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

Bart De Bruyn

The electronic journal of combinatorics, Tome 16 (2009) no. 1

Résumé

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.

Détail

DOI : 10.37236/90

Classification : 51A45, 51A50
Mots-clés : symplectic dual polar space, Hermitian dual polar space, hyperplane, Grassmann-embedding, ovoid