On extensions of hyperplanes of dual polar spaces

Let Delta be a thick dual polar space and F a convex subspace of diameter at least 2 of Delta. Every hyperplane G of the subgeometry (F) over tilde of Delta induced on F will give rise to a hyperplane H of Delta, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of Delta and if e(F) is the full embedding of (F) over tilde. induced by e(F) does the fact that G arises from the embedding e(F) imply that H arises from the embedding e? We will study this problem in the cases that e is an absolutely universal embedding, a minimal full polarized embedding or a Grassmann embedding of a symplectic dual polar space. Our study will allow us to prove that if e is absolutely universal, then also e(F) is absolutely universal. (C) 2010 Elsevier Inc. All rights reserved.