The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces

Abstract

Abstract Let Δ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of Δ into a projective space ∑. For every point x of Δ and every i ∈ {0, …, n }, let T i ( x ) denote the subspace of ∑ generated by all points e ( y ) with d( x , y ) ≤ i . We show that T i ( x ) does not contain points e ( z ) with d( x , z ) ≥ i + 1. We also show that there exists a well-defined map e i x from the set of ( i – 1)-dimensional subspaces of the residue Res Δ ( x ) of Δ at the point x (which is a projective space of dimension n – 1) to the set of points of the quotient space T i ( x )/T i –1 ( x ). In this paper we study the structure of the maps e i x and the subspaces T i ( x ) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.