Hyperelliptic graphs and metrized complexes

TL;DR

This paper extends classical theorems like Clifford's and Martens' to the setting of metrized complexes and hyperelliptic graphs, providing new structural insights and classifications in tropical geometry.

Contribution

It proves a Clifford's theorem analogue for metrized complexes, classifies hyperelliptic metrized complexes, and discusses a tropical Martens' theorem.

Findings

Metrized complexes with certain divisors also have simpler divisors of degree 2 and rank 1.

Structural theorem for hyperelliptic metrized complexes established.

A tropical version of Martens' theorem for metric graphs is discussed.

Abstract

We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0<r<g-1$) also carries a divisor of degree $2$ and rank $1$. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens' theorem for metric graphs.