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{\bf Bart De Bruyn}
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{\bf On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces}
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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.



\bye