Multiprojective spaces and the arithmetically Cohen-Macaulay property

Abstract

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, more recently, 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 ${\mathbb{P}}^m\times {\mathbb{P}}^n$. In such an ambient space it is equivalent to the so-called $(\star )$-property. Moreover, we start an investigation of the ACM property in ${\mathbb{P}}^1\times {\mathbb{P}}^n.$ We give a new construction that highlights how different the behavior of the ACM property is in this setting.