Heavy hyperplanes in multiarrangements and their freeness


Only few categories of free arrangements are known in which Terao’s conjecture holds. One such category consists of 3-arrangements with unbalanced Ziegler restrictions. In this paper, we generalize this result to arbitrary dimensional arrangements in terms of flags by introducing unbalanced multiarrangements. For that purpose, we generalize several freeness criterions for simple arrangements, including Yoshinaga’s freeness criterion, to unbalanced multiarrangements.

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  1. 1.

    Abe, T.: Divisionally free arrangements of hyperplanes. Math. Invent. 204, 317–346 (2016). https://doi.org/10.1007/s00222-015-0615-7

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Abe, T.: Restrictions of free arrangements and the division theorem arXiv:1603.03863. In: Proceedings of the Intensive Period “Perspectives in Lie Theory” (2016)

  3. 3.

    Abe, T., Numata, Y.: Exponents of 2-multiarrangements and multiplicity lattices. J. Algebr. Comb. 35, 1–17 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Abe, T., Terao, H., Wakefield, M.: The characteristic polynomial of a multiarrangement. Adv. Math. 215, 825–838 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Abe, T., Terao, H., Wakefield, M.: The Euler multiplicity and addition-deletion theorems for multiarrangements. J. Lond. Math. Soc. 77(2), 335–348 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Abe, T., Terao, H., Yoshinaga, M.: Totally free arrangements of hyperplanes. Proc. Am. Math. Soc. 137, 1405–1410 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Abe, T., Yoshigana, M.: Splitting criterion for reflexive sheaves. Proc. Am. Math. Soc. 136(6), 1887–1891 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Abe, T., Yoshigana, M.: Free arrangements and coefficients of characteristic polynomials. Math. Z. 275(3), 911–919 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Edelman, P., Reiner, V.: A counterexample to Orlik’s conjecture. Proc. Am. Math. Soc. 118(3), 927–929 (1993)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kühne, L.: Freeness of hyperplane arrangements with multiplicities, Bachelor Thesis at TU Kaiserslautern (2014)

  11. 11.

    Mustata, M., Schenck, H.: The module of logarithmic \(p\)-forms of a locally free arrangement. J. Algebra 241, 699–719 (2001)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 265–291 (1980)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Solomon, L., Terao, H.: A formula for the characteristic polynomial of an arrangement. Adv. Math. 64(3), 305–325 (1987)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Terao, H.: Multiderivations of Coxeter arrangements. Invent. Math. 148, 659–674 (2002)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Terao, H.: Combinatorial and algebro-geometric properties of free arrangements. In: A lecture at Workshop on “Algebra and Geometry of Configuration Spaces and related structures,” Centro matematica De Giorgi, Pisa, Italy (2010)

  16. 16.

    Wakamiko, A.: On the exponents of 2-multiarrangements. Tokyo J. Math. 30(1), 99–116 (2007)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Wakefield, M., Yuzvinsky, S.: Derivations of an effective divisor on the complex projective line. Trans. Am. Math. Soc. 359(9), 4389–4403 (2007)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Yoshinaga, M.: On the freeness of 3-arrangements. Bull. Lond. Math. Soc. 37(1), 126–134 (2005)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Yoshinaga, M.: Characterization of a free arrangement and conjecture of Edelman and Reiner. Invent. Math. 157(2), 449–454 (2004)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ziegler, G.M.: Multiarrangements of hyperplanes and their freeness. In: Singularities (Iowa City, IA, 1986), 345–359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI (1989)

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The first author is partially supported by JSPS Grant-in-Aid for Young Scientists (B) No. 24740012. The main part of this work was done during a research stay of the second author at Kyoto University. He wishes to thank the German National Academic Foundation for funding this stay and Professor Mathias Schulze for the supervision of his Bachelor thesis at the University of Kaiserslautern.

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Correspondence to Lukas Kühne.

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Abe, T., Kühne, L. Heavy hyperplanes in multiarrangements and their freeness. J Algebr Comb 48, 581–606 (2018). https://doi.org/10.1007/s10801-017-0806-y

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  • Hyperplane arrangements
  • Freeness
  • Multiarrangements
  • Supersolvable arrangements