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arXiv · 2307.06160

$q$-bic hypersurfaces and their Fano schemes

Raymond Cheng
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Abstract

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.

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Raymond Cheng. 2023-07-12. $q$-bic hypersurfaces and their Fano schemes. https://doi.org/10.4310/pamq.250402030943

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