Computing Linear Systems on Metric Graphs

TL;DR

This paper introduces anchor divisors and cells to efficiently compute the structure of linear systems on metric graphs, including their extremal generators and cell complexes, with applications to small trivalent graphs.

Contribution

It presents a novel approach using anchor divisors to compute the cell complex structure and extremal generators of linear systems on metric graphs.

Findings

Successfully computed the f-vector of the cell complex.

Identified extremal generators of the linear systems.

Applied methods to canonical linear systems of small trivalent graphs.

Abstract

The linear system $|D|$ of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute its extremal generators using the landmarks. We apply these methods to some examples - namely the canonical linear systems of some small trivalent graphs.