Global Brill--Noether Theory over the Hurwitz Space
Eric Larson, Hannah Larson, Isabel Vogt
TL;DR
The paper develops a global Brill--Noether theory for curves with fixed gonality by degenerating to a chain of elliptic curves and proving a regeneration theorem that transports central-fiber combinatorics to the general fiber. It identifies $\vec{e}$-positive limit line bundles with fillings of a $k$-staircase diagram $\Gamma(\vec{e})$, linking components to reduced words in the affine symmetric group, and proves that $W^{\vec{e}}(C)$ has normal, Cohen--Macaulay, and irreducible structure when the expected dimension condition $\rho'=g-u(\vec{e})\ge 0$ holds. The authors establish connectedness for $g>u(\vec{e})$, prove Cohen--Macaulayness and reducedness of splitting loci, and obtain a regeneration-based dimension formula via a detailed analysis of limit linear series and their monodromy. As a corollary, they verify a conjecture of Eisenbud and Schreyer on versal deformation spaces, and illuminate the monodromy structure of the splitting loci through explicit two-point ramification analyses on open Hurwitz strata. Overall, the work provides a comprehensive, dimensionally sharp picture of Brill--Noether theory for curves of fixed gonality and connects geometric degeneration with affine Weyl combinatorics.
Abstract
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on $\mathbb{P}^1$.