Free divisors and rational cuspidal plane curves

A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of rational cuspidal curves which are free. Some stronger properties are stated as conjectures.Comment: version 4: a misprint in the conjecture added yesterday is correcte

Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.Comment: v3: minor changes, final versio

Rational cuspidal curves with four cusps on Hirzebruch surfaces

The purpose of this article is to shed light on the question of how many and what kind of cusps a rational cuspidal curve on a Hirzebruch surface can have. We use birational transformations to construct rational cuspidal curves with four cusps on the Hirzebruch surfaces and find associated results for these curves.Comment: 26 pages, 1 figur

Saturation of Jacobian ideals: Some applications to nearly free curves, line arrangements and rational cuspidal plane curves

International audienceIn this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Some applications to rational cuspidal plane curves are given, and a natural related question is raised