$q$-bic hypersurfaces and their Fano schemes

TL;DR

This paper studies $q$-bic hypersurfaces in positive characteristic, revealing their structure as moduli spaces, analyzing their Fano schemes, and establishing connections with Deligne--Lusztig varieties and intermediate Jacobians.

Contribution

It identifies $q$-bics as moduli spaces of isotropic vectors, analyzes their Fano schemes, and relates their invariants to finite group varieties and conjectural Jacobians.

Findings

Fano scheme of $m$-planes is a smooth projective variety of general type.

Betti numbers are computed via Deligne--Lusztig varieties.

Albanese variety is purely inseparably isogenous to a conjectural intermediate Jacobian.

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.