The Hilbert functions of ACM sets of points in [formula omitted

Abstract

If X is a set of points in P n 1 ×⋯× P n k , then the associated coordinate ring R/I X is an N k -graded ring. The Hilbert function of X , defined by H X ( i):= dim k (R/I X ) i for all i∈ N k , is studied. Since the ring R/I X 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 P 1× P 1 , 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.