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Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1 pp 117–125Cite as

Applications

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Abstract

In the previous chapter we described how to compute the bigraded minimal free resolution of I(Z) when Z is a set of double points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) with the property that X = Supp(Z) is ACM. In this situation the bigraded minimal free resolution of I(Z) is a function of the tuple α X  = (α 1, …, α h ) associated with X.

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References

  1. M. Baczyńska, M. Dumnicki, A. Habura, G. Malara, P. Pokora, T. Szemberg, J. Szpond, H. Tutaj-Gasińska, Points fattening on \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and symbolic powers of bi-homogeneous ideals. J. Pure Appl. Algebra 218(8), 1555–1562 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Bocci, B. Harbourne, Comparing powers and symbolic power of ideals. J. Algebraic Geom. 19(3), 399–417 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. C. Bocci, B. Harbourne, The resurgence of ideals of points and the containment problem. Proc. Am. Math. Soc. 138(4), 1175–1190 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Bocci, E. Carlini, B. Harbourne, E. Guardo, Mini-workshop: ideals of linear subspaces, their symbolic powers and waring problems. Oberwolfach Rep., vol. 12, no. 1 (2015)

    Google Scholar 

  5. L. Ein, R. Lazarsfeld, K.E. Smith, Uniform bounds and symbolic powers on smooth varieties. Invent. Math. 144(2), 241–252 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Fløystad, Boij-Söderberg theory: introduction and survey, in Progress in Commutative Algebra 1 (de Gruyter, Berlin, 2012), pp. 1–54

    MATH  Google Scholar 

  7. E. Guardo, Fat point schemes on a smooth quadric. J. Pure Appl. Algebra 162(2–3), 183–208 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Guardo, A. Van Tuyl, 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Guardo, B. Harbourne, A. Van Tuyl, Fat lines in \(\mathbb{P}^{3}\): powers versus symbolic powers. J. Algebra 390, 221–230 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Guardo, B. Harbourne, A. Van Tuyl, Symbolic powers versus regular powers of ideals of general points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). Can. J. Math. 65(4), 823–842 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Harbourne, C. Huneke, Are symbolic powers highly evolved? J. Ramanujan Math. Soc. 28A, 247–266 (2013)

    MathSciNet  MATH  Google Scholar 

  12. J. Herzog, H. Srinivasan, Bounds for multiplicities. Trans. Am. Math. Soc. 350(7), 2879–2902 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Hochster, C. Huneke, Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147(2), 349–369 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Morey, Stability of associated primes and equality of ordinary and symbolic powers of ideals. Commun. Algebra 27(7), 3221–3231 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Peterson, Quasicomplete intersections, powers of ideals, and deficiency modules. J. Algebra 204(1), 1–14 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Römer, Betti numbers and shifts in minimal graded free resolutions. Ill. J. Math. 54(2), 449–467 (2010)

    MathSciNet  MATH  Google Scholar 

  17. O. Zariski, P. Samuel, Commutative Algebra, vol. II. The University Series in Higher Mathematics (D. Van Nostrand, Princeton, 1960)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Catania, Catania, Italy

    Elena Guardo

  2. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    Adam Van Tuyl

Authors
  1. Elena Guardo
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  2. Adam Van Tuyl
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Guardo, E., Van Tuyl, A. (2015). Applications. 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_8

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