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

TL;DR

This paper explores the relationship between level structures on the primitive cohomology of certain cyclic covers of projective space and their hypersurfaces, with applications to cubic surfaces and threefolds.

Contribution

It establishes a connection between level structures on covers and their hypersurfaces, providing new insights into the geometry of Fermat hypersurfaces and related varieties.

Findings

Relates level structures on covers to those on hypersurfaces.

Provides a method to determine level structures on cubic threefolds from cubic surfaces.

Answers a question by Beauville regarding cubic surfaces and threefolds.

Abstract

Let $Z'\subset \mathbb{P}^{n}$ be a smooth projective hypersurface of degree $d>1$ and let $Z\to \mathbb{P}^n$ be the $\mu_d$-cover totally ramified along $Z'$. We relate full level $d$ structures on the primitive cohomology $Z'$ 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.