On some conjectures about free and nearly free divisors

TL;DR

This paper constructs infinite families of complex plane curves that are free or nearly free but not rational, providing counterexamples to certain conjectures by Dimca and Sticlaru, while leaving open the question about rational cuspidal curves.

Contribution

The paper introduces new infinite families of irreducible free and nearly free curves with arbitrary singularities, challenging existing conjectures in algebraic geometry.

Findings

Counterexamples to some conjectures by Dimca and Sticlaru

Existence of non-rational free and nearly free curves with complex singularities

Examples do not resolve the conjecture on rational cuspidal plane curves

Abstract

In this paper infinite families of examples of irreducible free and nearly free curves in the complex projective plane which are not rational curves and whose local singularites can have an arbitrary number of branches are given. All these examples answer negatively to some conjectures proposed by A. Dimca and G. Sticlaru. Our examples say nothing about the most remarkable conjecture by A. Dimca and G. Sticlaru, i.e. every rational cuspidal plane curve is either free or nearly free.