Global Brill–Noether theory over the Hurwitz space

Larson, Eric¹ ; Larson, Hannah² ; Vogt, Isabel¹

¹ Department of Mathematics, Brown University, Providence, RI, United States
² Department of Mathematics, University of California, Berkeley, Berkeley, CA, United States

Geometry & topology, Tome 29 (2025) no. 1, pp. 193-257.

Résumé

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 ${\mathbb{P}}^1$.

Détail

DOI : 10.2140/gt.2025.29.193

Keywords: Brill–Noether theory, Hurwitz space

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