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arXiv · 1609.02091

A note on Brill--Noether existence for graphs of low genus

Stanislav Atanasov·Dhruv Ranganathan
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Abstract

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence conjecture for special divisors. For $g\leq 5$ and $\rho(g,r,d)$ non-negative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$. Moreover, the conjecture is shown to hold in rank $1$ for a number of families of highly connected combinatorial types of graphs of arbitrarily high genus. 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.

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Stanislav Atanasov, Dhruv Ranganathan. 2016-09-07. A note on Brill--Noether existence for graphs of low genus. https://arxiv.org/abs/1609.02091

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