대한수학회지 (Journal of the Korean Mathematical Society)

  • 제55권4호
  • /
  • Pages.763-783
  • /
  • 2018
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

AN ARTINIAN POINT-CONFIGURATION QUOTIENT AND THE STRONG LEFSCHETZ PROPERTY

  • Kim, Young Rock (Major in Mathematics Education Graduate School of Education Hankuk University of Foreign Studies) ;
  • Shin, Yong-Su (Department of Mathematics Sungshin Women's University)
  • 투고 : 2017.01.07
  • 심사 : 2018.03.27
  • 발행 : 2018.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we study an Artinian point-configuration quotient having the SLP. We show that an Artinian quotient of points in $\mathbb{p}^n$ has the SLP when the union of two sets of points has a specific Hilbert function. As an application, we prove that an Artinian linear star configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ and $\mathbb{Y}$ are linear starconfigurations in $\mathbb{p}^2$ of type s and t for $s{\geq}(^t_2)-1$ and $t{\geq}3$. We also show that an Artinian $\mathbb{k}$-configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ is a $\mathbb{k}$-configuration of type (1, 2) or (1, 2, 3) in $\mathbb{p}^2$, and $\mathbb{X}{\cup}\mathbb{Y}$ is a basic configuration in $\mathbb{p}^2$.

키워드

과제정보

연구 과제 주관 기관 : NRF (Korea)

참고문헌

  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. Korean Math. Soc. 49 (2012), no. 2, 405-417. https://doi.org/10.4134/JKMS.2012.49.2.405
  2. E. Carlini, E. Guardo, and A. Van Tuyl, Star configurations on generic hypersurfaces, J. Algebra 407 (2014), 1-20. https://doi.org/10.1016/j.jalgebra.2014.02.013
  3. E. Carlini and A. Van Tuyl, Star configuration points and generic plane curves, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4181-4192. https://doi.org/10.1090/S0002-9939-2011-11204-8
  4. M. V. Catalisano, A. V. Geramita, A. Gimigliano, B. Habourne, J. Migliore, U. Nagel, and Y. S. Shin, Secant varieties to the varieties of reducible hypersurfaces in ${\mathbb{P}^n}$, J. of Alg. Geo. submitted.
  5. M. V. Catalisano, A. V. Geramita, A. Gimigliano, and Y. S. Shin, The secant line variety to the varieties of reducible plane curves, Ann. Mat. Pura Appl. (4) 195 (2016), no. 2, 423-443. https://doi.org/10.1007/s10231-014-0470-y
  6. A. V. Geramita, B. Harbourne, and J. Migliore, Star configurations in ${\mathbb{P}^n}$, J. Algebra 376 (2013), 279-299. https://doi.org/10.1016/j.jalgebra.2012.11.034
  7. A. V. Geramita, B. Harbourne, J. C. Migliore, and U. Nagel, Matroid configurations and symbolic powers of their ideals, In preparation.
  8. 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.
  9. 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
  10. A. V. Geramita, J. Migliore, and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in ${\mathbb{P}^2}$, J. Algebra 298 (2006), no. 2, 563-611. https://doi.org/10.1016/j.jalgebra.2006.01.035
  11. A. V. Geramita and Y. S. Shin, k-configurations in ${\mathbb{P}^3}$ all have extremal resolutions, J. Algebra 213 (1999), no. 1, 351-368. https://doi.org/10.1006/jabr.1998.7651
  12. T. Harima, Some examples of unimodal Gorenstein sequences, J. Pure Appl. Algebra 103 (1995), no. 3, 313-324. https://doi.org/10.1016/0022-4049(95)00109-A
  13. T. Harima, T. Maeno, H. Morita, Y. Numata, A.Wachi, and J.Watanabe, The Lefschetz Properties, Lecture Notes in Mathematics, 2080, Springer, Heidelberg, 2013.
  14. T. Harima, J. Migliore, U. Nagel, and J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003), no. 1, 99-126. https://doi.org/10.1016/S0021-8693(03)00038-3
  15. 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
  16. J. Migliore and R. Miro-Roig, Ideals of general forms and the ubiquity of the weak Lefschetz property, J. Pure Appl. Algebra 182 (2003), no. 1, 79-107. https://doi.org/10.1016/S0022-4049(02)00314-6
  17. J. Migliore and R. Miro-Roig, On the strong Lefschetz problem for uniform powers of general linear forms in k[x, y, z], Proc. Amer. Math. Soc. 146 (2018), no. 2, 507-523.
  18. J. Migliore and U. Nagel, Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329-358. https://doi.org/10.1216/JCA-2013-5-3-329
  19. 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), no. 6, 2124-2133. https://doi.org/10.1016/j.jpaa.2014.07.026
  20. Y. S. Shin, Secants to the variety of completely reducible forms and the Hilbert function of the union of star-configurations, J. Algebra Appl. 11 (2012), no. 6, 1250109, 27 pp.
  21. Y. S. Shin, Star-configurations in ${\mathbb{P}^2}$ having generic Hilbert function and the weak Lefschetz property, Comm. Algebra 40 (2012), no. 6, 2226-2242. https://doi.org/10.1080/00927872.2012.656783
  22. Y. S. Shin, Some application of the union of two ${\mathbb{k}}$-configurations in ${\mathbb{P}^2}$, J. of Chungcheong Math. Soc. 27 (2014), no. 3, 413-418. https://doi.org/10.14403/jcms.2014.27.3.413