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\title{Local transversely product singularities}
\alttitle{Feuilletages à structure produit le long du lieu singulier}

\subjclass{37F75, 34M15}
\keywords{foliation, locally product}

\author[\initial{A.} \lastname{Lins-Neto}]{\firstname{Alcides} \lastname{Lins-Neto}}
\address{IMPA, Est. D. Castorina,\\
110, 22460-320, Rio de Janeiro, \\
RJ, (Brazil)}
\email{alcides@impa.br}

\begin{abstract}
In the main result of this paper we prove that a codimension one foliation of $\p^n$, which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.
\end{abstract}

\begin{altabstract}
Nous démontrons qu'un feuilletage de codimension un de $\p^n$ qui est localement un produit autour de tous les points d'une composante de codimension~$2$ de l'ensemble singulier, a une composante de Kupka. En particulier, nous obtenons une généralisation d'un résultat déjà connu de Calvo Andrade et Brunella sur les feuilletages avec une composante de Kupka.
\end{altabstract}



%\thanks{}

\datereceived{2019-05-02}
\dateaccepted{2020-05-28}

\editor{J. V. Pereira}
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\dateposted{2021-05-27}
\begin{document}

\maketitle


\section{Basic definitions and results}

It is known that a holomorphic codimension one foliation on $\p^n$, $n\ge3$, with a Kupka component in the singular set has a rational first integral, which in homogeneous coordinates is of the form $P^k/Q^\ell$, where $P$ and $Q$ are generic homogeneous polynomials with $k.\,deg(P)=\ell.\,deg(Q)$. The Kupka component for this specific example is the set $\Pi(P=Q=0)$, where $\Pi\colon\C^{n+1}\setminus\{0\}\to\p^n$ is the canonical projection (cf.~\cite{br,ca1,ca2}). The aim of this paper is to generalize this result for codimension one foliations with a \emph{local transversely product component} in the singular set. We will define this concept in a more general situation.

Let $\fa$ be a holomorphic foliation of dimension $k\ge2$ on a complex manifold $M$ of dimension $n\ge k+1$, with singular set $Sing(\fa)$. We say that $\fa$ is \emph{a transversely product} at a point $p\in Sing(\fa)$ if the germ $\fa_p$ of $\fa$ at $p$ is holomorphically equivalent to a product of a germ of singular foliation of dimension one with an isolated singularity by a regular foliation of dimension $k-1$. In other words, we can say that there exists a germ of submersion $\var\colon(M,p)\to(\C^{n-k+1},0)$ and a germ of a one dimensional foliation $\Gcal$ at $0\in\C^{n-k+1}$, with $Sing(\Gcal)=\{0\}$, such that $\fa_p=\var^*(\Gcal)$. In particular, the germ of the singular set of $\fa$ at $p$ is smooth of dimension $k-1$: $Sing(\fa_p)=\var^{-1}(0)$.

\begin{defi}\label{def:1}
We say that $\Ga$ is a \emph{local transversely product component} (briefly l.t.p component) of $Sing(\fa)$ if $\Ga$ is an irreducible component of $Sing(\fa)$ and $\fa$ is a transversely product at all points of $\Ga$.
\end{defi}

\begin{rema}\label{r:11}
If $\Ga$ is a l.t.p. component of $Sing(\fa)$ then it follows from the definition that:
\begin{enumerate}\alphenumi
\item \label{rem1.2.a} $\Ga$ is smooth. Let $dim_\C(\Ga)=m$.
\item \label{rem1.2.b}There exists a singular one dimensional foliation $\Gcal$, on a polydisc $V$ of $\C^{n-m}$, with an isolated zero at $0\in V$, such that for any $p\in\Ga$ there exists a local chart $(U,z)$ around $p\in U$ satisfying the following conditions:
\begin{enumerate}\beenumii
\item \label{rem1.2.b.1} $z=(x,y)\colon U\to\C^{n-m}\times \C^{m}$ with $x(U)=V$.
\item \label{rem1.2.b.2} $\fa|_U=x^*(\Gcal)$.
\end{enumerate}
\end{enumerate}
In the chart $z=(x,y)$ the submersion of the definition is $\var=x\colon U\to V$ and the leaves of the non-singular foliation are the levels~$x^{-1}(a)$, $a\in V$. Moreover, $\Ga\cap U=x^{-1}(0)$.

