## The Michigan Mathematical Journal

### 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 $g\leq5$ 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.

#### Article information

Source
Michigan Math. J., Volume 67, Issue 1 (2018), 175-198.

Dates
Revised: 3 November 2016
First available in Project Euclid: 20 February 2018

https://projecteuclid.org/euclid.mmj/1519095622

Digital Object Identifier
doi:10.1307/mmj/1519095622

Mathematical Reviews number (MathSciNet)
MR3770859

Zentralblatt MATH identifier
06965595

#### Citation

Atanasov, Stanislav; Ranganathan, Dhruv. A Note on Brill–Noether Existence for Graphs of Low Genus. Michigan Math. J. 67 (2018), no. 1, 175--198. doi:10.1307/mmj/1519095622. https://projecteuclid.org/euclid.mmj/1519095622

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