Mathematische Zeitschrift

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

Linear systems on tropical curves

  • Christian HaaseEmail author
  • Gregg Musiker
  • Josephine Yu


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.


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

Mathematics Subject Classification (2010)

Primary 14T05 Secondary 14H99 14C20 05C57 


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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

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