Stability of foliations induced by rational maps

Cukierman, Fernando; Pereira, Jorge Vitorio; Vainsencher, Israel

doi:10.5802/afst.1221

Cited by 30 publications

(35 citation statements)

References 13 publications

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“…The study of the behavior of ϕ ω and the information attached to its Hilbert polynomial took us to a deeper knowledge of first order unfoldings and deformations that we present in this paper. ⋆ Rational and logarithmic foliations define irreducible components of the space of codimension one foliations, as it is shown by X. Gómez Mont and A. Lins-Neto in [GMLN91] and later by F. Cukierman, J. V. Pereira and I. Vainsencher in [CPV09] for rational foliations and O. Calvo Andrade in [CA94] for logarithmic foliations. Rational and logarithmic foliations in P n can be given, respectively, by differential forms of the type…”

Section: Introductionmentioning

confidence: 92%

“…First order deformations of rational foliations are studied in the works [GMLN91] and [CPV09]. The latter, takes into account the scheme structure of codimension one foliations and proves, among other things, that F 1 (P n )(e) is generycally reduced at a rational foliation.…”

Section: Rational and Logarithmic Foliationsmentioning

confidence: 99%

“…In [CPV09] the authors proves the infinitesimal stability of a generic rational foliation ω R , showing that D(ω R ) is generated by perturbations of the parameters F and G. On the other side, using [CA94] one can only show that the perturbations of the parameters f i and {λ i } generate all the deformations of the space of logarithmic foliations as a set, i.e., disregarding the sheaf structure along with any algebraic multiplicity that may arise from the equation ω ∧ dω = 0. This way, we get a partial description of D(ω L ) in terms of the parameters defining ω L , as we will see in Corollary 2.4.5.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Unfoldings and deformations of rational and logarithmic foliations

Molinuevo¹

2016

Annales De l'Institut Fourier

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We study codimension one foliations in projective space P n over C by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type.For a differential form ω defining a codimension one foliation, we present a graded module U(ω), related to the first order unfoldings of ω. If ω is a generic form of rational or logarithmic type, as a first application of the construction of U(ω), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of ω, in terms of a Hilbert polynomial associated to U(ω).We review the notion of regularity of ω in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, ω is regular if and only if every unfolding is trivial up to isomorphism.

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Section: Introductionmentioning

confidence: 92%

Section: Rational and Logarithmic Foliationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unfoldings and deformations of rational and logarithmic foliations

Molinuevo¹

2016

Annales De l'Institut Fourier

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“…As it is shown in [2], the Zariski tangent space to the space of such foliations can be parameterized by…”

Section: Geometric Motivationmentioning

confidence: 99%

Deformations of the exterior algebra of differential forms

Molinuevo

2016

Beitr Algebra Geom

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Let D : Ω → Ω be a differential operator defined in the exterior algebra Ω of differential forms over the polynomial ring S in n variables. In this work we give conditions for deforming the module structure of Ω over S induced by the differential operator D, in order to make D an S-linear morphism while leaving the C-vector space structure of Ω unchanged. One can then apply the usual algebraic tools to study differential operators: finding generators of the kernel and image, computing a Hilbert polynomial of these modules, etc.Taking differential operators arising from a distinguished family of derivations, we are able to classify which of them allow such deformations on Ω. Finally we give examples of differential operators and the deformations that they induce.

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“…There are a few other known components for d ≥ 3, such as the pullback, the rational and the logarithmic components, see [4], [5] and [1]. In [5] the authors managed to compute the degree of some rational components.…”

Section: Introductionmentioning

confidence: 99%

Degree of the exceptional component of foliations of degree two and codimension one in $${\mathbb {P}}^{3}$$P3

Rossini

Vainsencher

2019

RACSAM

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The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P 3 . This component is the closure of the orbit of the foliation defined by the differential form1 2 under the natural action of the group of automorphisms of P 3 . Our first task is to unravel a geometric characterization of the pair g, f . This leads us to the construction of a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.

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Stability of foliations induced by rational maps

Cited by 30 publications

References 13 publications

Unfoldings and deformations of rational and logarithmic foliations

Unfoldings and deformations of rational and logarithmic foliations

Deformations of the exterior algebra of differential forms

Degree of the exceptional component of foliations of degree two and codimension one in $${\mathbb {P}}^{3}$$P3

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