A Note on Brill–Noether Existence for Graphs of Low Genus

Abstract

In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For $g\le 5$ and $\rho (g,r,d)$ nonnegative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$. As further evidence, the conjecture is shown to hold in rank $1$ for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence theorem for metric graphs, giving a partial answer to a related question of Baker.