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

2024/05/08 by Changho Han, Han, Changho, Jesse Leo Kass +3
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2405.05199

openalex publication_date 2024/05/08 · openalex created_date 2024/05/11 · openalex updated_date 2026/07/28

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 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 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.

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