On the Artal--Carmona--Cogolludo construction

TL;DR

This paper derives explicit equations for certain complex plane sextic curves with specific singularities, computes their fundamental groups, and constructs examples of Zariski pairs, advancing understanding of algebraic curve topology.

Contribution

It provides explicit equations for irreducible maximizing sextic curves with double singular points and computes their abelian fundamental groups, including Zariski pair examples.

Findings

Explicit equations for several irreducible maximizing sextics.

Fundamental groups of the complements are all abelian.

Constructed examples of Zariski pairs.

Abstract

We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian. As a by-product, examples of Zariski pairs in the strongest possible sense are constructed.