# Brill-Noether theory for curves of a fixed gonality

## TL;DR

This paper generalizes the Brill-Noether theorem for curves with fixed gonality, providing a formula for the dimension of special divisor varieties and introducing logarithmic stable maps as a new tool in the field.

## Contribution

It proves a conjecture relating tropical divisor theory to algebraic curves, and develops new regeneration theorems using logarithmic stable maps in Brill-Noether theory.

## Key findings

- Proves the conjecture that tropical upper bounds are achieved on general curves.
- Introduces a new realizability theorem for tropical stable maps in obstructed geometries.
- Provides a new derivation of lifting for special divisors on chains of cycles.

## Abstract

We prove a generalization of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve $C$ of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$. We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill--Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realizability theorem for tropical stable maps in obstructed geometries, generalizing a well-known theorem of Speyer on genus one curves to arbitrary genus.