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 Δ 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 Δ has a polarized embedding e : Δ → P G ( V ) , namely a lax embedding satisfying the following: for every point p of Δ , the set H p of points at non-maximal distance from p is mapped by e into a hyperplane of P G ( V ) . Finally, we consider the case n = 2 and determine all the possible sizes of minimal scattered sets of finite classical generalized quadrangles.