  In this note we show that Harbourne's conjecture is true for symbolic powers
of ideals of points, we check that the stable version of this conjecture is
valid for ideals of very general points (resp. generic points) in $\mathbb
P_{\mathbb K}^N$ (resp. $\mathbb P_{\mathbb K(\underline{z})}^N$). We also show
that this conjecture and the Harbourne-Huneke conjecture are true for a class
of ideals $I$ defining fat points obtained from line arrangements in $\mathbb
P_{\mathbb K}^2$.
