Maximal Variation in the Moduli of Curves

TL;DR

This paper introduces the concept of maximal variation in families of projective curves, analyzing how degenerations and singularities, especially non-Gorenstein ones, affect the behavior of canonical differentials in moduli spaces.

Contribution

It defines maximal variation via conductor-level balancing, proves stability results, and characterizes degeneracy loci governed by non-Gorenstein singularities.

Findings

Maximal variation holds for all smooth and nodal curves.

Degeneracy loci are characterized by non-Gorenstein singularities.

Expected codimension and structure of degeneracy loci are computed.

Abstract

We introduce and study the maximal-variation locus in families and moduli spaces of projective curves, defined via conductor-level balancing of meromorphic differentials on the normalization. This notion captures precisely when the space of canonical differentials behaves with the expected dimension under degeneration. We prove semicontinuity and openness results showing that maximal variation is stable in flat families, identify a natural determinantal degeneracy locus where maximal variation fails, and establish that this failure is governed entirely by the presence of non-Gorenstein singularities. In particular, all smooth and nodal curves satisfy maximal variation, while every non-Gorenstein singularity contributes explicitly and additively to degeneracy. We compute the expected codimension of degeneracy loci, describe their closure and adjacency relations in moduli, and explain how non-Gorenstein defects give rise to additional Hodge-theoretic phenomena in degenerations. This framework provides a uniform, intrinsic, and deformation-theoretically meaningful classification of degeneracy in spaces of canonical differentials.