Maximal Variation in the Moduli of Curves

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.

💡 Research Summary

The paper introduces a precise notion of “maximal variation” for families of projective curves, including singular ones, by exploiting the conductor ideal of the normalization. For a reduced curve C with normalization ν: \tilde C → C, the conductor c ⊂ 𝒪_C measures the failure of normality and determines exactly which principal parts of meromorphic differentials on \tilde C descend to sections of the dualizing sheaf ω_C. The authors define conductor‑level balancing as the imposition of precisely those linear constraints dictated by c and no others. When these constraints are the only ones, the space H⁰(C, ω_C) has dimension equal to the arithmetic genus g, just as for a smooth curve; this situation is called maximal variation.

The first major result (Theorem 1.1) shows that for a flat, proper family π: X → S of reduced projective curves, the function s ↦ dim H⁰(X_s, ω_{X_s}) is upper‑semicontinuous, and the conductor‑level constraints vary flatly in the family. Consequently maximal variation is an open condition in any flat family, yielding a canonical open substack 𝓜^{mv}{g,n} ⊂ 𝓜{g,n}. All smooth curves and all nodal stable curves lie in this locus; for nodal curves the conductor‑level balancing coincides with the classical residue condition, so no extra constraints appear.

When the curve possesses singularities worse than nodes, the conductor alone may not capture all obstructions. The authors model the failure of maximal variation as a determinantal degeneracy locus. Locally one considers the vector space V of all admissible principal parts and the subspace W ⊂ V annihilated by the conductor. The difference dim V − dim W measures the “extra” linear relations. Globally this gives a morphism of vector bundles φ: 𝔙 → 𝔚 over the base; the locus where φ drops rank is closed, its expected codimension being the sum of local contributions. This provides a concrete, computable description of where maximal variation fails.

A striking classification follows: Theorem 1.2 and 1.3 prove that a reduced curve singularity contributes to the degeneracy locus if and only if it is non‑Gorenstein. All Gorenstein singularities satisfy maximal variation under conductor‑level balancing; every non‑Gorenstein singularity yields a non‑trivial contribution, and contributions add under disjoint unions. Hence a reduced projective curve lies outside the maximal‑variation locus precisely when it contains at least one non‑Gorenstein point, and the codimension of the degeneracy locus equals the sum of the local non‑Gorenstein defects.

The paper also contains a residue‑theoretic result (Theorem 1.4): for a connected Cohen–Macaulay curve C with δ singular points, if δ ≥ g then the δ residue functionals span the full dual space H⁰(C, ω_C)^{∨}. This links conductor‑level balancing to classical residue theory.

Finally, the authors discuss Hodge‑theoretic consequences. In a degeneration from a smooth curve to a singular one, each non‑Gorenstein defect contributes extra (n,0)‑classes to the limiting mixed Hodge structure on cohomology. These classes do not extend holomorphically across the central fiber, reflecting the same rigidity detected by the conductor‑level analysis.

Overall, the work provides a unified, deformation‑theoretically natural framework: the conductor ideal gives the exact linear algebra governing canonical differentials; maximal variation is an open, semicontinuous condition; failure is precisely captured by a determinantal locus whose geometry is dictated by non‑Gorenstein singularities. This yields clear criteria for the behavior of canonical differentials in moduli, informs compactifications of strata of abelian differentials, and connects singularity theory with Hodge theory.