Counting maps from curves to projective space via graph theory

TL;DR

This survey explores the connections between Brill--Noether theory on algebraic curves and metric graphs, discussing recent results and conjectures about linear series on specialized graph classes.

Contribution

It synthesizes existing research on Brill--Noether theory for curves and metric graphs, highlighting new conjectures about linear series on decomposable metric graphs.

Findings

Summarizes key results in Brill--Noether theory for curves and graphs

Proposes conjectures on linear series behavior on certain metric graphs

Connects algebraic and combinatorial perspectives in the field

Abstract

In this (mostly) survey article, we give a synopsis of a number of results relating to Brill--Noether theory on curves and metric graphs, together with some speculations about the behavior of one-dimensional linear series on a class of metric graphs that admit decompositions as triples of trees rooted on a common vertex set.