Certain generalized quadrangles inside polar spaces of rank $4$

Abstract

Let $\mathrm{\Delta }$ be the dual of a thick polar space $\mathrm{\Pi }$ of rank $4$. The points, lines, quads, and hexes of $\mathrm{\Delta }$ correspond with the singular $3$-spaces, planes, lines, respectively points of $\mathrm{\Pi }$. Pralle and Shpectorov have investigated ovoidal hyperplanes of $\mathrm{\Delta }$ which intersect every hex in the extension of an ovoid of a quad. With every ovoidal hyperplane there corresponds a unique generalized quadrangle $\mathrm{\Gamma }$. In the finite case, $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }\cong S{p}_4\left(ℍ\right)$ for some field $ℍ$.

In this paper, we construct an embedding projective space for the generalized quadrangle $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ is a hermitian generalized quadrangle over some division ring $ℍ$.

Moreover we show that an ovoidal hyperplane $H$ arises from the universal embedding of $\mathrm{\Delta }$, 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.