Linearity and virtual poly-freeness of the fundamental group of plane curves of degree at most five

Abstract

We prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group ${\pi }_1(\mathbb{C}{P}^2\setminus C)$ is linear and virtually polyfree. As a consequence, we answer positively the open question on the residual finiteness of these groups for all plane curves of degree at most $5$.