Multiprojective spaces and the arithmetically Cohen-Macaulay property

TL;DR

This paper investigates the arithmetically Cohen-Macaulay property for point sets in multiprojective spaces, extending known results and introducing new properties and constructions to understand its behavior in various settings.

Contribution

It generalizes the inclusion property characterization of ACM in multiprojective spaces and explores the ACM property in \\mathbb{P}^1 imes \\mathbb{P}^n, revealing new behaviors.

Findings

Inclusion property implies ACM in \\mathbb{P}^m \\times \\mathbb{P}^n.

Equivalence between inclusion property and \\star-property in \\mathbb{P}^m \\times \\mathbb{P}^n.

Distinct ACM behavior in \\mathbb{P}^1 \\times \\mathbb{P}^n.

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.