Innovations in Incidence Geometry
- Innov. Incidence Geom.
- Volume 4, Number 1 (2006), 109-130.
Certain generalized quadrangles inside polar spaces of rank $4$
Let be the dual of a thick polar space of rank . The points, lines, quads, and hexes of correspond with the singular -spaces, planes, lines, respectively points of . Pralle and Shpectorov have investigated ovoidal hyperplanes of which intersect every hex in the extension of an ovoid of a quad. With every ovoidal hyperplane there corresponds a unique generalized quadrangle . In the finite case, has been classified combinatorially, and it has been shown that only the symplectic and elliptic dual polar spaces and of Witt index have ovoidal hyperplanes. For over an arbitrary field , it holds for some field .
In this paper, we construct an embedding projective space for the generalized quadrangle arising from an ovoidal hyperplane of the orthogonal dual polar space for a field . Assuming when is infinite, we prove that is a hermitian generalized quadrangle over some division ring .
Moreover we show that an ovoidal hyperplane arises from the universal embedding of , if the ovoids of all ovoidal quads are classical. This condition is satisfied for the finite dual polar spaces and by Pralle and Shpectorov.
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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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. https://projecteuclid.org/euclid.iig/1551323228