Transitive ovoids of the Hermitian surface of PG\((3,q ^{2}\)), \(q\) even
type: Item
instance of: scholarly article
title: Transitive ovoids of the Hermitian surface of PG\((3,q ^{2}\)), \(q\) even
zbMATH Open document ID: 1019.51006
DOI: 10.1016/S0097-3165(02)00021-3
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: An ovoid of the Hermitian surface \(H=H(3,q^2)\) in \(PG(3,q^2)\) is a set of \(q^3+1\) points of \(H\) intersecting every line of \(H\) in exactly one point. The classical ovoid of \(H\) is a planar section \(H(2,q^2)\) of \(H\) with a non-tangent plane. The authors study the ovoids of \(H\) stabilized by a subgroup of PGU\((4,q^2)\), acting transitively on the ovoid. Their results show that there are two such transitive ovoids. Apart from the classical ovoid, there is one other example. This latter example is obtained by using a cyclic spread of a Hermitian curve \(H(2,q^2)\). A cyclic Singer group of \(PG(2,q^2)\) contains a cyclic subgroup of order \(q^2-q+1\). Using this cyclic group, it is possible to partition the \(q^3+1\) points of \(H(2,q^2)\) into \(q^2-q+1\) disjoint Baer sublines, which is called a cyclic spread. By embedding \(H(2,q^2)\) into \(H\), it is then possible to construct the other example of a transitive ovoid of \(H\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 1888441
zbMATH Keywords: ovoid
zbMATH Keywords: Hermitian surface
zbMATH Keywords: Hermitian curve
zbMATH Keywords: Singer cycle
author: Antonio Cossidente
author: Gabor Korchmáros
reviewed by: Leo Storme
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1888441