arXiv subjects

Samir Bedrouni

Publications and source records attributed to Samir Bedrouni.

12 recordsLinked to original sources

arXiv2026

On the holomorphy of the curvature of planar webs along an invariant curve

Let $\mathcal{W}=\mathcal{W}_{n}\boxtimes\mathcal{W}_{d-n}$ be a $d$-web on $(\mathbb{C}^2,0)$, where $\mathcal{W}_n$ is an $n$-web with a totally invariant irreducible curve~$C$, and $\mathcal{W}_{d-n}$ is a regular $(d-n)$-web transverse to $C$. We show that the curvature of $\mathcal{W}$ is holomorphic along $C$ if and only if the curvature of $\mathcal{W}_n$ is holomorphic along $C$. When $\mathcal{W}_n$ is non-degenerate along $C$, we prove that $K(\mathcal{W}_n)$, and hence $K(\mathcal{W})$, is holomorphic along $C.$ We deduce that, if $\mathcal{W}_n$ is irreducible and $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1),$ then $K(\mathcal{W})$ is holomorphic along $C.$ This generalizes a result of \textsc{Mar\'{\i}n} and \textsc{Pereira}, obtained in the case where $C$ has minimal multiplicity $n-1$ in the discriminant $\Delta(\mathcal{W}_n).$ If $n$ is prime or $n=4$, the condition $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<3(n-1)$ can be weakened to $\mathrm{mult}\left(\Delta(\mathcal{W}_n),C\right)<n(n-1).$ Moreover, we describe a natural decomposition of $\mathcal{W}_n$ as the product of two subwebs $\mathcal{W}_n=\mathcal{W}_{n}^{\rm{str}}\boxtimes\mathcal{W}_{n}^{\rm{wk}}.$ Under the assumption that $\mathcal{W}_{n}^{\rm{wk}}$ is non-degenerate along $C$, we show that the holomorphy of $K(\mathcal{W})$ on $C$ is equivalent to that of $K(\mathcal{W}_{n}^{\rm{str}}).$

math.DS↗

arXiv2025

Le tissu dual d'un pré-feuilletage convexe réduit sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat

A holomorphic pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a reduced complex projective curve $\mathcal{C}$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$. When the foliation $\mathcal{F}$ is convex (resp. reduced convex) and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we say that the pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ is convex (resp. reduced convex). We prove that the dual web of a reduced convex pre-foliation on $\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This generalizes our previous result obtained in the case where the associated curve consists only of invariant lines.

math.CV↗

arXiv2024

Platitude des tissus duaux de certains pré-feuilletages convexes du plan projectif complexe

A holomorphic pre-foliation $\mathscr{F}=\mathcal{C}\boxtimes\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a reduced complex projective curve $\mathcal{C}$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$. When the foliation $\mathcal{F}$ is convex and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we speak of convex pre-foliation. In a previous paper, we showed that if a foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is reduced convex or homogeneous convex and if $\mathcal{C}$ is an invariant line of $\mathcal{F}$, then the dual web of the convex pre-foliation $\mathcal{C}\boxtimes\mathcal{F}$ is flat. In this paper, we propose to extend this result to the case of a curve $\mathcal{C}$ consisting of several invariant lines.

math.DS↗

arXiv2024

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

A holomorphic pre-foliation $\mathscr{F}=\ell\boxtimes\mathcal{F}$ of co-degree $1$ and degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is the data of a line $\ell$ of $\mathbb{P}^{2}_{\mathbb{C}}$ and a holomorphic foliation $\mathcal{F}$ on $\mathbb{P }^{2}_{\mathbb{C}}$ of degree $d-1.$ We study pre-foliations of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{ C}}$ with a flat Legendre transform (dual web). After having established some general results on the flatness of the dual $d$-web of a homogeneous pre-foliation of co-degree $1$ and degree $d$, we describe some explicit examples and we show that up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ there are two families and six examples of homogeneous pre-foliations of co-degree $1$ and degree $3$ on $\mathbb {P}^{2}_{\mathbb{C}}$ with a flat dual web. This allows us to prove an analogue for pre-foliations of co-degree $1$ and degree~$3$ of a result, obtained in collaboration with D. Mar\'ın, on foliations of degree $3$ with non-degenerate singularities and a flat Legendre transform. We also show that the dual web of a reduced convex pre-foliation of co-degree $1$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is flat. This is an analogue of a result on foliations of $\mathbb{P}^{2}_{\mathbb{C}}$ due to D. Mar\'ın and J. V. Pereira.

