minimal scattered sets and polarized embeddings of dual polar spaces

We introduce the notion of scattered sets of points of a dual polar space, focusing on minimal ones. We prove that a dual polar space Delta of rank n always admits minimal scattered sets of size 2(n). We also prove that the size of a minimal scattered set is a lower bound for dim(V) if the dual polar space Delta has a polarized embedding e : Delta -> PG(V), namely a lax embedding satisfying the following: for every point p of Delta, the set HI, of points at non-maximal distance from p is mapped by e into a hyperplane of PG(V). Finally, we consider the case n = 2 and determine all the possible sizes of minimal scattered sets of finite classical generalized quadrangles. (c) 2007 Published by Elsevier Ltd.