Vol. 6, No. 1, 2008

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1-polarized pseudo-hexagons

Joseph A. Thas and Hendrik J. van Maldeghem

Vol. 6 (2008), No. 1, 307–325
DOI: 10.2140/iig.2008.6.307
Abstract

In this paper we continue our study begun in “Generalized hexagons and Singer geometries” (2008), aiming at characterizing the embedding of the split Cayley hexagons H(q), q even, in PG(5,q) by intersection numbers with respect to their lines. We prove that, for q3, every pseudo-hexagon (i.e. a set of lines of PG(5,q) with the properties that (1) every plane contains 0, 1 or q + 1 elements of , (2) every solid contains no more than q2 + q + 1 and no less than q + 1 elements of , and (3) every point of PG(5,q) is on q + 1 members of ) which is 1-polarized at some point x (i.e., the lines of through x do not span PG(5,q)) is either the line set of the standard embedding of H(q) in PG(5,q), or q = 2 (in the latter case all pseudo-hexagons are classified in the paper cited).

Keywords
generalized hexagons, embedding
Mathematical Subject Classification 2000
Primary: 51E12, 51E20
Milestones
Received: 21 January 2008
Accepted: 17 March 2008
Authors
Joseph A. Thas
Department of Mathematics
Ghent University
Krijgslaan 281
S25
9000 Ghent
Belgium
Hendrik J. van Maldeghem
Vakgroep Zuivere Wiskunde en Computeralgebra
University of Ghent
9000 Gent
Belgium