BRAID MONODROMY OF COMPLEX LINE ... - Project Euclid
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BRAID MONODROMY OF COMPLEX LINE ... - Project Euclid
for the case when # is an arrangement stf of complex lines in C2 , using the notion of labyrinth of an arrangement. As a corollary we get the braid monodromy ...
V D. NGUYEN KODAI MATH. J. 22 (1999), 46-55
BRAID MONODROMY
OF COMPLEX LINE ARRANGEMENTS
Dedicated to the memory of Professor N. Sasakura NGUYEN VIET DUNG*
Abstract Let V be the complex vector space C 7 , s/ an arrangement in V, i.e. a finite family of hyperplanes in V In [11], Moishezon associated to any algebraic plane curve B(n), where Fs is a free group, B(ή) is the Artm braid group. In this paper, we will determine the braid monodromy for the case when # is an arrangement stf of complex lines in C 2 , using the notion of labyrinth of an arrangement. As a corollary we get the braid monodromy presentation for the fundamental group of the complement to the arrangement.
1.
Introduction
Let ^ = {/(x, y) = 0} e C2 be a plane algebraic curve. From the 1930's, it is well known (see [9], [17]) that the fundamental group of the complement to #, π i ( C 2 \ ^ ) 5 can be computed using the van Kampen's method. In [11], Moishezon introduced the notion of braid monodromy of c€. Suppose that the projection on the X-axis, pr\ : C 2 —> C 1 , is generic with respect to the curve #. Let S() are usually called the multiple points of the arrangement sd. By definition, a multiple point P of the arrangement &0 is the nonempty intersection of two or more hyperplanes of stf. The assumption on the genericity of the projection pr\ implies that the multiple points of the arrangement si are distinct by their x- coordinates. In other words, the images of multiple points of si on the x-plane C 1 are pairwise distinct. Let x/c e C 1 be the image of a multiple point P& = (x^, yk) of J / under the projection pr\. Suppose that Pk = f]r=ιHtj. Then it is clear that xk belongs to the lines Lϊj)lr, 1 < s < t < r of the labyrinth if(j^). However, there might be another line L e 3?{srf), which does not belong to {Lls^t\ 1 < s < t < r}, going through this point xk. 1
