  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.
