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

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