On a class of hyperplanes of the symplectic and Hermitian dual polar spaces.

Abstract

Let $\Delta$ be a symplectic dual polar space (2n-1,K)$or a Hermitian dual polar space (2n-1,K,\theta)$, \geq 2$. We define a class of hyperplanes of$\Delta$arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space (2n-1,K,\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta variety in$\PG(n-1,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