Skip to main content

Linear systems on tropical curves


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.

This is a preview of subscription content, access via your institution.


  1. Ardila F., Klivans C.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96, 38–49 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  2. Baker M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  4. Baker, M., Faber, X.: Metric properties of the tropical Abel-jacobi Map. eprint arXiv:0905.1679 (2009)

  5. Baker M., Norine S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  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)

  7. Biggs N.: Chip-firing and the critical group of a graph. J. Algebraic Combin. 9(1), 25–45 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  8. Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill-Noether Theorem. Preprint (2010). arXiv:1001.2774.

  9. Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  10. Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27, erratum 205–206 (2004)

  11. Gathmann A., Kerber M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  12. Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York, Heidelberg-Berlin (1977), Corr. 3rd printing (1983)

  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)

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  16. Speyer D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Christian Haase.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Haase, C., Musiker, G. & Yu, J. Linear systems on tropical curves. Math. Z. 270, 1111–1140 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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

Mathematics Subject Classification (2010)

  • Primary 14T05
  • Secondary 14H99
  • 14C20
  • 05C57