## Innovations in Incidence Geometry

### Certain generalized quadrangles inside polar spaces of rank $4$

Harm Pralle

#### Abstract

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 $Γ ≅ S p 4 ( ℍ )$ for some field $ℍ$.

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 $ℍ$.

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

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

Dates
Accepted: 6 December 2006
First available in Project Euclid: 28 February 2019

https://projecteuclid.org/euclid.iig/1551323228

Digital Object Identifier
doi:10.2140/iig.2006.4.109

Mathematical Reviews number (MathSciNet)
MR2334649

Zentralblatt MATH identifier
1129.51004

#### Citation

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

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