On Some Conjectures About Free and Nearly Free Divisors

  • Enrique Artal Bartolo
  • Leire Gorrochategui
  • Ignacio Luengo
  • Alejandro Melle-HernándezEmail author


In this paper we provide infinite families of non-rational irreducible free divisors or nearly free divisors in the complex projective plane. Moreover, their corresponding local singularities can have an arbitrary number of branches. All these examples contradict some of the conjectures proposed by Dimca and Sticlaru. Our examples say nothing about the most remarkable conjecture by A. Dimca and G. Sticlaru, which predicts that every rational cuspidal plane curve is either free or nearly free.


Free divisors Nearly free curves 

Subject Classifications:

14A05 14R15 



The first author is partially supported by the Spanish grant MTM2013-45710-C02-01-P and Grupo Geometría of Gobierno de Aragón/Fondo Social Europeo. The last three authors are partially supported by the Spanish grant MTM2013-45710-C02-02-P.


  1. 1.
    Artal, E.: Sur les couples de Zariski. J. Algebraic Geom. 3 (2), 223–247 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Artal, E., Carmona, J.: Zariski pairs, fundamental groups and Alexander polynomials. J. Math. Soc. Jpn. 50 (3), 521–543 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artal, E., Cogolludo-Agustín, J., Ortigas-Galindo, J.: Kummer covers and braid monodromy. J. Inst. Math. Jussieu 13 (3), 633–670 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Artal, E., Dimca, A.: On fundamental groups of plane curve complements. Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2), 255–262 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cogolludo-Agustín, J.: Fundamental group for some cuspidal curves. Bull. Lond. Math. Soc. 31 (2), 136–142 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cogolludo-Agustín, J., Kloosterman, R.: Mordell-Weil groups and Zariski triples. In: Faber, G.F.C., de Jong, R. (eds.) Geometry and Arithmetic, EMS Congress Reports. European Mathematical Society (2012). Also available at arXiv:1111.5703 [math.AG] Google Scholar
  7. 7.
    Daigle, D., Melle-Hernández, A.: Linear systems of rational curves on rational surfaces. Mosc. Math. J. 12 (2), 261–268, 459 (2012)Google Scholar
  8. 8.
    Daigle, D., Melle-Hernández, A.: Linear systems associated to unicuspidal rational plane curves. Osaka J. Math. 51 (2), 481–511 (2014). MathSciNetzbMATHGoogle Scholar
  9. 9.
    Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 4-0-2 — a computer algebra system for polynomial computations (2015).
  10. 10.
    Dimca, A.: Freeness versus maximal global Tjurina number for plane curves. Math. Proc. Camb. Philos. Soc. (2016). Preprint available at arXiv:1508.04954 [math.AG]. doi: 10.1017/S0305004116000803
  11. 11.
    Dimca, A., Sernesi, E.: Syzygies and logarithmic vector fields along plane curves. J. Éc. Polytech. Math. 1, 247–267 (2014). doi: 10.5802/jep.10.
  12. 12.
    Dimca, A., Sticlaru, G.: Free divisors and rational cuspidal plane curves (2015). Preprint available at arXiv:1504.01242v4 [math.AG] Google Scholar
  13. 13.
    Dimca, A., Sticlaru, G.: Nearly free divisors and rational cuspidal curves (2015). Preprint available at arXiv:1505.00666v3 [math.AG] Google Scholar
  14. 14.
    Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A., Némethi, A.: On rational cuspidal projective plane curves. Proc. Lond. Math. Soc. (3) 92 (1), 99–138 (2006). doi: 10.1017/S0024611505015467.
  15. 15.
    Fernández de Bobadilla, J., Luengo, I., Melle-Hernández, A., Némethi, A.: Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair. In: Real and Complex Singularities. Trends in Mathematics, pp. 31–45. Birkhäuser, Basel (2007). doi: 10.1007/978-3-7643-7776-2_4.
  16. 16.
    Flenner, H., Zaĭdenberg, M.: Rational cuspidal plane curves of type (d, d − 3). Math. Nachr. 210, 93–110 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hirano, A.: Construction of plane curves with cusps. Saitama Math. J. 10, 21–24 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lindner, N.: Cuspidal plane curves of degree 12 and their Alexander polynomials. Master’s thesis, Humboldt Universität zu Berlin, Berlin (2012)Google Scholar
  19. 19.
    Orlik, P., Terao, H.: Arrangements of hyperplanes. In: Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)Google Scholar
  20. 20.
    du Plessis, A., Wall, C.T.C.: Application of the theory of the discriminant to highly singular plane curves. Math. Proc. Camb. Philos. Soc. 126 (2), 259–266 (1999). doi: 10.1017/S0305004198003302.
  21. 21.
    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
  22. 22.
    Schenck, H., Tohǎneanu, Ş.: Freeness of conic-line arrangements in \(\mathbb{P}^{2}\). Comment. Math. Helv. 84 (2), 235–258 (2009). doi: 10.4171/CMH/161.
  23. 23.
    Sernesi, E.: The local cohomology of the Jacobian ring. Doc. Math. 19, 541–565 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Stein, W., et al.: Sage Mathematics Software (Version 6.7). The Sage Development Team (2015).
  25. 25.
    Sticlaru, G.: Invariants and rigidity of projective hypersurfaces. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 58 (106)(1), 103–116 (2015)Google Scholar
  26. 26.
    Tono, K.: On rational unicuspidal plane curves with \(\overline{\kappa } = 1\). In: Newton Polyhedra and Singularities (Japanese) (Kyoto, 2001). RIMS Kokyuroku, vol. 1233, pp. 82–89. Kyoto University, Kyoto (2001)Google Scholar
  27. 27.
    Tsunoda, S.: The complements of projective plane curves. In: Commutative Algebra and Algebraic Geometry. RIMS Kokyuroku, vol. 446, pp. 48–56. Kyoto University, Kyoto (1981)Google Scholar
  28. 28.
    Uludağ, A.: More Zariski pairs and finite fundamental groups of curve complements. Manuscripta Math. 106 (3), 271–277 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vallès, J.: Free divisors in a pencil of curves. J. of Singularities 11, 190–197 (2015). Preprint available at arXiv:1502.02416v1 [math.AG] Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Enrique Artal Bartolo
    • 1
  • Leire Gorrochategui
    • 2
  • Ignacio Luengo
    • 3
  • Alejandro Melle-Hernández
    • 3
    Email author
  1. 1.IUMA, Departamento de Matemáticas, Facultad de CienciasUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de Álgebra, Facultad de Ciencias MatemáticasUniversidad ComplutenseMadridSpain
  3. 3.ICMAT (CSIC-UAM-UC3M-UCM), Departamento de Álgebra, Facultad de Ciencias MatemáticasUniversidad ComplutenseMadridSpain

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