Convex foliations of degree 5 on the complex projective plane

TL;DR

This paper classifies convex foliations of degree 5 on the complex projective plane, showing there are exactly 14 up to automorphism, and identifies two canonical forms for reduced convex foliations.

Contribution

It provides a complete classification of degree 5 convex foliations on the complex projective plane, answering a question from 2013 and establishing properties of specific notable foliations.

Findings

14 convex foliations of degree 5 up to automorphism

Reduced convex foliations are linearly conjugate to two specific types

Properties of Fermat and Hilbert modular foliations of degree 5

Abstract

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\'{\i}n 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 $\mathbb P^2_{\mathbb C}.$

Tables

Table 1. Table 1: Types and Camacho-Sad polynomials of the homogeneous foliations  ℋ 1 , … , ℋ 14 . subscript ℋ 1 … subscript ℋ 14 \mathcal{H}_{1},\ldots,\mathcal{H}_{14}.

$i$	${𝒯}_{{\mathscr{H}}_i}$	${\mathrm{CS}}_{{\mathscr{H}}_i}\left(\lambda \right)$
$1$	$2\cdot R_4$	${\left(\lambda -1\right)}^2{\left(\lambda +\frac{1}{4}\right)}^4$
$2$	$1\cdot R_1+1\cdot R_3+1\cdot R_4$	$\frac{1}{491}{\left(\lambda -1\right)}^3\left(491{\lambda }^3+982{\lambda }^2+463\lambda +64\right)$
$3$	$2\cdot R_2+1\cdot R_4$	${\left(\lambda -1\right)}^3{\left(\lambda +\frac{3}{7}\right)}^2\left(\lambda +\frac{8}{7}\right)$
$4$	$1\cdot R_2+2\cdot R_3$	${\left(\lambda -1\right)}^3{\left(\lambda +\frac{9}{11}\right)}^2\left(\lambda +\frac{4}{11}\right)$
$5$	$2\cdot R_1+1\cdot R_2+1\cdot R_4$	${\left(\lambda -1\right)}^4{\left(\lambda +\frac{3}{2}\right)}^2$
$6$	$2\cdot R_1+1\cdot R_2+1\cdot R_4$	$\frac{1}{59}{\left(\lambda -1\right)}^4\left(59{\lambda }^2+177\lambda +64\right)$
$7$	$2\cdot R_1+2\cdot R_3$	${\left(\lambda -1\right)}^4\left({\lambda }^2+3\lambda +1\right)$
$8$	$2\cdot R_1+2\cdot R_3$	${\left(\lambda -1\right)}^4{\left(\lambda +\frac{3}{2}\right)}^2$
$9$	$1\cdot R_1+2\cdot R_2+1\cdot R_3$	$\frac{1}{197}{\left(\lambda -1\right)}^4\left(197{\lambda }^2+591\lambda +302-10\sqrt{61}\right)$
$10$	$1\cdot R_1+2\cdot R_2+1\cdot R_3$	$\frac{1}{197}{\left(\lambda -1\right)}^4\left(197{\lambda }^2+591\lambda +302+10\sqrt{61}\right)$
$11$	$4\cdot R_2$	${\left(\lambda -1\right)}^4{\left(\lambda +\frac{3}{2}\right)}^2$
$12$	$2\cdot R_1+3\cdot R_2$	${\left(\lambda -1\right)}^5\left(\lambda +4\right)$
$13$	$4\cdot R_1+1\cdot R_4$	${\left(\lambda -1\right)}^5\left(\lambda +4\right)$
$14$	$3\cdot R_1+1\cdot R_2+1\cdot R_3$	${\left(\lambda -1\right)}^5\left(\lambda +4\right)$

Equations

$${\text{mspace}2.0mu\overline{\text{mspace}-1.4mu\omega \text{mspace}-1.4mu}\text{mspace}2.0mu}_0^d=(x^d-x)\mathrm{d}y-(y^d-y)\mathrm{d}x,$$

$$\omega_{\scalebox{0.64}{$H$}}^{4}=(2x^{3}-y^{3}-1)y\mathrm{d}x+(2y^{3}-x^{3}-1)x\mathrm{d}y,$$

$$\omega_{\scalebox{0.64}{$H$}}^{5}=(y^{2}-1)(y^{2}-(\sqrt{5}-2)^{2})(y+\sqrt{5}x)\mathrm{d}x-(x^{2}-1)(x^{2}-(\sqrt{5}-2)^{2})(x+\sqrt{5}y)\mathrm{d}y,$$

$$\omega_{\scalebox{0.64}{$H$}}^{7}=(y^{3}-1)(y^{3}+7x^{3}+1)y\mathrm{d}x-(x^{3}-1)(x^{3}+7y^{3}+1)x\mathrm{d}y.$$

$${\text{mspace}2.0mu\underline{\text{mspace}-2.0mu\mathcal{G}\text{mspace}-2.0mu}\text{mspace}2.0mu}_{\mathcal{H}}([x:y])=[-A(x,y):B(x,y)].$$

$$\displaystyle\mathrm{CS}_{\mathcal{H}_{\hskip 0.56905pti}}(\lambda)=\prod\limits_{s\in\mathrm{Sing}(\mathcal{H}_{\hskip 0.56905pti})\cap\scalebox{0.9}{$\ell$}_{\infty}}(\lambda-\mathrm{CS}(\mathcal{H}_{\hskip 0.56905pti},\scalebox{1.1}{$\ell$}_{\infty},s)).$$

$$\omega =A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,A,B\in \mathbb{C}[x,y{]}_5,\gcd (A,B)=1.$$

$${\mathcal{T}}_{\mathcal{H}}=2\cdot {\mathrm{R}}_4;$$

$${\mathcal{T}}_{\mathcal{H}}=1\cdot {\mathrm{R}}_2+2\cdot {\mathrm{R}}_3;$$

$${\mathcal{T}}_{\mathcal{H}}=1\cdot {\mathrm{R}}_1+2\cdot {\mathrm{R}}_2+1\cdot {\mathrm{R}}_3;$$

$${\mathcal{T}}_{\mathcal{H}}=4\cdot {\mathrm{R}}_1+1\cdot {\mathrm{R}}_4;$$

$${\omega }_1^d=y^d\mathrm{d}x-x^d\mathrm{d}y,\text{resp}.{\omega }_3^{d,\nu }=\sum _{i=\nu +1}^d\left(\genfrac{}{}{0ex}{}{d}{i}\right)x^{d-i}{y}^i\mathrm{d}x-\sum _{i=0}^{\nu }\left(\genfrac{}{}{0ex}{}{d}{i}\right)x^{d-i}{y}^i\mathrm{d}y.$$

$${\omega }_1^5=y^5\mathrm{d}x-x^5\mathrm{d}y={\omega }_1,$$

$$\text{resp}.{\omega }_3^{5,1}$$

$$\text{resp}.{\omega }_3^{5,2}$$

$$B(x,y)=b{x}^5,$$

$$a_0=\frac{5a_2\alpha }{3},$$

$$\omega =y^3(20\alpha x^2-15(\alpha +1)xy+12y^2)\mathrm{d}x-(5\alpha -3)x^5\mathrm{d}y,(\alpha +1)(3{\alpha }^2-5\alpha +3)=0.$$

$$\omega_{6}=\frac{121(15\alpha-16)}{81(3\alpha-8)^{5}}\varphi^{*}\omega,\quad\text{where}\hskip 4.2679pt\varphi=\Big{(}(3\alpha-8)x,-3y\Big{)}.$$

$$z\mapsto -\frac{z^4(3z+4cz-5c-4)}{z+c},$$

$$\omega =y^4(3y+4cy-5cx-4x)\mathrm{d}x+x^4(y+cx)\mathrm{d}y,c\in \left\{-\frac{1}{2}\pm \frac{\sqrt{5}}{10},-\frac{3}{10}\pm \frac{\sqrt{21}}{10}\right\}.$$

$${\omega }_7=-(25+11\sqrt{5}){\phi }^{\ast }\omega ,\text{where}\phi =\left(\frac{3-\sqrt{5}}{2}x,y\right),$$

