ACM sets of points in multiprojective space

If$$\mathbb{X}$$ is a finite set of points in a multiprojective space$$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then$$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM