Infinitesimal Variations of Hodge Structure for Singular Curves I
Mounir Nisse
TL;DR
This work extends infinitesimal variations of Hodge structure to families of singular curves, showing that the IVHS decomposes into local contributions supported at singular points, with nodes contributing a rank-one residue term and cusps and higher ADE singularities contributing only higher-order data. It provides sharp maximal-variation criteria, notably that a nodal cuspidal plane curve with δ ≥ g attains maximal IVHS rank g^2, while ADE singularities beyond A2 obstruct maximality for large genus. The analysis blends deformation theory of the normalization, residue calculus, and adjunction, and extends the results from plane curves to curves on very general surfaces, using Cayley–Bacharach type properties to control adjoint linear systems and residue span. The findings illuminate why nodal singularities drive maximal IVHS and how more complicated singularities impose intrinsic linear constraints, connecting to Noether–Lefschetz phenomena and equisingular families with broad implications for Hodge theoretic Torelli questions in singular settings.
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 $δ\ge g$, 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.