Freeness and invariants of rational plane curves

Abstract

Given a parameterization φ of a rational plane curve C, we study some invariants of C via φ. 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 φ, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via φ, 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.

Deformations of plane curves and Jacobian syzygies

“…Using our algorithm to decide the freeness of a rational curve given by a parametrization described in [3], we get the following. The equation for  4 given above does not correspond to the given parametrization.…”

“…Using our algorithm to decide the freeness of a rational curve given by a parametrization described in , we get the following. Corollary The curve is nearly free with exponents (2,2).…”