Details
| Original language | English |
|---|---|
| Pages (from-to) | 1721-1773 |
| Number of pages | 53 |
| Journal | Pure and Applied Mathematics Quarterly |
| Volume | 21 |
| Issue number | 4 |
| Publication status | Published - 1 Apr 2025 |
Abstract
Keywords
- math.AG, 14J70 (primary), 14N25, 14J10, 14G17, 14G10, 20C33 (secondary)
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Pure and Applied Mathematics Quarterly, Vol. 21, No. 4, 01.04.2025, p. 1721-1773.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - q-bic hypersurfaces and their Fano schemes
AU - Cheng, Raymond
N1 - Publisher Copyright: © 2005 International Press, Inc. All rights reserved.
PY - 2025/4/1
Y1 - 2025/4/1
N2 - 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.
AB - 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.
KW - math.AG
KW - 14J70 (primary), 14N25, 14J10, 14G17, 14G10, 20C33 (secondary)
UR - http://www.scopus.com/inward/record.url?scp=105003415028&partnerID=8YFLogxK
U2 - 10.4310/PAMQ.250402030943
DO - 10.4310/PAMQ.250402030943
M3 - Article
VL - 21
SP - 1721
EP - 1773
JO - Pure and Applied Mathematics Quarterly
JF - Pure and Applied Mathematics Quarterly
SN - 1558-8599
IS - 4
ER -