Multiprojective spaces and the arithmetically Cohen-Macaulay property

In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for P 1 × P 1 and, more recently, in (P 1 ) r . In P 1 × P 1 the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in P m × P n . In such an ambient space it is equivalent to the so-called (∗)-property. Moreover, we start an investigation of the ACM property in P 1 × P n . We give a new construction that highlights how different the behavior of the ACM property is in this setting.