The nonabelian Brill–Noether divisor on ℳ13 and the Kodaira dimension of ℛ13

Farkas, Gavril¹ ; Jensen, David² ; Payne, Sam³

¹ Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany
² Department of Mathematics, University of Kentucky, Lexington, KY, United States
³ Department of Mathematics, University of Texas at Austin, Austin, TX, United States

Geometry & topology, Tome 28 (2024) no. 2, pp. 803-866.

Résumé

We highlight several novel aspects of the moduli space of curves of genus $13$, the first genus $g$ where phenomena related to $K3$ surfaces no longer govern the birational geometry of ${\overline{\mathcal{ℳ}}}_g$. We compute the class of the nonabelian Brill–Noether divisor on ${\overline{\mathcal{ℳ}}}_{13}$ of curves that have a stable rank-two vector bundle with canonical determinant and many sections. This provides the first example of an effective divisor on ${\overline{\mathcal{ℳ}}}_g$ with slope less than $6+10∕g$. Earlier work on the slope conjecture suggested that such divisors may not exist. The main geometric application of our result is a proof that the Prym moduli space ${\overline{\mathcal{ℛ}}}_{13}$ is of general type. Among other things, we also prove the Bertram–Feinberg–Mukai and the strong maximal rank conjectures on ${\overline{\mathcal{ℳ}}}_{13}$.

Détail

DOI : 10.2140/gt.2024.28.803

Keywords: strong maximal rank conjecture, genus 13, Mukai–Petri divisor, Prym moduli space

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