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 Δ n associated to a non-singular quadratic form q of Witt index n ⩾ 2 in V = V ( 2 n + 1 , F ) . Three such embeddings are considered, namely the Grassmann embedding ε n gr which maps a maximal singular subspace 〈 v 1 , … , v n 〉 of V (namely a point of Δ n ) to the point 〈 ⋀ i = 1 n v i 〉 of PG ( ⋀ n V ) , the composition ε n vs : = ν 2 n ∘ ε n spin of the spin (projective) embedding ε n spin of Δ n in PG ( 2 n − 1 , F ) with the quadric veronesean map ν 2 n : V ( 2 n , F ) → V ( ( 2 n + 1 2 ) , F ) , and a third embedding ε ˜ n defined algebraically in the Weyl module V ( 2 λ n ) , where λ 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 ε ˜ n and ε n vs are isomorphic. If char ( F ) ≠ 2 then V ( 2 λ n ) is irreducible and ε ˜ n is isomorphic to ε n gr while if char ( F ) = 2 then ε n gr is a proper quotient of ε ˜ 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 ε 2 sv is relatively universal. On the contrary, if char ( F ) = 2 then ε 2 vs is not universal. We also prove that if F is a perfect field of characteristic 2 then ε n vs is not universal, for any n ⩾ 2 .