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On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces. (English) Zbl 1388.13032
A characterization of finite sets with the arithmetically Cohen-Macaulay (ACM) property is only known in \(\mathbb{P}^1\times\mathbb{P}^1\). Several classifications of ACM sets of reduced and fat points in \(\mathbb{P}^1\times\mathbb{P}^1\) are known, in terms of the Hilbert function, separators, and combinatorial properties.
In the paper under review, the authors study the ACM property for finite sets of points in multiprojective spaces, especially \((\mathbb{P}^1)^n\). In \(\mathbb{P}^1\times\mathbb{P}^1\) the \((\star)\)-property (Remark 2.8 of this paper) is equivalent to the inclusion property (Definition 2.5). They show that the inclusion property does imply the ACM property in \((\mathbb{P}^1)^n\). Finally, the authors introduce for sets of points in \((\mathbb{P}^1)^n\) the \(({\star}_s)\)-property for \(2\leq s\leq n\) (Definition 3.6), a generalization of the \((\star)\)-property and they show that for \(s=n\) this characterizes the ACM property.

13C40 Linkage, complete intersections and determinantal ideals
13C14 Cohen-Macaulay modules
13A15 Ideals and multiplicative ideal theory in commutative rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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