Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces

TL;DR

This paper advances the understanding of line bundle stratification on smooth curves in the plane and Hirzebruch surfaces, providing dimension and smoothness results for specific stratifications inspired by algebraic number theory.

Contribution

It introduces a refined stratification of line bundles on these curves based on splitting types, extending Brill--Noether theory with explicit dimension and smoothness results.

Findings

Determines dimensions of stratified line bundle spaces.

Proves smoothness of strata in characteristic zero.

Provides a detailed stratification framework inspired by Wood's parameterization.

Abstract

In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of their pushforward along projection maps $C \to \mathbb{P}^1$. Inspired by Wood's parameterization of ideal classes in rings associated to binary forms, we further refine the stratification of line bundles $L$ on $C$ by fixing the splitting types of both $L$ and $L(\Delta)$, where $\Delta \subset C$ is the intersection of $C$ with the directrix of the Hirzebruch surface. Our main theorem determines the dimensions of these locally closed strata and, if the characteristic of the ground field is zero, proves that they are smooth.