Abstract
If\(\mathbb{X}\) is a finite set of points in a multiprojective space\(\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } \) withr ≥ 2, then\(\mathbb{X}\) may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM
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S. Abrescia, L. Bazzotti, and L. Marino, Conductor degree and socle degree,Matematiche (Catania) 56 (2003), 129–148.
L. Bazzotti and M. Casanellas, Separators of points on algebraic surfaces,J. Pure Appl. Algebra 207 (2006), 319–326.
L. Bazzotti, Sets of points and their conductor,J. Algebra 283 (2005), 799–820.
J. Chan, C. Cumming, and H.T. Hà, CohenMacaulay multigraded modules, (preprint 2007), arXiv:0705.1839, to appear inIllinois Journal of Mathematics.
CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it
A.V. Geramita, P. Maroscia, and L. Roberts, The Hilbert function of a reducedk-algebra,J. London Math. Soc. (2)28 (1983), 443–452.
S. Giuffrida, R. Maggioni, and A. Ragusa, On the postulation of 0-dimensional subschemes on a smooth quadric,Pacific J. Math. 155 (1992), 251–282.
S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of 0-dimensional subschemes of a smooth quadric,Zero-dimensional schemes (Ravello, 1992), 191–204, de Gruyter, Berlin, 1994.
S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of generic points lying on a smooth quadric,Manuscripta Math. 91 (1996), 421–444.
E. Guardo,Schemi di „Fat Points”, Ph.D. thesis, Università di Messina, 2000.
E. Guardo, Fat points schemes on a smooth quadric,J. Pure Appl. Algebra 162 (2001), 183–208.
E. Guardo, A survey on fat points on a smooth quadric,Algebraic structures and their representations, 61–87, Contemp. Math.376, Amer. Math. Soc., Providence, RI, 2005.
H.T. Hà and A. Van Tuyl, The regularity of points in multiprojective spaces,J. Pure Appl. Algebra 187 (2004), 153–167.
L. Marino, Conductor and separating degrees for sets of points in ℙr and in ℙ1 × ℙ1,Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)9 (2006), 397–421.
L. Marino, The minimum degree of a surface that passes through all the points of a 0dimensional scheme but a pointP, Algebraic structures and their representations, 315–332, Contemp. Math.376, Amer. Math. Soc., Providence, RI, 2005.
L. Marino,A characterization of ACM 0-dimensional subschemes of ℙ1 × ℙ1, to appear.
F. Orecchia, Points in generic position and conductors of curves with ordinary singularities,J. London Math. Soc. (2)24 (1981), 85–96.
J. Sidman and A. Van Tuyl, Multigraded regularity: syzygies and fat points,Beiträge Algebra Geom.47 (2006), 67–87.
A. Van Tuyl,Sets of points in multiprojective spaces and their Hilbert function, Ph.D. thesis, Queen’s University, 2001.
A. Van Tuyl, The border of the Hilbert function of a set of points in\(\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_k } \),J. Pure Appl. Algebra 176 (2002), 223–247.
A. Van Tuyl, The Hilbert functions of ACM sets of points in\(\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_k } \),J. Algebra 264 (2003), 420–441.
A. Van Tuyl, The defining ideal of a set of points in multiprojective space,J. London Math. Soc. (2)72 (2005), 73–90.
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Guardo, E., Van Tuyl, A. ACM sets of points in multiprojective space. Collect. Math. 59, 191–213 (2008). https://doi.org/10.1007/BF03191367
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DOI: https://doi.org/10.1007/BF03191367