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Freeness and invariants of rational plane curves

Authors: Laurent Busé, Alexandru Dimca and Gabriel Sticlaru
Journal: Math. Comp. 89 (2020), 1525-1546
MSC (2010): Primary 14H50; Secondary 14H20, 14H45
Published electronically: December 16, 2019
MathSciNet review: 4063327
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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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Additional Information

Laurent Busé
Affiliation: Université Côte d’Azur; and Inria, Sophia Antipolis, France

Alexandru Dimca
Affiliation: Université Côte d’Azur, Laboratoire Jean-Alexandre Dieudonné; and Inria, Nice, France

Gabriel Sticlaru
Affiliation: Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania

Received by editor(s): April 26, 2018
Received by editor(s) in revised form: February 26, 2019
Published electronically: December 16, 2019
Additional Notes: This work was partially supported by the French government through the UCA\tsup{JEDI} Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01
Article copyright: © Copyright 2019 American Mathematical Society