We introduce a new class of Hodge cycles with nonreduced 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. They provide examples of algebraic cycles not generated by their periods in the sense of Movasati and Sertöz (2021). 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 Villaflor Loyola.

我们引入了一类新的霍奇循环，其具有非还原相关霍奇基因座；我们称它们为假线性循环。我们对所有费马变体进行了表征，并表明它们仅存在于度数上$d=3,4,6$，其中霍奇循环的空间中有无穷多个。这些循环是病态的，因为它们相关的霍奇轨迹的扎里斯基切线空间具有最大维度，这与莫瓦萨蒂的猜想相反。它们提供了 Movasati 和 Sertöz (2021) 意义上的代数循环的例子，这些循环不是由其周期生成的。为了研究它们，我们计算它们的上同调伽罗瓦作用和 IVHS 的二阶不变量。我们的结论是，对于任何学位$d\ge 2+\frac{6}{n}$，穿过费马簇的霍奇轨迹的最小余维分量是包含维数线性子簇的一个参数化超曲面$\frac{n}{2}$，扩展了 Green、Voisin、Otwinowska 和 Villaflor Loyola 的结果。