Veronesean embeddings of dual polar spaces of orthogonal type

摘要

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(2n+1,F). Three such embeddings are considered, namely the Grassmann embedding εngr which maps a maximal singular subspace 〈v1,...,vn〉 of V (namely a point of δ_n) to the point 〈{n-ary logical and}i=1nvi〉 of PG({n-ary logical and}~nV), the composition εnvs:=ν2n{ring operator}εnspin of the spin (projective) embedding εnspin of δ_n in PG(2n-1,F) with the quadric veronesean map ν2n:V(2n,F)→V((2n+12),F), and a third embedding ε~n defined algebraically in the Weyl module V(2λn), where λ_n is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that ε~n and εnvs are isomorphic. If char(F)≠2 then V(2λ_n) is irreducible and ε~n is isomorphic to εngr while if char(F)=2 then εngr 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 ε2sv is relatively universal. On the contrary, if char(F)=2 then ε2vs is not universal. We also prove that if F is a perfect field of characteristic 2 then εnvs is not universal, for any n≥2.