On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

Abstract

Let $\Delta$ be a symplectic dual polar space $DW(2n-1,{\Bbb K})$ or a Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$, $n \geq 2$. We define a class of hyperplanes of $\Delta$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta$-Hermitian variety in ${\rm PG}(n-1,{\Bbb K})$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.

The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings

“…If H does not admit ovoidal quads, then by De Bruyn and Pralle [15,Proposition 4.2], H is either a singular hyperplane, the extension of a (q + 1) × (q + 1)-grid in a quad or a so-called hexagonal hyperplane (which only exists if q is even). All these hyperplanes arise from the Grassmannembedding of DW (5, q), see De Bruyn [11], [14] and Shult and Thas [29]. In the sequel, we therefore suppose that there exists a quad Q which is ovoidal with respect to H. Put O := Q ∩ H. Let e : DW (5, q) → Σ denote the Grassmann-embedding of DW (5, q).…”