Special divisors on marked chains of cycles

TL;DR

This paper provides a complete description of Brill-Noether loci on metric graphs formed by chains of cycles, using displacement tableaux, and offers a tropical proof of the generalized Brill-Noether theorem for marked curves.

Contribution

It introduces a novel combinatorial framework using displacement tableaux to analyze Brill-Noether loci on chains of cycles, extending previous work to marked graphs and tropical geometry.

Findings

Complete classification of Brill-Noether loci on chain of cycles graphs.

Introduction of displacement tableaux as a tool for analysis.

Tropical proof of the generalized Brill-Noether theorem for marked curves.

Abstract

We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill-Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.