On Some Conjectures About Free and Nearly Free Divisors

Bartolo, Enrique Artal; Gorrochategui, Leire; Luengo, I.; Melle-Hernández, A.

doi:10.1007/978-3-319-28829-1_1

Cited by 15 publications

(18 citation statements)

References 21 publications

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“…Proof. When C is a plus-one generated curve, then the formulas for the exponents and for the total Tjurina number follow from Proposition 3.7 and Proposition 2.1 (4). We assume now that C is not a plus-one generated curve, which in view of Theorem 2.3 means exactly that d 1 + d 2 > d. We prove that this inequality implies that the Hilbert function of the Jacobian module N(f ) has not the form described in Corollary 3.10 and that ν(C) ≥ 3, unless C is as in (ii) above.…”

Section: Remark 32mentioning

confidence: 99%

“…On the other hand, for a reduced degree d curve C, one has m ≤ d + 1, see [18,Proposition 2.1], while for the case C a line arrangement, we have the slightly stronger inequality m ≤ d − 1, see [16,Corollary 1.3], as well as its generalization in Corollary 5.2 below. The curve C is free when m = 2, since then AR(f ) is a free module of rank 2, see for such curves [4,9,13,28,29,30]. In this note we consider the next simplest possibility for this resolution, namely we start the study of the 3-syzygy curves.…”

Section: Introductionmentioning

confidence: 99%

“…(i) The nearly free curves, introduced in [14] and studied in [3,4,9,10,24] correspond to the 3-syzygy curves satisfying d 3 = d 2 and d 1 + d 2 = d.…”

Section: Introductionmentioning

confidence: 99%

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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: Remark 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Dimca

Sticlaru

2019

Geom Dedicata

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“…We also show that there exist rational free curves and nearly free curves that are not cuspidal. Examples showing that there exist free curves and nearly free curves that are not rational curves are given in [ABGLMH17].…”

Section: 2mentioning

confidence: 99%

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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“…Recall that C is a free curve if