# zbMATH — the first resource for mathematics

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.

##### MSC:
 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)
Full Text:
##### References:
 [1] Baczy\'nska, Magdalena; Dumnicki, Marcin; Habura, Agata; Malara, Grzegorz; Pokora, Piotr; Szemberg, Tomasz; Szpond, Justyna; Tutaj-Gasi\'nska, Halszka, Points fattening on $$\mathbb{P}^1×\mathbb{P}^1$$ and symbolic powers of bi-homogeneous ideals, J. Pure Appl. Algebra, 218, 8, 1555-1562, (2014) · Zbl 1291.14015 [2] Geramita, Anthony V.; Migliore, Juan C., A generalized liaison addition, J. Algebra, 163, 1, 139-164, (1994) · Zbl 0798.14026 [3] Giuffrida, S.; Maggioni, R.; Ragusa, A., On the postulation of $$0$$-dimensional subschemes on a smooth quadric, Pacific J. Math., 155, 2, 251-282, (1992) · Zbl 0723.14035 [4] Guardo, Elena, Fat points schemes on a smooth quadric, J. Pure Appl. Algebra, 162, 2-3, 183-208, (2001) · Zbl 1044.14025 [5] Guardo, Elena; Van Tuyl, Adam, Fat points in $$\mathbb{P}^1×\mathbb{P}^1$$ and their Hilbert functions, Canad. J. Math., 56, 4, 716-741, (2004) · Zbl 1092.14057 [6] Guardo, Elena; Van Tuyl, Adam, Separators of points in a multiprojective space, Manuscripta Math., 126, 1, 99-13, (2008) · Zbl 1145.13007 [7] Guardo, Elena; Van Tuyl, Adam, ACM sets of points in multiprojective space, Collect. Math., 59, 2, 191-213, (2008) · Zbl 1146.13012 [8] Guardo, Elena; Van Tuyl, Adam, Classifying ACM sets of points in $$\mathbb{P}^1 ×\mathbb{P}^1$$ via separators, Arch. Math. (Basel), 99, 1, 33-36, (2012) · Zbl 1263.13011 [9] Guardo, Elena; Van Tuyl, Adam, Separators of arithmetically Cohen-Macaulay fat points in $$\mathbf{P}^1×\mathbf{P}^1$$, J. Commut. Algebra, 4, 2, 255-268, (2012) · Zbl 1285.13019 [10] Guardo, Elena; Van Tuyl, Adam, Arithmetically Cohen-Macaulay sets of points in $$\mathbb{P}^{1} ×\mathbb{P}^{1}$$, SpringerBriefs in Mathematics, viii+134 pp., (2015), Springer, Cham · Zbl 1346.13001 [11] Guardo, Elena; Van Tuyl, Adam, On the Hilbert functions of sets of points in $$\mathbb{P}^1×\mathbb{P}^1×\mathbb{P}^1$$, Math. Proc. Cambridge Philos. Soc., 159, 1, 115-123, (2015) · Zbl 1371.13021 [12] Herzog, J\"urgen, A generalization of the Taylor complex construction, Comm. Algebra, 35, 5, 1747-1756, (2007) · Zbl 1121.13013 [13] Kalai, Gil; Meshulam, Roy, Intersections of Leray complexes and regularity of monomial ideals, J. Combin. Theory Ser. A, 113, 7, 1586-1592, (2006) · Zbl 1105.13026 [14] Migliore, Juan C., Introduction to liaison theory and deficiency modules, Progress in Mathematics 165, xiv+215 pp., (1998), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0921.14033 [15] Kleppe, Jan O.; Migliore, Juan C.; Mir\'o-Roig, Rosa; Nagel, Uwe; Peterson, Chris, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc., 154, 732, viii+116 pp., (2001) · Zbl 1006.14018 [16] Migliore, J.; Nagel, U., Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math., 180, 1, 1-63, (2003) · Zbl 1053.13006 [17] Rauf, Asia, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38, 2, 773-784, (2010) · Zbl 1193.13025 [18] Schenzel, Peter, Notes on liaison and duality, J. Math. Kyoto Univ., 22, 3, 485-498, (1982/83) · Zbl 0506.13012 [19] Villarreal, Rafael H., Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics 238, x+455 pp., (2001), Marcel Dekker, Inc., New York · Zbl 1002.13010 [20] Vasconcelos, Wolmer V., Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series 195, viii+329 pp., (1994), Cambridge University Press, Cambridge · Zbl 0813.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.