The strong maximal rank conjecture and moduli spaces of curves

TL;DR

This paper proves two cases of the strong maximal rank conjecture for genus 22 and 23, advancing the understanding of the moduli spaces of curves and their classification as general type.

Contribution

It introduces new degeneration techniques combining limit linear series and linked linear series to verify cases of the conjecture.

Findings

Proved the strong maximal rank conjecture for genus 22 and 23

Established that moduli spaces of these curves are of general type

Developed new methods using degenerations to chains of elliptic 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 $22$ 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.