CAT(0) geometry of complex curve complements and families

TL;DR

This paper explores the CAT(0) properties of fundamental groups of complex plane curve complements and related universal families, revealing conditions under which these groups are or are not CAT(0).

Contribution

It proves that fundamental groups of complements of certain generic plane curve branch loci are CAT(0), while those of specific universal families with singularities are not.

Findings

Complement groups of generic projections are CAT(0).

Universal family groups with E6, E7, E8 singularities are not CAT(0).

Provides insights into the geometric group theory of complex algebraic curves.

Abstract

Motivated by the question of whether braid groups are CAT(0), we investigate the CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If $C$ is the branch locus of a generic projection of a smooth, complete intersection surface to $\PP^2$, we show that $\pi_1(\PP^2\setminus C)$ is CAT(0). In the other direction, we prove that the fundamental group of the universal family associated with the singularities of type $E_6$, $E_7$, and $E_8$ is not CAT(0).