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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 \(\mathbb X\).

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
Full Text: DOI
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