$$\text{resp}.{\omega }_8=5(87+19\sqrt{21}){\psi }^{\ast }\omega ,\text{where}\psi =\left(\frac{\sqrt{21}-5}{2}x,y\right).$$

$$z\mapsto -\frac{z^3(z^2-5z+5)}{5z^2-10z+4}.$$

$$\omega =y^3(5x^2-5xy+y^2)\mathrm{d}x+x^3(4x^2-10xy+5y^2)\mathrm{d}y.$$

$${\omega }_{11}=\frac{1}{8}{\phi }^{\ast }\omega ,\text{where}\phi =(x+y,2y).$$

$$B(x,y)=b{x}^5,$$

$$(z-\alpha {)}^2\text{divides}Q(z),$$

$$b=\frac{a_3{\alpha }^2(\alpha -1{)}^2}{2({\alpha }^2-\alpha +1)},$$

$$\beta =\frac{(\alpha +1)(3{\alpha }^2-5\alpha +3)}{5({\alpha }^2-\alpha +1)},$$

$$\displaystyle\omega=-y^{2}\Big{(}\alpha(\alpha+1)(3\alpha^{2}-5\alpha+3)x^{3}+(\alpha+1)(4\alpha^{2}-5\alpha+4)xy^{2}-2(\alpha^{2}-\alpha+1)y^{3}\Big{)}\mathrm{d}x$$

$$+2({\alpha }^4+2{\alpha }^3-3{\alpha }^2+2\alpha +1)x^2{y}^3\mathrm{d}x+{\alpha }^2(\alpha -1{)}^2{x}^5\mathrm{d}y,$$

$${\omega }_{13}=-\frac{(\alpha +1)(3{\alpha }^2-5\alpha +3)}{5{\alpha }^3(\alpha -1{)}^4}{\phi }^{\ast }\omega ,\text{where}\phi =\left(x,\frac{5\alpha {\left(\alpha -1\right)}^2}{\left(\alpha +1\right)\left(3{\alpha }^2-5\alpha +3\right)}y\right).$$

$$\displaystyle\hskip 19.91692pt\omega=\omega(\alpha)=y^{3}\Big{(}\big{(}\alpha^{2}-3\alpha+1\big{)}\big{(}\alpha^{2}+5\alpha+1\big{)}x^{2}-2\big{(}\alpha+1\big{)}\big{(}\alpha^{2}-5\alpha+1\big{)}xy+\big{(}\alpha^{2}-7\alpha+1\big{)}y^{2}\Big{)}\mathrm{d}x$$

Full text

2010 Mathematics Subject Classification. — 37F75, 32S65, 32M25.

1 Introduction and statements of results

This article is part of a series of works by the authors [2, 3, 4] on holomorphic foliations on the complex projective plane. For the definitions and notations used (radial singularities, Camacho-Sad index \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s), homogeneous foliations, etc.) we refer to [3, Sections 1 and 2].

Following [10] a foliation on the complex projective plane is said to be convex if its leaves other than straight lines have no inflection points. Notice (see [11]) that if $\mathcal{F}$ is a foliation of degree $d\geq 1$ on $\mathbb{P}^{2}_{\mathbb{C}},$ then $\mathcal{F}$ cannot have more than $3d$ (distinct) invariant lines. Moreover, if this bound is reached, then $\mathcal{F}$ is necessarily convex; in this case $\mathcal{F}$ is said to be reduced convex.

To our knowledge the only reduced convex foliations known in the literature are those presented in [10, Table 1.1]: the Fermat foliation $\mathcal{F}_{0}^{d}$ of degree $d$, the Hesse pencil \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{4} of degree $4$, the Hilbert modular foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5} of degree $5$ and the Hilbert modular foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{7} of degree $7$ defined in affine chart respectively by the $1$-forms

[TABLE]

[TABLE]

[TABLE]

[TABLE]

D. Marín and J.V. Pereira [10, Problem 9.1] asked the following question: are there other reduced convex foliations? The answer in degree $2$, resp. $3$, resp. $4$, to this question is negative, thanks to [8, Proposition 7.4], resp. [3, Corollary 6.9], resp. [4, Theorem B]. In this paper we show that the answer in degree $5$ to [10, Problem 9.1] is also negative. To do this, we follow the same approach as that described in degree $4$ in [4]. It mainly consists in using Proposition 3.2 of [4] which allows to associate to every pair (\mathcal{F},\scalebox{1.1}{\ell}), where $\mathcal{F}$ is a reduced convex foliation of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and

$\ell$

an invariant line of $\mathcal{F},$ a homogeneous convex foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ belonging to the Zariski closure of the $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$-orbit of $\mathcal{F},$ and then to study for $d=5$ the set of foliations \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} where

$\ell$

runs through the invariant lines of $\mathcal{F}.$

A homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is given, for a suitable choice of affine coordinates $(x,y),$ by a homogeneous $1$-form $\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,$ where $A,B$ are complex homogeneous polynomials of degree $d$ with $\gcd(A,B)=1.$ By [3] to such a foliation is associated the rational map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ defined by

[TABLE]

Notice (see [3]) that a homogeneous foliation $\mathcal{H}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is convex if and only if its associated map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ is critically fixed, i.e. every critical point of ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ is a fixed point of ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$. More precisely, a homogeneous foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is convex of type $\mathcal{T}_{\mathcal{H}}=\sum_{k=1}^{d-1}r_{k}\cdot\mathrm{R}_{k}$ (i.e. having $r_{1}$, resp. $r_{2},\ldots,$ resp. $r_{d-1}$ radial singularities of order $1$, resp. $2,\ldots,$ resp. $d-1$, the $\mathrm{R}_{k}$’s being just symbols) if and only if the map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ possesses $r_{1}$, resp. $r_{2},\ldots,$ resp. $r_{d-1}$ fixed critical points of multiplicity $1$, resp. $2\ldots,$ resp. $d-1,$ with $\sum_{k=1}^{d-1}kr_{k}=2d-2.$

Using results of [6, pages 79–80] on critically fixed rational maps of degree $5$ from $\mathbb{P}^{1}_{\mathbb{C}}$ to itself and studying the convexity of a homogeneous foliation $\mathcal{H}$ of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$ according to the shape of its type $\mathcal{T}_{\mathcal{H}},$ we obtain the classification, up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$, of homogeneous convex foliations of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}.$

Theorem A**.**

Up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$ there are $14$ homogeneous convex foliations $\mathcal{H}_{1},\ldots,\mathcal{H}_{14}$ of degree $5$ on the complex projective plane. They are respectively described in affine chart by the following $1$-forms

