Lefschetz properties for complete intersection ideals generated by products of linear forms

Juhnke-Kubitzke, Martina; Miró-Roig, Rosa; Murai, Satoshi; Wachi, Akihito

doi:10.1090/proc/14009

Lefschetz properties for complete intersection ideals generated by products of linear forms
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by Martina Juhnke-Kubitzke, Rosa M. Miró-Roig, Satoshi Murai and Akihito Wachi PDF

Proc. Amer. Math. Soc. 146 (2018), 3249-3256 Request permission

Abstract:

In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.

References

Abed Abedelfatah, On the Eisenbud-Green-Harris conjecture, Proc. Amer. Math. Soc. 143 (2015), no. 1, 105–115. MR 3272735, DOI 10.1090/S0002-9939-2014-12216-7
C. Almeida and A.V. Andrade, Lefschetz property and powers of linear forms in $K[x,y,z]$, Forum Math., to appear.
D. Grayson and M. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu.Macaulay2/
Brian Harbourne, Hal Schenck, and Alexandra Seceleanu, Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712–730. MR 2855798, DOI 10.1112/jlms/jdr033
Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, and Junzo Watanabe, The Lefschetz properties, Lecture Notes in Mathematics, vol. 2080, Springer, Heidelberg, 2013. MR 3112920, DOI 10.1007/978-3-642-38206-2
J. Herzog and D. Popescu, The strong Lefschetz property and simple extensions, preprint. Available on the arXiv at http://front.math.ucdavis.edu/0506.5537.
Hidemi Ikeda, Results on Dilworth and Rees numbers of Artinian local rings, Japan. J. Math. (N.S.) 22 (1996), no. 1, 147–158. MR 1394376, DOI 10.4099/math1924.22.147
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Amer. Math. Soc. 363 (2011), no. 1, 229–257. MR 2719680, DOI 10.1090/S0002-9947-2010-05127-X
Juan C. Migliore, Rosa M. Miró-Roig, and Uwe Nagel, On the weak Lefschetz property for powers of linear forms, Algebra Number Theory 6 (2012), no. 3, 487–526. MR 2966707, DOI 10.2140/ant.2012.6.487
Juan Migliore and Uwe Nagel, Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra 5 (2013), no. 3, 329–358. MR 3161738, DOI 10.1216/JCA-2013-5-3-329
Rosa M. Miró-Roig, Harbourne, Schenck and Seceleanu’s conjecture, J. Algebra 462 (2016), 54–66. MR 3519499, DOI 10.1016/j.jalgebra.2016.05.020
Les Reid, Leslie G. Roberts, and Moshe Roitman, On complete intersections and their Hilbert functions, Canad. Math. Bull. 34 (1991), no. 4, 525–535. MR 1136655, DOI 10.4153/CMB-1991-083-9
Hal Schenck and Alexandra Seceleanu, The weak Lefschetz property and powers of linear forms in $\Bbb K[x,y,z]$, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2335–2339. MR 2607862, DOI 10.1090/S0002-9939-10-10288-3
Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021
Junzo Watanabe, The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 303–312. MR 951211, DOI 10.2969/aspm/01110303
Attila Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (2004), no. 12, 4601–4611. MR 2111103, DOI 10.1081/AGB-200036809