Zariski $k$-plets via dessins d'enfants

TL;DR

This paper constructs large collections of irreducible plane curves with identical singularities and abelian fundamental groups, expanding understanding of their deformation families.

Contribution

It introduces a method to generate exponentially many distinct equisingular deformation families of plane curves with abelian fundamental groups.

Findings

Exponential growth in the number of deformation families.

All constructed curves have abelian fundamental groups.

Curves share the same singularity sets.

Abstract

We construct exponentially large collections of pairwise distinct equisingular deformation families of irreducible plane curves sharing the same sets of singularities. The fundamental groups of all curves constructed are abelian.