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The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$

Gavril Farkas, David Jensen, Sam Payne

Abstract

We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to linear series of rank 6 with quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors.

The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$

Abstract

We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to linear series of rank 6 with quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors.

Paper Structure

This paper contains 94 sections, 128 theorems, 345 equations, 40 figures.

Table of Contents

  1. Introduction
  2. Divisors of small slope
  3. Strong Maximal Rank Conjecture
  4. Effectivity of the virtual divisor
  5. Tropical independence
  6. Tropicalization of rational functions
  7. Tropicalization of linear relations
  8. A characterization of tropical independence
  9. From tropical independence to diagonal specialization
  10. Chains of loops
  11. Further constructions of virtual divisors of small slope
  12. Preliminaries
  13. Slopes of divisors
  14. Test curves and intersection numbers
  15. A vanishing condition for effective divisors
  16. ...and 79 more sections

Key Result

Theorem 1.1

The moduli spaces $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$ are of general type.

Figures (40)

  • Figure 1: The functions $\psi_0$ and $\psi_1$, when the $(0,2)$ region is nonempty.
  • Figure 2: The functions $\psi_A$, $\psi_B$, and $\psi_C$ when the $(0,2)$ region is empty.
  • Figure 3: The tropical dependence that shows $t=t'$.
  • Figure 4: The chain of loops $\Gamma$.
  • Figure 5: The slopes $s_k$ and $s'_k$.
  • ...and 35 more figures

Theorems & Definitions (323)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 313 more