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Arrangements of lines and monodromy of plane curves. (English) Zbl 0661.14038
Let \(C=\{f(x,y)=0\}\) be an algebraic curve in \({\mathbb{C}}{\mathbb{P}}^ 2\) such that the projection \(\pi_ x:\quad {\mathbb{C}}^ 2\to {\mathbb{C}}_ x\) onto the x-axis is generic for C. Set \(S(C,\pi)=\{p\in C| (\partial f/\partial y(p)=0\}\), \(D(C,\pi)=\pi (S(C,\pi))\), and choose \(M\in {\mathbb{C}}_ x\setminus D(C,\pi)\). The braid monodromy of C is the homomorphism \(\theta:\quad \pi_ 1({\mathbb{C}}_ x\setminus D(C,\pi),M)\to B[\pi^{-1}(M),C\cap \pi^{-1}(M)],\) this last being the group of homotopy classes of compact-supported homeomorphisms of \(\pi^{-1}(M)\) which preserve \(C\cap \pi^{-1}(M).\)
In the paper we give an a priori construction of the braid monodromy for certain classes of curves with ordinary singularities of branch points. In particular we obtain that the monodromy of an arrangement of real lines is determined by its “dual graph”.
Some applications to the study of the fundamental group of the complement are given. In the last part, we exploit the above result to conclude that the complement to certain arrangements is not a K(\(\pi\),1).
Reviewer: M.Salvetti

MSC:
14N05 Projective techniques in algebraic geometry
14H20 Singularities of curves, local rings
14H30 Coverings of curves, fundamental group
14E20 Coverings in algebraic geometry
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