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Geometriae Dedicata
Article . 1995 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1995
Data sources: zbMATH Open
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A geometric construction of a class of twisted field planes by Ree-Tits unitals

Authors: de Smet, Veerle; van Maldeghem, Hendrik;

A geometric construction of a class of twisted field planes by Ree-Tits unitals

Abstract

Let \(H(q)\) denote the generalized quadrangle arising from Dickson's simple group \(G_2 (q)\) [see \textit{J. Tits}, Inst. Haut. Etud. Sci. Publ. Math. No. 2, 13-60 (1959; Zbl 0088.37204)]. It is well known that \(H(q)\) admits a polarity if and only if \(q\) is an odd power of 3, and it is assumed \(q = 3^{2h + 1}\) for some nonnegative integer \(h\) throughout this review. In this case it is also known that the set of absolute points forms an ovoid in \(H(q)\); that is, a set of \(q^3 + 1\) points such that every other point is collinear with exactly one of them. This collection of points in \(H(q)\) is called a Ree-Tits unital, whose blocks are defined to be the fixed sets of the involutions in \(\Aut (H(q)) \cong^2 G_2 (q)\). Any two Ree-Tits unitals are isomorphic, and there are \(q^3 (q - 1)\) Ree-Tits unitals containing a particular flag (point-line incident pair). In the paper under review it is shown that if \(h = 0\) (hence \(q = 3)\), then any two distinct Ree-Tits unitals have 0,1 or \(q + 1\) flags in common, while if \(h \geq 1\), then any two distinct Ree-Tits unitals of \(H(q)\) have 0,1,2 or \(q + 1\) flags in common. From this result it is observed that whenever two Ree-Tits unitals meet in \(q + 1\) flags, the points of these flags constitute a block in both unitals. It is then shown how to construct projective plane of order \(q\) from a certain class of Ree-Tits unitals containing a specified flag. This plane turns out to be nondesarguesian if and only if \(h \neq 0\) \((q > 3)\). In fact, these planes are isomorphic to twisted field planes. Moreover, ovals are constructed in all these planes.

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Keywords

generalized quadrangle, Generalized quadrangles and generalized polygons in finite geometry, Ree-Tits unitals, twisted field planes

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selected citations
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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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