Advertisement

Collectanea Mathematica

, Volume 70, Issue 2, pp 283–294 | Cite as

Complete intersections of quadrics and the Weak Lefschetz Property

  • Alberto Alzati
  • Riccardo ReEmail author
Article
  • 15 Downloads

Abstract

We consider graded artinian complete intersection algebras \(A=\mathbb {C}[x_0,\ldots ,x_m]/I\) with I generated by homogeneous forms of degree \(d\ge 2\). We show that the general multiplication by a linear form \(\mu _L:A_{d-1}\rightarrow A_d\) is injective. We prove that the Weak Lefschetz Property for holds for any c.i. algebra A as above with \(d=2\) and \(m\le 4\), previously known for \(m\le 3\).

Keywords

Weak Lefschetz Property Artinian algebra Complete intersection 

Mathematics Subject Classification

Primary 13A02 Secondary 14N05 

References

  1. 1.
    Boji, M., Migliore, J., Miró-Roig, R.M., Nagel, U.: The non-Lefschetz locus. arXiv:1609.00952 [math.AC]
  2. 2.
    Geramita, A.: Inverse systems of fat points: Waring problem, secant varieties of Veronese varities and parameter spaces for Gorenstein ideals. The Curves Seminar at Queen’s, vol. X (Kingston, ON, 1995), pp. 2–114, Queen’s Papers in Pure and Applied Mathematics, 102, Queen’s University, Kingston, ON (1996)Google Scholar
  3. 3.
    Gondim, R., Zappalà, G.: Lefschetz properties for Artinian Gorenstein algebras presented by quadrics. Proc. Am. Math. Soc. 146, 993–1003 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience Publ., New York (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    Harima, T., Migliore, J., Nagel, U., Watanabe, J.: The weak and strong Lefschetz Properties for Artinian K-algebras. J. Algebra 262, 99–126 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mezzetti, E., Miró-Roig, R., Ottaviani, G.: Laplace equations and the Weak Lefschetz Property. Can. J. Math. 65(3), 634–654 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Migliore, J., Nagel, U.: Gorenstein algebras presented by quadrics. Collect. Math. 64, 211–233 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Migliore, J., Nagel, U.: Survey article: a tour of the Weak and Strong Lefschetz Properties. J. Commut. Algebra 5(3), 329–358 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Stanley, R.: Weyl groups, the Hard Lefschetz Theorem, and the Sperner property. SIAM J. Algebr. Discrete Methods 1, 168–184 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Watanabe, J.: The Dilworth Number of Artinian Rings and Finite Posets with Rank Function, Commutative Algebra and Combinatorics, Advanced Studies in Pure Mathematics, vol. 11, pp. 303–312. Kinokuniya Co., Amsterdam (1987)Google Scholar

Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.Dipartimento di Matematica F.EnriquesUniversità di MilanoMilanItaly
  2. 2.Dipartimento di Scienza e Alta TecnologiaUniversità dell’ InsubriaComoItaly

Personalised recommendations