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

TL;DR

This paper explores Brill-Noether existence on graphs using polyhedral geometry and lattices, formulating conjectures for real-divisor analogues and providing approximate existence results along with bounds on graph gonality.

Contribution

It introduces geometric reformulations of Brill-Noether conjectures for real-divisors on graphs and supports a weak covering radius conjecture, leading to approximate existence results.

Findings

Formulated Brill-Noether conjectures for $ ext{R}$-divisors on graphs.

Established a weak version of the covering radius conjecture.

Derived upper bounds on graph gonality and its real-divisor analogue.

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.