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Global Brill–Noether theory over the Hurwitz space

Eric Larson, Hannah Larson and Isabel Vogt

Geometry & Topology 29 (2025) 193–257
Abstract

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps C 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 1.

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Keywords
Brill–Noether theory, Hurwitz space
Mathematical Subject Classification
Primary: 14H51
References
Publication
Received: 8 January 2023
Accepted: 25 February 2023
Published: 1 January 2025
Proposed: Jim Bryan
Seconded: Richard P Thomas, Mark Gross
Authors
Eric Larson
Department of Mathematics
Brown University
Providence, RI
United States
Hannah Larson
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Isabel Vogt
Department of Mathematics
Brown University
Providence, RI
United States