On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially (ℙ1)n. A combinatorial characterization, the (⋆)-property, is known in ℙ1× ℙ1. We propose a combinatorial property, (⋆s) with 2 ≤ s ≤ n, that directly generalizes the (⋆)-property to (ℙ1)nfor 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.