The Brill-Noether rank of a tropical curve

TL;DR

This paper extends the concept of Brill-Noether rank to tropical curves, constructing a space of divisor classes, proving upper semicontinuity of the rank function, and relating tropical and algebraic Brill-Noether properties.

Contribution

It introduces a new definition of Brill-Noether rank for tropical curves and proves its upper semicontinuity in families, linking tropical and algebraic Brill-Noether theory.

Findings

Constructed a space classifying divisor classes on tropical curves.

Proved the rank function is upper semicontinuous.

Established a specialization lemma relating tropical and algebraic Brill-Noether ranks.

Abstract

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill-Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill-Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether locus of an algerbaic curve.