Freeness and invariants of rational plane curves

Busé, Laurent; Dimca, Alexandru; Sticlaru, Gabriel

doi:10.1090/mcom/3495

Freeness and invariants of rational plane curves
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Abstract:

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

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