Separators of fat points in Pn

Abstract

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 ⊆ 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 ν = 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 ν ) . 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.