#### Vol. 4, No. 1, 2006

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Certain generalized quadrangles inside polar spaces of rank $4$

### Harm Pralle

Vol. 4 (2006), No. 1, 109–130
DOI: 10.2140/iig.2006.4.109
##### Abstract

Let $\Delta$ be the dual of a thick polar space $\Pi$ of rank $4$. The points, lines, quads, and hexes of $\Delta$ correspond with the singular $3$-spaces, planes, lines, respectively points of $\Pi$. Pralle and Shpectorov have investigated ovoidal hyperplanes of $\Delta$ which intersect every hex in the extension of an ovoid of a quad. With every ovoidal hyperplane there corresponds a unique generalized quadrangle $\Gamma$. In the finite case, $\Gamma$ has been classified combinatorially, and it has been shown that only the symplectic and elliptic dual polar spaces $DS{p}_{8}\left(q\right)$ and $D{O}_{10}^{-}\left(q\right)$ of Witt index $4$ have ovoidal hyperplanes. For $DS{p}_{8}\left(\mathbb{K}\right)$ over an arbitrary field $\mathbb{K}$, it holds $\Gamma \cong S{p}_{4}\left(ℍ\right)$ for some field $ℍ$.

In this paper, we construct an embedding projective space for the generalized quadrangle $\Gamma$ arising from an ovoidal hyperplane of the orthogonal dual polar space $D{O}_{10}^{-}\left(\mathbb{K}\right)$ for a field $\mathbb{K}$. Assuming $char\left(\mathbb{K}\right)\ne 2$ when $\mathbb{K}$ is infinite, we prove that $\Gamma$ is a hermitian generalized quadrangle over some division ring $ℍ$.

Moreover we show that an ovoidal hyperplane $H$ arises from the universal embedding of $\Delta$, if the ovoids $Q\cap H$ of all ovoidal quads $Q$ are classical. This condition is satisfied for the finite dual polar spaces $DS{p}_{8}\left(q\right)$ and $D{O}_{10}^{-}\left(q\right)$ by Pralle and Shpectorov.

##### Mathematical Subject Classification 2000
Primary: 51A50, 51E12, 51E23