The fundamental group of a $\mathcal{CP}^2$ complement of a branch curve as an extension of a solvable group by a symmetric group

Abstract

The main result in this paper is as follows: Theorem. Let S be the branch curve in CP 2 of a generic projection of a Veronese surface. Then �1(CP 2 S) is an extension of a solvable group by a symmetric group. A group with the property mentioned in the theorem is "almost solvable" in the sense that it contains a solvable normal subgroup of finite index. We pose the following question. Question. For which families of simply connected algebraic surfaces of general type is the fundamental group of the complement of the branch curve of a generic projec- tion to CP 2 an extension of a solvable group by a symmetric group?