Multiprojective spaces and the arithmetically Cohen–Macaulay property

Abstract

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 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.