2024/08/22 by Hannah Larson, Larson, Hannah, Sameera Vemulapalli +1
Engineering · Mathematics · #14H51 (Primary) 14H10 (Secondary) #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2408.12678
openalex publication_date 2024/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → ℙ1. 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(Δ), where Δ⊂ 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.