The strong maximal rank conjecture and moduli spaces of curves

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

doi:10.2140/ant.2024.18.1403

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The strong maximal rank conjecture and moduli spaces of curves

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

Vol. 18 (2024), No. 8, 1403–1464

DOI: 10.2140/ant.2024.18.1403

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 Osserman.

Keywords

maximal rank, moduli space, algebraic curve, general type

Mathematical Subject Classification 2010

Primary: 14H10

Secondary: 14H51, 14D06

Milestones

Received: 8 October 2019

Revised: 19 August 2022

Accepted: 11 September 2023

Published: 18 September 2024

Authors

Fu Liu
Department of Mathematics University of California, Davis Davis, CA United States

Brian Osserman
Department of Mathematics University of California, Davis Davis, CA United States

Montserrat Teixidor i Bigas
Department of Mathematics Tufts University Medford, MA United States

Naizhen Zhang
Fairleigh Dickinson University Vancouver Campus Vancouver, BC Canada

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Vol. 18, No. 8, 2024