Classification of Rational Unicuspidal Projective Curves whose Singularities Have one Puiseux Pair

Bobadilla, Javier Fernández de; Luengo, I.; Hernández, Adolfo; Némethi, András

doi:10.1007/978-3-7643-7776-2_4

Cited by 31 publications

(84 citation statements)

References 6 publications

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“…In [10] we prove that if C is a unicuspidal rational plane curve of degree d and if the singular point p of C has only one characteristic pair (a, b), then the triple (d, a, b) is one of the above cases. This classification is coordinated by the following integer.…”

Section: Remarksmentioning

confidence: 97%

See 1 more Smart Citation

On Rational Cuspidal Projective Plane Curves

Bobadilla

Luengo-Velasco²,

Melle-Hernández³

et al. 2005

Proc. Lond. Math. Soc.

111

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In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

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Section: Remarksmentioning

confidence: 97%

“…Then (via Step 4) I l ∩ Γ is the disjoint union of the image of s l and P l . Therefore, it is enough to show that #P l = ϕ j −2 (10) for any ϕ j −2 l < d.…”

Section: Remarksmentioning

confidence: 99%

On Rational Cuspidal Projective Plane Curves

Bobadilla

Luengo-Velasco²,

Melle-Hernández³

et al. 2005

Proc. Lond. Math. Soc.

111

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“…Recently, Borodzik and Livingston studied the rational case, i.e. when the resolution of C is a sphere, and obtained a strong constraint on some coefficients of the Alexander polynomials of the link of the singularity [3], proving a conjecture of Fernández de Bobadilla, Luengo, Melle-Hernandez and Némethi [4].…”

Section: Introductionmentioning

confidence: 99%

Cuspidal curves and Heegaard Floer homology

Bodnár¹,

Celoria²,

Golla³

2016

Proc. London Math. Soc.

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Abstract. We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.

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“…Remark The HN‐types

{scriptOR}_{1} false(1 false)

and

{scriptOR}_{2} false(1 false)

written in a standard way are

((), \binom{22}{3})

and

((), \binom{43}{6})

, respectively. The main result of asserts that the Orevkov curves

C_{4}

and

C_{4}^{*}

, which realize these HN‐types, are the only unicuspidal planar curves with complement of log general type and with one HN‐pair. Using the bounds obtained by Borodzik and Livingstone in via Heegard‐Floer homology methods, Liu [, Theorem 1.1] extended this result by describing possible types of singularities of unicuspidal curves under the assumption that they have at most two Puiseux pairs (see Section 2.4 for definitions).…”

Section: Realization Of Hn‐typesmentioning

confidence: 99%

Classification of planar rational cuspidal curves I. C∗∗-fibrations

Palka

Pełka

2017

Proc. London Math. Soc.

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To classify planar complex rational cuspidal curves E⊆double-struckP2 it remains to classify the ones with complement of log general type, that is, the ones for which κ(KX+D)=2, where (X,D) is a log resolution of (double-struckP2,E). It is conjectured that κ(KX+12D)=−∞ and hence double-struckP2∖E is C∗∗‐fibered, where double-struckC∗∗=double-struckC1∖{0,1}, or −(KX+12D) is ample on some minimal model of (X,12D). Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is C∗∗‐fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.

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Classification of Rational Unicuspidal Projective Curves whose Singularities Have one Puiseux Pair

Cited by 31 publications

References 6 publications

On Rational Cuspidal Projective Plane Curves

On Rational Cuspidal Projective Plane Curves

Cuspidal curves and Heegaard Floer homology

Classification of planar rational cuspidal curves I. C∗∗-fibrations

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