Separators of fat points in Pn

In this paper we extend the definition of a separator of a point P in Pn to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z=m1P1+⋯+miPi+⋯+msPs⊆Pn and Z′=m1P1+⋯+(mi−1)Pi+⋯+msPs. We associate to Pi a tuple of positive integers of length . We call this tuple the degree of the minimal separators of Pi of multiplicity mi, and we denote it by degZ(Pi)=(d1,…,dν). We show that if one knows degZ(Pi) and the Hilbert function of Z, one will also know the Hilbert function of Z′. We also show that the entries of degZ(Pi) are related to the shifts in the last syzygy module of IZ. Both results generalize well-known results about reduced sets of points and their separators.