Level structures on cyclic covers of $\mathbb{P}^n$ and the homology of Fermat hypersurfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911455308152832 |
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| author | Looijenga, Eduard |
| author_facet | Looijenga, Eduard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Level structures on cyclic covers of $\mathbb{P}^n$ and the homology of Fermat hypersurfaces Looijenga, Eduard Algebraic Geometry 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. |
| title | Level structures on cyclic covers of $\mathbb{P}^n$ and the homology of Fermat hypersurfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2602.16599 |