##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Gregg Musiker

#### MIT \\ Department of Mathematics

## Linear Systems on Tropical Curves

##### Abstract:

A tropical curve 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 analogously to the classical counterpart. Due to work of Baker and Norine, there is a rank function $r(D)$ on such linear systems, as well a canonical divisor $K$. Completely analogous to the classical case, this rank function satisfies Riemann-Roch and analogues of Riemann-Hurwitz. After an introduction to these tropical analogues, this talk will describe joint work with Josephine Yu and Christian Haase investigating the structure of $|D|$ as a cell complex. We 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 the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as an embedded polyhedral complex.

Host: Jeff Remmel

### November 24, 2009

### 3:00 PM

### AP&M 7321

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