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arXiv · 2008.10765

Global Brill--Noether Theory over the Hurwitz Space

Eric Larson·Hannah Larson·Isabel Vogt
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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 $\mathbb{P}^1$.

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Eric Larson, Hannah Larson, Isabel Vogt. 2020-08-25. Global Brill--Noether Theory over the Hurwitz Space. https://doi.org/10.2140/gt.2025.29.193

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