Dihedral coverings of algebraic surfaces and their application

Tokunaga, Hiro-o

doi:10.1090/S0002-9947-00-02524-1

Dihedral coverings of algebraic surfaces and their application
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by Hiro-o Tokunaga

Trans. Amer. Math. Soc. 352 (2000), 4007-4017

DOI: https://doi.org/10.1090/S0002-9947-00-02524-1

Published electronically: March 15, 2000

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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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