Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci

TL;DR

This paper classifies components of the Hodge locus in parameter spaces of smooth projective varieties using determinantal varieties derived from IVHS, providing new insights into the structure and minimal codimension of these components.

Contribution

It introduces a classification method for Hodge locus components via determinantal varieties and establishes minimal codimension results for hypersurfaces, extending known theorems.

Findings

Minimum codimension achieved by hypersurfaces passing through a linear space

Classification of Hodge locus components using IVHS and determinantal varieties

Implication for the Harris-Voisin conjecture under computational hypotheses

Abstract

We give a classification of components of the Hodge locus in any parameter space of smooth projective varieties. This is done using determinantal varieties constructed from the infinitesimal variation of Hodge structures (IVHS) of the underlying family. As a corollary we prove that the minimum codimension for the components of the Hodge locus in the parameter space of $m$-dimensional hypersurfaces of degree $d$ with $d\geq 2+\frac{4}{m}$ and in a Zariski neighborhood of the point representing the Fermat variety, is obtained by the locus of hypersurfaces passing through an $\frac{m}{2}$-dimensional linear projective space. In the particular case of surfaces in the projective space of dimension three, this is a theorem of Green and Voisin. In this case our classification under a computational hypothesis on IVHS implies a weaker version of the Harris-Voisin conjecture which says that the set of special components of the Noether-Lefschetz locus is not Zariski dense in the parameter space.