$q$-bic hypersurfaces and their Fano schemes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908324847419392 |
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| author | Cheng, Raymond |
| author_facet | Cheng, Raymond |
| contents | A $q$-bic hypersurface is a hypersurface in projective space of degree $q+1$, where $q$ is a power of the positive ground field characteristic, whose equation consists of monomials which are products of a $q$-power and a linear power; the Fermat hypersurface is an example. I identify $q$-bics as moduli spaces of isotropic vectors for an intrinsically defined bilinear form, and use this to study their Fano schemes of linear spaces. Amongst other things, I prove that the scheme of $m$-planes in a smooth $(2m+1)$-dimensional $q$-bic hypersurface is an $(m+1)$-dimensional smooth projective variety of general type which admits a purely inseparable covering by a complete intersection; I compute its Betti numbers by relating it to Deligne--Lusztig varieties for the finite unitary group; and I prove that its Albanese variety is purely inseparably isogenous via an Abel--Jacobi map to a certain conjectural intermediate Jacobian of the hypersurface. The case $m = 1$ may be viewed as an analogue of results of Clemens and Griffiths regarding cubic threefolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_06160 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $q$-bic hypersurfaces and their Fano schemes Cheng, Raymond Algebraic Geometry 14J70 (primary), 14N25, 14J10, 14G17, 14G10, 20C33 (secondary) A $q$-bic hypersurface is a hypersurface in projective space of degree $q+1$, where $q$ is a power of the positive ground field characteristic, whose equation consists of monomials which are products of a $q$-power and a linear power; the Fermat hypersurface is an example. I identify $q$-bics as moduli spaces of isotropic vectors for an intrinsically defined bilinear form, and use this to study their Fano schemes of linear spaces. Amongst other things, I prove that the scheme of $m$-planes in a smooth $(2m+1)$-dimensional $q$-bic hypersurface is an $(m+1)$-dimensional smooth projective variety of general type which admits a purely inseparable covering by a complete intersection; I compute its Betti numbers by relating it to Deligne--Lusztig varieties for the finite unitary group; and I prove that its Albanese variety is purely inseparably isogenous via an Abel--Jacobi map to a certain conjectural intermediate Jacobian of the hypersurface. The case $m = 1$ may be viewed as an analogue of results of Clemens and Griffiths regarding cubic threefolds. |
| title | $q$-bic hypersurfaces and their Fano schemes |
| topic | Algebraic Geometry 14J70 (primary), 14N25, 14J10, 14G17, 14G10, 20C33 (secondary) |
| url | https://arxiv.org/abs/2307.06160 |