Braid Group techniques in Complex Geometry V: The fundamental group of the complement of a Veronese generic projection

Details Braid Group techniques in Complex Geometry V: The fundamental group of the complement of a Veronese generic projection Braid Group techniques in Complex Geometry V: The fundamental group of the complement of a Veronese generic projection

1995-02-14

arxiv.org/abs/alg-geom/9502009v1

Mathematics

Algebraic Geometry

125 pages, amstex, figures by request. To be published in "Communications in Analysis and Geometry"

Scientific paper

Computation of the fundamental group of the complement in the complex plane of the branch curve S , of a generic projection of the Veronese surface to the plane is presented. This paper is a continuation of our previous papers: Braid Group Technique I - IV. In I and II we developed algorithms to compute braid monodromy of a brunch curve, provided there exist a degeneration of the surface to union of planes in a configuration where the associated branch curve is partial to a line arrangement dual to generic. In III we constructed a degeneration of the Veronese surface of order 3 to union of planes with the desired property and in IV we used I -III to compute the braid monodromy of the associated branch curve. Here we use the Van-Kampen method to compute the fundamental group of the complement from a braid monodromy factorization and some extra properties of the factorization , namely invariant properties, also proven in IV. The group is presented using a certain quotient of the Braid group defined by transversal half-twists.

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