We introduce a new class of Hodge cycles with expected highly non-reduced associated Hodge loci, we call them fake linear cycles. We characterize them for Fermat varieties and show that there are infinitely many in the space of Hodge cycles of Fermat varieties of degree d=3,4,6. These cycles are pathological in the sense that their associated Artin Gorenstein ideals are complete intersections with the same Hilbert function as the Artin Gorenstein ideal of a linear cycle. In particular, the Zariski tangent space of their associated Hodge locus is of maximal dimension, so they provide counterexamples to a conjecture of Movasati and complete the missing cases of [Vil21]. Our main tool to characterize them is the explicit description of the Galois action of the cyclotomic field of 2d-th roots of unities on the set of totally decomposable Hodge monomials in the sense of Shioda. This is a work in progress joint with Jorge Duque.

[Vil21] R. Villaflor. "Small codimension components of the Hodge locus containing the Fermat variety". Communications in Contemporary Mathematics, 2021.