Residues and Infinitesimal Torelli for Equisingular Curves

TL;DR

This paper develops a residue-based approach to study infinitesimal Torelli problems for families of curves, including singular and equisingular cases, demonstrating that maximal Hodge structure variation persists under strong geometric constraints.

Contribution

It introduces a uniform residue calculus framework for infinitesimal Torelli theorems applicable to singular, equisingular, and constrained curves, extending Green--Voisin philosophy.

Findings

Proves infinitesimal Torelli for high-degree equisingular plane curves.

Constructs IVHS exact sequences for curves on threefolds.

Shows maximal Hodge variation persists under strong geometric conditions.

Abstract

We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincar\'e residue calculus, allowing a uniform treatment of smooth, singular, and equisingular situations. In particular, we prove infinitesimal Torelli theorems for general equisingular plane curves of sufficiently high degree and construct relative IVHS exact sequences for curves lying on smooth projective threefolds. Our results show that maximal infinitesimal variation of Hodge structure persists even after imposing strong extrinsic conditions, such as fixed degree and prescribed singularities, and in the presence of isolated planar singularities. The methods presented here provide a concrete and geometric realization of Jacobian-type constructions and extend the Green--Voisin philosophy to singular and equisingular settings and provide a unified residue--theoretic framework for Torelli--type problems across dimensions and codimensions.