Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Position of singularities and fundamental group of the complement of a union of lines


Author: Kwai-Man Fan
Journal: Proc. Amer. Math. Soc. 124 (1996), 3299-3303
MSC (1991): Primary 14H30; Secondary 14H20
MathSciNet review: 1343691
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give two examples of complex line arrangements in $CP^{2}$ with 7 lines, that both have 3 triple points and 12 double points, and their complements have nonisomorphic global fundamental groups. These two line arrangements are, in some sense, a much simpler example of a pair of plane algebraic curves that have the same local topology but have complements with different global topology---compare with the example given by Zariski, or the recent example given by Artal-Bartolo.


References [Enhancements On Off] (What's this?)

  • 1. E. Artal-Bartolo, Sur le premier nombre de Betti de la fibre de Milnor du cone sur une courne projective plane et son rapport avec la position des points singuliers, University of Wisconsin, Madison (preprint 1990).
  • 2. Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180 (94b:32058)
  • 3. E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260.
  • 4. Mutsuo Oka, Some plane curves whose complements have non-abelian fundamental groups, Math. Ann. 218 (1975), no. 1, 55–65. MR 0396556 (53 #419)
  • 5. Mutsuo Oka and Koichi Sakamoto, Product theorem of the fundamental group of a reducible curve, J. Math. Soc. Japan 30 (1978), no. 4, 599–602. MR 513072 (81h:14019), http://dx.doi.org/10.2969/jmsj/03040599
  • 6. Richard Randell, Correction: “The fundamental group of the complement of a union of complex hyperplanes” [Invent. Math. 69 (1982), no. 1, 103–108; MR0671654 (84a:32016)], Invent. Math. 80 (1985), no. 3, 467–468. MR 791670 (87e:32010), http://dx.doi.org/10.1007/BF01388726
  • 7. O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curves, Amer. J. Math. 51 (1929), 305-328.
  • 8. O. Zariski, The topological discriminant group of a Riemann surface of genus $p$, Amer. J. Math. 59 (1937), 335-358.
  • 9. O. Zariski, On the irregularity of cyclic multiple plane, Ann. of Math. 32 (1931), 485--511.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14H30, 14H20

Retrieve articles in all journals with MSC (1991): 14H30, 14H20


Additional Information

Kwai-Man Fan
Affiliation: Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan
Email: kmfan@math.ccu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03487-9
PII: S 0002-9939(96)03487-9
Received by editor(s): May 1, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society