Maximal Infinitesimal Variation of Hodge Structure for Singular Curves

TL;DR

This paper extends the classical theory of infinitesimal variation of Hodge structure to singular algebraic curves, analyzing how singularities affect the maximality and decomposition of the variation in both equisingular and non-equisingular families.

Contribution

It introduces a singular analogue of maximal infinitesimal variation of Hodge structure and characterizes its behavior in families of singular and non-planar curves, linking it to invariants like the $oldsymbol{ extdelta}$-invariant.

Findings

Maximal infinitesimal variation is achieved for equisingular families with planar Gorenstein singularities.

The rank decomposes into normalization and singularity contributions, with the latter measured by $oldsymbol{ extdelta}$-invariants.

In non-equisingular degenerations, the rank defect relates to vanishing cycles and mixed Hodge structures.

Abstract

We study the infinitesimal variation of Hodge structure for families of algebraic curves and extend the classical theory from smooth curves to singular and non--planar settings. Using the deformation space $\mathrm{Ext}^1(\Omega_X,\mathcal O_X)$ and the dualizing sheaf, we define a singular analogue of maximal infinitesimal variation. For equisingular families of plane curves with planar Gorenstein singularities, we prove that the infinitesimal variation attains maximal rank equal to the arithmetic genus. We show that the rank decomposes into a geometric contribution from the normalization and a singular contribution measured by the $\delta$--invariants. For non--equisingular degenerations, the rank defect equals the drop of the total $\delta$--invariant and admits an interpretation in terms of vanishing cycles and mixed Hodge structures. We further extend the results to non--planar curves under suitable Petri and deformation conditions.