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The nonabelian Brill–Noether divisor on $\overline{\mathcal{M}}_{13}$ and the Kodaira dimension of $\overline{\mathcal{R}}_{13}$

Gavril Farkas, David Jensen and Sam Payne

Geometry & Topology 28 (2024) 803–866
Abstract

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 ¯g. We compute the class of the nonabelian Brill–Noether divisor on ¯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  ¯g with slope less than 6 + 10g. 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 ¯13 is of general type. Among other things, we also prove the Bertram–Feinberg–Mukai and the strong maximal rank conjectures on ¯13.

Keywords
strong maximal rank conjecture, genus 13, Mukai–Petri divisor, Prym moduli space
Mathematical Subject Classification
Primary: 14H10
Secondary: 14T20
References
Publication
Received: 31 October 2021
Revised: 17 February 2022
Accepted: 1 July 2022
Published: 13 March 2024
Proposed: Dan Abramovich
Seconded: Marc Levine, Mark Gross
Authors
Gavril Farkas
Institut für Mathematik
Humboldt-Universität zu Berlin
Berlin
Germany
David Jensen
Department of Mathematics
University of Kentucky
Lexington, KY
United States
Sam Payne
Department of Mathematics
University of Texas at Austin
Austin, TX
United States

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