Partial flocks of non-singular quadrics in PG\((2r+1,q)\)
type: Item
instance of: scholarly article
title: Partial flocks of non-singular quadrics in PG\((2r+1,q)\)
zbMATH Open document ID: 1067.51005
author: Matthew Brown
author: Christine M. O'Keefe
author: Cristina Tonesi
published in: Journal of Algebraic Combinatorics
publication date: 2005-03-15T00:00:00Z
review text: The authors generalize the notions of flocks and partial flocks, known for cones in PG\((3,q)\) and for non-singular quadrics in PG\((3,q)\), to non-singular quadrics in PG\((2r+1,q)\). This generalization for quadratic cones in PG\((2r+1,q)\) with a point vertex was already done by \textit{Christine M. O'Keefe} and \textit{J. A. Thas} [J. Algebr. Comb. 6, No. 4, 377--392 (1997; Zbl 0897.51007)]. Now the analogous results for non-singular quadrics in PG\((2r+1,q)\) are presented. Let \(Q_{2r+1}\) be a non-singular quadric of elliptic or hyperbolic character in PG\((2r+1,q)\). A partial flock of \(Q_{2r+1}\) of cardinality \(s\) is a set of \(s\) hyperplanes of PG\((2r+1,q)\) intersecting this quadric in non-singular parabolic quadrics, and such that two distinct such hyperplanes intersect in a \((2r-1)\)-dimensional space sharing an elliptic quadric with \(Q_{2r+1}\). Among the results presented are: upper bounds on the size of such partial flocks, a generalization of a construction of Thas, and a characterization of linear partial flocks. Moreover, partial flocks of \(Q_{2r+1}\) of size larger than the size of flocks of the non-singular quadrics in PG\((3,q)\) are constructed.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 2146019
zbMATH Keywords: flock
zbMATH Keywords: partial flock
zbMATH Keywords: quadric
zbMATH Keywords: exterior set
zbMATH Keywords: Thas flock
reviewed by: Leo Storme
MaRDI profile type: Publication
full work available at URL: b:jaco.0000048518.34713.fc
OpenAlex ID: W2049258608
DOI: 10.1023/B:JACO.0000048518.34713.FC
Recommended article: Partial flocks of the quadratic cone.
Recommended article: Extending partial flocks containing linear subflocks
Recommended article: \(k\)-arcs and partial flocks
Recommended article: Transitive conical flocks of even order
Recommended article: Some remarks on flocks
Recommended article: Q4276304
Recommended article: Flock generalized quadrangles and tetradic sets of elliptic quadrics of PG\((3,q)\)
Recommended article: A geometrical construction of the oval(s) associated with an α-flock
Recommended article: Classification of flocks of the quadratic cone over fields of order at most 29
Recommended article: Generalized quadrangles and flocks of cones
description: scientific article; zbMATH DE number 2146019