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Effective invariants of braid monodromy

Authors: Enrique Artal Bartolo, Jorge Carmona Ruber and José Ignacio Cogolludo Agustín
Journal: Trans. Amer. Math. Soc. 359 (2007), 165-183
MSC (2000): Primary 14D05, 14H30, 14H50, 68W30
Published electronically: August 16, 2006
MathSciNet review: 2247887
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Abstract: In this paper we construct new invariants of algebraic curves based on (not necessarily generic) braid monodromies. Such invariants are effective in the sense that their computation allows for the study of Zariski pairs of plane curves. Moreover, the Zariski pairs found in this work correspond to curves having conjugate equations in a number field, and hence are not distinguishable by means of computing algebraic coverings. We prove that the embeddings of the curves in the plane are not homeomorphic. We also apply these results to the classification problem of elliptic surfaces.

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  • 1. H. Abelson, Topologically distinct conjugate varieties with finite fundamental group, Topology 13 (1974), 161-176. MR 0349679 (50:2172)
  • 2. E. Artal, J. Carmona, and J.I. Cogolludo, On sextic curves with big Milnor number, Trends in Singularities (A. Libgober and M. Tibar, eds.), Trends in Mathematics, Birkhäuser Verlag Basel/Switzerland, 2002, pp. 1-29. MR 1900779 (2003d:14034)
  • 3. -, Braid monodromy and topology of plane curves, Duke Math. J. 118 (2003), no. 2, 261-278. MR 1980995 (2004k:14015)
  • 4. E. Artal, J. Carmona, J.I. Cogolludo, and H. Tokunaga, Sextics with singular points in special position, J. Knot Theory Ramifications 10 (2001), no. 4, 547-578. MR 1831676 (2002c:14047)
  • 5. J. S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974, Annals of Mathematics Studies, No. 82.MR 0375281 (51:11477)
  • 6. E. Brieskorn, Automorphic sets and braids and singularities, Braids (Santa Cruz, CA, 1986), Amer. Math. Soc., Providence, RI, 1988, pp. 45-115. MR 0975077 (90a:32024)
  • 7. J. Carmona, Monodromía de trenzas de curvas algebraicas planas, Ph.D. thesis, Universidad de Zaragoza, 2003.
  • 8. A.I. Degtyarëv, Isotopic classification of complex plane projective curves of degree $ 5$, Leningrad Math. J. 1 (1990), no. 4, 881-904. MR 1027461 (91b:14041)
  • 9. M. Fukae, Monodromies of rational elliptic surfaces and extremal elliptic $ {K}3$ surfaces, Preprint available at arXiv:math.AG/0205062.
  • 10. The GAP Group, Aachen, St Andrews, GAP - Groups, Algorithms, and Programming, Version 4.2, 2000, (
  • 11. V. Kharlamov and V. Kulikov, Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 9, 855-859. MR 1873224 (2003c:14047)
  • 12. A. Libgober, Invariants of plane algebraic curves via representations of the braid groups, Invent. Math. 95 (1989), no. 1, 25-30. MR 0969412 (90a:14038)
  • 13. -, Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), Kluwer Acad. Publ., Dordrecht, 2001, pp. 215-254. MR 1866902 (2003e:14008)
  • 14. R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537-558. MR 0867347 (88a:14044)
  • 15. U. Persson, Double sextics and singular $ K$-$ 3$ surfaces, Algebraic geometry, Sitges (Barcelona), 1983, Lecture Notes in Math., vol. 1124, Springer, Berlin, 1985, pp. 262-328. MR 0805337 (87i:14036)
  • 16. J. P. Serre, Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris Sér. I Math. 258 (1964), 4194-4196.MR 0166197 (29:3474)
  • 17. I. Shimada and D.-Q. Zhang, Classification of extremal elliptic $ {K}3$ surfaces and fundamental groups of open $ {K}3$ surfaces, Nagoya Math. J. 161 (2001), 23-54.MR 1820211 (2002d:14056)
  • 18. J.-G. Yang, Sextic curves with simple singularities, Tohoku Math. J. (2) 48 (1996), no. 2, 203-227. MR 1387816 (98e:14026)

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Additional Information

Enrique Artal Bartolo
Affiliation: Departamento de Matemáticas, Campus Plaza de San Francisco s/n, E-50009 Zara- goza, Spain

Jorge Carmona Ruber
Affiliation: Departamento de Sistemas Informáticos y Programación, Universidad Complutense, Ciudad Universitaria s/n, E-28040 Madrid, Spain

José Ignacio Cogolludo Agustín
Affiliation: Departamento de Matemáticas, Campus Plaza de San Francisco s/n, E-50009 Zara- goza, Spain

Keywords: Braid monodromy, plane curve, group representations
Received by editor(s): January 26, 2004
Received by editor(s) in revised form: October 13, 2004
Published electronically: August 16, 2006
Additional Notes: The first and third authors were partially supported by MTM2004-08080-C02-02. The second author was partially supported by MTM2004-08080-C02-01
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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