Title: Small codimension components of the Hodge locus and periods of the Fermat variety
Speaker: Roberto Villaflor (IMPA)
Date: 10:30 12/06/2020
The study of the components of the Noether-Lefschetz locus (corresponding to surfaces in $\mathbb{P}^3$ with Picard number bigger than 1) was leaded by Green, Harris and Voisin during the 80's and beginning of the 90's. One of the central questions answered by Voisin and independently by Green was to show that the smallest codimension component corresponds to the locus of surfaces containing a line. This result relies on the development of the infinitesimal variations of Hodge structures and the study of the Hilbert function of a special Artinian Gorenstein ideal associated to each algebraic cycle. The problem in higher dimension was considered in 2002 by Otwinowska. Performing a fine analysis of the Hilbert function of the ideal associated to a Hodge cycle, she showed that the smallest codimension component of the Hodge locus of hypersurfaces of $\mathbb{P}^{n+1}$ of even dimension $n$ and degree $d$ corresponds to the locus of hypersurfaces containing a linear subvariety of dimension $\frac{n}{2}$ for $d>>n$. In this talk we will explain how we can get Otwinowska's result for small degrees analyzing the periods of the corresponding Hodge cycle. We are able to do this at the Fermat variety for degree $d\neq 3,4,6$.