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Virtual complete intersections in \(\mathbb{P}^1\times\mathbb{P}^1\). (English) Zbl 1441.13031
Summary: The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces \(\mathbb{P}^1\times\mathbb{P}^1\), we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.
MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Software:
Macaulay2
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[1] Ahlfors, Lars, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (1979), McGraw-Hill Inc.: McGraw-Hill Inc. New York · Zbl 0395.30001
[2] Berkesch, Christine; Erman, Daniel; Smith, Gregory G., Virtual resolutions for a product of projective spaces (2017), available at
[3] Cox, David A.; Little, John B.; Schenck, Henry K., Toric Varieties, Graduate Studies in Mathematics, vol. 124 (2011), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1223.14001
[4] Favacchio, Giuseppe; Guardo, Elena; Migliore, Juan, On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces, Proc. Am. Math. Soc., 146, 7, 2811-2825 (2018) · Zbl 1388.13032
[5] Giuffrida, S.; Maggioni, R.; Ragusa, A., On the postulation of 0-dimensional subschemes on a smooth quadric, Pac. J. Math., 155, 2, 251-282 (1992) · Zbl 0723.14035
[6] Giuffrida, S.; Maggioni, R.; Ragusa, A., Resolutions of 0-dimensional subschemes of a smooth quadric, (Zero-Dimensional Schemes. Zero-Dimensional Schemes, Ravello, 1992 (1994), de Gruyter: de Gruyter Berlin), 191-204 · Zbl 0826.14029
[7] Guardo, Elena, Fat points schemes on a smooth quadric, J. Pure Appl. Algebra, 162, 2-3, 183-208 (2001) · Zbl 1044.14025
[8] Guardo, Elena; Van Tuyl, Adam, ACM sets of points in multiprojective space, Collect. Math., 59, 2, 191-213 (2008) · Zbl 1146.13012
[9] Guardo, Elena; Van Tuyl, Adam, Arithmetically Cohen-Macaulay Sets of Points in \(\mathbb{P}^1 \times \mathbb{P}^1\), SpringerBriefs in Mathematics (2015), Springer: Springer Cham · Zbl 1346.13001
[10] Guardo, Elena; Van Tuyl, Adam, Fat points in \(\mathbb{P}^1 \times \mathbb{P}^1\) and their Hilbert functions, Can. J. Math., 56, 4, 716-741 (2004) · Zbl 1092.14057
[11] Guardo, Elena; Van Tuyl, Adam, Separators of arithmetically Cohen-Macaulay fat points in \(\mathbf{P}^1 \times \mathbf{P}^1\), J. Commut. Algebra, 4, 2, 255-268 (2012) · Zbl 1285.13019
[12] Guardo, Elena; Van Tuyl, Adam, Separators of points in a multiprojective space, Manuscr. Math., 126, 1, 99-113 (2008) · Zbl 1145.13007
[13] Guardo, Elena; Van Tuyl, Adam, The minimal resolutions of double points in \(\mathbb{P}^1 \times \mathbb{P}^1\) with ACM support, J. Pure Appl. Algebra, 211, 3, 784-800 (2007) · Zbl 1126.13013
[14] Grayson, Daniel R.; Stillman, Michael E., Macaulay2, a software system for research in algebraic geometry, available at
[15] Peeva, Irena, Graded Syzygies, Algebra and Applications, vol. 14 (2011), Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd. London · Zbl 1213.13002
[16] Shafarevich, Igor R., Basic Algebraic Geometry I: Varieties in Projective Space (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg
[17] Van Tuyl, Adam, The Hilbert functions of ACM sets of points in \(\mathbb{P}^{n_1} \times \ldots \times \mathbb{P}^{n_k} \), J. Algebra, 264, 2, 420-441 (2003) · Zbl 1039.13008
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