Maximal Variation in the Moduli of Curves

TL;DR

This work introduces maximal variation for canonical differentials on singular curves via conductor-level balancing on the normalization, and shows that MV is open and deformation-invariant in flat families. The authors prove that the complement of MV is a determinantal degeneracy locus whose expected codimension equals the sum of local non-Gorenstein defects, thereby classifying degeneracy entirely by non-Gorenstein singularities. They give a complete local-to-global picture: smooth and nodal curves satisfy MV, while any non-Gorenstein singularity yields extra descent constraints, with explicit codimension contributions. The paper also connects these degeneracy phenomena to Hodge theory, period maps, and moduli-theoretic compactifications, and provides detailed explicit computations for non-descent of $\frac{dt}{t}$ in non-Gorenstein monomial curves. Overall, the framework offers an intrinsic, deformation-theoretic classification of degeneracy in spaces of canonical differentials and clarifies how singularities shape the geometry of moduli spaces.

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.

Theorems & Definitions (143)

• Theorem 1.1: Semicontinuity of conductor-level differentials
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Scheme-theoretic residue span
• Theorem 3.1: Semicontinuity of conductor-level differentials
• proof
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof