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On the Lefschetz theorem for the complement of a curve in \(\mathbb{P}^ 2\). (English. Russian original) Zbl 0798.14008

Russ. Acad. Sci., Izv., Math. 41, No. 1, 169-184 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 889-906 (1992).
Summary: Let \(E\) be an irreducible plane curve over the field \(\mathbb{C}\) of complex numbers, let \(\widetilde \nu : \widetilde E \to E \subset \mathbb{P}^ 2\) be the normalization morphism, and let \(\overline D\) be an arbitrary curve in \(\mathbb{P}^ 2\) such that \(\overline E \not \subset \overline D\). The main result of this paper says that if \(\overline E\) and \(\overline D\) intersect transversely, then \[ \widetilde \nu_ * : \pi_ 1 (\widetilde E \backslash \widetilde \nu^{-1} (\overline E \cap \overline D)) \to \pi_ 1 (\mathbb{P}^ 2 \overline D) \] is an epimorphism.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
57M05 Fundamental group, presentations, free differential calculus
14H99 Curves in algebraic geometry
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