On a sharp bound for the regularity index of any set of fat points

We propose an upper bound for the regularity index of fat points of image with no geometric conditions on the points. Whenever the conjecture is true, the bound is sharp. It is, in fact, reached when there are points with high multiplicities either on a line or on some rational curve. Besides giving an easy proof of the conjecture in image, we prove it in image, by using some preliminary results which hold, more generally, in image.