Effective divisor classes on metric graphs

We introduce the notion of semibreak divisors on metric graphs 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). 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.