Powers of complete intersections: Graded betti numbers and applications

Let I = (F-1,...,F-r) be a homogeneous ideal of the ring R = k[x(0),..., x(n)] generated by a regular sequence of type (d(1),...,d(r)). We give an elementary proof for an explicit description of the graded Betti numbers of I-s for any s >= 1. These numbers depend only upon the type and s. We then use this description to: (1) write H-R/Is, the Hilbert function of R/I-s, in terms of H-R/I; (2) verify that the k-algebra R/I-s satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in PI whose support is a complete intersection or a complete intersection minus a point.