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Position of singularities and fundamental group of the complement of a union of lines
Author(s):
Kwai-Man
Fan
Journal:
Proc. Amer. Math. Soc.
124
(1996),
3299-3303.
MSC (1991):
Primary 14H30;
Secondary 14H20
MathSciNet review:
1343691
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Abstract:
In this paper we give two examples of complex line arrangements in  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:
- 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.
- A. Dimca, Singularities and Topology of Hypersurfaces, Springer-Verlag, New York (1992). MR 94b:32058
- 3.
- E. R. van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260.
- 4.
- M. Oka, Some plane curves whose complements have non-abelian fundamental groups, Math. Ann. 218 (1975), 55-65. MR 53:419
- 5.
- M. Oka and K. Sakamoto, Product theorem of the fundamental group of a reducible curve, J. Math. Soc. Japan 30 (1978), 599-602. MR 81h:14019
- 6.
- R. Randell, The fundamental group of the complement of a union of a complex hyperplanes, Invent. Math. 80 (1985), 467-468, Correction: Invent. Math. 80 (1985), 467--468. MR 87e:32010
- 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
 , Amer. J. Math. 59 (1937), 335-358. - 9.
- O. Zariski, On the irregularity of cyclic multiple plane, Ann. of Math. 32 (1931), 485--511.
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Additional Information:
Kwai-Man
Fan
Affiliation:
Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan
Email:
kmfan@math.ccu.edu.tw
DOI:
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
Copyright of article:
Copyright
1996,
American Mathematical Society
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