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Given a point-line geometry P and a pappian projective space S,a veroneseanembedding of P in S is an injective map e from the point-set of P to the set ofpoints of S mapping the lines of P onto non-singular conics of S and such thate(P) spans S. In this paper we study veronesean embeddings of the dual polarspace \Delta_n 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 theGrassmann embedding gr_n,the composition vs_n of the spin (projective)embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n;F) and a third embedding w_n defined algebraically in the Weyl module V(2\lambda_n),where \lambda_n is the fundamental dominant weight associated tothe n-th simple root of the root system of type Bn. We shall prove that w_n andvs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) isirreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is aproper quotient of w_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 ifF is a finite field of odd order q > 3 then sv_2 is relatively universal. Onthe contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if Fis a perfect field of characteristic 2 then vs_n is not universal,for any n>=2.

Veronesean embeddings of dual polar spaces of orthogonal type