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

If X is a set of points in ℙⁿ¹ x ⋯ x ℙ^nk, then the associated coordinate ring R/IX is an ℕ^k-graded ring. The Hilbert function of X, defined by HX (i) := dimk (R/IX)i for all i ∈ ℕ²k, is studied. Since the ring R/IX may or may not be Cohen-Macaulay, we consider only those X that are ACM. Generalizing the case of k = 1 to all k, we show that a function is the Hilbert function of an ACM set of points if and only if its first difference function is the Hilbert function of a multi-graded Artinian quotient. We also give a new characterization of ACM sets of points in ℙ¹ × ℙ¹, and show how the graded Betti numbers (and hence, Hilbert function) of ACM sets of points in this space can be obtained solely through combinatorial means. © 2003 Elsevier Science (USA). All rights reserved.