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On the strong Lefschetz problem for uniform powers of general linear forms in $ k[x,y,z]$

Authors: Juan C. Migliore and Rosa María Miró-Roig
Journal: Proc. Amer. Math. Soc. 146 (2018), 507-523
MSC (2010): Primary 13D40; Secondary 13D02, 13E10
Published electronically: September 6, 2017
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Abstract: Schenck and Seceleanu proved that if $ R = k[x,y,z]$, where $ k$ is an infinite field and $ I$ is an ideal generated by any collection of powers of linear forms, then multiplication by a general linear form $ L$ induces a homomorphism of maximal rank from any component of $ R/I$ to the next. That is, $ R/I$ has the weak Lefschetz property. Considering the more general strong Lefschetz problem of when $ \times L^j$ has maximal rank for $ j \geq 2$, we give the first systematic study of this problem. We assume that the linear forms are general and that the powers are all the same, i.e. that $ I$ is generated by uniform powers of general linear forms. We prove that for any number of such generators, $ \times L^2$ always has maximal rank. We then specialize to almost complete intersections, i.e. to four generators, and we show that for $ j = 3,4,5$ the behavior depends on the uniform exponent and on $ j$ in a way that we make precise. In particular, there is always at most one degree where $ \times L^j$ fails maximal rank. Finally, we note that experimentally all higher powers of $ L$ fail maximal rank in at least two degrees.

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  • [1] J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201-222. MR 1311347
  • [2] C. Almeida and A. Andrade, Lefschetz property and powers of linear forms in $ \mathbb{K}[x,y,z]$, preprint 2017, arXiv:1703.07598.
  • [3] David J. Anick, Thin algebras of embedding dimension three, J. Algebra 100 (1986), no. 1, 235-259. MR 839581,
  • [4] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at
  • [5] D. Cook II, B. Harbourne, J. Migliore, and U. Nagel, Line arrangements and configurations of points with an unusual geometric property, preprint 2017, arXiv:1602.02300v2.
  • [6] S. Cooper, G. Fatabbi, E. Guardo, A. Lorenzini, J. Migliore, U. Nagel, A. Seceleanu, J. Szpond, and A. Van Tuyl, Symbolic powers of codimension two Cohen-Macaulay ideals, preprint, 2016, arXiv:1606.00935v1.
  • [7] E. D. Davis, A. V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), no. 4, 593-597. MR 776185,
  • [8] Cindy De Volder and Antonio Laface, On linear systems of $ \mathbb{P}^3$ through multiple points, J. Algebra 310 (2007), no. 1, 207-217. MR 2307790,
  • [9] Roberta Di Gennaro, Giovanna Ilardi, and Jean Vallès, Singular hypersurfaces characterizing the Lefschetz properties, J. Lond. Math. Soc. (2) 89 (2014), no. 1, 194-212. MR 3174740,
  • [10] Marcin Dumnicki, An algorithm to bound the regularity and nonemptiness of linear systems in $ \mathbb{P}^n$, J. Symbolic Comput. 44 (2009), no. 10, 1448-1462. MR 2543429,
  • [11] J. Emsalem and A. Iarrobino, Inverse system of a symbolic power. I, J. Algebra 174 (1995), no. 3, 1080-1090. MR 1337186,
  • [12] 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,
  • [13] Tadahito Harima, Juan C. Migliore, Uwe Nagel, and Junzo Watanabe, The weak and strong Lefschetz properties for Artinian $ K$-algebras, J. Algebra 262 (2003), no. 1, 99-126. MR 1970804,
  • [14] C. Huneke and B. Ulrich, Powers of licci ideals, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 339-346. MR 1015526,
  • [15] Antonio Laface and Luca Ugaglia, On a class of special linear systems of $ \mathbb{P}^3$, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5485-5500. MR 2238923,
  • [16] J. Migliore and R. M. Miró-Roig, On the minimal free resolution of $ n+1$ general forms, Trans. Amer. Math. Soc. 355 (2003), no. 1, 1-36. MR 1928075,
  • [17] 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,
  • [18] 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,
  • [19] J. Migliore and U. Nagel, The Lefschetz question for ideals generated by powers of arbitrarily many linear forms in $ K[x,y,z]$, preprint 2017, arXiv:1703.07456.
  • [20] Masayoshi Nagata, On the fourteenth problem of Hilbert, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 459-462. MR 0116056
  • [21] Hal Schenck and Alexandra Seceleanu, The weak Lefschetz property and powers of linear forms in $ \mathbb{K}[x,y,z]$, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2335-2339. MR 2607862,
  • [22] 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,
  • [23] 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

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Additional Information

Juan C. Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Rosa María Miró-Roig
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barce-lona, Spain

Received by editor(s): November 18, 2016
Received by editor(s) in revised form: March 28, 2017
Published electronically: September 6, 2017
Additional Notes: The first author was partially supported by Simons Foundation grant #309556
Communicated by: Irena Peeva
Article copyright: © Copyright 2017 American Mathematical Society

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