1. $\omega_{1}\hskip 2.84526pt=y^{5}\mathrm{d}x-x^{5}\mathrm{d}y$;
2. $\omega_{2}\hskip 2.84526pt=y^{2}(10\hskip 0.56905ptx^{3}+10x^{2}y+5xy^{2}+y^{3})\mathrm{d}x-x^{4}(x+5y)\mathrm{d}y$;
3. $\omega_{3}\hskip 2.84526pt=y^{3}(10\hskip 0.56905ptx^{2}+5xy+y^{2})\mathrm{d}x-x^{3}(x^{2}+5xy+10y^{2})\mathrm{d}y$;
4. $\omega_{4}\hskip 2.84526pt=y^{4}(5x-3y)\mathrm{d}x+x^{4}(3x-5y)\mathrm{d}y$;
5. $\omega_{5}\hskip 2.84526pt=y^{3}(5x^{2}-3y^{2})\mathrm{d}x-2x^{5}\mathrm{d}y$;
6. $\omega_{6}\hskip 2.84526pt=y^{3}(220\hskip 0.56905ptx^{2}-165xy+36y^{2})\mathrm{d}x-121x^{5}\mathrm{d}y$;
7. \omega_{7}\hskip 2.84526pt=y^{4}\Big{(}\big{(}5-\sqrt{5}\big{)}x-2y\Big{)}\mathrm{d}x+x^{4}\Big{(}\big{(}7-3\sqrt{5}\big{)}x-2\big{(}5-2\sqrt{5}\big{)}y\Big{)}\mathrm{d}y;
8. \omega_{8}\hskip 2.84526pt=y^{4}\Big{(}5\big{(}3-\sqrt{21}\big{)}x+6y\Big{)}\mathrm{d}x+x^{4}\Big{(}3\big{(}23-5\sqrt{21}\big{)}x-10\big{(}9-2\sqrt{21}\big{)}y\Big{)}\mathrm{d}y;
9. \omega_{9}\hskip 2.84526pt=y^{3}\Big{(}2\big{(}5+a\big{)}x^{2}-\big{(}15+a\big{)}xy+6y^{2}\Big{)}\mathrm{d}x-x^{4}\Big{(}\big{(}1-a\big{)}x+2ay\Big{)}\mathrm{d}y, where a=\sqrt{\scalebox{0.83}{5\big{(}4\sqrt{61}-31\big{)}}};
10. \omega_{10}=y^{3}\Big{(}2\big{(}5+\mathrm{i}b\big{)}x^{2}-\big{(}15+\mathrm{i}b\big{)}xy+6y^{2}\Big{)}\mathrm{d}x-x^{4}\Big{(}\big{(}1-\mathrm{i}b\big{)}x+2\mathrm{i}by\Big{)}\mathrm{d}y, where b=\sqrt{\scalebox{0.83}{5\big{(}4\sqrt{61}+31\big{)}}};
11. $\omega_{11}=y^{3}(5x^{2}-y^{2})\mathrm{d}x+x^{3}(x^{2}-5y^{2})\mathrm{d}y$;
12. $\omega_{12}=y^{3}(20\hskip 0.56905ptx^{2}-5xy-y^{2})\mathrm{d}x+x^{3}(x^{2}+5xy-20y^{2})\mathrm{d}y$;
13. $\omega_{13}=y^{2}(5x^{3}-10\hskip 0.56905ptx^{2}y+10\hskip 0.56905ptxy^{2}-4y^{3})\mathrm{d}x-x^{5}\mathrm{d}y$;
14. \omega_{14}=y^{3}\Big{(}u(\sigma)x^{2}+v(\sigma)xy+w(\sigma)y^{2}\Big{)}\mathrm{d}x+\sigma\hskip 0.85358ptx^{4}\Big{(}2\sigma\big{(}\sigma^{2}-\sigma+1\big{)}x-\big{(}\sigma+1\big{)}\big{(}3\sigma^{2}-5\sigma+3\big{)}y\Big{)}\mathrm{d}y,
15. where $u(\sigma)=(\sigma^{2}-3\sigma+1)(\sigma^{2}+5\sigma+1),$   $v(\sigma)=-2(\sigma+1)(\sigma^{2}-5\sigma+1),$   $w(\sigma)=(\sigma^{2}-7\sigma+1),$
16. $\sigma=\rho+\mathrm{i}\sqrt{\frac{1}{6}-\frac{4}{3}\rho-\frac{1}{3}\rho^{2}}$  and $\rho$ is the unique real number satisfying $8\rho^{3}-52\rho^{2}+134\rho-15=0.$

During the proof of this theorem in §2 we also obtain the following dual result.

Theorem B**.**

Up to conjugation by a Möbius transformation there are $14$ critically fixed rational maps of degree $5$ from the Riemann sphere to itself, namely the maps ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}_{\hskip 0.28453pt1}},\ldots,{\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}_{\hskip 0.28453pt14}}.$ **

To every foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and to every integer $d\geq 2,$ we associate respectively the following two subsets of $\mathbb{C}\setminus\{0,1\}$:

• —

$\mathcal{CS}(\mathcal{F})$ is, by definition, the set of $\lambda\in\mathbb{C}\setminus\{0,1\}$ for which there is a line

$\ell$

invariant by $\mathcal{F}$ and a non-degenerate singular point s\in\scalebox{1.1}{\ell} of $\mathcal{F}$ such that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s)=\lambda;

• —

$\mathcal{HCS}_{d}$ is defined as the set of $\lambda\in\mathbb{C}\setminus\{0,1\}$ for which there exist two homogeneous convex foliations $\mathcal{H}$ and $\mathcal{H}^{\prime}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ having respective singular points $s$ and $s^{\prime}$ on the line at infinity \scalebox{1.1}{\ell}_{\infty} such that \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=\lambda and \mathrm{CS}(\mathcal{H}^{\prime},\scalebox{1.1}{\ell}_{\infty},s^{\prime})=\frac{1}{\lambda}.

The following proposition, which will be proved in §2, motivates the introduction of the sets $\mathcal{CS}(\mathcal{F})$ and $\mathcal{HCS}_{d}.$

Proposition C**.**

Let $\mathcal{F}$ be a reduced convex foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Then

• (a)

$\emptyset\neq\mathcal{CS}(\mathcal{F})\subset\mathcal{HCS}_{d};$

• (b)

$\forall\hskip 2.84526pt\lambda\in\mathcal{CS}(\mathcal{F})$, $\frac{1}{\lambda}\in\mathcal{CS}(\mathcal{F}).$

Remark 1.1*.*

In particular, for the foliations \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5} and $\mathcal{F}_{0}^{d},$ we have

• $\bullet$

\{-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\}=\mathcal{CS}(\mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5})\subset\mathcal{HCS}_{5}, cf. [9, Theorem 2];

• $\bullet$

$\{(1-d)^{\pm 1}\}=\mathcal{CS}(\mathcal{F}_{0}^{d})\subset\mathcal{HCS}_{d}$ for any $d\geq 2,$ cf. [3, Example 6.5].

The following theorem gives equivalent conditions for a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ to be conjugated to the Fermat foliation $\mathcal{F}_{0}^{d}.$

Theorem D**.**

Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ The following assertions are equivalent:

• (1)

$\mathcal{F}$ is linearly conjugated to the Fermat foliation $\mathcal{F}_{0}^{d}$;

• (2)

$\mathcal{F}$ is reduced convex and $\mathcal{CS}(\mathcal{F})=\{(1-d)^{\pm 1}\}$;

• (3)

$\mathcal{F}$ possesses three radial singularities of maximal order $d-1,$ necessarily non-aligned.

In this theorem, the implication $\textit{(3)}\Rightarrow\textit{(1)}$ is a slight generalization of our previous result [3, Proposition 6.3], where we had obtained the same conclusion but with the additional hypothesis that the three radial singularities of $\mathcal{F}$ are not aligned.

Corollary E**.**

If $\mathcal{HCS}_{d}=\{(1-d)^{\pm 1}\}$ then, up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$, the Fermat foliation $\mathcal{F}_{0}^{d}$ is the unique reduced convex foliation in degree $d.$**

The following theorem gives equivalent conditions for a foliation of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$ to be conjugated to the Hilbert modular foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5}.

Theorem F**.**

Let $\mathcal{F}$ be a foliation of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ The following assertions are equivalent:

• (1)

$\mathcal{F}$ is linearly conjugated to the Hilbert modular foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5};

• (2)

$\mathcal{F}$ is reduced convex and $\mathcal{CS}(\mathcal{F})=\{-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\}$;

• (3)

$\mathcal{F}$ possesses three radial singularities $m_{1},m_{2},m_{3}$ of order $3$ (necessarily non-aligned) and two radial singularities of order $1$ on each invariant line $(m_{j}m_{l}),1\leq j<l\leq 3.$

Using essentially Theorems A, D, F and Proposition C, we establish the following theorem.

Theorem G**.**

Up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$ the Fermat foliation $\mathcal{F}_{0}^{5}$ and the Hilbert modular foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5} are the only reduced convex foliations of degree five on $\mathbb{P}^{2}_{\mathbb{C}}.$**

2 Proof of the main results

We need to know the numbers  $r_{ij}$  of radial singularities of order $j$ of the homogeneous foliations $\mathcal{H}_{\hskip 0.56905pti},$   $i=1,\ldots,14$, $j=1,\ldots,4,$ and the values of the Camacho-Sad indices \mathrm{CS}(\mathcal{H}_{\hskip 0.56905pti},\scalebox{1.1}{\ell}_{\infty},s), s\in\mathrm{Sing}(\mathcal{H}_{\hskip 0.56905pti})\cap\scalebox{1.1}{\ell}_{\infty}, $i=1,\ldots,14$. For this reason, we have computed, for each $i=1,\ldots,14$, the type $\mathcal{T}_{\mathcal{H}_{\hskip 0.56905pti}}$ of $\mathcal{H}_{\hskip 0.56905pti}$ and the following polynomial (called Camacho-Sad polynomial of the homogeneous foliation $\mathcal{H}_{\hskip 0.56905pti}$)

[TABLE]

Table 1 below summarizes the types and Camacho-Sad polynomials of the foliations $\mathcal{H}_{\hskip 0.56905pti}$, $i=1,\ldots,14$.

