Symbolic powers of planar point configurations

In [12] we began to study the initial sequences α(I(m)), m=1,2,3…, of radical ideals I of finite sets of points in the projective plane. In the present note we complete results obtained in [12] by answering a number of questions left open in the previous note and we extend our considerations to the asymptotic setting of Waldschmidt constants. The concept of the Bezout decomposition introduced in Definition 2.7 might be of independent interest.