On the Arithmetically Cohen-Macaulay property for sets of points in Multiprojective Spaces

We study the arithmetically Cohen-Macaulay (ACM) property for finite setsof points in multiprojective spaces, especially (P_1)^n. A combinatorial characterization, the (⋆)-property, is known in P1 × P1. We propose a combinatorial property, (⋆n), thatdirectly generalizes the (⋆)-property to (P1^)n for larger n. We show that X is ACM if and only if it satisfies the (⋆n)-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.