# Tropical linear series and matroids

## TL;DR

This paper introduces a new concept of tropical linear series on metric graphs, linking algebraic geometry, combinatorics, and matroid theory, and characterizes their properties and realizability.

## Contribution

It defines tropical linear series combining key properties of algebraic tropicalizations and relates them to matroid and valuated matroid geometry, providing new characterizations and examples.

## Key findings

- Bergman fan of any matroid appears as a local fan of a tropical linear series
- Characterization of when tropicalization of canonical linear series matches realizable tropical canonical divisors
- Establishment of connections between tropical linear series and matroid combinatorics

## Abstract

We study a notion of tropical linear series on metric graphs that combines two essential properties of tropicalizations of linear series on algebraic curves: the Baker-Norine rank and the independence rank. Our main results relate the local and global geometry of these tropical linear series to the combinatorial geometry of matroids and valuated matroids, respectively. As an application, we characterize exactly when the tropicalization of the canonical linear series on a single curve is equal to the locus of realizable tropical canonical divisors determined by M\"oller, Ulirsch, and Werner. We also illustrate our results with a wealth of examples; in particular, we show that the Bergman fan of every matroid appears as the local fan of a tropical linear series on a metric graph. The paper concludes with a list of ten open questions for future investigation.