On supersolvable and nearly supersolvable line arrangements


We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian syzygy for the defining equation of the line arrangement, which is a subtle algebraic invariant, is determined in this case by the combinatorics. When such a line arrangement is nearly free, we discuss the splitting types and the jumping lines of the associated rank two vector bundle, as well as the corresponding jumping points, introduced recently by S. Marchesi and J. Vallès. As a by-product of our results, we get a version of the Slope Problem, valid over the real and the complex numbers as well.

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  1. 1.

    Abe, T., Dimca, A.: On the splitting types of bundles of logarithmic vector fields along plane curves. Internat. J. Math. 29, 1850055 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Anzis, B., Tohăneanu, S.O.: On the geometry of real and complex supersolvable line arrangements. J. Combin. Theory Ser. A 140, 76–96 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cook, D., Harbourne, B., Migliore, J., Nagel, U.: Line arrangements and configurations of points with an unexpected geometric property. Compos. Math. 154, 2150–2194 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Di Marca, M., Malara, G., Oneto, A.: Unexpected curves arising from special line arrangements. arXiv:1804.02730

  5. 5.

    Dimca, A.: Hyperplane Arrangements: An Introduction. Universitext. Springer, New York (2017)

    Google Scholar 

  6. 6.

    Dimca, A.: Freeness versus maximal global Tjurina number for plane curves. Math. Proc. Cambridge Philos. Soc. 163, 161–172 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dimca, A.: Curve arrangements, pencils, and Jacobian syzygies. Michigan Math. J. 66, 347–365 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dimca, A., Ibadula, D., Măcinic, A.: Numerical invariants and moduli spaces for line arrangements. arXiv:1609.06551

  9. 9.

    Dimca, A., Sernesi, E.: Syzygies and logarithmic vector fields along plane curves. J. Éc. polytech. Math. 1, 247–267 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Dimca, A., Sticlaru, G.: On the exponents of free and nearly free projective plane curves. Rev. Mat. Complut. 30, 259–268 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dimca, A., Sticlaru, G.: Free divisors and rational cuspidal plane curves. Math. Res. Lett. 24, 1023–1042 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dimca, A., Sticlaru, G.: Free and nearly free curves vs. rational cuspidal plane curves. Publ. Res. Inst. Math. Sci. 54, 163–179 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Dimca, A., Sticlaru, G.: On the jumping lines of bundles of logarithmic vector fields along plane curves. arXiv:1804.06349

  14. 14.

    du Plessis, A.A., Wall, C.T.C.: Application of the theory of the discriminant to highly singular plane curves. Math. Proc. Cambridge Philos. Soc. 126, 259–266 (1999)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Faenzi, D., Vallès, J.: Logarithmic bundles and line arrangements, an approach via the standard construction. J. Lond. Math. Soc. 90, 675–694 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math. 52, 248–258 (1984)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Marchesi, S., Vallès, J.: Nearly free curves and arrangements: a vector bundle point of view. arXiv:1712.04867

  18. 18.

    Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. With an Appendix by S. I. Gelfand. Modern Birkhäuser Classics. Birkhäuser, Basel (1980)

    Google Scholar 

  19. 19.

    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin, Heidelberg, New York (1992)

    Google Scholar 

  20. 20.

    Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Scott, P.: On the sets of directions determined by n points. Amer. Math. Monthly 77, 502–505 (1970)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tohăneanu, S.O.: A computational criterion for supersolvability of line arrangements. Ars Combin. 117, 217–223 (2014)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Tohăneanu, S.O.: Projective duality of arrangements with quadratic logarithmic vector fields. Discrete Math. 339, 54–61 (2016)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ungar, P.: 2N noncollinear points determine at least 2N directions. J. Combin. Theory Ser. A 33, 343–347 (1982)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Yoshinaga, M.: Freeness of hyperplane arrangements and related topics. Ann. Fac. Sci. Toulouse Math. (6) 23(2), 483–512 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ziegler, G.: Combinatorial construction of logarithmic differential forms. Adv. Math. 76, 116–154 (1989)

    MathSciNet  Article  Google Scholar 

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Correspondence to Alexandru Dimca.

Additional information

A. Dimca: This work has been supported by the French government, through the \(\mathrm{UCA}^{\mathrm{JEDI}}\) Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.

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Dimca, A., Sticlaru, G. On supersolvable and nearly supersolvable line arrangements. J Algebr Comb 50, 363–378 (2019). https://doi.org/10.1007/s10801-018-0859-6

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  • Jacobian syzygy
  • Tjurina number
  • Free line arrangement
  • Nearly free line arrangement
  • Slope Problem
  • Terao’s conjecture

Mathematics Subject Classification

  • Primary 14H50
  • Secondary 14B05
  • 13D02
  • 32S22