Plane curves with a big fundamental group of the complement - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Amer. Math. Soc. Transl. Année : 1998

Plane curves with a big fundamental group of the complement

Gerd Dethloff
  • Fonction : Auteur
  • PersonId : 932798
Stepan Orevkov
M. Zaidenberg

Résumé

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 $\pi_1(\pr^2 \se C)$ 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 $C$.

Dates et versions

hal-00467722 , version 1 (28-03-2010)

Identifiants

Citer

Gerd Dethloff, Stepan Orevkov, M. Zaidenberg. Plane curves with a big fundamental group of the complement. Amer. Math. Soc. Transl., 1998, 184, pp.63-84. ⟨hal-00467722⟩
160 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More