# Towards Brill--Noether theory for cuspidal curves

## TL;DR

This paper explores the extension of Brill--Noether theory to singular curves with cusps, analyzing how the geometry of smooth curves degenerates to local singularities using numerical semigroups.

## Contribution

It provides a foundational analysis of Brill--Noether theory for cuspidal singularities, connecting global curve properties with local semigroup invariants.

## Key findings

- Brill--Noether theory is extended to unibranch cuspidal curves.
- Semigroup invariants encode local singularity geometry.
- Results suggest specialization of smooth curve properties to cusps.

## Abstract

Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill--Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill--Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve $C_0$, one expects certain aspects of the global geometry of the smooth fibers to ``specialize" to the local geometry of the singularities of $C_0$. Making this expectation quantitatively precise involves analyzing the arithmetic and combinatorics of semigroups ${\rm S}$ attached to discrete valuations defined on (the local rings of) these singularities. In this largely-expository note we focus primarily on Brill--Noether-type results for curves with {\it cusps}, i.e., unibranch singularities; in this setting, the associated semigroups are {\it numerical} semigroups with finite complement in $\mathbb{N}$.