Plane curves with a big fundamental group of the complement

Abstract

Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group $π_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.

23 pages LaTeX. A revised version. The unnecessary restriction $d\ge 2g-1$ of the previous version has been removed, and the main result has taken its final form

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