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 $\mathbb P^1\times \mathbb P^1$and,morerecently,in$(\mathbb P^1)^r.$In$\mathbb P^1\times \mathbb P^1$ the so called inclusion property characterizes the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in $\mathbb P^m\times \mathbb P^n$\mathrm{.}Insuchanambientspaceitisequivalenttotheso-called$(\star)$-property\mathrm{.}Moreover,westartaninvestigationoftheACMpropertyin$\mathbb P^1\times \mathbb P^n.$ We give a new construction that highlights how different the behavior of the ACM property is in this setting.