The germ of $\Gcal$ at $0\in Q$ is called the \emph{normal type} of $\fa$ along $\Ga$. Remark that, if $T$ is a germ at $p\in\Ga$ of $n-m$ manifold transverse to $\Ga$ then the restricted foliation $\fa|_T$ is holomorphically equivalent to the normal type of $\fa$ along $\Ga$.

Moreover, since $\Gcal$ is one dimensional we can assume that it is defined by a holomorphic vector field $X=\sum_{j=1}^{n-m}A_j(x)\frac{\pa}{\pa x_j}$, or by the $(n-m-1)$-form
\begin{equation}\label{eq:1}
\eta=i_Xdx_1\wedge\dots\wedge dx_{n-m}=\sum_{j=1}^{n-m}(-1)^{j-1}A_j(x)\,dx_1\wedge\dots\wedge\wh{dx_j}\wedge\dots\wedge dx_{n-m}\,,
\end{equation}
where in~\eqref{eq:1} $\wh{dx_j}$ means omission of $dx_j$ in the product. The form $\eta$, considered as a form on $U$ in the coordinates $(x,y)$, defines $\fa|_U$.


Let us see some examples:
\end{rema}

\begin{exem}\label{ex:1}
 Recall that a point $p\in M$ is a Kupka singularity of the foliation $\fa$ if $p\in Sing(\fa)$ and $\fa$ is represented in a neighborhood of $p$ by an integrable $(n-k)$-form $\eta$ such that $d\eta(p)\ne0$. The form $d\eta$ defines a $k+1$ distribution $\de=ker(d\eta)$ in a neighborhood of $p$, where
\[
\de(q)=ker(d\eta(q)):=\left\{v\in T_qM\,\middle|\,i_v(d\eta(q))=0\right\}\,.
\]


The distribution $\de$ is integrable and defines a regular foliation of dimension $k-1$ in a neighborhood of $p$. There exists a local chart $(U,z=(x,y))$, where\linebreak$x=(x_1,\,\dots,\,x_{n-k+1})\colon U\to \C^{n-k+1}$, $y\colon U\to\C^{k-1}$ and $z(p)=(0,0)$, such that $d\eta=dx_1\wedge\,\dots\,\wedge dx_{n-k+1}$. In this case, the form $\eta$ can be written as $\eta=i_Xdx_1\wedge\,\dots\,\wedge dx_{n-k+1}$, where
\[
X=\sum_{j=1}^{n-k+1}A_j(x)\,\frac{\pa}{\pa x_j}
\]
defines the normal type of $\fa_p$.

Note that $d\eta=\De(X)\,dx_1\wedge\dots\wedge dx_{n-k+1}$, where
\[
\De(X)=div(X)=\sum_{j=1}^{n-k+1}\frac{\pa A_j}{\pa x_j}\,,
\]
so that $\De(X)\equiv1$.
\end{exem}

\begin{defi}\label{def:2}
We say that $K$ is a Kupka component of a foliation $\fa$ (of dimension $k\ge2$) if $K$ is a l.t.p. component of $Sing(\fa)$ and the normal type of $\fa$ along $K$ is of Kupka type.
\end{defi}

\begin{exem}\label{ex:2}
Let $P$ and $Q$ be homogeneous polynomials on $\C^{n+1}$, $n\ge3$, where $deg(P)=p$ and $deg(Q)=q$. The levels of the rational function $f=\tfrac{P^q}{Q^p}$ define a singular foliation of $\p^n$, that will be denoted by $\fa(P,Q)$. We say that $P$ and $Q$ are transverse if the set
\[
\left\{z\in \C^{n+1}\,\middle|\,P(z)=Q(z)=0\,\text{ and }\,\,dP(z)\wedge dQ(z)=0\right\}
\]
is either $\{0\}$, or empty (if $p=q=1$). If $P$ and $Q$ are transverse then the subset $\Ga$ of $\p^n$ defined in homogeneous coordinates by $(P=Q=0)$ is a Kupka component on $\fa(P,Q)$. The normal type of $\fa(P,Q)$ at the points of $\Ga$ is given by the linear vector field $X=p.\,x\tfrac{\pa}{\pa x}+q.\,y\tfrac{\pa}{\pa y}$.

