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

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