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The strong maximal rank conjecture and moduli spaces of curves

Fu Liu, Brian Osserman, Montserrat Teixidor i Bigas, Naizhen Zhang

Abstract

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus $22$ and $23$ are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.

The strong maximal rank conjecture and moduli spaces of curves

Abstract

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus and . This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus and are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.

Paper Structure

This paper contains 8 sections, 42 theorems, 79 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Nondegeneracy calculations
  3. Background on limit linear series and linked linear series
  4. Degenerate linked linear series
  5. General setup
  6. An independence criterion
  7. The $r=6$ case
  8. Proofs in the degenerate case

Key Result

Theorem 1.2

In characteristic $0$, the loci ${\mathcal{D}}_{22}$ and ${\mathcal{D}}_{23}$ are proper subsets of ${\mathcal{M}}_{22}$ and ${\mathcal{M}}_{23}$ respectively.

Theorems & Definitions (95)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3: Aprodu-Farkas
  • Remark 1.4
  • Proposition 2.3
  • Lemma 2.4
  • proof : Proof of Proposition \ref{['prop:nondegenerate']}
  • Proposition 2.6
  • proof
  • Definition 3.1
  • ...and 85 more