Discrete Mathematics & Theoretical Computer Science |

- 1 Institute of Mathematics [Berlin]
- 2 Department of Mathematics [MIT]
- 3 School of Mathematics - Georgia Institute of Technology

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.

Source: HAL:hal-01186275v1

Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

Section: Proceedings

Published on: January 1, 2010

Imported on: January 31, 2017

Keywords: tropical curves,divisors,linear systems,canonical embedding,chip-firing games,tropical convexity,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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