Mathematische Zeitschrift

, Volume 270, Issue 3–4, pp 1111–1140 | Cite as

Linear systems on tropical curves

Article

Abstract

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to deg(D) when |D| is base point free. The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a \({\mathbb{Q}}\) -tropical curve is a direct limit of critical groups of finite graphs converging to the curve.

Keywords

Tropical curves Divisors Linear systems Canonical embedding Chip-firing games Tropical convexity 

Mathematics Subject Classification (2010)

Primary 14T05 Secondary 14H99 14C20 05C57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ardila F., Klivans C.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96, 38–49 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baker M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Baker M., Faber X.: Metrized graphs, laplacian operators, and electrical networks. quantum graphs and their applications. Contemp. Math., Am. Math. Soc., Providence, RI 415, 15–33 (2006)MathSciNetGoogle Scholar
  4. 4.
    Baker, M., Faber, X.: Metric properties of the tropical Abel-jacobi Map. eprint arXiv:0905.1679 (2009)Google Scholar
  5. 5.
    Baker M., Norine S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Benjamin, A., Quinn, J.: Proofs that really count. The art of combinatorial proof. In: The Dolciani Mathematical Expositions, vol. 27. Mathematical Association of America, Washington, DC (2003)Google Scholar
  7. 7.
    Biggs N.: Chip-firing and the critical group of a graph. J. Algebraic Combin. 9(1), 25–45 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill-Noether Theorem. Preprint (2010). arXiv:1001.2774.Google Scholar
  9. 9.
    Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27, erratum 205–206 (2004)Google Scholar
  11. 11.
    Gathmann A., Kerber M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York, Heidelberg-Berlin (1977), Corr. 3rd printing (1983)Google Scholar
  13. 13.
    Holroyd, A., Levine, L., Meszaros, K., Peres, Y., Propp, J., Wilson, D.: Chip-firing and rotor-routing on directed graphs. In: Sidoravicius, V., Eullia Vares, M. (eds.) In and Out of Equilibrium 2, Progress in Probability, vol. 60, pp. 331–364. Birkhuser, Switzerland (2008)Google Scholar
  14. 14.
    Mikhalkin G., Zharkov I.: Tropical curves, their Jacobians and theta functions. Curves and abelian varieties. Contemp. Math., Am. Math. Soc., Providence, RI. 465, 203–230 (2008)MathSciNetGoogle Scholar
  15. 15.
    Richter-Gebert J., Sturmfels B., Theobald T.: First steps in tropical geometry. Idempotent mathematics and mathematical physics. Contemp. Math., Am. Math. Soc., Providence, RI. 377, 289–317 (2005)MathSciNetGoogle Scholar
  16. 16.
    Speyer D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Goethe-Universität, Institute of MathematicsFrankfurt/MainGermany
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations