Minimal full polarized embeddings of dual polar spaces

Abstract

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 $DW(2n-1,q)$ ($q$\ud 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 $DW(2n-1,q)$

Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

“…Following Cardinali, De Bruyn and Pasini [5], we call e polarized if through every point e(x) of the image of e there exists a (necessarily unique) hyperplane T x of Σ such that for every point y of ∆, e(y) ∈ T x if and only if x and y are not opposite.…”

“…Since R contains no point of the image of e, a quotient embedding e/R of ∆ T can be defined in the quotient projective space of PG(7 + 6n, K) determined by R. Following the terminology of Cardinali, De Bruyn and Pasini [5], the embedding e/R is precisely the minimal full polarized embedding of ∆ T (up to isomorphism).…”

“…If |K| = 2, then ∆ T is isomorphic to one of the dual polar spaces DW (5, 2) or DH (5,4), and e is isomorphic to the Grassmann embedding. The Grassmann embeddings of these two dual polar spaces are known not to be absolutely universal, see for instance Yoshiara [30].…”

Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

“…The singular hyperplanes of DW (2n − 1, K) arise from the Grassmann embedding of DW (2n − 1, K), see e.g. Cardinali, De Bruyn and Pasini [7,Section 4.3] or De Bruyn [15,Proposition 2.15].…”

“…This embedding is isomorphic to the Grassmann embedding of DW (3, K), see e.g. Cardinali, De Bruyn and Pasini [7,Proposition 4.10]. (Although the discussion there was limited to the finite case, the arguments work as well for the infinite case.)…”