Innovations in Incidence Geometry

The classification of spreads of $T_2({\mathcal O})$ and $\alpha$-flocks over small fields

Matthew R. Brown, Christine M. O’Keefe, Stanley E. Payne, Tim Penttila, and Gordon F. Royle

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Abstract

We classify spreads of the Tits quadrangles T 2 ( O ) , for O an oval in PG ( 2 , q ) , for q = 2 , 4 , 8 , 1 6 and 3 2 , using a computer for the last three cases. Along the way, we classify α -flocks of PG ( 3 , 3 2 ) , and so flocks of the quadratic cone in PG ( 3 , 3 2 ) . Perhaps our most striking results are that, for many ovals O in PG ( 2 , 3 2 ) , including all 12 O’Keefe-Penttila ovals, T 2 ( O ) has no spreads, and that T 2 ( O ) is a proper subGQ of a GQ of order ( s , 3 2 ) for precisely 6 of the 35 ovals O of PG ( 2 , 3 2 ) , all of which were previously known to be subquadrangles of a (flock or dual Tits) GQ of order ( 1 0 2 4 , 3 2 ) . Also T 2 ( O ) is not a proper subGQ of a GQ of order ( s , q ) or of a GQ of order ( q , t ) for O a pointed conic in PG ( 2 , q ) , for q = 1 6 , 3 2 .

Article information

Source
Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 111-126.

Dates
Received: 27 December 2007
Accepted: 4 March 2008
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323173

Digital Object Identifier
doi:10.2140/iig.2008.6.111

Mathematical Reviews number (MathSciNet)
MR2515261

Zentralblatt MATH identifier
1175.51003

Subjects
Primary: 51B15: Laguerre geometries 51E12: Generalized quadrangles, generalized polygons 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx] 51E21: Blocking sets, ovals, k-arcs 51E23: Spreads and packing problems

Keywords
generalized quadrangle spread flock subquadrangle oval

Citation

Brown, Matthew R.; O’Keefe, Christine M.; Payne, Stanley E.; Penttila, Tim; Royle, Gordon F. The classification of spreads of $T_2({\mathcal O})$ and $\alpha$-flocks over small fields. Innov. Incidence Geom. 6-7 (2007), no. 1, 111--126. doi:10.2140/iig.2008.6.111. https://projecteuclid.org/euclid.iig/1551323173


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