The Hilbert functions of ACM sets of points in Pⁿ¹ x... x Pⁿᵏ

الأصل · EN

The Hilbert functions of sets of distinct points in Pⁿ have been characterized. We show that if we restrict to sets of distinct of points in Pⁿ¹ x... x Pⁿᵏ that are also arithmetically Cohen-Macaulay (ACM for short), then there is a natural generalization of this result. We begin by determining the possible values for the invariants K-dim R/Ix and depth R/Ix, where R/Ix is the coordinate ring associated to a set of distinct points X in Pⁿ¹ x... x Pⁿᵏ. At the end of this paper we give a new characterization of ACM sets of points in P¹ x P¹.