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 2n. 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→PG(V), namely a lax embedding satisfying the following: for every point p of Δ, the set Hp 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.