Extending the Torelli map to alternative compactifications of the moduli space of curves

TL;DR

This paper introduces a new compactification of the moduli space of curves with axis-like singularities, extending the Torelli map where previous compactifications failed, and analyzes its properties.

Contribution

It constructs an alternative compactification including axis-like singularities and proves the Torelli map extends over it, expanding understanding of degenerations.

Findings

Torelli map extends over the new compactification with axis-like singularities.

Identifies an axis-like locus where the Torelli map extends for Smyth's compactifications.

Provides a framework for studying degenerations of Jacobians in new singularity contexts.

Abstract

Determining the limiting behaviour of the Jacobian as the underlying curve degenerates has been the subject of much interest. For nodal singularities, there are beautiful constructions of Caporaso as well as Pandharipande of compactified universal Jacobians over the moduli space of stable curves. Alexeev later obtained a canonical such compactification by extending the Torelli map out of the Deligne-Mumford compactification of $\mathcal{M}_{g,n}$. In contrast, Alexeev and Brunyate proved that the Torelli map does not extend over the cuspidal locus in Schubert's alternative compactification of pseudostable curves. In this paper, we consider curves with singularities that locally look like the axes in $m$-space, which we call axis-like singularities. We construct an alternative compactification of $\mathcal{M}_{g,n}$ consisting of curves with such singularities and prove that the Torelli map extends out of this compactification. Furthermore, for every alternative compactification in the sense of Smyth, we identify an axis-like locus over which the Torelli map extends.