Position of singularities and fundamental group of the complement of a union of lines
HTML articles powered by AMS MathViewer
- by Kwai-Man Fan PDF
- Proc. Amer. Math. Soc. 124 (1996), 3299-3303 Request permission
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
- 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).
- Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
- E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260.
- Mutsuo Oka, Some plane curves whose complements have non-abelian fundamental groups, Math. Ann. 218 (1975), no. 1, 55–65. MR 396556, DOI 10.1007/BF01350067
- 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, DOI 10.2969/jmsj/03040599
- 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, DOI 10.1007/BF01388726
- 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.
- O. Zariski, The topological discriminant group of a Riemann surface of genus $p$, Amer. J. Math. 59 (1937), 335-358.
- O. Zariski, On the irregularity of cyclic multiple plane, Ann. of Math. 32 (1931), 485–511.
Additional Information
- Kwai-Man Fan
- Affiliation: Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan
- Email: kmfan@math.ccu.edu.tw
- Received by editor(s): May 1, 1995
- Communicated by: Peter Li
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3299-3303
- MSC (1991): Primary 14H30; Secondary 14H20
- DOI: https://doi.org/10.1090/S0002-9939-96-03487-9
- MathSciNet review: 1343691