Configurations of points and topology of real line arrangements



A central question in the study of line arrangements in the complex projective plane \(\mathbb {C}\mathbb {P}^2\) is the following: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, called the chamber weight. This invariant is based on the weight counting over the points of the associated dual configuration, located in particular chambers of the real projective plane \(\mathbb {R}\mathbb {P}^2\). Using this dual setting, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in \(\mathbb {C}\mathbb {P}^2\) (i.e. Zariski pairs) which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over \(\mathbb {Q}\) and containing only double and triple points. For each one of our examples, we derive some degenerations containing points of multiplicity 2, 3 and 5, which are also Zariski pairs. We compute explicitly the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain three geometric characterizations of these components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.

Mathematics Subject Classification

52C30 52C35 32Q55 54F65 32S22 



The main part of this work was carried out during the second author’s visit to Japan. He would like to thank Tokyo Gakugei University, as well as the first author and his wife for their hospitality and the grant MTM2013-45710-C02-01-P for the travel support. Both authors would like to thank Hokkaido University and the organizers of the Summer Conference on Hyperplane Arrangements in Sapporo 2016, as important advances of the present work were made during this week. In particular, we are grateful to Prof. Yoshinaga and Prof. Falk for rewarding discussions. We also would like to thank Prof. Artal and Prof. Cogolludo for the subsequent fruitful comments, in particular those about the properties of these new Zariski pairs.


