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

Eduard Looijenga

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 $μ_d$-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.

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

Abstract

Let be a smooth projective hypersurface of degree and let be the -cover totally ramified along . We relate full level structures on the primitive cohomology with full level structures on the primitive cohomology of . In the special case, this makes a marking of a smooth cubic surface determine a level -structure on the associated cubic threefold, thereby answering a question by Beauville. We expect many more such applications.
Paper Structure (4 sections, 8 theorems, 27 equations)

This paper contains 4 sections, 8 theorems, 27 equations.

Table of Contents

  1. Classical Lefschetz theory with integral coefficients
  2. A pair of maps associated with a module over a group ring
  3. Cyclic covers
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Lemma 1.1

Let $Z'\subset Z$ be smooth hyperplane section and put $\mathring Z:=Z\smallsetminus Z'$. Then $\mathring Z$ is a smooth affine hypersurface which has the homotopy type of a wedge of $(d-1)^{n+1}$$n$-spheres and we have a short exact sequences fitting in the (self-dual) commutative diagram of which the vertical maps are the natural ones (defining the intersection pairing on $\tilde{H}_n(\mathring

Theorems & Definitions (19)

  • Lemma 1.1
  • proof
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • ...and 9 more