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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Dihedral coverings of algebraic surfaces and their application


Author: Hiro-o Tokunaga
Journal: Trans. Amer. Math. Soc. 352 (2000), 4007-4017
MSC (2000): Primary 14E20; Secondary 14E15
Published electronically: March 15, 2000
MathSciNet review: 1675238
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Abstract | References | Similar Articles | Additional Information

Abstract:

In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following:

Let $B$ be a reduced plane curve of even degree $d$ having only $a$ nodes and $b$ cusps. If $2a + 6b > 2d^2 - 6d + 6$, then $\pi_1(\mathbf{P}^2 \setminus B)$ is non-abelian.

Note that Nori's result implies that $\pi_1(\mathbf{P}^2 \setminus B)$ is abelian, provided that $2a + 6b < d^2$.


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Additional Information

Hiro-o Tokunaga
Affiliation: Department of Mathematics and Information Science, Kochi University, Kochi 780-8520, Japan
Address at time of publication: Department of Mathematics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397 Japan
Email: tokunagamath.kochi-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02524-1
PII: S 0002-9947(00)02524-1
Received by editor(s): June 20, 1998
Published electronically: March 15, 2000
Additional Notes: This research is partly supported by the Grant-in-Aid for Encouragement of Young Scientists 09740031 from the Ministry of Education, Science and Culture.
Article copyright: © Copyright 2000 American Mathematical Society