On a conjecture on Hodge loci of linear combinations of linear subvarieties

Abstract

For each $k\ge 5$ we give a counterexample to a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in ${\mathbf{P}}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-3$. We give similar examples for Hodge loci of cubic hypersurfaces in ${\mathbf{P}}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$ and for quartic hypersurfaces in ${\mathbf{P}}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$. Moreover, we present new evidence for Movasati's conjecture for the values of $k$ for which our type of counterexamples cannot exist, i.e., for $k=3,4$.

v2: Minor corrections and improved presentation