Infinitesimal Variations of Hodge Structure for Singular Curves I

TL;DR

This paper investigates how the infinitesimal variation of Hodge structure behaves for families of singular algebraic curves, providing criteria for maximal variation and extending classical results to more general singularities and ambient surfaces.

Contribution

It generalizes the understanding of infinitesimal Hodge variation to curves with various singularities and on different surfaces, beyond the classical nodal case.

Findings

Maximal variation criteria depend on the number of nodes and genus.

Nodal singularities contribute nontrivially to the variation.

Higher ADE singularities may not affect the infinitesimal variation at first order.

Abstract

We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves lying on smooth projective surfaces as well as families with more general isolated singularities. Using deformation-theoretic and residue-theoretic methods, we describe how the infinitesimal period map decomposes into local contributions supported at singular points, together with global constraints arising from the geometry of the normalization. While nodal singularities give rise to nontrivial rank-one contributions, other singularities may contribute only through higher-order local data or may be invisible at the infinitesimal level. As a consequence, we obtain sharp criteria for maximal infinitesimal variation in terms of numerical invariants of the curve, notably when the number of nodes satisfies the inequality $\delta \ge g$, where $g$ denotes the genus of the normalization. We extend these results to curves on very general surfaces in projective three-space, showing that maximal variation persists on Picard-rank-one surfaces but fails for sufficiently large genus in the presence of higher ADE singularities. These results extend classical maximality phenomena in infinitesimal Hodge theory to a broader singular and geometric setting.