Vol. 6+7, No. 1, 2008

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1-polarized pseudo-hexagons

Joseph A. Thas and Hendrik J. van Maldeghem

Vol. 6+7 (2008), No. 1, 307–325
DOI: 10.2140/iig.2008.6.307

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).

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