Veronesean embeddings of dual polar spaces of orthogonal type

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 $n \geq 2$.