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). The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. © 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.