Proof of Theorem A.

Let $\mathcal{H}$ be a homogeneous convex foliation of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$, defined in the affine chart $(x,y)$, by the $1$-form

[TABLE]

By [2, Remark 2.5] the foliation $\mathcal{H}$ cannot have $5+1=6$ distinct radial singularities; in other words it cannot be of one of the two types $5\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{3}$ or $4\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{2}.$ We are then in one of the following situations:

[TABLE]

The proof consists in to analyze these $11$ possibilities either by applying some results in [3] or by appealing to a specific classification taken from [6].

• $\bullet$

We know from [3, Propositions 4.1, 4.2] that if a homogeneous convex foliation of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is of type $2\cdot\mathrm{R}_{d-1},$  resp. $1\cdot\mathrm{R}_{\nu}+1\cdot\mathrm{R}_{d-\nu-1}+1\cdot\mathrm{R}_{d-1}$ with $\nu\in\{1,2,\ldots,d-2\},$ then it is linearly conjugated to the foliation $\mathcal{H}_{1}^{d}$, resp. $\mathcal{H}_{3}^{d,\nu},$ given by

[TABLE]

It follows that if the foliation $\mathcal{H}$ is of type $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{4}$,  resp. $\mathcal{T}_{\mathcal{H}}=1\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{3}+1\cdot\mathrm{R}_{4}$,  resp. $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{4}$, then the $1$-form $\omega$ is linearly conjugated to

[TABLE]

[TABLE]

[TABLE]

• $\bullet$

Assume that $\mathcal{T}_{\mathcal{H}}=1\cdot\mathrm{R}_{2}+2\cdot\mathrm{R}_{3}$. This means that the rational map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$, ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}(z)=-\dfrac{A(1,z)}{B(1,z)},$ possesses three fixed critical points, one of multiplicity $2$ and two of multiplicity $3.$ By [6, page 79], ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ is conjugated by a Möbius transformation to $z\mapsto-\dfrac{z^{4}(3z-5)}{5z-3}$. As a result, $\omega$ is linearly conjugated to $\omega_{4}$.

• $\bullet$

Let us study the possibility $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{4}.$ Up to linear conjugation we can assume that, for some $\alpha\in\mathbb{C}\setminus\{0,1\},$ the points $[1:0:0],\,[0:1:0],\,[1:1:0],\,[1:\alpha:0]\in\mathbb{P}^{2}_{\mathbb{C}}$ are radial singularities of $\mathcal{H}$ with respective orders $4,2,1,1,$ or equivalently that the points $\infty=[1:0],\,[0:1],\,[1:1],\,[1:\alpha]\in\mathbb{P}^{1}_{\mathbb{C}}$ are fixed and critical for ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ with respective multiplicities $4,2,1,1.$ By [3, Lemma 3.9], there exist constants $a_{0},a_{2},b\in\mathbb{C}^{*},a_{1}\in\mathbb{C}$ such that

[TABLE]

where $P(z):=A(1,z)+B(1,z)$ and $Q(z):=A(1,z)+\alpha B(1,z).$ A straightforward computation leads to

[TABLE]

Replacing $\omega$ by \dfrac{\raisebox{-2.27621pt}{12}}{\raisebox{2.84526pt}{a_{2}}}\omega, we reduce it to

[TABLE]

This $1$-form is linearly conjugated to one of the two $1$-forms $\omega_{5}$ or $\omega_{6}$. Indeed, on the one hand, if $\alpha=-1$, then $\omega_{5}=-\frac{1}{4}\omega.$ On the other hand, if $3\alpha^{2}-5\alpha+3=0$, then

[TABLE]

• $\bullet$

Assume that $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{3}$. Then the rational map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ admits four fixed critical points, two of multiplicity $1$ and two of multiplicity $3.$ This implies, by [6, page 79], that up to conjugation by a Möbius transformation, ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ writes as

[TABLE]

where $c=-1/2\pm\sqrt{5}/10$  or  $c=-3/10\pm\sqrt{21}/10.$ Thus, up to linear conjugation

[TABLE]

In the case where $c=-1/2\pm\sqrt{5}/10$,  resp. $c=-3/10\pm\sqrt{21}/10$, the $1$-form $\omega$ is linearly conjugated to $\omega_{7}$, resp. $\omega_{8}$. Indeed, on the one hand, if $c=-1/2+\sqrt{5}/10$,  resp. $c=-3/10+\sqrt{21}/10$, then $\omega_{7}=-2(5-2\sqrt{5})\omega$, resp. $\omega_{8}=-10(9-2\sqrt{21})\omega$. On the other hand, if $c=-1/2-\sqrt{5}/10$,  resp. $c=-3/10-\sqrt{21}/10$, then

[TABLE]

• $\bullet$

We know from [6, page 79] that up to Möbius transformation there are two rational maps of degree $5$ from the Riemann sphere to itself having four distinct fixed critical points, one of multiplicity $1$, two of multiplicity $2$ and one of multiplicity $3$; thus up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$ there are two homogeneous convex foliations of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$ having type $1\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{3}.$ Now, by Table 1, we have on the one hand $\mathrm{CS}_{\mathcal{H}_{\hskip 0.56905pt9}}\neq\mathrm{CS}_{\mathcal{H}_{\hskip 0.56905pt10}},$ so that the foliations $\mathcal{H}_{\hskip 0.56905pt9}$ and $\mathcal{H}_{\hskip 0.56905pt10}$ are not linearly conjugated, and on the other hand $\mathcal{T}_{\mathcal{H}_{9}}=\mathcal{T}_{\mathcal{H}_{10}}=1\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{3}.$ It follows that if the foliation $\mathcal{H}$ is of type $\mathcal{T}_{\mathcal{H}}=1\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{3},$ then $\mathcal{H}$ is linearly conjugated to one of the two foliations $\mathcal{H}_{\hskip 0.56905pt9}$ or $\mathcal{H}_{\hskip 0.56905pt10}.$

• $\bullet$

Assume that $\mathcal{T}_{\mathcal{H}}=4\cdot\mathrm{R}_{2}$. The rational map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ has therefore four different fixed critical points of multiplicity $2$. By [6, page 80], up to conjugation by a Möbius transformation, ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ writes as

[TABLE]

As a consequence, up to linear conjugation

[TABLE]

This $1$-form is linearly conjugated to

[TABLE]

• $\bullet$

Assume that $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{1}+3\cdot\mathrm{R}_{2}$. Then the rational map ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ possesses five fixed critical points, two of multiplicity $1$ and three of multiplicity $2.$ By [6, page 80], ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ is conjugated by a Möbius transformation to $z\mapsto-\dfrac{z^{3}(z^{2}+5z-20)}{20z^{2}-5z-1}$, which implies that $\omega$ is linearly conjugated to $\omega_{12}$.

