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

If X is a set of points in Pn1×⋯×Pnk, then the associated coordinate ring R/IX is an Nk-graded ring. The Hilbert function of X, defined by HX(i):=dimk(R/IX)i for all i∈Nk, 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.