Giuliana Fatabbi

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

Abstract We propose an upper bound for the regularity index of fat points of P n 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 P 2 , we prove it in P 3 , by using some preliminary results which hold, more generally, in P n .

Segre's Bound and the Case of n+ 2 Fat Points of ℙ n

First we prove that the locus of all sets of fat points with assigned multiplicities whose regularity index is bounded by the so called Segres bound is open and strictly contains the non-empty open set given by the locus of the same number of fat points with the same multiplicities and support in general position. Then we prove that the general Segres bound holds for any set of n + 2 fat points of ℙ n .