Freeness and invariants of rational plane curves

Abstract

Given a parameterization $\varphi$ of a rational plane curve C, we study some invariants of C via $\varphi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\varphi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\varphi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

Mathematics of Computation, American Mathematical Society, In press

References in corpus (4)

• Freeness versus maximal global Tjurina number for plane curves
• On some conjectures about free and nearly free divisors
• Rational Cuspidal Curves
• On rational cuspidal plane curves, and the local cohomology of Jacobian rings