Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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THE ARTINIAN COMPLETE INTERSECTION QUOTIENT AND THE STRONG LEFSCHETZ PROPERTY

  • Shin, Yong-Su (Department of Mathematics, Sungshin Women's University)
  • Received : 2019.03.17
  • Accepted : 2019.05.08
  • Published : 2019.05.15
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Abstract

It has been little known when an Artinian (point) quotient has the strong Lefschetz property. In this paper, we find the Artinian complete intersection quotient having the SLP. More precisely, we prove that if ${\mathbb{X}}$ is a complete intersection in ${\mathbb{P}}^2$ of type (2, 2) and ${\mathbb{Y}}$ is a finite set of points in ${\mathbb{P}}^2$ such that ${\mathbb{X}}{\cup}{\mathbb{Y}}$ is a basic configuration of type (2, a) with $a{\geq}3$ or (3, a) with a = 3, 4, 5, 6, then $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP. We also show that if ${\mathbb{X}}$ is a complete intersection in ${\mathbb{P}}^2$ of type (3, 2) and ${\mathbb{Y}}$ is a finite set of points in ${\mathbb{P}}^2$ such that ${\mathbb{X}}{\cup}{\mathbb{Y}}$ is a basic configuration of type (3, 3) or (3, 4), then $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP.

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References

  1. J. Ahn and Y. S. Shin. The Minimal Free Resolution of A Star-Configuration in ${\mathbb{P}}^n$ and The Weak-Lefschetz Property, J. of Korean Math. Soc. 49 (2012), no.2, 405-417. https://doi.org/10.4134/JKMS.2012.49.2.405
  2. A. V. Geramita, T. Harima, and Y. S. Shin. Some Special Configurations of Points in ${\mathbb{P}}^n$, J. Algebra, 268 (2003), no. 2, 484-518. https://doi.org/10.1016/S0021-8693(03)00118-2
  3. A. V. Geramita, T. Harima, J. C. Migliore, and Y. S. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc. 186 (2007), no. 872, vi+139 pp.
  4. A. V. Geramita, T. Harima, and Y. S. Shin. Extremal point sets and Gorenstein ideals, Adv. Math. 152 (2000), no. 1, 78-119. https://doi.org/10.1006/aima.1998.1889
  5. T. Harima, Some examples of unimodal Gorenstein sequences, Journal of Pure and Applied Algebra, 103 (1995) 313-324. https://doi.org/10.1016/0022-4049(95)00109-A
  6. T. Harima, A note on Artinian Gorenstein algebras of codimension three, J. Pure Appl. Algebra, 135 (1999) 45-56. https://doi.org/10.1016/S0022-4049(97)00162-X
  7. T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, and J. Watanabe, The Lefschetz properties, Lecture Notes in Mathematics, 2080. Springer, Heidelberg, 2013.
  8. T. Harima, J. Migliore, U. Nagel, and J. Watanabe, The Weak and Strong Lefschetz Properties for Artinian K-Algebras, J. Algebra, 262 (2003), 99-126. https://doi.org/10.1016/S0021-8693(03)00038-3
  9. A. Iarrobino, P. M. Marques, and C. MaDaniel, Jordan type and the Associated graded algebra of an Artinian Gorenstein algebra, arXiv:1802.07383 (2018).
  10. Y. R. Kim and Y. S. Shin, Star-configurations in ${\mathbb{P}}^n$ and The Weak-Lefschetz Property, Comm. Algebra, 44 (2016), no. 9, 3853-3873. https://doi.org/10.1080/00927872.2015.1027373
  11. Y. R. Kim and Y. S. Shin, An Artinian point-configuration quotient and the strong Lefschetz property, J. Korean Math. Soc. 55 (2018), no. 4, 763-783. https://doi.org/10.4134/JKMS.J170035
  12. J. Migliore, R. Miro-Roig, Ideals of general forms and the ubiquity of the Weak Lefschetz Property, J. Pure Appl. Algebra, 182 (2003), 79-107. https://doi.org/10.1016/S0022-4049(02)00314-6
  13. J. P. Park and Y. S. Shin, The Minimal Free Graded Resolution of A Star-configuration in ${\mathbb{P}}^n$, J. Pure Appl. Algebra, 219 (2015), 2124-2133. https://doi.org/10.1016/j.jpaa.2014.07.026
  14. Y. S. Shin, Secants to The Variety of Completely Reducible Forms and The Union of Star-Configurations, J. Algebra Appl. 11 (2012), 27 pages.
  15. Y. S. Shin, Star-Configurations in ${\mathbb{P}}^2$ Having Generic Hilbert Functions and The Weak-Lefschetz Property, Comm. in Algebra, 40 (2012), 2226-2242. https://doi.org/10.1080/00927872.2012.656783
  16. J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Adv. Stud. Pure Math. 11 (1987), 303-312. https://doi.org/10.2969/aspm/01110303