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Linear Systems on Tropical Curves

Abstract : A tropical curve $\Gamma$ 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 $\Gamma$ 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 $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex.
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Christian Haase, Gregg Musiker, Josephine Yu. Linear Systems on Tropical Curves. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.295-306. ⟨hal-01186275⟩

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