• $\bullet$

Let us consider the eventuality $\mathcal{T}_{\mathcal{H}}=4\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{4}.$ Up to isomorphism, we can assume that, for some $\alpha,\beta\in\mathbb{C}\setminus\{0,1\}$ with $\alpha\neq\beta,$ the points $\infty=[1:0],\,[0:1],\,[1:1],\,[1:\alpha],\,[1:\beta]\in\mathbb{P}^{1}_{\mathbb{C}}$ are fixed and critical for ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}},$ with respective multiplicities $4,1,1,1,1.$ By [3, Lemma 3.9], there exist constants $a_{0},a_{3},b\in\mathbb{C}^{*},a_{1},a_{2}\in\mathbb{C}$ such that

[TABLE]

where $P(z):=A(1,z)+B(1,z)$, $Q(z):=A(1,z)+\alpha B(1,z)$ and $R(z):=A(1,z)+\beta B(1,z).$ A straightforward computation gives us

[TABLE]

Multiplying $\omega$ by \dfrac{\raisebox{-2.27621pt}{2}}{\raisebox{2.84526pt}{a_{3}}}(\alpha^{2}-\alpha+1), we reduce it to

[TABLE]

with $(\alpha^{2}-2\alpha+2)(2\alpha^{2}-2\alpha+1)(\alpha^{2}+1)=0.$ This $1$-form $\omega$ is linearly conjugated to

[TABLE]

• $\bullet$

Finally let us examine the case $\mathcal{T}_{\mathcal{H}}=3\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{2}+1\cdot\mathrm{R}_{3}.$ Up to linear conjugation we can assume that the points $\infty=[1:0],\,[0:1],\,[1:1],\,[1:\alpha],\,[1:\beta]\in\mathbb{P}^{1}_{\mathbb{C}},$ where $\alpha\beta\in\mathbb{C}\setminus\{0,1\}$ and $\alpha\neq\beta,$ are fixed and critical for ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}},$ with respective multiplicities $3,2,1,1,1.$ A similar reasoning as in the previous case leads to

[TABLE]

with $P(\alpha)=0$ where $P(z):=3z^{6}-39z^{5}+194z^{4}-203z^{3}+194z^{2}-39z+3.$ The $1$-form $\omega$ is linearly conjugated to

[TABLE]

where $\sigma=\rho+\mathrm{i}\sqrt{\frac{1}{6}-\frac{4}{3}\rho-\frac{1}{3}\rho^{2}}$  and $\rho$ is the unique real number satisfying $8\rho^{3}-52\rho^{2}+134\rho-15=0.$ Indeed, on the one hand, it is easy to see that $\sigma$ is a root of the polynomial $P$, so that $\omega_{14}=\omega(\sigma).$ On the other hand, a straightforward computation shows that if $\alpha_{1}$ and $\alpha_{2}$ are any two roots of $P$ then

[TABLE]

with $\mu=195\alpha_{2}^{4}-202\alpha_{2}^{3}+233\alpha_{2}^{2}-42\alpha_{2}+3$,  $\varphi=\left(x,-\dfrac{\lambda}{43200}y\right)$ where {Small}

[TABLE]

The foliations $\mathcal{H}_{1},\ldots,\mathcal{H}_{14}$ are not linearly conjugated because we have $\mathcal{T}_{\mathcal{H}_{\hskip 0.56905pti}}\leavevmode\nobreak\ \neq\leavevmode\nobreak\ \mathcal{T}_{\mathcal{H}_{j}}$ or $\mathrm{CS}_{\mathcal{H}_{\hskip 0.56905pti}}\neq\mathrm{CS}_{\mathcal{H}_{j}}$ for all $i,j\in\{1,\ldots,14\}$ with $i\neq j$ (see Table 1). This ends the proof Theorem A. ∎

Let $\mathcal{F}$ be a reduced convex foliation of degree $d\geq 1$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and let

$\ell$

be one of its $3d$ invariant lines. To the pair (\mathcal{F},\scalebox{1.1}{\ell}) we can associate thanks to [4] a homogeneous convex foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}},$ called homogeneous degeneration of $\mathcal{F}$ along \scalebox{1.1}{\ell}, as follows. Let us fix homogeneous coordinates $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ such that \scalebox{1.1}{\ell}=(z=0); since

$\ell$

is $\mathcal{F}$-invariant, $\mathcal{F}$ is described in the affine chart $z=1$ by a $1$-form $\omega$ of type

[TABLE]

where $A_{i},\,B_{i}$ are homogeneous polynomials of degree $i$. By [4, Proposition 3.2] we have $\gcd(A_{d},B_{d})=1$ which allows us to define the foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} by the $1$-form

[TABLE]

It is easy to check that this definition is intrinsic, i.e. it does not depend on the choice of the homogeneous coordinates $[x:y:z]$ nor on the choice of the $1$-form $\omega$ describing $\mathcal{F}.$

The following result, taken from [4, Proposition 3.2], will be very useful to us.

Proposition 2.1** ([4]).**

With the previous notations, the foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} has the following properties:

• (i)

\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} belongs to the Zariski closure of the $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$-orbit of $\mathcal{F};$

• (ii)

$\ell$

is invariant by \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}};

• (iii)

\mathrm{Sing}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}})\cap\scalebox{1.1}{\ell}=\mathrm{Sing}(\mathcal{F})\cap\scalebox{1.1}{\ell};

• (iv)

every singular point of \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} on

$\ell$

is non-degenerate;

• (v)

a point s\in\scalebox{1.1}{\ell} is a radial singularity of order $k\leq d-1$ for \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} if and only if it is for $\mathcal{F}$;

• (vi)

\forall\hskip 2.84526pts\in\mathrm{Sing}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}})\cap\scalebox{1.1}{\ell},\hskip 2.84526pt\mathrm{CS}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}},\scalebox{1.1}{\ell},s)=\mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s).

Proof of Proposition C.

Since by hypothesis $\mathcal{F}$ is reduced convex, all its singularities are non-degenerate ([3, Lemma 6.8]). Let

$\ell$

be an invariant line of $\mathcal{F}.$ By [1, Proposition 2.3] it follows thseeat $\mathcal{F}$ possesses exactly $d+1$ singularities on \scalebox{1.1}{\ell}. The Camacho-Sad formula (see [5]) \sum\limits_{s\in\mathrm{Sing}(\mathcal{F})\cap\scalebox{0.9}{\ell}}\mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s)=1 then implies the existence of s\in\mathrm{Sing}(\mathcal{F})\cap\scalebox{1.1}{\ell} such that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s)\in\mathbb{C}\setminus\{0,1\}; as a result $\mathcal{CS}(\mathcal{F})\neq\emptyset.$

Let $\lambda\in\mathcal{CS}(\mathcal{F})\subset\mathbb{C}\setminus\{0,1\}$; there is a line \scalebox{1.1}{\ell}_{1} invariant by $\mathcal{F}$ and a singular point s\in\scalebox{1.1}{\ell}_{1} of $\mathcal{F}$ such that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{1},s)=\lambda. By [4, Lemma 3.1] through the point $s$ passes a second $\mathcal{F}$-invariant line \scalebox{1.1}{\ell}_{2}. Since \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{1},s)\mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{2},s)=1, we have \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{2},s)=\frac{1}{\lambda}; thus $\frac{1}{\lambda}\in\mathcal{CS}(\mathcal{F}).$ Moreover, by [4, Proposition 3.2] (cf. assertion (vi) of Proposition 2.1 above), we have

[TABLE]

which shows that $\lambda\in\mathcal{HCS}_{d},$ hence $\mathcal{CS}(\mathcal{F})\subset\mathcal{HCS}_{d}.$ ∎

An immediate consequence of Table 1 is the following:

Corollary 2.2**.**

\mathcal{HCS}_{5}=\{-4^{\pm 1},-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\}=\mathcal{CS}(\mathcal{F}_{0}^{5})\cup\mathcal{CS}(\mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5}). **

The proof of Theorem D uses Lemma 2.3 and Lemma 2.4 stated below.

Lemma 2.3**.**

Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ having two radial singularities $m_{1},m_{2}$ of maximal order $d-1.$ Then the line $(m_{1}m_{2})$ cannot contain a third radial singularity of $\mathcal{F}.$ **

Proof.

