On the homotopy type of the complement to plane algebraic curves.

*(English)*Zbl 0576.14019We 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 |