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arXiv · 1911.11514

Brill-Noether Existence on Graphs via $\mathbb{R}$-Divisors, Polytopes and Lattices

Madhusudan Manjunath
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Abstract

We study Brill-Noether existence on a finite graph using methods from polyhedral geometry and lattices. We start by formulating analogues of the Brill-Noether conjectures (both the existence and non-existence parts) for $\mathbb{R}$-divisors, i.e. divisors with real coefficients, on a graph. We then reformulate the Brill-Noether existence conjecture for $\mathbb{R}$-divisors on a graph in geometric terms, that we refer to as the covering radius conjecture and we show a weak version, in support of it. Using this, we show an approximate version of the Brill-Noether existence conjecture for divisors on a graph. As applications, we derive upper bounds on the gonality of a graph and its $\mathbb{R}$-divisor analogue.

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BibTeXRIS

Madhusudan Manjunath. 2019-11-26. Brill-Noether Existence on Graphs via $\mathbb{R}$-Divisors, Polytopes and Lattices. https://doi.org/10.1007/s00029-021-00728-0

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