  In this paper, we study the zero loci of local systems of the form
$\delta\Pi$, where $\Pi$ is the period sheaf of the universal family of CY
hypersurfaces in a suitable ambient space $X$, and $\delta$ is a given
differential operator on the space of sections $V^\vee=\Gamma(X,K_X^{-1})$.
Using earlier results of three of the authors and their collaborators, we give
several different descriptions of the zero locus of $\delta\Pi$. As
applications, we prove that the locus is algebraic and in some cases,
non-empty. We also give an explicit way to compute the polynomial defining
equations of the locus in some cases. This description gives rise to a natural
stratification to the zero locus.