In fact, the following result is known (cf.~\cite{br, ca1,ca2,lc}):

\begin{theo}\label{t:11}
Let $\fa$ be a holomorphic foliation of codimension one on $\p^n$, $n\ge3$. If $\fa$ has a Kupka component then $\fa=\fa(P,Q)$, where $P$ and $Q$ are transverse polynomials.
\end{theo}


\end{exem}

\begin{exem}\label{ex:3}
Example~\ref{ex:2} admits the following generalization: let $P_1,\,\dots,\,P_m$ be homogeneous polynomials on $\C^{n+1}$ with $deg(P_j)=d_j$, $1\le j\le m$. Assume that $n\ge m+1\ge4$ and that $P_1,\,\dots,\,P_m$ are transverse; i.e. the set
\[
\left\{z\in\C^{n+1}\,\middle|\,P_1(z)=\dots=P_m(z)=0\,\text{ and }\,dP_1(z)\wedge\,\dots\,\wedge dP_m(z)=0\right\}
\]
is either $\{0\}$, or empty (if $d_1=\dots=d_m=1$). Let $(k_1,\,\dots,\,k_m)\in\N^m$ be such that $gcd(k_1,\,\dots,\,k_m)=1$ and $k_1.\,d_1=\dots=k_m.\,d_m$. The levels of the rational map $\Pcal\colon\p^n\to\p^{m-1}$, defined by
\[
\Pcal:=\left[P_1^{k_1}:\,\dots\,:P_m^{k_m}\right]\,,
\]
define a foliation of codimension $m-1$ on $\p^n$, denoted by $\fa(P_1,\,\dots,\,P_m)$. If $P_1,\,\dots,\,P_m$ are transverse then the set $\Ga\sub\p^n$, defined in homogeneous coordinates by $(P_1=\dots=P_m=0)$, is a Kupka component of $\fa(P_1,\dots,P_m)$. The normal type of $\fa(P_1,\,\dots,\,P_m)$ at the points of $\Ga$ is given by the linear vector field $S=\sum_{j=1}^m d_j\,x_j\tfrac{\pa}{\pa x_j}$\,.
\end{exem}
A natural problem is the following:

\begin{enonce}{Problem}
Let $\fa$ be a holomorphic foliation of codimension $m-1$ on $\p^n$, where $n\ge m+1\ge4$. Assume that $\fa$ has a Kupka component $\Ga$. Are there transverse homogeneous polynomials $P_1,\,\dots,\,P_m$ on $\C^{n+1}$ such that $\fa=\fa(P_1,\,\dots,\,P_m)$ and $\Ga$ is defined by $(P_1=\dots=P_m=0)$?
\end{enonce}

Some partial results about this problem were proved (see for instance~\cite{ca3, moa}).


In this paper we generalize Theorem~\ref{t:11}:

\begin{theo}\label{t:1}
Let $\fa$ be a holomorphic foliation of codimension one on $\p^n$, $n\ge3$. Assume that $\fa$ has a l.t.p. component $\Ga$. Then $\Ga$ is a Kupka component of $\fa$. In particular, $\fa$ is like in Example~\ref{ex:2}.
\end{theo}

Let us state some consequences of Theorem~\ref{t:1}.

\begin{coro}\label{c:1}
Let $\fa$ be a codimension one holomorphic foliation on $\p^n$, $n\ge4$. Assume that there is a linear embedding $i\colon\p^3\to\p^n$ such that $i^*(\fa)$ has a l.t.p component. Then $\fa$ has a rational first integral that can be written in homogeneous coordinates as $P^q/Q^p$, where $P$ and $Q$ are homogeneous polynomials on $\C^{n+1}$ with $deg(P)=p$ and $deg(Q)=q$.
\end{coro}

The proof of Corollary~\ref{c:1} is based in the fact that if there exists a linear embedding $i\colon\p^3\to\p^n$ such that $i^*(\fa)$ has a first integral then $\fa$ has also a first integral (see~\cite{lc3}).

