Bilinear flocks of quadratic cones
type: Item
instance of: scholarly article
title: Bilinear flocks of quadratic cones
zbMATH Open document ID: 0928.51007
DOI: 10.1007/BF01229210
author: Mauro Biliotti
author: Norman L. Johnson
published in: Journal of Geometry
publication date: 1999-05-26T00:00:00Z
review text: The authors investigate the translation planes, isomorphisms, the derived flocks and the generalized quadrangles of a class of bilinear flocks of a quadratic cone over ordered fields, whose construction is modelled on the construction of a flock of the sphere by considering all planes through a line \(L\) (with \(L\) lying in the plane of the equator) meeting the sphere in the north half and all planes through a line \(M\neq L\) (also \(M\) lies in the plane of the equator) meeting the sphere in the south half (since the field is ordered, ``north'' and ``south'' make sense -- as ``positive'' and ``negative'', respectively). So ``bilinear'' means that all planes of the flock contain one of two fixed lines, and these lines are contained in at least two planes of the flock. It turns out that the associated projective planes admit \(SL_2\) as a collineation group, and the authors also investigate, in a more general setting, this situation more closely. There is also an interesting characterization of bilinear flock planes as translation planes over ordered fields (in which all positive elements are squares) whose associated spread is doubly covered by reguli.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 1292215
zbMATH Keywords: net replacement
zbMATH Keywords: translation planes
zbMATH Keywords: generalized quadrangles
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
cites work: Derivation of Flocks of Quadratic Cones
cites work: Q4358762
cites work: On flocks of infinite quadratic cones
cites work: Q5554833
cites work: Q3228640
cites work: The translation planes of order \(q_ 2\) that admit SL(2,q) as a collineation group. I: Even order
cites work: A characterization of the Desarguesian planes of order \(q^ 2\) by \(SL(2,q)\)
cites work: Q3827166
cites work: Flocks of quadratic cones, generalized quadrangles and translation planes
cites work: Quasifibrations
cites work: Infinite flocks of a quadratic cone
cites work: Q3972344
cites work: Q5285568
cites work: Flocks of infinite hyperbolic quadrics
cites work: Extending partial flocks containing linear subflocks
cites work: Q3891315
cites work: Vector spaces and construction of finite projective planes
cites work: Conical flocks, partial flocks, derivation, and generalized quadrangles
cites work: Generalized quadrangles and flocks of cones
cites work: Q4129398
description: scientific article; zbMATH DE number 1292215