Mathematics Colloquium — Special Lecture
Spring 2010
Josephine Yu
Mathematical Sciences Research Institute, Berkeley
Metric Graphs and Tropical Geometry
An abstract tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define linear equivalence and linear systems on a tropical curve analogously to the classical counterparts. We investigate the structure of the complete linear system |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 the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. This is based on joint work with Christian Haase and Gregg Musiker.