\begin{coro}\label{c:2}
Let $\fa$ be a codimension one foliation on $\p^n$, $n\ge3$. Assume that all components of its singular set are l.t.p. Then $\fa$ has degree zero: the first integral of Corollary~\ref{c:1} is of the form $L_2/L_1$, where $L_1$ and $L_2$ are linear.
\end{coro}

\begin{coro}\label{c:3}
Let $\eta$ be an integrable 2-form on $\C^n$, $n\ge4$, with homogeneous coefficients of the same degree $d\ge1$. Then $dim_\C(sing(\eta))\ge1$.
\end{coro}

\begin{rema}
Corollary~\ref{c:3} was proved in~\cite{lc1} in the case $n=4$. We would like to observe that the assertion is not true in the case of distributions of $\C^4$. The following example, due to Krishanu and Nagaraj~\cite{KN}: define a 2-form $\tet$ on $\C^4$ by
\begin{gather*}
\tet=x_3^2\,dx_2\wedge dx_3-x_1^2\,dx_3\wedge dx_1+\left(x_1\,x_2+x_3\,x_4\right)\,dx_1\wedge dx_2+
\\
\left[x_4^2\,dx_1+x_2^2\,dx_2+\left(x_1\,x_2-x_3\,x_4\right)\,dx_3\right]\wedge dx_4
\end{gather*}
has $Sing(\tet)=\{0\}$ and satisfies $\tet\wedge\tet=0$. Hence, it generates a distribution of codimension two on $\C^4\setminus\{0\}$. This distribution is not integrable.
\end{rema}

Theorem~\ref{t:1} motivates the following problem:

\begin{enonce}[plain]{Problem}
Let $\fa$ be a holomorphic foliation on $\p^n$ of codimension $\ge2$ and dimension $\ge2$. Assume that $\fa$ has a l.t.p. component $\Ga$. Is $\Ga$ a Kupka component of $\fa$?
\end{enonce}

A crucial point of our proof of Theorem~\ref{t:1} is the Camacho--Sad theorem on the existence of a separatrix for germs of holomorphic vector fields on $(\C^2,0)$~\cite{cs}. The same type of argument cannot be used in the general case: there are examples of germs of vector fields on $(\C^m,0)$, $m\ge3$, without separatrices~\cite{gl}.

%\vskip.1in

The proof of Theorem~\ref{t:1} will be done in Section~\ref{ss:2}. Since this proof is technical, in Section~\ref{ss:21} we give an idea of the proof by stating the main objects and results that will be used. In Sections~\ref{ss:22} and~\ref{ss:23} we will prove the main auxiliary results used in the proof and stated in Section~\ref{ss:21}. Section~\ref{ss:3} is dedicated to the proof of Corollaries~\ref{c:2} and~\ref{c:3}.

%\vskip.1in

\subsection*{Acknowledgement}
I would like to thank J. Vitório Pereira for helpful conversations that suggested me a simplification of the proof of Lemma~\ref{l:23}.

\section{Proof of Theorem~\ref{t:1}}\label{ss:2}

\subsection{Preliminaries and idea of the proof}\label{ss:21}
Let $\fa$ be a codimension one foliation on $\p^n$, $n\ge3$, with a l.t.p. component $\Ga\sub Sing(\fa)$. The definition implies that $cod_\C(\Ga)=2$, so that the transversal type of $\fa$ at the points of $\Ga$ is a germ of singular foliation at $(\C^2,0)$ with an isolated singularity at $0\in\C^2$ (see Remark~\ref{r:11} and Example~\ref{ex:1}). We can assume that this transversal type is given by germ at $0\in\C^2$ of vector field $X=X_1(x,y)\tfrac{\pa}{\pa x}+X_2(x,y)\tfrac{\pa}{\pa y}$, where $X_1$, $X_2\in\Ocal_2$ and $X_1(0,0)=X_2(0,0)=0$. Recall that $\Ga$ is a Kupka component if, and only if, we have $Tr(DX(0))\ne0$, where
\[
Tr(DX(0)):=\frac{\pa X_1}{\pa x}(0)+\frac{\pa X_2}{\pa y}(0)
\]
is the trace of the linear part $DX(0)$ of $X$ at $0\in\C^2$. In this case, as we have pointed out before, $\fa$ is like in Example~\ref{ex:2} (see Theorem~\ref{t:11}).

