Noether-Lefschetz locus and a special case of the variational Hodge conjecture

Abstract: For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satisfies the variational Hodge conjecture. The interesting part is that we use techniques from commutative algebra to prove this statement (after certain identifications coming from Hodge theory). In the case $n=1$, we use similar methods to characterize all non-reduced components of the Noether-Lefschetz locus of codimension less than or equal to $3d$.