Multiprojective spaces and the arithmetically Cohen-Macaulay property

2017/07/24 by Giuseppe Favacchio, Favacchio, Giuseppe, Juan Migliore +1
Mathematics · #13A15 #13C14 #13C40 #13H10 #14M05 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1707.07417

openalex publication_date 2017/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

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