Another useful ingredient is the normal Baum--Bott index of the component $\Ga$, that we will denote as $BB(\fa,\Ga)$. Since $\Ga$ is a l.t.p. component of $Sing(\fa)$ then $BB(\fa,\Ga)$ coincides with the Baum--Bott index of $X$ at the singularity $0$ of $X$, denoted by $BB(X,0)$ (see~\cite{lc2}) (for the definition of $BB(X,0)$ see~\cite{br1}). In~\cite[Lemma~3.4, Section~3.2]{lc2} it is proven that if $BB(\fa,\Ga)\ne0$ and $DX(0)\not\equiv0$ then $\Ga$ is a Kupka component and we are done.

One of the tools used in the proof of~\cite[Lemma~3.4]{lc2} is the existence of a smooth analytic separatrix along $\Ga$. Below we define the concept of separatrix in a way that will be used in the proof of Theorem~\ref{t:1}.

\begin{defi}\label{def:3}
 Let $\fa$ be a holomorphic foliation of dimension $k$ on a n dimensional compact complex manifold, $2\le k<n$, and $\Ga$ be l.t.p. component of $\fa$ (recall that $dim(\Ga)=k-1$). A \emph{separatrix $\Si$ of dimension $\ell$ along} $\Ga$ of $\fa$, where $k\le\ell<n$, is a germ of $\ell$ analytic manifold along $\Ga$ which is $\fa$-invariant in the sense that:
\begin{enumerate}\alphenumi
\item \label{def2.1.a}$\Si\sup\Ga$.
\item \label{def2.1.b}$\Si\setminus\Ga$ is contained in an union of leaves of $\fa$.
\end{enumerate}
\end{defi}

\begin{rema}\label{r:21}
Let $\Si$ be a separatrix of $\fa$ of dimension $\ell$ along $\Ga$, as in Definition~\ref{def:3}. Fix $p\in\Ga$ and $(x,y)\colon U\to\C^{n-k+1}\times\C^{k-1}$, a local coordinate system around $p$ as in Remark~\ref{r:11}. It follows from the definition that $x^{-1}(x(\Si\cap U))=\Si\cap U$.

Let $T$ be a germ at $p$ of a $n-k+1$ dimensional manifold transverse to $\Ga$. As we have observed before, $\fa|_T$ is equivalent to the normal type of $\fa$ along $\Ga$. In particular, the intersection $\Si\cap T$ is invariant by $\fa|_T$.

In the case of Theorem~\ref{t:1}, where $\fa$ has codimension one, then $dim(T)=2$ and the normal type is a germ $\Gcal$ of one dimensional foliation on $(\C^2,0)$. In this case $\Si\cap T$ is a finite number of analytic separatrices of $\Gcal$ as considered in~\cite{cs}. The next result will be used in proof of Theorem~\ref{t:1}.
\end{rema}

\begin{lemm}\label{l:21}
Let $\fa$ be a holomorphic codimension one foliation on a compact complex manifold $M$, where $dim(M)=n\ge3$, and $\Ga$ be a l.t.p. component of $Sing(\fa)$. If the normal type of $\fa$ along $\Ga$ is not equivalent to the radial foliation of $(\C^2,0)$ then $\fa$ admits an irreducible separatrix $\Si$ along $\Ga$ with $dim(\Si)=n-1$.
\end{lemm}

Recall that the radial foliation of $\C^2$ is defined by the form $x\,dy-y\,dx$ and its leaves are the straight lines through $0$. Lemma~\ref{l:21} will be proved in Section~\ref{ss:22}.

%\vskip.1in

From now on, in this section, we will assume that $\fa$ is a codimension one holomorphic foliation on the compact manifold $M$, $dim_\C(M)\ge3$, with a l.t.p. component $\Ga$ and with a separatrix $\Si$ along $\Ga$, $dim(\Si)=n-1$. Next we will introduce the \emph{normal bundle of $\Si$ along $\Ga$}.

