On some conjectures about free and nearly free divisors

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.

20 pages

References in corpus (4)

• Freeness versus maximal global Tjurina number for plane curves
• Nearly free divisors and rational cuspidal curves
• Free and nearly free surfaces in $P^3$
• Controllable spiking patterns in long-wavelength VCSELs for neuromorphic photonics systems

Cited by in corpus (7)

• Conic-line arrangements in the complex projective plane
• Maximizing curves viewed as free curves
• Jacobian syzygies and plane curves with maximal global Tjurina numbers
• Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves
• Freeness and invariants of rational plane curves
• Ramblings on the freeness of affine hypersurfaces
• Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves