Free and Nearly Free Curves vs. Rational Cuspidal Plane Curves

Abstract

We define a class of plane curves that are close to the free divisors in terms of the local cohomology of their Jacobian algebras and such that, conjecturally, any rational cuspidal curve $C$ is either free or belongs to this class. We prove this conjecture when the degree of $C$ is either even or a prime power, or when the group of $C$ is abelian.