Let us choose homogeneous coordinates $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ such that $m_{1}=[0:1:0]$ and $m_{2}=[1:0:0].$ Thanks to [1, Proposition 2.2] (cf. [2, Remark 1.2]) the line $\ell=(m_{1}m_{2})$ must be invariant by $\mathcal{F}.$ Then the foliation $\mathcal{F}$ is given in the affine chart $z=1$ by a $1$-form $\omega$ of type $\omega=\omega_{0}+\omega_{1}+\cdots+\omega_{d},$ where, for $0\leq i\leq d,\omega_{i}=A_{i}(x,y)\mathrm{d}x+B_{i}(x,y)\mathrm{d}y,$ with $A_{i},B_{i}$ homogeneous polynomials of degree $i.$

Writing explicitly that the points $m_{j},j=1,2,$ are radial singularities of maximal order $d-1$ of $\mathcal{F}$ (see [3, Proposition 6.3]), we obtain that the highest degree homogeneous part $\omega_{d}$ of $\omega$ is of the form $\omega_{d}=ay^{d}\mathrm{d}x+bx^{d}\mathrm{d}y,$ with $a,b\in\mathbb{C}^{*}.$ Thus, $\omega_{d}$ defines a homogeneous convex foliation $\mathcal{H}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ of type $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{d-1}.$ If we would know that $\mathcal{F}$ was a convex reduced foliation then $\mathcal{H}=\mathcal{H}^{\ell}_{\mathcal{F}}$ for the invariant line $\ell=(m_{1}m_{2})$ and we could apply Proposition 2.1 to conclude. Anyway, reasoning as in the proof of [3, Proposition 6.4], we see that $\mathcal{F}$ and $\mathcal{H}$ have the same singularities on the line $(m_{1}m_{2})$ and that every singularity $s$ of $\mathcal{F}$ on $(m_{1}m_{2})$ distinct from $m_{1}$ and $m_{2}$ is non-degenerate and has Camacho-Sad index $\mathrm{CS}(\mathcal{F},(m_{1}m_{2}),s)=\mathrm{CS}(\mathcal{H},(m_{1}m_{2}),s)=\frac{1}{1-d}\neq 1,$ hence the lemma. ∎

Lemma 2.4**.**

Let $\mathcal{H}$ be a homogeneous convex foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Assume that every non radial singularity $s$ of $\mathcal{H}$ on \scalebox{1.1}{\ell}_{\infty} has Camacho-Sad index \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\in\{(1-d)^{\pm 1}\}. Denote by $\kappa_{0}$ the number of (distinct) radial singularities of $\mathcal{H}$ and by $\kappa_{1}$ (resp. $\kappa_{2}$) the number of singularities s\in\scalebox{1.1}{\ell}_{\infty} of $\mathcal{H}$ such that \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=1-d (resp. \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=\frac{1}{1-d}). Then

• –

either $(\kappa_{0},\kappa_{1},\kappa_{2})=(d,1,0)$;

• –

or $(\kappa_{0},\kappa_{1},\kappa_{2})=(2,0,d-1),$ in which case $\mathcal{T}_{\mathcal{H}}=2\cdot\mathrm{R}_{d-1}.$

Before proving this lemma let us make two remarks:

Remark 2.5*.*

By [7, Theorem 4.3] every homogeneous convex foliation of degree $d$ on the complex projective plane has exactly $d+1$ singularities on the line at infinity, necessarily non-degenerate.

Remark 2.6*.*

A straightforward computation shows that if a homogeneous foliation $\mathcal{H}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ possesses a non-degenerate singularity s\in\scalebox{1.1}{\ell}_{\infty} such that \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=1, then $s$ is necessarily radial. In particular, when $\mathcal{H}$ is convex, a singularity s\in\scalebox{1.1}{\ell}_{\infty} of $\mathcal{H}$ is radial if and only if it has Camacho-Sad index \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=1.

Proof of Lemma 2.4.

The Camacho-Sad formula \sum\limits_{s\in\mathrm{Sing}(\mathcal{H})\cap\scalebox{0.9}{\ell}_{\infty}}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=1 (see [5]) and Remarks 2.5 and 2.6 imply that

[TABLE]

From these two equations we obtain that $\kappa_{0}=2+\kappa_{1}(d-2)$  and  $\kappa_{2}=(d-1)(1-\kappa_{1})\geq 0,$ so that $\kappa_{1}\in\{0,1\},$ hence the lemma. ∎

Proof of Theorem D.

The implication $\textit{(3)}\Rightarrow\textit{(1)}$ follows from [3, Proposition 6.3] and from Lemma 2.3.

The fact that (1) implies (2) follows from the reduced convexity of the foliation $\mathcal{F}_{0}^{d}$ and from the equality $\mathcal{CS}(\mathcal{F}_{0}^{d})=\{(1-d)^{\pm 1}\}$ (Remark 1.1).

Let us show that (2) implies (3). Assume that $\mathcal{F}$ is reduced convex and that $\mathcal{CS}(\mathcal{F})=\{(1-d)^{\pm 1}\}.$ Let $m$ be a non radial singular point of $\mathcal{F}$; through $m$ pass exactly two $\mathcal{F}$-invariant lines \scalebox{1.1}{\ell}_{m}^{(1)} and \scalebox{1.1}{\ell}_{m}^{(2)} ([4, Lemma 3.1]). It follows that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{m}^{(i)},m)=(1-d)^{\pm 1} for $i=1,2.$ Up to renumbering the \scalebox{1.1}{\ell}_{m}^{(i)}, we can assume that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{m}^{(1)},m)=\frac{1}{1-d} and \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}_{m}^{(2)},m)=1-d for any choice of the non radial singularity $m\in\mathrm{Sing}\mathcal{F}.$ Moreover, according to Proposition 2.1, for any invariant line

$\ell$

of $\mathcal{F}$ and for any non radial singularity s\in\scalebox{1.1}{\ell} of the homogeneous degeneration \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} of $\mathcal{F}$ along \scalebox{1.1}{\ell}, we have \mathrm{CS}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}},\scalebox{1.1}{\ell},s)=\mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},s)\in\mathbb{C}\setminus\{0,1\} and therefore \mathrm{CS}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}},\scalebox{1.1}{\ell},s)\in\{(1-d)^{\pm 1}\}. It follows by Lemma 2.4 that \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}_{m}^{(1)}} is of type $2\cdot\mathrm{R}_{d-1}$. This implies, according to assertion (v) of Proposition 2.1, that $\mathcal{F}$ possesses two radial singularities $m_{1},m_{2}$ of maximal order $d-1$ on the line \scalebox{1.1}{\ell}_{m}^{(1)}. Let $m^{\prime}$ be another non radial singular point of $\mathcal{F}$ not belonging to the line \scalebox{1.1}{\ell}_{m}^{(1)}. For any $s\in\mathrm{Sing}\mathcal{F}$ let us denote, as in [3, Section 1], by $\tau(\mathcal{F},s)$ the tangency order of $\mathcal{F}$ with a generic line passing through $s.$ For $i=1,2$ we have $\tau(\mathcal{F},m^{\prime})+\tau(\mathcal{F},m_{i})=1+d>\deg\mathcal{F},$ which implies (cf. [1, Proposition 2.2]) that the lines $(m^{\prime}m_{i})$ are invariant by $\mathcal{F}.$ Thus, the line \scalebox{1.1}{\ell}_{m^{\prime}}^{(1)} is one of the lines $(m^{\prime}m_{1})$ or $(m^{\prime}m_{2})$ and it in turn contains another radial singularity $m_{3}$ of maximal order $d-1$ of $\mathcal{F}.$ ∎

The proof of Theorem F uses the following lemma for $d=5$ which we state in arbitrary degree $d$ as it could be used in other situations. It can be proved in the same way as in [3, Proposition 6.3].

Lemma 2.7**.**

Let $\mathcal{F}$ be a foliation of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Assume that the points $m_{1}=[0:0:1],\,m_{2}=[1:0:0]$  and  $m_{3}=[0:1:0]$ are radial singularities of order $d-2$ of $\mathcal{F}.$ Let $\omega$ be a $1$-form defining $\mathcal{F}$ in the affine chart $z=1.$ Then $\omega$ is of the form

[TABLE]

where $\lambda_{i,j},a_{i,j},b_{i,j}\in\mathbb{C}$ with $\lambda_{0,0}\neq 0.$ **

Proof of Theorem F.

The implication $\textit{(1)}\Rightarrow\textit{(2)}$ follows from the reduced convexity of the foliation \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5} and from the equality \mathcal{CS}(\mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5})=\{-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\} (Remark 1.1).

