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.
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References
Ardila F., Klivans C.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96, 38–49 (2006)
Baker M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)
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)
Baker, M., Faber, X.: Metric properties of the tropical Abel-jacobi Map. eprint arXiv:0905.1679 (2009)
Baker M., Norine S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)
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)
Biggs N.: Chip-firing and the critical group of a graph. J. Algebraic Combin. 9(1), 25–45 (1999)
Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill-Noether Theorem. Preprint (2010). arXiv:1001.2774.
Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)
Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27, erratum 205–206 (2004)
Gathmann A., Kerber M.: A Riemann-Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)
Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, vol. 52. Springer, New York, Heidelberg-Berlin (1977), Corr. 3rd printing (1983)
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)
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)
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)
Speyer D.E.: Tropical linear spaces. SIAM J. Discrete Math. 22(4), 1527–1558 (2008)
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Haase, C., Musiker, G. & Yu, J. Linear systems on tropical curves. Math. Z. 270, 1111–1140 (2012). https://doi.org/10.1007/s00209-011-0844-4
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DOI: https://doi.org/10.1007/s00209-011-0844-4