Geodesic
The fundamental group of the scomplement to a hypersurface in $\mathbf C^n$
Izvestiya. Mathematics , Tome 38 (1992) no. 2, pp. 399-418.

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Let $D$ be a complex algebraic hypersurface in $\mathbf C^n$ not passing through the point $o\in\mathbf C^n$. The generators of the fundamental group $\pi_1(\mathbf C^n\setminus D,o)$ and the relations among them are described in terms of the real cone over $D$ with apex at $o$. This description is a generalization to the algebraic case of Wirtinger's corepresentation of the fundamental group of a knot in $\mathbf R^3$. A new proof of Zariski's conjecture about commutativity of the fundamental group $\pi_1(\mathbf P^2\setminus C)$ for a projective nodal curve $C$ is given in the second part of the paper based on the description of the generators and the relations in the group $\pi_1(\mathbf C^n\setminus D)$ obtained in the first part. 
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Vik. S. Kulikov. The fundamental group of the scomplement to a hypersurface in $\mathbf C^n$. Izvestiya. Mathematics , Tome 38 (1992) no. 2, pp. 399-418. http://geodesic.mathdoc.fr/item/IM2_1992_38_2_a8/

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