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On fake linear cycles inside Fermat varieties

Jorge Duque Franco and Roberto Villaflor Loyola

Vol. 17 (2023), No. 10, 1847–1865

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 2 + 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 n 2 , extending results of Green, Voisin, Otwinowska and Villaflor Loyola.

fake linear cycles, algebraic cycles, Hodge locus, Noether–Lefschetz locus, Galois cohomology, second-order invariant IVHS
Mathematical Subject Classification
Primary: 14C25, 14C30, 14D07
Received: 6 January 2022
Revised: 6 August 2022
Accepted: 28 November 2022
Published: 19 September 2023
Jorge Duque Franco
Departamento de Matemáticas
Universidad de Chile
Roberto Villaflor Loyola
Facultad de Matemáticas
Pontificia Universidad Católica de Chile

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