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

TL;DR

This paper proves that the fundamental groups of plane curves of degree at most five are linear and virtually polyfree, confirming their residual finiteness and advancing understanding of their algebraic properties.

Contribution

It establishes the linearity and virtual polyfreeness of fundamental groups of plane curves up to degree five, a previously unresolved question in algebraic geometry.

Findings

Fundamental groups are linear for degree ≤ 5 curves.

Fundamental groups are virtually polyfree for degree ≤ 5 curves.

Residual finiteness is confirmed for these groups.

Abstract

We prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ 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 $5$.