Since $dim(\Si)=n-1$ we can find a Leray covering $\Ucal=(U_\alp)_{\alp\,\in\,A}$ of $\Ga$ by open sets and two collections $f=(f_\alp)_{\alp\,\in\,A}$ and $g=(g_{\alp\,\be})_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$ with the following properties:
\begin{enumerate}\alphenumi
\item \label{lem2.3.a} $f_\alp\in\Ocal(U_\alp)$, $\forall\alp\in A$, and $f_a=0$ is a reduced equation of $\Si\cap U_\alp$.
\item \label{lem2.3.b}$g_{\alp\,\be}\in\Ocal^*(U_\alp\cap\,U_\be)$ and $f_\alp=g_{\alp\,\be}.\,f_\be$ on $U_\alp\,\cap\,U_\be\,\ne\,\emp$.
\end{enumerate}

Of course $g=(g_{\alp\,\be})_{U_\alp\,\cap\,U_\be\,\ne\emp}$ is a multiplicative cocycle. We define the normal bundle of $\Si$ along $\Ga$ as the line bundle on $Pic(\Ga)$ induced on a tubular neighborhood $U\sub\bigcup_\alp U_\alp$ by the cocycle $g=(g_{\alp\,\be})_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$.

It will be denoted by $N_\Si$. Let $c_1(N_\Si)$ be the first Chern class of $N_\Si$, considered as an element of $H^2(U,\R)$ via the homomorphism $H^2(U,\,\Z)\to H^2(U,\,\R)\simeq H^2_{DR}(U)$ induced by the inclusion $\Z\to\R$.

As we have seen in Remark~\ref{r:11}, the normal type of $\fa$ along $\Ga$ can be represented by a germ at $0\in\C^2$ of holomorphic vector field $X=A_1(x,y)\,\frac{\pa}{\pa x}+A_2(x,y)\,\frac{\pa}{\pa y}$ with an isolated at $0$. When we intersect $\Si$ with a germ of transversal section $T\simeq(\C^2,0)$ we obtain a separatrix of $X$, say $\g:=\Si\cap T$ (in general $\g$ is not irreducible). Let $f\in\Ocal_2$ be a reduced analytic equation of $\g$. Since $\g$ is $X$-invariant we can write
\begin{equation}\label{eq:}
X(f)=h.\,f,\quad\text{where}\quad h\in\Ocal_2\,.
\end{equation}

\begin{lemm}\label{l:22}
In the above situation, if $h(0)=0$ then $c_1(N_\Si)=0$.
\end{lemm}

On the other hand, we have the following:

\begin{lemm}\label{l:23}
If the ambient space is $M=\p^n$, $n\ge3$, then $c_1(N_\Si)\ne0$. In particular, if $X(f)=h.\,f$ then $h(0)\ne0$.
\end{lemm}

As a consequence of Lemma~\ref{l:23} we get the following:

\begin{cor}\label{c:21}
If $M=\p^n$, $n\ge3$, then $\Si$ is a Kupka component of $\fa$.
\end{cor}

In particular, Theorem~\ref{t:11} will imply Theorem~\ref{t:1}. Lemma~\ref{l:21} will be proved in the next section.

%\vskip.1in

\subsection{Proof of Lemma~\ref{l:21}.}\label{ss:22}

Let $\fa$ be a holomorphic codimension one foliation on a compact complex manifold $M$ with $dim(M)\ge3$. Assume that $\fa$ has a l.t.p. component $\Ga$ with normal type $\Gcal$, where $\Gcal$ is a germ of foliation on $(\C^2,0)$ with an isolated singularity at $0\in\C^2$. As before, we will assume that $\Gcal$ is the foliation defined by a germ at $(\C^2,0)$ of vector field $X=X_1\frac{\pa}{\pa x}+X_2\frac{\pa}{\pa y}$ with an isolated singularity at the origin of $\C^2$. The germ of foliation $\Gcal$ can be defined also by the 1-form
\[
\om=i_X(dx\wedge dy)=X_1\,dy-X_2\,dx\,,
\]
so that, $d\om(0)=Tr(DX(0))\,dx\wedge dy$. We can assume that $\om$ has a representative, denoted by $\td\om$, defined in the polydisc $Q=\D^2$ with an isolated singularity at $0\in\D^2$.

