Level structures on cyclic covers of ${\mathbb{P}}^n$ and the homology of Fermat hypersurfaces

Let $Z^{\mathrm{\prime }}\subset {\mathbb{P}}^n$ be a smooth projective hypersurface of degree $d>1$ and let $Z\to {\mathbb{P}}^n$ be the ${\mu }_d$-cover totally ramified along $Z^{\mathrm{\prime }}$. We relate full level $d$ structures on the primitive cohomology $Z^{\mathrm{\prime }}$ with full level $d$ structures on the primitive cohomology of $Z$. In the special case, $d=n=3$ this makes a marking of a smooth cubic surface determine a level $3$-structure on the associated cubic threefold, thereby answering a question by Beauville. We expect many more such applications.