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$q$-bic hypersurfaces and their Fano schemes

Raymond Cheng

TL;DR

$q$-bic hypersurfaces in positive characteristic are studied by treating them as moduli spaces of isotropic lines for a canonical $q$-bic form $eta$, enabling a moduli-theoretic approach to their Fano schemes. The paper develops a Hermitian-structure framework and relates the Fano schemes to Deligne–Lusztig theory for finite unitary groups, yielding explicit Betti numbers, canonical bundles, and a rich stratification of the $m$-plane Fano variety; in the odd-dimensional case these schemes are irreducible, smooth, of general type, and admit a purely inseparable complete-intersection cover. For $m=1$ a Clemens–Griffiths–type analogue emerges: the Albanese of the Fano surface of lines is essentially the intermediate Jacobian of $X$, via a purely inseparable Abel–Jacobi map, with deeper links to Prym-type structures in positive characteristic. Altogether, the work forges deep connections between $q$-bic geometry, Deligne–Lusztig theory, unitary groups, and algebraic representatives, providing explicit structural results and guiding principles for further study in characteristic $p>0$.

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.

$q$-bic hypersurfaces and their Fano schemes

TL;DR

-bic hypersurfaces in positive characteristic are studied by treating them as moduli spaces of isotropic lines for a canonical -bic form , enabling a moduli-theoretic approach to their Fano schemes. The paper develops a Hermitian-structure framework and relates the Fano schemes to Deligne–Lusztig theory for finite unitary groups, yielding explicit Betti numbers, canonical bundles, and a rich stratification of the -plane Fano variety; in the odd-dimensional case these schemes are irreducible, smooth, of general type, and admit a purely inseparable complete-intersection cover. For a Clemens–Griffiths–type analogue emerges: the Albanese of the Fano surface of lines is essentially the intermediate Jacobian of , via a purely inseparable Abel–Jacobi map, with deeper links to Prym-type structures in positive characteristic. Altogether, the work forges deep connections between -bic geometry, Deligne–Lusztig theory, unitary groups, and algebraic representatives, providing explicit structural results and guiding principles for further study in characteristic .

Abstract

A -bic hypersurface is a hypersurface in projective space of degree , where is a power of the positive ground field characteristic, whose equation consists of monomials which are products of a -power and a linear power; the Fermat hypersurface is an example. I identify -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 -planes in a smooth -dimensional -bic hypersurface is an -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 may be viewed as an analogue of results of Clemens and Griffiths regarding cubic threefolds.
Paper Structure (24 sections, 53 theorems, 159 equations)

This paper contains 24 sections, 53 theorems, 159 equations.

Table of Contents

  1. Setup
  2. q-bic forms
  3. Canonical endomorphisms
  4. Hermitian subspaces
  5. q-bic hypersurfaces
  6. Classification
  7. Fano schemes
  8. Numerical invariants
  9. Smoothness and connectedness
  10. Tangent sheaves
  11. Nonsmooth locus
  12. Fano correspondences
  13. Action of the Fano correspondence
  14. Incidence schemes
  15. Hermitian structures
  16. ...and 9 more sections

Key Result

Theorem A

For each $0 \leq r < \frac{n}{2}$, the Fano scheme $\mathbf{F}$ of $r$-planes in a $q$-bic hypersurface $X \subset \mathbf{P}^n$ is In particular, if $X$ is smooth, then $\mathbf{F}$ is smooth of dimension $(r+1)(n-2r-1)$ and is irreducible whenever $\mathbf{F}$ is positive-dimensional. Furthermore, in this case, $\mathbf{F}$

Theorems & Definitions (92)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 1.3
  • proof
  • proof : Proof of equivalences
  • Corollary 1.5
  • Lemma 1.9
  • Lemma 1.11
  • proof
  • ...and 82 more