Classification of foliations of degree three on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>ℙ</mml:mi> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup></mml:math> with a flat Legendre transform

Bedrouni, Samir; Marı́n, David

doi:10.5802/aif.3431

Annales De l'Institut Fourier

2022

DOI: 10.5802/aif.3431

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Classification of foliations of degree three on ℙ ℂ 2 with a flat Legendre transform

Samir Bedrouni¹,

David Marı́n²

Abstract: Classi cation of foliationsof degree three on P 2C with a at Legendre transform Article à paraître, mis en ligne le décembre , p.© Association des Annales de l'institut Fourier, , Certains droits réservés.Cet article est mis à disposition selon les termes de la licence C C -. F .

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2024

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A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$

Bedrouni,

Marín

2024

Mathematische Nachrichten

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Let be an integer. For a holomorphic ‐web on a complex surface , smooth along an irreducible component of its discriminant , we establish an effective criterion for the holomorphy of the curvature of along , generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) of a homogeneous foliation of degree on , generalizing some of our previous results. This then allows us to study the flatness of the ‐web in the particular case where the foliation is Galois. When the Galois group of is cyclic, we show that is flat if and only if is given, up to linear conjugation, by one of the two 1‐forms , . When the Galois group of is noncyclic, we obtain that is always flat.

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A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$

Bedrouni,

Marín

2024

Mathematische Nachrichten

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Pre-foliations of co-degree one on $$\mathbb {P}^{2}_{\mathbb {C}}$$ with a flat Legendre transform

Bedrouni

2024

Geom Dedicata

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Classification of foliations of degree three on ℙ ℂ 2 with a flat Legendre transform

Abstract: Classi cation of foliationsof degree three on P 2C with a at Legendre transform Article à paraître, mis en ligne le décembre , p.© Association des Annales de l'institut Fourier, , Certains droits réservés.Cet article est mis à disposition selon les termes de la licence C C -. F .

Cited by 2 publications

References 17 publications

A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$

A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on PC2$\mathbb {P}^{2}_{\mathbb {C}}$

Pre-foliations of co-degree one on $$\mathbb {P}^{2}_{\mathbb {C}}$$ with a flat Legendre transform

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