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 ℙ¹ × ℙ¹ and, more recently, in (ℙ¹) ^r . In ℙ¹ × ℙ¹ 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 × ℙⁿ. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ¹ × ℙⁿ. We give a new construction that highlights how different the behavior of the ACM property is in this setting. [ABSTRACT FROM AUTHOR]