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Lefschetz property and powers of linear forms in 𝕂[x,y,z]

  • Charles Almeida and Aline V. Andrade EMAIL logo
Published/Copyright: November 12, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 4

Abstract

In [9], Migliore, Miró-Roig and Nagel proved that if R=𝕂⁢[x,y,z], where 𝕂 is a field of characteristic zero, and I=(L1a1,…,L4a4) is an ideal generated by powers of four general linear forms, then the multiplication by the square L2 of a general linear form L induces a homomorphism of maximal rank in any graded component of R/I. More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if I=(L1a1,…,Lrar) is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square L2 of a general linear form L induces a homomorphism of maximal rank in any graded component of R/I.

Keywords: Strong Lefschetz property; weak Lefschetz property; power linear forms
MSC 2010: 14C20; 13D40

Communicated by Jan Bruinier

Funding source: Fundação de Amparo à Pesquisa do Estado de São Paulo

Award Identifier / Grant number: 2016/14376-0

Award Identifier / Grant number: 2014/08306-4

Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Award Identifier / Grant number: 99999.000282/2016-02

Funding statement: The first author was supported by FAPESP process numbers 2016/14376-0 and 2014/08306-4, and the second author was supported by CAPES process number 99999.000282/2016-02.

Acknowledgements

This paper was written while we were under supervision of Professor Rosa Maria Miró-Roig at IMUB University of Barcelona. We would like to thank Professor Rosa Maria Miró-Roig for the close support that she provided and for the several suggestions that helped improve this work. We would also like to thank her and IMUB for the warm hospitality during our visit. Furthermore, we would like to thank Darcy Camargo for his help in the proof of inequality (3.3), and the referee for his valuable comments on the paper.

References

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Received: 2017-03-24
Revised: 2017-09-24
Published Online: 2017-11-12
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Distinct distances between points and lines in 𝔽q2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁢(B)-conjecture for π-blocks of π-separable groups
https://doi.org/10.1515/forum-2017-0059
Keywords for this article
Strong Lefschetz property; weak Lefschetz property; power linear forms

Articles in the same Issue

  1. Frontmatter
  2. Distinct distances between points and lines in 𝔽q2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁢(B)-conjecture for π-blocks of π-separable groups
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