The Locus of Brill-Noether General Graphs is Not Dense

TL;DR

This paper presents an example of a 3-connected, trivalent graph that remains Brill-Noether special regardless of how its metric is chosen, challenging assumptions about the density of Brill-Noether general graphs.

Contribution

It introduces a specific graph example demonstrating that Brill-Noether generality is not dense among metric graphs.

Findings

A trivalent, 3-connected graph exists with all metrics Brill-Noether special.

Brill-Noether general graphs are not dense in the space of metric graphs.

Abstract

We provide an example of a trivalent, 3-connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.