Let us show that (2) implies (3). Assume that $\mathcal{F}$ is reduced convex and that $\mathcal{CS}(\mathcal{F})=\{-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\}.$ Let 

$\ell$

 be an invariant line of $\mathcal{F}.$ The homogeneous foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} – homogeneous degeneration of $\mathcal{F}$ along

$\ell$

– being convex of degree $5$, it must be linearly conjugated to one of the fourteen homogeneous foliations given by Theorem A. Moreover, let $m$ be a non radial singular point of $\mathcal{F}$ on

$\ell$

; then we have \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},m)=-\frac{3}{2}\pm\frac{\sqrt{5}}{2}. According to Proposition 2.1, the point $m$ is also a non radial singularity for \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} and we have \mathrm{CS}(\mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}},\scalebox{1.1}{\ell},m)=\mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},m)=-\frac{3}{2}\pm\frac{\sqrt{5}}{2}. It then follows from Table 1 that \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} is of type $2\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{3}.$ This implies, according to assertion (v) of Proposition 2.1, that $\mathcal{F}$ has exactly four radial singularities on the line \scalebox{1.1}{\ell}; two of them $m_{1},m_{2}$ are of order $3$ and the other two are of order $1.$ Let us consider another $\mathcal{F}$-invariant line \scalebox{1.1}{\ell}^{\prime}\neq\scalebox{1.1}{\ell} passing through $m_{1},$ whose existence is guaranteed by [4, Lemma 3.1]. Then \scalebox{1.1}{\ell}^{\prime} contains another radial singularity $m_{3}$ of order $3$ of $\mathcal{F}$ and two radial singularities of order $1$ of $\mathcal{F}.$ By [1, Proposition 2.2], the fact that $\tau(\mathcal{F},m_{2})+\tau(\mathcal{F},m_{3})=4+4>\deg\mathcal{F}$ ensures the $\mathcal{F}$-invariance of the line \scalebox{1.1}{\ell}^{\prime\prime}=(m_{2}m_{3}). Therefore \scalebox{1.1}{\ell}^{\prime\prime} in turn contains two radial singularities of order $1$ of $\mathcal{F}.$

Finally, let us prove that (3) implies (1). Assume that (3) holds. Then there is a homogeneous coordinate system $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ in which $m_{1}=[0:0:1],\,m_{2}=[1:0:0]$  and  $m_{3}=[0:1:0].$ Moreover, in this coordinate system the lines $x=0$, $y=0$, $z=0$ must be invariant by $\mathcal{F}$ and there exist $x_{0},y_{0},z_{0},x_{1},y_{1},z_{1}\in\mathbb{C}^{*},$ $x_{1}\neq x_{0},y_{1}\neq y_{0},z_{1}\neq z_{0},$ such that the points $m_{4}=[x_{0}:0:1],$  $m_{5}=[1:y_{0}:0],$  $m_{6}=[0:1:z_{0}],$  $m_{7}=[x_{1}:0:1],$  $m_{8}=[1:y_{1}:0],$  $m_{9}=[0:1:z_{1}]$ are radial singularities of order $1$ of $\mathcal{F}.$ Let us set $\xi=\frac{x_{1}}{x_{0}},\,\rho=\frac{y_{1}}{y_{0}},\,\sigma=\frac{z_{1}}{z_{0}},\,w_{0}=x_{0}y_{0}z_{0}$; then $w_{0}\in\mathbb{C}^{*}$, $\xi,\rho,\sigma\in\mathbb{C}\setminus\{0,1\}$ and, up to renumbering the $x_{i},y_{i},z_{i},$ we can assume that $\xi,$ $\rho$ and $\sigma$ are all of modulus greater than or equal to $1.$ Let $\omega$ be a $1$-form defining $\mathcal{F}$ in the affine chart $z=1.$ By conjugating $\omega$ by the diagonal linear transformation $(x_{0}\hskip 0.56905ptx,\hskip 0.56905ptx_{0}y_{0}y)$, we reduce ourselves to $m_{4}=[1:0:1],$  $m_{5}=[1:1:0],$  $m_{6}=[0:1:w_{0}],$  $m_{7}=[\xi:0:1],$  $m_{8}=[1:\rho:0],$  $m_{9}=[0:1:\sigma w_{0}].$ Since $m_{1}$, $m_{2}$ and $m_{3}$ are radial singularities of order $3$, $\omega$ can be written as in the expression given in Lemma 2.7 in the case $d=5$. Then, as in the proof of [4, Theorem B], by writing explicitly that the points $m_{j},4\leq j\leq 9,$ are radial singularities of order $1$ of $\mathcal{F}$ we obtain that $w_{0}=\pm(\sqrt{5}-2)$ and

{Small}

[TABLE]

with $a_{0,2}\neq 0.$ Thus $\omega$ is of the form

[TABLE]

The $1$-form $\omega$ is linearly conjugated to

[TABLE]

Indeed, if $w_{0}=\sqrt{5}-2$, resp. $w_{0}=2-\sqrt{5}$, then

[TABLE]

∎

Proof of Theorem G.

Let $\mathcal{F}$ be a reduced convex foliation of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ By assertion (a) of Proposition C and Corollary 2.2 we have $\emptyset\neq\mathcal{CS}(\mathcal{F})\subset\mathcal{HCS}_{5}=\left\{-4^{\pm 1},-\tfrac{3}{2}\pm\tfrac{\sqrt{5}}{2}\right\}.$ Hence, according to assertion (b) of Proposition C, one of the following three possibilities does occur:

• (i)

$\mathcal{CS}(\mathcal{F})=\{-4^{\pm 1}\};$

• (ii)

$\mathcal{CS}(\mathcal{F})=\{-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\};$

• (iii)

$\mathcal{CS}(\mathcal{F})=\{-4^{\pm 1},-\frac{3}{2}\pm\frac{\sqrt{5}}{2}\}.$

In the case (i) (resp. (ii)) the foliation $\mathcal{F}$ is linearly conjugated to $\mathcal{F}_{0}^{5}$ (resp. \mathcal{F}_{\raisebox{-0.56905pt}{\tiny{H}}}^{5}), thanks to Theorem D (resp. Theorem F). To establish the theorem, it therefore suffices to exclude the possibility (iii). Let us assume by contradiction that (iii) happens. Then $\mathcal{F}$ possesses two invariant lines \scalebox{1.1}{\ell},\scalebox{1.1}{\ell}^{\prime} and two non radial singularities m\in\scalebox{1.1}{\ell},m^{\prime}\in\scalebox{1.1}{\ell}^{\prime} such that \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell},m)=-\frac{1}{4} and \mathrm{CS}(\mathcal{F},\scalebox{1.1}{\ell}^{\prime},m^{\prime})=-\frac{3}{2}\pm\frac{\sqrt{5}}{2}. According to Proposition 2.1, the point $m$ (resp. $m^{\prime}$) is also a non radial singularity for the homogeneous foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} (resp. \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}^{\prime}}) and we have

[TABLE]

Moreover, as in the proof of Theorem F, each of the foliations \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} and \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}^{\prime}} is linearly conjugated to one of the fourteen homogeneous foliations given by Theorem A. It then follows from Table 1 that \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}} and \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}^{\prime}} are respectively of types $2\cdot\mathrm{R}_{4}$ and $2\cdot\mathrm{R}_{1}+2\cdot\mathrm{R}_{3}.$ This implies, according to assertion (v) of Proposition 2.1, that $\mathcal{F}$ admits two radial singularities of order $4$ on the line

$\ell$

and four radial singularities on the line \scalebox{1.1}{\ell}^{\prime}, two of order $1$ and two of order $3.$ Let $m_{1}$ (resp. $m_{2}$) be a radial singularity of order $4$ (resp. $3$) of $\mathcal{F}$ on the line

$\ell$

(resp. \scalebox{1.1}{\ell}^{\prime}). Since $\tau(\mathcal{F},m_{1})+\tau(\mathcal{F},m_{2})=5+4>\deg\mathcal{F},$ the line \scalebox{1.1}{\ell}^{\prime\prime}=(m_{1}m_{2}) is invariant by $\mathcal{F}$ (cf. [1, Proposition 2.2]). The homogeneous foliation \mathcal{H}_{\mathcal{F}}^{\scalebox{0.9}{\ell}^{\prime\prime}} being convex of degree $5$, it must therefore be of type $1\cdot\mathrm{R}_{1}+1\cdot\mathrm{R}_{3}+1\cdot\mathrm{R}_{4}$ so that it possesses a non radial singularity $m^{\prime\prime}$ on the line \scalebox{1.1}{\ell}^{\prime\prime} satisfying (see Table 1)

