Hyperovals arising from a Singer group action on \(\mathcal{H}(3, q^2)\), \(q\) even
type: Item
instance of: scholarly article
title: Hyperovals arising from a Singer group action on \(\mathcal{H}(3, q^2)\), \(q\) even
zbMATH Open document ID: 1426.51002
author: Antonio Cossidente
author: Oliver King
author: Giuseppe Marino
publication date: 2018-03-29T00:00:00Z
review text: Let \(H(3,q^2)\) be a generalized quadrangle associated with a non-singular Hermitian surface in \(\mathrm{PG}(3,q^2)\). A hyperoval (or local subspace) of \(H(3,q^2)\) is a set of points with the property that each line of the generalized quadrangle meets the set in either 0 or 2 points. The hyperovals of \(H(3, 4)\) have been classified by \textit{A. A. Makhnev} [Discrete Math. Appl. 7, No. 4, 419--435 (1997; Zbl 0964.51004); translation from Diskretn. Mat. 9, No. 3, 101--116 (1997)]. Suppose \(q\) is even. Let \(W_1=\mathrm{PG}(3,q)\cap H(3,q^2)\) be a symplectic subquadrangle of \(H(3,q^2)\). If \(\mathcal{S}\) is a partial spread of \(\mathrm{PG}(3,q^2)\) such that \(\mathcal{S}\) induces on \(W_1\) a Desarguesian spread, then there are \(q-1\) symplectic subquadranlges \(W_1,W_2,\dots ,W_{q-1}\) of \(H(3,q^2)\) such that \(\mathcal{S}\) induces on \(W_i\) (\(i=1,2,\dots,q-1\)) a Desarguesian spread. Let us choose two symplectic subquadrangles \(W\) and \(W'\) among the subquadrangles \(W_1,W_2,\dots,W_{q-1}\). If \(G\) is a Singer cyclic group of \( \mathrm{PO}^-(4,q)<\mathrm{PSp}(4,q)\), we can suppose that \(\mathcal{S}\) is \(G\)-invariant. Let now \(\Omega\) and \(\Omega'\) be two elliptic quadrics in \(W\) and \(W'\), respectively, which are invariant under \(G\). The authors prove that \(\mathcal{I}=\Omega\cap \Omega'\) is an hyperoval of \(H(3,q^2)\). This implies that there exists an infinite family of hyperovals of \(H(3, q^2)\), \(q\) even, admitting a Singer cyclic group of order \(q^2+ 1\).
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 6854775
zbMATH Keywords: Hermitian surface
zbMATH Keywords: hyperoval
zbMATH Keywords: symplectic subquadrangle
reviewed by: Guglielmo Lunardon
MaRDI profile type: Publication
full work available at URL: advgeom-2016-0021
OpenAlex ID: W2338090077
cites work: Complete spans on Hermitian varieties
cites work: Intersection pattern of the classical ovoids in symplectic 3-space of even order
cites work: On ovoids of parabolic quadrics
cites work: Tight sets and \(m\)-ovoids of generalised quadrangles
cites work: Locally polar spaces and related rank 3 groups
cites work: Hyperovals on \(\mathcal H(3,q^2)\)
cites work: On \(m\)-ovoids of \({\mathcal W}_3 (q)\)
cites work: On Some Maximal Subgroups of Unitary Groups
cites work: Hyperovals of \(H(3,q^2)\) when \(q\) is even
cites work: On hyperovals of polar spaces
cites work: Q4028429
cites work: Proper 2-covers of \(PG (3,q)\), \(q\) even
cites work: Partitioning Projective Geometries into Caps
cites work: Extensions of GQ(4,2), the description of hyperovals
cites work: New families of \(Q\)-polynomial association schemes
cites work: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two
cites work: Forme e geometrie hermitiane, con particolare riguardo al caso finito
DOI: 10.1515/ADVGEOM-2016-0021
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published in: advg
description: scientific article; zbMATH DE number 6854775