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The Hilbert functions of ACM sets of points in \({\mathbb P}^{n_1} {\times}\dots{\times}{\mathbb P}^{n_k}\). (English) Zbl 1039.13008
Given a set of points \(\mathbb X\) in a multiprojective space \(\mathbb P^{n_1}\times\cdots\times \mathbb P^{n_k}\), one can form its homogeneous coordinate ring \(R/I_{\mathbb X}\), where \(R= k[x_{10},\dots, x_{1n_1},\dots, x_{k0},\dots, x_{kn_k}]\). This ring is graded by \(\mathbb N^k\) and we can define the Hilbert function \(H_{\mathbb X}: \mathbb N^k\to \mathbb N\) by \(H_{\mathbb X}(i)= \dim_k(R/I_{\mathbb X})_i\) for \(i\in \mathbb N^k\). The author continues his study of these Hilbert functions [see A. Van Tuyl, J. Pure Appl. Algebra 176, No. 2–3, 223–247 (2002; Zbl 1019.13008)]. Except for the case \(k= 1\), a complete classification is not known, partly because the homogeneous coordinate ring \(R/I_X\) need not be a Cohen-Macaulay ring, i.e. \(\mathbb X\) need not be arithmetically Cohen-Macaulay (aCM).
Under the additional hypothesis that \(\mathbb X\) is aCM, the author succeeds in generalizing the proof for the case \(k= 1\) as given by A. V. Geramita, D. Gregory and L. Roberts [J. Pure Appl. Algebra 40, 33–62 (1986; Zbl 0586.13015)] and proves that \(\mathbb H\) is the Hilbert function of an aCM set of points in \(\mathbb P^{n_1}\times\cdots\times \mathbb P^{n_k}\) if and only if \(\Delta\mathbb H\) is the Hilbert function of an \(\mathbb N^k\)-graded Artinian quotient of \(R\). Unfortunately, a satisfactory characterization of those functions \(\Delta\mathbb H\) is not known either. At least, the author is able to finish his project in the case \(n_1=\cdots= n_k= 1\). In particular, for aCM point sets \(\mathbb X\subseteq \mathbb P^1\times \mathbb P^1\) he is able to characterize all possible Hilbert functions and to compute the graded Betti numbers from certain numerical information about the set \(\mathbb X\).

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13C14 Cohen-Macaulay modules
14N05 Projective techniques in algebraic geometry
Software:
CoCoA
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References:
[1] Bruns, W.; Herzog, J., Cohen – macaulay rings, (1998), Cambridge Univ. Press New York · Zbl 0909.13005
[2] A. Capani, G. Niesi, L. Robbiano, \scCoCoA, a System for Doing Computations in Commutative Algebra, Available via anonymous ftp from: http://cocoa.dima.unige.it
[3] Geramita, A.V.; Gregory, D.; Roberts, L., Monomial ideals and points in projective space, J. pure appl. algebra, 40, 33-62, (1986) · Zbl 0586.13015
[4] Geramita, A.V.; Maroscia, P.; Roberts, L., The Hilbert function of a reduced k-algebra, J. London math. soc., 28, 443-452, (1983) · Zbl 0535.13012
[5] Giuffrida, S.; Maggioni, R.; Ragusa, A., On the postulation of 0-dimensional subschemes on a smooth quadric, Pacific J. math., 155, 251-282, (1992) · Zbl 0723.14035
[6] E. Guardo, A. Van Tuyl, Fat points in \(P\^{}\{1\}×P\^{}\{1\}\) and their Hilbert functions, Canad. J. Math. (2003), to appear
[7] Macaulay, F.S., Some properties of enumeration in the theory of modular systems, Proc. London math. soc., 26, 531-555, (1927) · JFM 53.0104.01
[8] Migliore, J., Introduction to liaison theory and deficiency modules, (1998), Birkhäuser Boston · Zbl 0921.14033
[9] Migliore, J.; Nagel, U., Lifting monomial ideals, Comm. algebra, 28, 5679-5701, (2000) · Zbl 1003.13005
[10] Roberts, P., Multiplicities and Chern classes in local algebra, Cambridge tracts in math., 133, (1998), Cambridge Univ. Press Cambridge · Zbl 0917.13007
[11] Stanley, R., Hilbert functions of graded algebras, Adv. math., 28, 57-83, (1978) · Zbl 0384.13012
[12] A. Van Tuyl, Sets of points in multi-projective spaces and their Hilbert function, PhD thesis, Queen’s University, 2001
[13] Van Tuyl, A., The border of the Hilbert function of a set of points in \(P\^{}\{n1\}×⋯×P\^{}\{nk\}\), J. pure appl. algebra, 176, 223-247, (2002) · Zbl 1019.13008
[14] Weibel, C., An introduction to homological algebra, (1994), Cambridge Univ. Press New York · Zbl 0797.18001
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