Degrees of Hodge Loci

TL;DR

This paper establishes asymptotic estimates for the growth of degrees of Hodge loci, linking geometric properties to arithmetic data, with applications to Noether-Lefschetz loci and Torelli locus subvarieties.

Contribution

It provides general asymptotic formulas for Hodge locus degrees applicable to various Hodge structures, extending understanding of their arithmetic and geometric behavior.

Findings

Asymptotic formulas for Hodge locus degrees

Applications to Noether-Lefschetz loci in hypersurfaces

Results on degrees of Torelli locus subvarieties

Abstract

We prove asymptotic estimates for the growth in the degree of the Hodge locus in terms of arithmetic properties of the integral vectors that define it. Our methods are general and apply to most variations of Hodge structures for which the Hodge locus is dense. As applications we give asymptotic formulas controlling the degrees of Noether-Lefschetz loci associated to smooth projective hypersurfaces in $\mathbb{P}^3$, and the degrees of subvarieties of the Torelli locus parameterizing Jacobians split up to isogeny.