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On the homotopy type of the complement to plane algebraic curves. (English) Zbl 0576.14019
We present an algorithm for finding the homotopy type of the complement to an algebraic curve in $${\mathbb{C}}^ 2$$ with arbitrary singularities. More precisely we describe a presentation of the fundamental group of the complement [in terms of the braid monodromy introduced by B. G. Moishezon in Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 107-192 (1981; Zbl 0476.14005)] such that the associated 2-dimensional complex has the same homotopy type as the complement to the curve. As a corollary we describe how the homotopy type of the complement to a plane curve changes when this curve degenerates and acquires new singularities. In particular in a regeneration of a curve in which one cusp changes into a node from the homotopy point of view amounts to taking wedge with a 2-dimensional sphere. As an example we show that the complement in an affine plane to the sextic with 6 cusps on conic has the homotopy type of the wedge of the complement to the trefoil knot in 3- sphere and 13 copies of 2-sphere.

##### MSC:
 14F35 Homotopy theory and fundamental groups in algebraic geometry 14H20 Singularities of curves, local rings 55P15 Classification of homotopy type 14D15 Formal methods and deformations in algebraic geometry 14J17 Singularities of surfaces or higher-dimensional varieties 14E20 Coverings in algebraic geometry
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