Abstract
The coordinate ring of a finite sets of points in \(\mathbb{P}^{n}\) is always a Cohen-Macaulay ring. However, the multigraded coordinate ring of a set of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\), or more generally, a set of points in \(\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}}\), may fail to have this highly desirable property. This feature is one of the fundamental differences between sets of points in a single projective space and sets of points in a multiprojective space.
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References
W. Bruns, J. Herzog, Cohen-Macaulay Rings (Revised Version) (Cambridge University Press, New York, 1998)
S. Giuffrida, R. Maggioni, A. Ragusa, On the postulation of 0-dimensional subschemes on a smooth quadric. Pac. J. Math. 155(2), 251–282 (1992)
E. Guardo, Schemi di “Fat Points”. PhD Thesis, Università di Messina (2000)
E. Guardo, Fat point schemes on a smooth quadric. J. Pure Appl. Algebra 162(2–3), 183–208 (2001)
E. Guardo, A. Van Tuyl, ACM sets of points in multiprojective space. Collect. Math. 59(2), 191–213 (2008)
E. Guardo, A. Van Tuyl, Classifying ACM sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) via separators. Arch. Math. 99(1), 33–36 (2012)
L. Marino, A characterization of ACM 0-dimensional schemes in Q. Matematiche (Catania) 64(2), 41–56 (2009)
J. Migliore, U. Nagel, Liaison and related topics: notes from the Torino workshop-school. Rend. Sem. Mat. Univ. Politec. Torino 59(2), 59–126 (2001)
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(2), 420–441 (2003)
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Guardo, E., Van Tuyl, A. (2015). Classification of ACM sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) . In: Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-24166-1_4
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DOI: https://doi.org/10.1007/978-3-319-24166-1_4
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