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Linear systems on tropical curves

Mathematische Zeitschrift volume 270, pages 1111–1140 (2012)Cite this 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.

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Authors and Affiliations

  1. Goethe-Universität, Institute of Mathematics, Frankfurt/Main, Germany

    Christian Haase

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Gregg Musiker

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Josephine Yu

Authors
  1. Christian Haase
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  2. Gregg Musiker
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  3. Josephine Yu
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Correspondence to Christian Haase.

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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

Keywords

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

Mathematics Subject Classification (2010)

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