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Linear systems on tropical curves

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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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