Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds

TL;DR

This paper investigates the density of Noether-Lefschetz loci for surfaces within families on Fano and Calabi-Yau threefolds, applying advanced Hodge theory to understand their geometric distribution.

Contribution

It extends the understanding of Noether-Lefschetz loci by applying general Hodge locus results to specific threefold families, highlighting differences in their geometric behavior.

Findings

Density results for Noether-Lefschetz loci in Fano threefolds

Density results for Noether-Lefschetz loci in Calabi-Yau threefolds

Distinct behaviors of general and exceptional components

Abstract

In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of surfaces. In particular, we consider the family of surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold $Y$, in case $Y$ is a Fano or a Calabi-Yau threefold, and we discuss the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.