The fundamental group of the scomplement to a hypersurface in $\mathbf C^n$

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.