Fundamental Groups of a Class of Rational Cuspidal Plane Curves

TL;DR

This paper computes and analyzes the fundamental groups of complements of a specific class of rational cuspidal plane curves, revealing their algebraic properties and using advanced algebraic geometry techniques.

Contribution

It provides explicit presentations of these fundamental groups and investigates their properties, including abelianity, finiteness, and complexity, using the Zariski-Van Kampen algorithm and Cremona transformations.

Findings

Determined if the groups are abelian, finite, or big

Simplified and studied the group presentations

Analyzed quotients of the fundamental groups

Abstract

We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski-Van Kampen algorithm and exploit the Cremona transformations used in the construction of these curves. We simplify and study these group presentations so obtained and determine if they are abelian, finite or big, i.e. if they contain free non-abelian subgroups. We also study the quotients of these groups to some extend.