Effective divisor classes on metric graphs

TL;DR

This paper introduces semibreak divisors on metric graphs, proves their existence for effective divisor classes, and develops algorithms and properties analogous to classical Riemann surface results.

Contribution

It generalizes break divisors to semibreak divisors, provides an efficient computation method, and establishes foundational properties of effective loci in metric graph Picard groups.

Findings

Effective loci are pure-dimensional polyhedral sets.

Generic divisor classes have rank zero.

The Abel-Jacobi map is birational onto its image.

Abstract

We introduce the notion of semibreak divisors on metric graphs (tropical curves) and prove that every effective divisor class (of degree at most the genus) has a semibreak divisor representative. This appropriately generalizes the notion of break divisors (in degree equal to genus). Our method of proof is new, even for the special case of break divisors. We provide an algorithm to efficiently compute such semibreak representatives. Semibreak divisors provide the tool to establish some basic properties of effective loci inside Picard groups of metric graphs. We prove that effective loci are pure-dimensional polyhedral sets. We also prove that a `generic' divisor class (in degree at most the genus) has rank zero, and that the Abel-Jacobi map is `birational' onto its image. These are analogues of classical results for Riemann surfaces.