Revista Matemática Complutense

, Volume 30, Issue 2, pp 259–268 | Cite as

On the exponents of free and nearly free projective plane curves

  • Alexandru DimcaEmail author
  • Gabriel Sticlaru


We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be \(K(\pi ,1)\) spaces.


Jacobian ideal Tjurina number Free curve Nearly free curve 

Mathematics Subject Classification

Primary 14H50 Secondary 14B05 13D02 32S22 


  1. 1.
    Artal Bartolo, E., Cogolludo-Agustín, J.I., Matei, D.: Quasi-projectivity, Artin-Tits groups, and pencil maps. Topology of algebraic varieties and singularities, 113–136, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, (2011)Google Scholar
  2. 2.
    Artal Bartolo, E., Gorrochategui, L., Luengo, I., Melle-Hernández, A.: On some conjectures about free and nearly free divisors, In: Singularities and Computer Algebra, Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday, pp. 1–19, Springer (2017)Google Scholar
  3. 3.
    Damon, J.N.: Higher Multiplicities and Almost Free Divisors and Complete Intersections, vol. 589. Memoirs A.M.S, Providence (1996)zbMATHGoogle Scholar
  4. 4.
    Dimca, A.: Singularities and Topology of Hpersurfaces, vol. Universitext. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dimca, A.: Freeness versus maximal global Tjurina number for plane curves, to appear in Math. Proc. Cambridge Phil. SocGoogle Scholar
  6. 6.
    Dimca, A.: Curve arrangements, pencils, and Jacobian syzygies. arXiv:1601.00607, to appear in Michigan Math. J
  7. 7.
    Dimca, A., Sernesi, E.: Syzygies and logarithmic vector fields along plane curves. Journal de l’École polytechnique-Mathématiques 1, 247–267 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dimca, A., Sticlaru, G.: Free divisors and rational cuspidal plane curves. arXiv:1504.01242v4, to appear in Math. Res. Lett
  9. 9.
    Dimca, A., Sticlaru, G.: Nearly free divisors and rational cuspidal curves. arXiv:1505.00666v3
  10. 10.
    du Pleseis, A.A., Wall, C.T.C.: Application of the theory of the discriminant to highly singular plane curves. Math. Proc. Camb. Philos. Soc. 126, 259–266 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    du Plessis, A.A., Wall, C.T.C.: Curves in \(P^2(\mathbb{C})\) with 1-dimensional symmetry. Rev. Mat. Complut. 12, 117–132 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Falk, M.: \( K(\pi,1)\) arrangements. Topology 34(1), 141–154 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Falk, M., Randell, R.: On the homotopy theory of arrangements. II. Arrangements-Tokyo 1998. Adv. Stud. Pure Math. 27, Kinokuniya, Tokyo, 93–125 (2000)Google Scholar
  14. 14.
    Hirzebruch, F.: Arrangements of lines and algebraic surfaces, In: Arithmetic and Geometry. Progress in Mathematics, II (36), pp. 113-140. Birkhäuser, Boston (1983)Google Scholar
  15. 15.
    Jambu, M., Terao, H.: Free arrangements of hyperplanes and supersolvable lattices. Adv. Math. 52, 248–258 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mond, D.: Differential forms on free and almost free divisors. Proc. Lond. Math. Soc. 81, 587–617 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Orlik, P., Terao, H.: Arrangements of Hyperplanes. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Randell, R.: Lattice-isotopic arrangements are topologically isomorphic. Proc. Am. Math. Soc. 107, 555–559 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 265–291 (1980)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Simis, A., Tohaneanu, S.O.: Homology of homogeneous divisors. Israel J. Math. 200, 449–487 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yoshinaga, M.: Freeness of hyperplane arrangements and related topics. Annales de la Faculté des Sciences de Toulouse 23(2), 483–512 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRSLJADFrance
  2. 2.Faculty of Mathematics and InformaticsOvidius UniversityConstantaRomania

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