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| Original language | Undefined/Unknown |
|---|---|
| Publication status | E-pub ahead of print - 12 Jul 2023 |
Abstract
Keywords
- math.AG, 14J70 (primary), 14N25, 14J10, 14G17, 14G10, 20C33 (secondary)
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2023.
Research output: Working paper/Preprint › Preprint
}
TY - UNPB
T1 - q-bic hypersurfaces and their Fano schemes
AU - Cheng, Raymond
N1 - 32 pages, comments very welcome!
PY - 2023/7/12
Y1 - 2023/7/12
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)
M3 - Preprint
BT - q-bic hypersurfaces and their Fano schemes
ER -