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Freeness and invariants of rational plane curves

Laurent Busé 1 Alexandru Dimca 2, 1 Gabriel Sticlaru 3 
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
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.
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Submitted on : Tuesday, January 21, 2020 - 2:05:27 PM
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Laurent Busé, Alexandru Dimca, Gabriel Sticlaru. Freeness and invariants of rational plane curves. Mathematics of Computation, 2020, 89, pp.1525--1546. ⟨10.1090/mcom/3495⟩. ⟨hal-01767751v2⟩



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