Algebraic and combinatorial Brill-Noether theory

TL;DR

This paper explores the relationship between algebraic and combinatorial Brill-Noether theories, establishing conditions for the non-emptiness of Brill-Noether varieties for graphs and smooth curves using a refined specialization technique.

Contribution

It demonstrates the equivalence of non-emptiness conditions for Brill-Noether varieties in graphs and curves, extending Baker's lemma with a new refinement.

Findings

Non-empty Brill-Noether variety for graphs when the Brill-Noether number is non-negative

Existence of a graph with empty Brill-Noether variety implies the same for general curves

Refinement of Baker's Specialization Lemma as a key tool

Abstract

The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve. The main tool is a refinement of Baker's Specialization Lemma.