  1. 1.
    Artal Bartolo, E.: Combinatorics and topology of line arrangements in the complex projective plane. Proc. Am. Math. Soc. 121(2), 385–390 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artal Bartolo, E.: Sur les couples de Zariski. J. Algebraic Geom. 3(2), 223–247 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Artal Bartolo, E.: Topology of arrangements and position of singularities. Ann. Fac. Sci. Toulouse Math. 23(2), 223–265 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Artal Bartolo, E., Cogolludo-Agustín, J.I., Guerville-Ballé, B., Marco-Buzunáriz, M.: An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 111(2), 377–402 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Artal Bartolo, E., Cogolludo-Agustín, J.I., Tokunaga, H.-o.: A survey on Zariski pairs. In Algebraic geometry in East Asia—Hanoi 2005, vol. 50 of Adv. Stud. Pure Math., pp. 1–100. Math. Soc. Tokyo, Japan, (2008)Google Scholar
  6. 6.
    Artal Bartolo, E., Carmona Ruber, J., Cogolludo-Agustín, J.I., Marco Buzunáriz, M.: Topology and combinatorics of real line arrangements. Compos. Math. 141(6), 1578–1588 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Artal Bartolo, E., Florens, V., Guerville-Ballé, B.: A topological invariant of line arrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVII(3), 949–968 (2017)MathSciNetMATHGoogle Scholar
  8. 8.
    Artal Bartolo, E., Guerville-Ballé, B., Viu-Sos, J.: Fundamental groups of real arrangements and torsion in the lower central series quotients. Exp. Math. (2018).
  9. 9.
    Arvola, W.A.: The fundamental group of the complement of an arrangement of complex hyperplanes. Topology 31(4), 757–765 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bailet, P., Settepanella, S.: Homology graph of real arrangements and monodromy of Milnor fiber. Adv. in Appl. Math. 90, 46–85 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carmona-Ruber, J.: Monodromía de trenzas de curvas algebraicas planas. PhD thesis, Universidad de Zaragoza (2003)Google Scholar
  12. 12.
    Cohen, D.C., Suciu, A.I.: On Milnor fibrations of arrangements. J. London Math. Soc. 51(1), 105–119 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cohen, D.C., Suciu, A.I.: Characteristic varieties of arrangements. Math. Proc. Camb. Philos. Soc. 127(1), 33–53 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dimca, A.: Monodromy of triple point line arrangements. In Singularities in geometry and topology 2011, vol. 66 of Adv. Stud. Pure Math., pp. 71–80. Math. Soc. Tokyo, Japan (2015)Google Scholar
  15. 15.
    Dimca, A.: Hyperplane Arrangements. Universitext. Springer, Cham (2017). (an introduction)CrossRefMATHGoogle Scholar
  16. 16.
    Dimca, A., Papadima, S.: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments. Ann. of Math. 158(2), 473–507 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Guerville-Ballé, B.: An arithmetic Zariski 4-tuple of twelve lines. Geom. Topol. 20(1), 537–553 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Guerville-Ballé, B.: Multiplicativity of the \(\cal{I}\)-invariant and topology of glued arrangements. J. Math. Soc. Japan 70(1), 215–227 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Guerville-Ballé, B., Meilhan, J.-B.: A linking invariant for algebraic curves (2016). arXiv:1602.04916
  20. 20.
    Grünbaum, B.: Configurations of Points and Lines, Volume 103 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2009)Google Scholar
  21. 21.
    Hironaka, E.: Abelian coverings of the complex projective plane branched along configurations of real lines. Mem. Am. Math. Soc. 105(502), vi+85 (1993)MathSciNetMATHGoogle Scholar
  22. 22.
    Van Kampen, E.R.: On the fundamental group of an algebraic curve. Am. J. Math. 55(1–4), 255–260 (1933)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Libgober, A.: Eigenvalues for the monodromy of the Milnor fibers of arrangements. In Trends in singularities, Trends Math., pp. 141–150. Birkhäuser, Basel (2002)Google Scholar
  24. 24.
    Libgober, A.: On combinatorial invariance of the cohomology of the Milnor fiber of arrangements and the Catalan equation over function fields. In Arrangements of hyperplanes—Sapporo 2009, vol. 62 of Adv. Stud. Pure Math., pp. 175–187. Math. Soc. Tokyo, Japan (2012)Google Scholar
  25. 25.
    Moishezon, B.G.: Stable branch curves and braid monodromies. In Algebraic geometry (Chicago, Ill., 1980), vol. 862 of Lecture Notes in Math., pp. 107–192. Springer, Berlin-New York (1981)Google Scholar
  26. 26.
    Nazir, S., Yoshinaga, M.: On the connectivity of the realization spaces of line arrangements. Ann. Sc. Norm. Super. Pisa Cl. Sci. 11(4), 921–937 (2012)MathSciNetMATHGoogle Scholar
  27. 27.
    Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Invent. Math. 56(2), 167–189 (1980)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Orlik, P., Terao, H.: Arrangements of hyperplanes, Volume 300 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Springer, Berlin (1992)Google Scholar
  29. 29.
    Papadima, S., Suciu, A.I.: The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy. Proc. Lond. Math. Soc. 114(6), 961–1004 (2017)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rybnikov, G.L.: On the fundamental group of the complement of a complex hyperplane arrangement. Funktsional. Anal. i Prilozhen., 45(2):71–85 (2011). arXiv:math.AG/9805056
  31. 31.
    Suciu, A.I.: Fundamental groups of line arrangements: enumerative aspects. In Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), vol. 276 of Contemp. Math., pp. 43–79. Amer. Math. Soc., Providence (2001)Google Scholar
  32. 32.
    Yoshinaga, M.: Milnor fibers of real line arrangements. J. Singul. 7, 220–237 (2013)MathSciNetMATHGoogle Scholar
  33. 33.
    Yoshinaga, M.: Resonant bands, local systems and Milnor fibers of real line arrangements. In Combinatorial methods in topology and algebra, vol. 12 of Springer INdAM Ser., pp. 143–148. Springer, Cham (2015)Google Scholar
  34. 34.
    Zariski, O.: On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve. Am. J. Math. 51(2), 305–328 (1929)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zariski, O.: On the irregularity of cyclic multiple planes. Ann. Math. 32(3), 485–511 (1931)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zariski, O.: On the Poincaré group of rational plane curves. Am. J. Math. 58(3), 607–619 (1936)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de Ciências Matemáticas e de Computação University of São Paulo, Avenida Trabalhador SancarlenseSão CarlosBrazil

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