Separators of fat points in P{double-struck}n

Abstract: In this paper we extend the definition of a separator of a point P in P(n) to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m(1) P(1) + ... + m(i) P(i) + ... + m(s) P(s) subset of P(n) and Z' = m(1) P(1) + ... + (m(i) - 1) P(i) + ... + m(s) P(s). We associate to P(i) a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P(i) of multiplicity m(i), and we denote it by deg(Z)(P(i)) = (d(1) ,..., d(v)). We show that if one knows deg(Z)(P(i)) and the Hilbert function of Z. one will also know the Hilbert function of Z'. We also show that the entries of deg(Z) (P(i)) are related to the shifts in the last syzygy module of I(Z). Both results generalize well-known results about reduced sets of points and their separators. (C) 2010 Elsevier Inc. All rights reserved.