On supersolvable and nearly supersolvable line arrangements

  • Alexandru DimcaEmail author
  • Gabriel Sticlaru


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.


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 


  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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  6. 6.
    Dimca, A.: Freeness versus maximal global Tjurina number for plane curves. Math. Proc. Cambridge Philos. Soc. 163, 161–172 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dimca, A.: Curve arrangements, pencils, and Jacobian syzygies. Michigan Math. J. 66, 347–365 (2017)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dimca, A., Sticlaru, G.: Free divisors and rational cuspidal plane curves. Math. Res. Lett. 24, 1023–1042 (2017)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math. 52, 248–258 (1984)MathSciNetCrossRefGoogle 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)zbMATHGoogle Scholar
  19. 19.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin, Heidelberg, New York (1992)CrossRefGoogle Scholar
  20. 20.
    Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Scott, P.: On the sets of directions determined by n points. Amer. Math. Monthly 77, 502–505 (1970)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tohăneanu, S.O.: A computational criterion for supersolvability of line arrangements. Ars Combin. 117, 217–223 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Tohăneanu, S.O.: Projective duality of arrangements with quadratic logarithmic vector fields. Discrete Math. 339, 54–61 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ungar, P.: 2N noncollinear points determine at least 2N directions. J. Combin. Theory Ser. A 33, 343–347 (1982)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yoshinaga, M.: Freeness of hyperplane arrangements and related topics. Ann. Fac. Sci. Toulouse Math. (6) 23(2), 483–512 (2014)Google Scholar
  26. 26.
    Ziegler, G.: Combinatorial construction of logarithmic differential forms. Adv. Math. 76, 116–154 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, LJADUniversité Côte d’AzurNiceFrance
  2. 2.Faculty of Mathematics and InformaticsOvidius UniversityConstantaRomania

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