Speaker: Remke Kloosterman 
Date: 10:30 19/01/2024
Title: Hodge loci of linear combinations of linear subvarieties

Abstract: In the first part of the talk we will give for each $k \geq 5$ we give a counterexample to 
a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in $\Ps^{2k+1}$ containing 
two $k$-planes intersecting in dimension $k-3$. We give similar examples for  Hodge loci of cubic hypersurfaces 
in $\Ps^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$ and for quartic hypersurfaces in 
$\Ps^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$. Moreover, we show that the Hodge locus 
is reducible at the point corresponding to the Fermat hypersurface, which explains some of the evidence for 
Movasati's conjecture.

In the second part we will also discuss the case where the planes intersect in codimension 1. 
We will show that our method suggest that the Hodge locus contains an embedded component containing the 
Fermat point. This explain some results recently obtained by Duque and Villaflor.