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On Some Conjectures About Free and Nearly Free Divisors

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

Abstract

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.

Keywords

Free divisors Nearly free curves 

Subject Classifications:

14A05 14R15 

Notes

Acknowledgements

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.

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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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