(Hurwitz-)Brill-Noether general marked graphs via the Demazure product

TL;DR

This paper introduces a new proof technique linking permutations and Demazure products to analyze Brill-Noether properties of marked graphs, generalizing existing theorems and providing insights into their structure.

Contribution

It provides a novel, compact proof of Brill-Noether generality for chains of loops and extends results to graphs with two marked points using Demazure product and tropical matrix multiplication.

Findings

Chains of loops of torsion order 0 are Brill-Noether general.

Chains of loops of torsion order k are Hurwitz-Brill-Noether general.

Vertex gluing corresponds to Demazure product via tropical matrix multiplication.

Abstract

This paper gives a novel and compact proof that a metric graph consisting of a chain of loops of torsion order $0$ is Brill-Noether general (a theorem of Cools-Draisma-Payne-Robeva), and a finite or metric graph consisting of a chain of loops of torsion order $k$ is Hurwitz-Brill-Noether general in the sense of splitting loci (a theorem of Cook-Powell-Jensen). In fact, we prove a generalization to (metric) graphs with two marked points, that behaves well under vertex gluing. The key construction is a way to associate permutations to divisors on twice-marked graphs, simultaneously encoding the ranks of every twist of the divisor by the marked points. Vertex gluing corresponds to the Demazure product, which can be formulated via tropical matrix multiplication.