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.