# zbMATH — the first resource for mathematics

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
CoCoA
Full Text:
##### 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
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.