The h-vector of the union of two sets of points in the projective plane

Given two h-vectors, h and h', we study which are the possible h-vectors for the union of two disjoint sets of points in P^2 , respectively associated to h and h' and how they can be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.