The structure of full polarized embeddings of symplectic and hermitian dual polar spaces

Let $\Delta$ be a thick dual polar space of rank $n \geq 2$ and let $e$ be a full polarized embedding of $\Delta$ into a projective space $\Sigma$. For every point $x$ of $\Delta$ and every $i \in \{ 0,\ldots,n \}$, let $T_i(x)$ denote the subspace of $\Sigma$ generated by all points $e(y)$ with $\d(x,y) \leq i$. We show that $T_i(x)$ does not contain points $e(z)$ with $\d(x,z) \geq i+1$. We also show that there exists a well-defined map $e_i^x$ from the set of $(i-1)$-dimensional subspaces of the residue $Res_\Delta(x)$ of $\Delta$ at the point $x$ (which is a projective space of dimension $n-1$) to the set of points of the quotient space $T_i(x)/T_{i-1}(x)$. In this paper we study the structure of the maps $e_i^x$ and the subspaces $T_i(x)$ for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.