Translation generalized quadrangles of order \((s,s ^{2})\), \(s\) even, and eggs
type: Item
instance of: scholarly article
title: Translation generalized quadrangles of order \((s,s ^{2})\), \(s\) even, and eggs
zbMATH Open document ID: 1019.51009
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: An egg \(O(n,2n,q)\) of \(PG(4n-1,q)\) is a set of \(q^{2n}+1\) \((n-1)\)-dimensional subspaces \(PG^{(i)}(n-1,q)\), \(i=0,\ldots,q^{2n}\), any three of which generate a \(PG(3n-1,q)\) and such that every element \(PG^{(i)}(n-1,q)\) of \(O(n,2n,q)\) is contained in a \(PG^{(i)}(3n-1,q)\) having no point in common with any element \(PG^{(j)}(n-1,q)\), \(j \neq i\). Similarly, a set \(O(n,n,q)\) of \(PG(3n-1,q)\) is a set of \(q^{n}+1\) \((n-1)\)-dimensional subspaces \(PG^{(i)}(n-1,q)\), \(i=0,\ldots,q^{n}\), any three of which generate a \(PG(3n-1,q)\) and such that every element \(PG^{(i)}(n-1,q)\) of \(O(n,n,q)\) is contained in a \(PG^{(i)}(2n-1,q)\) having no point in common with any element \(PG^{(j)}(n-1,q)\), \(j \neq i\). The easiest examples of an egg and of an \(O(n,n,q)\) are obtained by considering an ovoid of \(PG(3,q^n)\) and by considering an oval of \(PG(2,q^n)\). When this latter ovoid of \(PG(3,q^n)\) is the elliptic quadric, we call the egg classical and when this latter oval of \(PG(2,q^n)\) is a conic, we call the set \(O(n,n,q)\) a pseudo-conic. An egg \(O(n,2n,q)\) is good at its element \(PG^{(i)}(n-1,q)\) if any \(PG(3n-1,q)\) containing \(PG^{(i)}(n-1,q)\) and at least two other elements of \(O(n,2n,q)\) contains exactly \(q^n+1\) elements of \(O(n,2n,q)\). The author presents two characterization results on classical eggs. He first of all proves that the egg \(O(n,2n,q)\) of \(PG(4n-1,q)\), \(q\) even, is classical if and only if it is good at some element and contains at least one pseudo-conic. And secondly, he proves that the egg \(O(n,2n,q)\) of \(PG(4n-1,q)\), \(q\) even, is classical if and only if it contains at least two intersecting pseudo-conics. As an application, three characterization results of the generalized quadrangle \(Q^-(5,q)\), \(q\) even, are given.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 1888399
zbMATH Keywords: egg
zbMATH Keywords: translation generalized quadrangle
author: Joseph A. Thas
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: jcta.2002.3254
OpenAlex ID: W1971347885
cites work: A new semifield flock
cites work: Generalized quadrangles of order \((q,q^ 2)\), \(q\) even, containing \(W(q)\) as a subquadrangle
cites work: The determination of ovoids of PG(3,\(q\)) containing a pointed conic
cites work: Ovoids of PG(3,q ), q Even, with a Conic Section
cites work: Subquadrangles of order \(s\) of generalized quadrangles of order (\(s,s^{2}\)). I
cites work: Q3788620
cites work: Q4131250
cites work: An essay on skew translation generalized quadrangles
cites work: Q3344217
cites work: Q4092574
cites work: Combinatorial characterizations of generalized quadrangles with parameters \(s=q\) and \(t=q^2\)
cites work: Generalized quadrangles and flocks of cones
cites work: Generalized quadrangles of order \((s,s^ 2)\). I
cites work: Generalized quadrangles of order \((s,s^ 2)\). II
cites work: Symplectic spreads in \(PG(3,q)\), inversive planes and projective planes
cites work: Generalized quadrangles of order \((s,s^2)\). III
cites work: Characterizations of translation generalized quadrangles
cites work: Spreads and ovoids in finite generalized quadrangles
cites work: Q4357355
DOI: 10.1006/JCTA.2002.3254
description: scientific article; zbMATH DE number 1888399