Veronesean embeddings of dual polar spaces of orthogonal type

Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S is an injective map e from the point-set of Γ to the set of points of S mapping the lines of Γ onto non-singular conics of S and such that e(Γ) spans S. In this paper we study veronesean embeddings of the dual polar space Δn associated to a non-singular quadratic form q of Witt index n⩾2 in V=V(2n+1,F). Three such embeddings are considered, namely the Grassmann embedding which maps a maximal singular subspace 〈v1,…,vn〉 of V (namely a point of Δn) to the point of PG(⋀nV), the composition of the spin (projective) embedding of Δn in PG(2n−1,F) with the quadric veronesean map , and a third embedding defined algebraically in the Weyl module V(2λn), where λn is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that and are isomorphic. If char(F)≠2 then V(2λn) is irreducible and is isomorphic to while if char(F)=2 then is a proper quotient of . 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 is relatively universal. On the contrary, if char(F)=2 then is not universal. We also prove that if F is a perfect field of characteristic 2 then is not universal, for any n⩾2.