Multiple Points with Increasing Multiplicities on a Fixed Projective Set

Take a finite subset S of an n-dimensional projective space. We study the Hilbert function of the multiples mS of S, mainly when S is general or at least very general. We recall several classical conjectures on this problem, raise new open questions, and prove some particular cases. An open question is if all mS have the expected Hilbert function. We find cases in which there are Zariski open subsets of sets S with maximal rank for all m and pairs (n,#S) for which no such open set exists. We start the study of the m-Terracini sets proving when the first one is nonempty for Veronese embeddings.