Collisions of fat points and applications to interpolation theory

We address the problem to determine the limit of the collision of fat points in $$\mathbb{P}$^n. We give a description of the limit scheme in many cases, in particular in low dimension and multiplicities. The problem turns out to be closely related with interpolation theory, and as an application we exploit collisions to prove some new cases of Laface-Ugaglia Conjecture.