[TABLE]

which is impossible. ∎

3 Conjectures

The notion of convex reduced foliation has an interesting relation with certain line arrangements in $\mathbb{P}^{2}_{\mathbb{C}}.$ Indeed, according to [12] we say that an arrangement $\mathcalsansmathptm{A}$ of $3d$ lines in $\mathbb{P}^{2}_{\mathbb{C}}$ has Hirzebruch’s property if each line of $\mathcalsansmathptm{A}$ intersects the other lines of $\mathcalsansmathptm{A}$ in exactly $d+1$ points. The $3d$ invariant lines of a reduced convex foliation of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ form a line arrangement which satisfies Hirzebruch’s property, thanks to [3, Lemma 6.8] and [4, Lemma 3.1]. The expected conjectural picture for the reduced convex foliations on $\mathbb{P}^{2}_{\mathbb{C}}$ is the following: besides the Fermat foliations $\mathcal{F}_{0}^{d}$, with $\mathcal{CS}(\mathcal{F}_{0}^{d})=\{(1-d)^{\pm 1}\}$, there exist special reduced convex foliations only for $d=4,5$ and $d=7$, namely, the Hesse pencil in degree $4$, and the two Hilbert modular foliations in degree $5$ and $7$ presented in the Introduction, for which

[TABLE]

i.e. we expect that there are no other convex reduced foliations on $\mathbb{P}^{2}_{\mathbb{C}}$ and for this reason we propose:

Conjecture 1**.**

We have

[TABLE]

This conjecture, combined with Corollary E, would imply a negative answer in degree $d\neq 7$ to [10, Problem 9.1] as we have already shown for $d\leq 5$.

To every rational map $f\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ and to every integer $d\geq 2,$ we associate respectively the following subsets of $\mathbb{C}\setminus\{0,1\}$:

• —

$\mathcal{M}(f)$ is, by definition, the set of $\mu\in\mathbb{C}\setminus\{0,1\}$ such that there is a fixed point $p$ of $f$ satisfying $f^{\prime}(p)=\mu;$

• —

$\mathcal{M}_{d}$ is defined as the set of $\mu\in\mathbb{C}\setminus\{0,1\}$ for which there exist critically fixed rational maps $f_{1},f_{2}\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ of degree $d$ having respective fixed points $p_{1}$ and $p_{2}$ such that $f_{1}^{\prime}(p_{1})=\mu$ and $f_{2}^{\prime}(p_{2})=\frac{\mu}{\mu-1}.$

The introduction of the sets $\mathcal{M}(f)$ and $\mathcal{M}_{d}$ is motivated by the following remark.

Remark 3.1*.*

Let $\mathcal{H}$ be a homogeneous foliation of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ According to [3, Section 2] the point s=[b:a:0]\in\scalebox{1.1}{\ell}_{\infty} is a non-degenerate singularity of $\mathcal{H}$ if and only if the point $p=[a:b]\in\mathbb{P}^{1}_{\mathbb{C}}$ is fixed by ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ with multiplier ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}^{\prime}(p)\neq 1,$ in which case the Camacho-Sad index \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s) coincides with the index $\imath({\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}},p)$ of ${\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}}$ at the fixed point $p$:

[TABLE]

Thus, the map $\mu\mapsto\frac{1}{1-\mu}$ sends $\mathcal{M}({\mspace{2.0mu}\underline{\mspace{-2.0mu}\mathcal{G}\mspace{-2.0mu}}\mspace{2.0mu}}_{\mathcal{H}})$ (resp. $\mathcal{M}_{d}$) bijectively onto $\mathcal{CS}(\mathcal{H})$ (resp. $\mathcal{HCS}_{d}$).

Using the above definition of the sets $\mathcal{M}(f),$ Theorem 4.3 of E. Crane [7] can be reformulated as follows:

Theorem 3.2** (Crane, [7]).**

Let $f\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ be a critically fixed rational map of degree $d\geq 2.$ Let $n\leq d$ denote the number of (distinct) critical points of $f.$ Then

1. $f$ has exactly $d+1$ fixed points, of which $d+1-n$ are non-critical;

2. the set $\mathcal{M}(f)$ is contained in $\mathbb{C}\setminus\left(\overline{\mathbb{D}}\left(0,1\right)\cup\mathbb{D}\left(1+\rho,\rho\right)\right),$ where $\overline{\mathbb{D}}(0,1)$ denotes the closed unit disk of $\mathbb{C}$ and $\mathbb{D}(1+\rho,\rho)\subset\mathbb{C}$ the open disk of radius $\rho=\frac{1}{d+n-2}$ and center $1+\rho.$ Moreover, $\mu\in\mathcal{M}(f)$ belongs to the boundary of the disk $\mathbb{D}(1+\rho,\rho)$ if and only if $n=d,$ in which case $\mu=\frac{d}{d-1}.$ **

This theorem translates in terms of homogeneous foliations as follows:

Corollary 3.3**.**

Let $\mathcal{H}$ be a homogeneous convex foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Let $n=\deg\mathcal{T}_{\mathcal{H}}$ denote the number of (distinct) radial singularities of $\mathcal{H}.$ Then

1. $\mathcal{H}$ has exactly $d+1$ singularities on the line at infinity, of which $d+1-n$ are non radial;

2. for any non radial singularity s\in\scalebox{1.1}{\ell}_{\infty} of $\mathcal{H},$ we have -\frac{1}{2}<-\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)}\leq\frac{d+n}{2}-1. This last inequality is an equality if and only if $n=d$, in which case \mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)=1-d. **

With the notations of Corollary 3.3, since $n\leq d$ we have in particular -\frac{1}{2}<-\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)}\leq d-1. According to Remark 1.1, the value $d-1$ is attained by (\mathcal{H},s)\mapsto-\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)}. However, after having checked many examples, we think that the lower bound $-\frac{1}{2}$ of -\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)} is not optimal and we propose the following conjecture with the value $\frac{1}{d-1}$ which is also attained by (\mathcal{H},s)\mapsto-\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)} (Remark 1.1).

Conjecture 2**.**

If $\mathcal{H}$ is a homogeneous convex foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}},$ then for any non radial singularity s\in\scalebox{1.1}{\ell}_{\infty} of $\mathcal{H}$ we have \frac{1}{d-1}\leq-\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)}. Alternatively, if $f\hskip 2.84526pt\colon\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ is a critically fixed rational map of degree $d\geq 2,$ then the set $\mathcal{M}(f)$ is contained in the closed disk $\overline{\mathbb{D}}\left(\frac{d+1}{2},\frac{d-1}{2}\right)\subset\mathbb{C}$ of center $\frac{d+1}{2}$ and radius $\frac{d-1}{2}.$ **

This conjecture is also motivated by the following remark:

Remark 3.4*.*

If Conjecture 2 is true, Conjecture 1 claims that in degree $2\leq d\neq 4,5,7$ the set $\mathcal{HCS}_{d}$ consists of the extreme values of -\mathrm{Re}\big{(}\mathrm{CS}(\mathcal{H},\scalebox{1.1}{\ell}_{\infty},s)\big{)} when $\mathcal{H}$ runs through the set of homogeneous convex foliations of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and $s$ runs through the set of non radial singularities of $\mathcal{H}$ on the line \scalebox{1.1}{\ell}_{\infty}.

Elementary computations, using the normal forms of homogeneous convex foliations of degree $d\in\{2,3,4,5\}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ presented in [8, Proposition 7.4], [2, Corollary C], [4, Theorem A] and in Theorem A, show the validity of Conjecture 1 for $d\in\{2,3\}$ and Conjecture 2 for $d\in\{2,3,4,5\}$. Moreover, very long computations carried out with Maple by the first author give $49$ normal forms for homogeneous convex foliations of degree $6$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and allow to verify the validity of Conjectures 1 and 2 for $d=6.$ The more difficult case $d=7$ is out of reach at this moment.