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Special arrangements of lines: codimension 2 ACM varieties in \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\). (English) Zbl 1411.13010

In the paper under review, the authors study the so called varieties of lines in \(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\). Let \(F, G \in R:= \mathbb{K}[x_{1,0}, x_{1,1}, x_{2,0}, x_{2,1}, x_{3,0}, x_{3,1}]\) be two homogeneous linear forms of different degree, where \(\mathbb{K}\) is an algebraically closed field of characteristic zero. In \(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\) the variety \(\mathcal{L}\) defined by the ideal \(\langle F, G \rangle \subseteq R\) is called a line of \(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\) and we denote it by \(\mathcal{L}(F, G)\). Now we say that \(X \subset \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\) is a variety of lines if it is given by a finite union of distinct lines in \(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\). The main result of the paper provides a nice description of arithmetically Cohen-Macaulay varieties of lines which is combinatorial in its nature. In order to do so, we need to define a special class of varieties of lines. Let \(X \subseteq \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\) be a variety of lines. Let \(n \geq 4\), \(n \in \mathbb{N}\), we say that \(X\) has the \(n\)-hyperplanes \((\star)\) property (for short, \(\text{Hyp}_{n}(\star)\)-property) if given \(n\) hyperplanes \(H_1, H_2,\dots,H_n\) such that \(\mathcal{L}(H_i, H_j ) \in X\) for any \(j \neq i - 1, i, i + 1\) then \(\mathcal{L}(H_u, H_{u+1}) \in X\) for some \(u \in \{1, 2,\dots,n\}\), where \(H_0 = H_n\) and \(H_{n+1} = H_{1}\).
Main result. Let \(X\) be a variety of lines. Then \(X\) is aCM if and only if \(X\) has the \(\text{Hyp}_{n}(\star)\)-property for \(n = 4, 5, 6\).
Let us also present, very briefly, another results contained in the paper: in Section 3 the authors introduce a numerical way to check the aCM property for any varieties of lines, in Section 4 the authors study special class of varieties of lines, namely Ferrers varieties of lines (this class, in principle, is a special case of aCM varieties of lines, and these are strictly related to Ferrers diagrams), and in the last section the authors characterize varieties of lines defined by a complete intersection ideal in \(\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}\).

MSC:

13C40 Linkage, complete intersections and determinantal ideals
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
14C20 Divisors, linear systems, invertible sheaves
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14N20 Configurations and arrangements of linear subspaces
13C14 Cohen-Macaulay modules
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References:

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