The Fundamental Group of a CP^2 Complement of a Branch Curve as an Extension of a Solvable Group by a Symmetric Group

TL;DR

This paper investigates the fundamental group of the complement of a branch curve from a generic projection of a Veronese surface, showing it forms an extension of a solvable group by a symmetric group, highlighting its 'almost solvable' nature.

Contribution

It establishes that the fundamental group of the complement of the branch curve of a Veronese surface is an extension of a solvable group by a symmetric group, introducing the concept of 'almost solvable' groups in this context.

Findings

The fundamental group is an extension of a solvable group by a symmetric group.

Such groups are 'almost solvable' with a solvable normal subgroup of finite index.

Raises questions about similar properties for other algebraic surfaces.

Abstract

The main result in this paper is as follows: Let S be the branch curve (in the projective plan) of a generic projection of a Veronese surface. Then the fundamental group of the complement of S is an extension of a solvable group by a symmetric group. A group with the property mentioned above is ``almost solvable'' in the sense that it contains a solvable normal subgroup of finite index. This raises the 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 projection to the complex plane an extension of a solvable group by a symmetric group?