Innovations in Incidence Geometry

Certain generalized quadrangles inside polar spaces of rank $4$

Harm Pralle

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Let Δ be the dual of a thick polar space Π of rank 4 . The points, lines, quads, and hexes of Δ correspond with the singular 3 -spaces, planes, lines, respectively points of Π . Pralle and Shpectorov have investigated ovoidal hyperplanes of Δ which intersect every hex in the extension of an ovoid of a quad. With every ovoidal hyperplane there corresponds a unique generalized quadrangle Γ . In the finite case, Γ has been classified combinatorially, and it has been shown that only the symplectic and elliptic dual polar spaces D S p 8 ( q ) and D O 1 0 ( q ) of Witt index 4 have ovoidal hyperplanes. For D S p 8 ( K ) over an arbitrary field K , it holds Γ S p 4 ( ) for some field .

In this paper, we construct an embedding projective space for the generalized quadrangle Γ arising from an ovoidal hyperplane of the orthogonal dual polar space D O 1 0 ( K ) for a field K . Assuming c h a r ( K ) 2 when K is infinite, we prove that Γ is a hermitian generalized quadrangle over some division ring .

Moreover we show that an ovoidal hyperplane H arises from the universal embedding of Δ , if the ovoids Q H of all ovoidal quads Q are classical. This condition is satisfied for the finite dual polar spaces D S p 8 ( q ) and D O 1 0 ( q ) by Pralle and Shpectorov.

Article information

Innov. Incidence Geom., Volume 4, Number 1 (2006), 109-130.

Received: 12 July 2006
Accepted: 6 December 2006
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51A50: Polar geometry, symplectic spaces, orthogonal spaces 51E12: Generalized quadrangles, generalized polygons 51E23: Spreads and packing problems

dual polar space generalized quadrangle hyperplane ovoid polar space spread symplectic spread embedding


Pralle, Harm. Certain generalized quadrangles inside polar spaces of rank $4$. Innov. Incidence Geom. 4 (2006), no. 1, 109--130. doi:10.2140/iig.2006.4.109.

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