Asymptotic integrals and Hodge structures

Varchenko, Alexander

doi:10.1007/bf01084820

Cited by 10 publications

(9 citation statements)

References 21 publications

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“…The above negative distribution property has some analogies with the criteria provided by the semicontinuity of the spectrum [41,42]. Namely, if {(C, p i )} i are the local singular points of the degree d curve C, then the multisingularity i (C, p i ) is in the deformation of the universal plane germ (U, 0) := (x d +y d , 0).…”

Section: Comparison With Varchenko's Criterionmentioning

confidence: 98%

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: Comparison With Varchenko's Criterionmentioning

confidence: 98%

On Rational Cuspidal Projective Plane Curves

Bobadilla

Luengo-Velasco²,

Melle-Hernández³

et al. 2005

Proc. Lond. Math. Soc.

111

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“…In the "Appendix", E. Shustin shows the existence of a plane curve of degree 11 with 30 cusps, which proves that k(11) ≥ 30, while k(11) ≤ 31 follows by Varchenko's results in [30].…”

Section: Introductionmentioning

confidence: 88%

Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin

Calabri

Paccagnan

Stagnaro

2012

Annali di Matematica

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The maximum number of cusps on a plane algebraic curve of degree d is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value of the maximum number of cusps on plane curves of degree d, divided by d^2, when d tends to infinity. In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The Appendix to this paper is devoted to the case of degree 11 and it is due to E. Shustin

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“…Spectral bound. Further necessary conditions can be obtained by applying the semicontinuity of the singularity spectrum (see [90]), which works in any dimension. The singularity spectrum of a hypersurface singularity f : C n+1 → C gathers the information about the eigenvalues of the monodromy operator T and about the Hodge filtration {F p } on its vanishing cohomology.…”

Section: Finally We Mention the Inequalitymentioning

confidence: 99%

“…The semicontinuity of the spectrum says that any half open interval (t, t+1] ⊂ R is a semicontinuity domain for F , that is, the sum M F −1 (s) of the frequencies of the elements of (t, t+1] in Σ F −1 (s) is upper semicontinuous for s ∈ S ([81, Theorem 2.4]). Before that Varchenko [90] had proved that for deformations F of low weight of a quasi-homogeneous f even every open interval (t, t + 1) is a semicontinuity domain.…”

Section: Finally We Mention the Inequalitymentioning

confidence: 99%

“…+ x d n are in the interval (t, t + 1], t ∈ R, then the sum of the frequencies of the spectral numbers in the interval (t, t + 1] of the singularities (close to 0) of a small deformation F of f can be at most M , i.e., M F −1 (s) ≤ M f −1 (0) = M (cf. [81,Theorem 2.4]; if the deformation is of low weight, we can use even the open interval (t, t + 1) by [90]).…”

Section: Finally We Mention the Inequalitymentioning

confidence: 99%

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Plane algebraic curves with prescribed singularities

Greuel¹,

Shustin²

2020

Preprint

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We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer d and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree d having singular points of the given type as its only singularities. The set of all such curves is a quasiprojective variety, which we call an equisingular family ESF . We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and T -smoothness (i.e., smooth of expected dimension) of the corresponding ESF . The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for T -smooth equisingular families if the degree goes to infinity.

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Asymptotic integrals and Hodge structures

Cited by 10 publications

References 21 publications

On Rational Cuspidal Projective Plane Curves

On Rational Cuspidal Projective Plane Curves

Plane algebraic curves with many cusps, with an appendix by Eugenii Shustin

Plane algebraic curves with prescribed singularities

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