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

Gavril Farkas, Dave Jensen, Sam Payne

Abstract

The paper is devoted to highlighting 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 M_g. We compute the class of the non-abelian Brill-Noether divisor on M_13 of curves that have a stable rank 2 vector bundle with many sections. This provides the first example of an effective divisor on 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 of genus 13 is of general type. Among other things, we also prove the Bertram-Feinberg-Mukai and the Strong Maximal Rank Conjectures on M_13

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

Abstract

The paper is devoted to highlighting 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 M_g. We compute the class of the non-abelian Brill-Noether divisor on M_13 of curves that have a stable rank 2 vector bundle with many sections. This provides the first example of an effective divisor on 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 of genus 13 is of general type. Among other things, we also prove the Bertram-Feinberg-Mukai and the Strong Maximal Rank Conjectures on M_13

Paper Structure

This paper contains 30 sections, 53 theorems, 231 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Kodaira dimension of the Prym moduli space $\overline{\mathcal{R}}_{13}$.
  3. The Strong Maximal Rank Conjecture on $\overline{\mathcal{M}}_{13}$.
  4. The divisor $\mathfrak{D}_{13}$ and rank two Brill-Noether loci.
  5. The failure locus of the Strong Maximal Rank Conjecture on $\widetilde{\mathcal{M}}_{13}$
  6. Stacks of limit linear series.
  7. A degeneracy locus inside $\widetilde{\mathfrak G}^5_{16}$.
  8. Test curves in $\widetilde{\mathcal{M}}_{13}$.
  9. Top Chern numbers on Jacobians.
  10. The class of the virtual divisor $\widetilde{\mathfrak{D}}_{13}$
  11. The Strong Maximal Rank Conjecture in genus $13$
  12. The unramified vertex avoiding case
  13. No switching loops
  14. Switching loops
  15. Step 1
  16. ...and 15 more sections

Key Result

Theorem 1.1

A general curve $X$ of genus $13$ carries precisely three stable vector bundles $E\in SU_X(2,\omega, 8)$. The closure in $\overline{\mathcal{M}}_{13}$ of the non-abelian Brill-Noether divisor on $\mathcal{M}_{13}$ has slope equal to

Figures (4)

  • Figure 1: The chain of loops $\Gamma$.
  • Figure 2: The slopes $s_k$ and $s'_k$.
  • Figure 3: The tableau corresponding to the divisor $D$.
  • Figure 4: The divisor $D' = 2D + \operatorname{div} (\theta)$. The function $\varphi_{ij}$ achieves the minimum uniquely on the region labeled $ij$ in $\Gamma \smallsetminus \mathrm{Supp}(D')$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Corollary 2.3
  • ...and 109 more