    Lucia Caporaso and Edoardo Sernesi: Recovering plane curves from their bitangents, J. Algebraic Geom. 12 (2003) 225–244.Google Scholar
  13. 13.
    Melody Chan: Tropical hyperelliptic curves, J. Algebraic Combin. 37 (2013) 331–359.Google Scholar
  14. 14.
    Melody Chan and Bernd Sturmfels: Elliptic curves in honeycomb form, in Algebraic and combinatorial aspects of tropical geometry, 87–107, Contemp. Math. 589, American Mathematical Society, Providence, RI, 2013.Google Scholar
  15. 15.
    Igor Dolgachev: Classical algebraic geometry: a modern view, Cambridge University Press, Cambridge, 2012.Google Scholar
  16. 16.
    Andreas Gathmann and Michael Kerber: A Riemann-Roch theorem in tropical geometry, Math. Z. 259 (2008) 217–230.Google Scholar
  17. 17.
    Corey Harris and Yoav Len: Tritangent planes to space sextics: the algebraic and tropical stories, in Combinatorial Algebraic Geometry, 47–63, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.Google Scholar
  18. 18.
    Daniel R. Grayson and Michael E. Stillman: Macaulay2, a software system for research in algebraic geometry, available at
  19. 19.
    Walter Gubler: A guide to tropicalizations, in Algebraic and combinatorial aspects of tropical geometry, 125–189, Contemp. Math. 589, American Mathematical Society, Providence, RI, 2013.Google Scholar
  20. 20.
    David Jensen and Yoav Len: Tropicalization of theta characteristics, double covers, and Prym varieties, arXiv:1606.02282 [math.AG].Google Scholar
  21. 21.
    David Jensen and Sam Payne: Tropical independence I: Shapes of divisors and a proof of the Gieseker-Petri Theorem, Algebra Number Theory 8 (2014) 2043–2066.Google Scholar
  22. 22.
    Eric Katz, Joseph Rabinoff, and David Zureick-Brown: Uniform bounds for the number of rational points on curves of small Mordell-Weil rank, Duke Math. J. 165 (2016) 3189–3240.Google Scholar
  23. 23.
    Allen Knutson and Terence Tao: The honeycomb model of \(\mathop{\mathrm{GL}}\nolimits _{n}(\mathbb{C})\) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999) 1055–1090.Google Scholar
  24. 24.
    David Lehavi: Any smooth plane quartic can be reconstructed from its bitangents, Israel J. Math. 146 (2005) 371–379.Google Scholar
  25. 25.
    Diane Maclagan and Bernd Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, RI, 2015.Google Scholar
  26. 26.
    Grigory Mikhalkin: Enumerative tropical algebraic geometry in \(\mathbb{R}^{2}\), J. Amer. Math. Soc. 18 (2005) 313–377.Google Scholar
  27. 27.
    Grigory Mikhalkin and Ilia Zharkov: Tropical curves, their Jacobians and theta functions, in Curves and abelian varieties, 203–230, Contemp. Math. 465, American Mathematical Society, Providence, RI, 2008.Google Scholar
  28. 28.
    Brian Osserman and Sam Payne: Lifting tropical intersections, Doc. Math. 18 (2013) 121–175.Google Scholar
  29. 29.
    Brian Osserman and Joseph Rabinoff: Lifting non-proper tropical intersections, in Tropical and Nonarchimedean geometry, 15–44, Contemp. Math. 605, American Mathematical Society, Providence, RI, 2013.Google Scholar
  30. 30.
    Marta Panizzut: Theta characteristics of hyperelliptic graphs, Arch. Math. (Basel) 106 (2016) 445–455.Google Scholar
  31. 31.
    David Speyer: Horn’s problem, Vinnikov curves, and the hive cone, Duke Math. J. 127 (2005) 395–427.Google Scholar
  32. 32.
    Amaury Thuillier: Théorie du potentiel sur les courbes en géométrie analytique non archimédienne, Applications à la théorie d’Arakelov, PhD thesis, Université Rennes, 2005.Google Scholar
  33. 33.
    Ilia Zharkov: Tropical theta characteristics, in Mirror symmetry and tropical geometry, 165–168, Contemp. Math. 527, American Mathematical Society, Providence, RI, 2010.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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