Given a non-singular quadratic form q of maximal Witt index on V := V (2n + 1, F), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1k . In this paper we give a new very easy proof of this fact. We also prove that if char(F) = 2 then dim(As a consequence, when 1 < k < n and char(F) = 2 the embedding ε k is not universal. Finally, we prove that if F is a perfect field of characteristic p > 2 or a number field, n > k and k = 2 or 3, then ε k is universal.

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 $DW(2n-1,q)$

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Given a point-line geometry $\Gamma$ and a pappian projective space $\cal S$, a veronesean embedding of $\Gamma$ in $\cal S$ is an injective map $e$ from the point-set of $\Gamma$ to the set of points of $\cal S$ mapping the lines of $\Gamma$ onto non-singular conics of $\cal S$ and such that $e(\Gamma)$ spans $\cal S$. In this paper we study veronesean embeddings of the dual polar space $\Delta_n$ associated to a non-singular quadratic form $q$ of Witt index $n \geq 2$ in $V = V(2n+1,\mathbb{F})$. Three such embeddings are considered, namely the Grassmann embedding $\varepsilon^{\mathrm{gr}}_n$ which maps a maximal singular subspace $\langle v_1,..., v_n\rangle$ of $V$ (namely a point of $\Delta_n$) to the point $\langle \wedge_{i=1}^nv_i\rangle$ of $\mathrm{PG}(\bigwedge^nV)$, the composition $\varepsilon^{\mathrm{vs}}_n := \nu_{2^n}\circ \varepsilon^{\mathrm{spin}}_n$ of the spin (projective) embedding $\varepsilon^{\mathrm{spin}}_n$ of $\Delta_n$ in $\mathrm{PG}(2^n-1,\mathbb{F})$ with the quadric veronesean map $\nu_{2^n}:V(2^n,\mathbb{F})\rightarrow V({{2^n+1}\choose 2}, \mathbb{F})$, and a third embedding $\tilde{\varepsilon}_n$ defined algebraically in the Weyl module $V(2\lambda_n)$, where $\lambda_n$ is the fundamental dominant weight associated to the $n$-th simple root of the root system of type $B_n$. We shall prove that $\tilde{\varepsilon}_n$ and $\varepsilon^{\mathrm{vs}}_n$ are isomorphic. If $\mathrm{char}(\F)\neq 2$ then $V(2\lambda_n)$ is irreducible and $\tilde{\varepsilon}_n$ is isomorphic to $\varepsilon^{\mathrm{gr}}_n$ while if $\mathrm{char}(\F)= 2$ then $\varepsilon^{\mathrm{gr}}_n$ is a proper quotient of $\tilde{\varepsilon}_n$. In this paper we shall study some of these submodules. Finally we turn to universality, focusing on the case of $n = 2$. We prove that if $\F$ is a finite field of odd order $q > 3$ then $\varepsilon^{\mathrm{sv}}_2$ is relatively universal. On the contrary, if $\mathrm{char}(\F)= 2$ then $\varepsilon^{\mathrm{vs}}_2$ is not universal. We also prove that if $\F$ is a perfect field of characteristic 2 then $\varepsilon^{\mathrm{vs}}_n$ is not universal, for any $n \geq 2$

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding ε gr k of an orthogonal Grassmannian k . In particular, we determine some of the parameters of the codes arising from the projective system determined by ε gr k ( k ). We also study special sets of points of k which are met by any line of k in at most 2 points and we show that their image under the Grassmann embedding ε gr k is a projective cap.

Let ∆ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of ∆ into a projective space Σ. For every point x of ∆ and every i ∈ {0, . . . , n}, let Ti(x) denote the subspace of Σ generated by all points e(y) with d(x, y) ≤ i. We show that Ti(x) does not contain points e(z) with d(x, z) ≥ i + 1. We also show that there exists a well-defined map e x i from the set of (i − 1)-dimensional subspaces of the residue Res∆(x) of ∆ at the point x (which is a projective space of dimension n − 1) to the set of points of the quotient space Ti(x)/Ti−1(x). In this paper we study the structure of the maps e x i and the subspaces Ti(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.

Polar Grassmann codes of orthogonal type have been introduced in [1]. They are punctured versions of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann codes of orthogonal type for q odd

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