On the Freeness of Rational Cuspidal Plane Curves

Dimca, Alexandru; Sticlaru, Gabriel

doi:10.17323/1609-4514-2018-18-4-659-666

Cited by 10 publications

(13 citation statements)

References 35 publications

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“…Proof. Exactly the same proof as for [15,Proposition 3.3] implies that n(f ) j ≤ 2 for j ≥ 2d − 3 − r 0 . Our assumption implies r = mdr(f ) < d/2 and hence case (ii) in Theorem 3.11 is excluded.…”

Section: Various Bounds On the Exponents And The Relation To Nearly Cmentioning

confidence: 68%

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

Dimca

Sticlaru

2019

Geom Dedicata

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We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

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Section: Various Bounds On the Exponents And The Relation To Nearly Cmentioning

confidence: 68%

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

Dimca

Sticlaru

2019

Geom Dedicata

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“…This class of curves is extremely rich and has been studied extensively. Conjecture 2.13 is proved in most of the cases; see [DS18a] and [DS18b] for the details.…”

Section: 2mentioning

confidence: 93%

Freeness and invariants of rational plane curves

2019

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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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“…This conjecture is known to hold when the degree of is even, or when ≤ 33, as well as in many other cases, see [10,17,18].…”

Section: Jacobian Ideal Jacobian Module and Free And Nearly Free Cumentioning

confidence: 98%

Deformations of plane curves and Jacobian syzygies

Dimca

Sticlaru

2019

Mathematische Nachrichten

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On the Freeness of Rational Cuspidal Plane Curves

Cited by 10 publications

References 35 publications

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

Plane curves with three syzygies, minimal Tjurina curves, and nearly cuspidal curves

Freeness and invariants of rational plane curves

Deformations of plane curves and Jacobian syzygies

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