math.DS↗

arXiv2024

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

Let $d\geq3$ be an integer. For a holomorphic $d$-web $\mathcal{W}$ on a complex surface $M$, smooth along an irreducible component $D$ of its discriminant $Δ(\mathcal{W}),$ we establish an effective criterion for the holomorphy of the curvature of $\mathcal{W}$ along $D,$ generalizing results on decomposable webs due to Mar\'ın, Pereira and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) $\mathrm{Leg}\mathcal{H}$ of a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ generalizing some of our previous results. This then allows us to study the flatness of the $d$-web $\mathrm{Leg}\mathcal{H}$ in the particular case where the foliation $\mathcal{H}$ is Galois. When the Galois group of $\mathcal{H}$ is cyclic, we show that $\mathrm{Leg}\mathcal{H}$ is flat if and only if $\mathcal{H}$ is given, up to linear conjugation, by one of the two 1-forms $ω_1^{\hspace{0.2mm}d}=y^d\mathrm{d}x-x^d\mathrm{d}y$, $ω_2^{\hspace{0.2mm}d}=x^d\mathrm{d}x-y^d\mathrm{d}y.$ When the Galois group of $\mathcal{H}$ is non-cyclic, we obtain that $\mathrm{Leg}\mathcal{H}$ is always flat.

math.DS↗

arXiv2021

Geometry of certain foliations on the complex projective plane

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim \mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, Déserti, Garba Belko and Meziani.

math.DS↗

arXiv2021

Convex foliations of degree 5 on the complex projective plane

We show that up to automorphisms of $\mathbb P^2_{\mathbb C}$ there are $14$ homogeneous convex foliations of degree $5$ on $\mathbb P^2_{\mathbb C}.$ We establish some properties of the Fermat foliation $\mathcal F_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_H^{5}$ of degree $5.$ As a consequence, we obtain that every reduced convex foliation of degree $5$ on $\mathbb P^2_{\mathbb C}$ is linearly conjugated to one of the two foliations $\mathcal F_{0}^{5}$ or $\mathcal{F}_H^{5},$ which is a partial answer to a question posed in $2013$ by D. Mar\'ın and J.V. Pereira. We end with two conjectures about the Camacho-Sad indices along the line at infinity at the non radial singularities of the homogeneous convex foliations of degree $d\geq2$ on $\mathbb P^2_{\mathbb C}.$

math.DS↗

arXiv2019

Une nouvelle démonstration de la classification des feuilletages convexes de degré deux sur $\mathbb P^2_{\mathbb C}$

A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves other than straight lines have no inflection points. The classification of the convex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has been established in $2015$ by C.~\textsc{Favre} and J.~\textsc{Pereira}. The main argument of this classification was a result obtained in~$2004$ by~D.~\textsc{Schlomiuk} and N.~\textsc{Vulpe} concerning the real polynomial vector fields of degree $2$ whose associated foliation on $\mathbb{P}^{2}_{\mathbb{R}}$ is convex. We present here a new proof of this classification, that is simpler, does not use this result and does not leave the holomorphic framework. It is based on the properties of certain models of convex foliations of $\mathbb P^2_{\mathbb C}$ of arbitrary degree and of the discriminant of the dual web of a foliation of $\mathbb P^2_{\mathbb C}$.

math.DS↗

arXiv2019

Classification of foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}$ with a flat Legendre transform

The set $\mathbf{F}(3)$ of foliations of degree three on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(3)$ of $\mathbf{F}(3)$ consisting of foliations of $\mathbf{F}(3)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. We classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree three with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreducible components. We also deduce that up to automorphism there are $4$ convex foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}.$

math.DS↗

arXiv2018

Convex foliations of degree 4 on the complex projective plane

We show that up to automorphisms of $\mathbb{P}^2_{\mathbb C}$ there are $5$ homogeneous convex foliations of degree four on $\mathbb{P}^2_{\mathbb C}.$ Using this result, we give a partial answer to a question posed in $2013$ by D. {Marín} and J. {Pereira} about the classification of reduced convex foliations on~$\mathbb{P}^2_{\mathbb C}.$

math.DG↗

arXiv2017

Feuilletages de degré trois du plan projectif complexe ayant une transformée de Legendre plate

The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $(d+2)^2-2$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(d)$ of $\mathbf{F}(d)$ consisting of foliations of $\mathbf{F}(d)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(d)$. In this dissertation we study foliations of $\mathbf{FP}(d)$ and we try to better understand the topological structure of $\mathbf{FP}(3)$. First, we establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. We will see also that it is possible, under certain assumptions, to bring the study of flatness of the dual web of a general foliation to the homogeneous framework. Second, we classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree $3$ with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreducible components.

math.DS↗

arXiv2016

Tissus plats et feuilletages homogènes sur le plan projectif

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. These results allow us to show that up to automorphism of $\mathbb{P}^2$ there are $11$ homogeneous foliations of degree $3$ with flat dual web. We will see also that it is possible, under certain assumptions, to bring the study of flatness of the dual web of a general foliation to the homogeneous framework. We get some classification results about foliations with non-degenerate singularities and flat \textsc{Legendre} transform.

math.DS↗