Minimal full polarized embeddings of dual polar spaces

Let $\Delta$ be a thick dual polar space of rank $n \geq 2$ admitting a full polarized embedding $e$ in a finite-dimensional projective space $\Sigma$, i.e., for every point $x$ of $\Delta$, $e$ maps the set of points of $\Delta$ at non-maximal distance from $x$ into a hyperplane $e^\ast(x)$ of $\Sigma$. Using a result of Kasikova and Shult , we are able the show that there exists up to isomorphisms a unique full polarized embedding of $\Delta$ of minimal dimension. We also show that $e^\ast$ realizes a full polarized embedding of $\Delta$ into a subspace of the dual of $\Sigma$, and that $e^\ast$ is isomorphic to the minimal full polarized embedding of $\Delta$. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces $DQ(2n,q)$, $DQ^-(2n+1,q)$, $DH(2n-1,q^2)$ and $DW(2n-1,q)$ ($q$ odd), but the latter only for $n\leq 5$. We shall prove that the minimal full polarized embeddings of $DQ(2n,q)$, $DQ^-(2n+1,q)$ and $DH(2n-1,q^2)$ are the `natural' ones, whereas this is not always the case for $DW(2n-1,q)$.