On fake linear cycles inside Fermat varieties

TL;DR

This paper introduces and characterizes a new class of Hodge cycles called fake linear cycles in Fermat varieties, revealing their existence only for degrees 3, 4, and 6, and exploring their algebraic and geometric properties.

Contribution

It identifies and characterizes fake linear cycles in Fermat varieties, showing their unique properties and their role in the structure of Hodge loci for specific degrees.

Findings

Fake linear cycles exist only for degrees 3, 4, and 6.

They have maximal Zariski tangent space dimension, contradicting a conjecture.

Minimal codimension components of Hodge loci relate to hypersurfaces containing linear subvarieties.

Abstract

We introduce a new class of Hodge cycles with non-reduced associated Hodge loci, we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees $d=3,4,6$, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension, contrary to a conjecture of Movasati. Moreover, they provide examples of algebraic cycles not generated by their periods in the sense of Movasati-Sert\"oz. To study them we compute their Galois action in cohomology and their second-order invariant of the IVHS. We conclude that for any degree $d\ge 2+\frac{6}{n}$, the minimal codimension component of the Hodge locus passing through the Fermat variety is the one parametrizing hypersurfaces containing linear subvarieties of dimension $\frac{n}{2}$, extending results of Green, Voisin, Otwinowska and the second author.