By the definition of l.t.p. component, we can find a covering $\Ucal=(U_\alp)_{\alp\,\in\,A}$ of $\La$ by open sets biholomorphic to polydiscs, a collection of local charts $((z_\alp,\,U_\alp))_{\alp\,\in \,A}$ and a multiplicative cocycle $(k_{\alp\,\be})_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$ with the following properties:
\begin{enumerate}
\item \label{sect2.2.1}$z_\alp=(x_\alp,y_\alp)\colon U_\alp\to\C^2\times\C^{n-2}$, where $x_\alp(U_\alp)=Q$ and $\Ga\cap U_\alp=x_\alp^{-1}(0)$, $\forall\alp\in A$.
\item \label{sect2.2.2}$\fa|_{U_\alp}$ is defined by the integrable 1-form $\td\om_\alp:=x_\alp^*(\td\om)$. The germ of $\td\om_\alp$ along $\Ga\cap U_\alp$ will be denoted by $\om_\alp$.
\item \label{sect2.2.3}$\td\om_a=k_{\alp\,\be}.\,\td\om_\be$ on $U_\alp\cap U_\be\ne\emp$.
\end{enumerate}
We will assume that $\Ucal$ satisfies the following:
\begin{enumerate}
\setcounter{enumi}{3}
\item \label{sect2.2.4} If $U_\alp\cap U_\be\ne\emp$ then $\Ga\cap U_\alp\cap U_\be\ne\emp$ and connected.
\end{enumerate}

\begin{rema}\label{r:22}
Given $\alp,\be\in A$ such that $\Ga\cap U_\alp\cap U_\be\ne\emp$ we can construct a germ $f_{\alp\,\be}\in \Diff(\C^2,0)$ as follows: fix $p\in\Ga\cap U_\alp\cap U_\be$ and a germ of plane $T=T_{\alp,\,\be}\simeq(\C^2,p)$ transverse to $\Ga$ at $p$. Note that $x_\alp|_T,x_\be|_T\colon (T,p)\to(\C^2,0)$ are biholomorphisms. Therefore, we define
\[
f_{\alp\,\be}=x_\alp\circ(x_\be|T)^{-1}=x_\alp|T\circ(x_\be|T)^{-1}\in \Diff\left(\C^2,0\right)\,.
\]
Since $\om_\alp|_T=(x_\alp|T)^*(\om)$, $\om_\be|_T=(x_\be|_T)^*(\om)$ and $\om_a=k_{\alp\be}.\,\om_\be$ we get $f_{\alp\be}^*(\om)=h_{\alp\be}.\,\om$, where
\[
h_{\alp\,\be}=k_{\alp\,\be}|_T\circ(x_\be|_T)^{-1}\in\Ocal_2^*\,.
\]
The biholomorphism $f_{\alp\,\be}$ can be interpreted as the \emph{glueing map} of $\fa|_{U_\be}$ with $\fa|_{U_\be}$.
\end{rema}

From now on, we fix a collection of germs $(f_{\alp\be})_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$ as above.

\begin{lemm}\label{l:24}
$\fa$ admits a separatrix $\Si$ along $\Ga$ if, and only if, $X$ (or $\om$) has a separatrix $\g$ (not necessarily irreducible) such that $f_{\alp\,\be}(\g)=\g$ for all $\Ga\cap U_\alp\cap U_\be\ne\emp$.
\end{lemm}


\subsubsection*{Terminology}
We will say that the separatrix $\g$ of $\Gcal$ generates the separatrix $\Si$ of $\fa$.


\begin{proof}
Assume that $X$ has a separatrix $\g$ such that $f_{\alp\,\be}(\g)=\g$ for all $\Ga\cap U_\alp\cap U_\be\ne\emp$. Given $\alp\in A$ define $\Si_\alp:=x_\alp^{-1}(\g)$. We assert that if $U_\alp\cap U_\be\ne\emp$ then $\Si_\alp\cap U_\be=\Si_\be\cap U_\alp$. In fact, let $(x_\alp,\,y_\alp)$, $(x_\be,\,y_\be)$ and $T$ be as before. Then
\[
\Si_\alp\cap T=x_\alp^{-1}(\g)\cap T=(x_\alp|_T)^{-1}(\g)=(x_\alp|_T)^{-1}(f_{\alp\be}(\g))=(x_\be|_T)^{-1}(\g)=\Si_\be\cap T\,.
\]
This, of course, implies the assertion. In particular, the local separatrices $\Si_a$ glue together forming a global separatrix $\Si$ along $\Ga$ such that $\Si\cap U_\alp=\Si_\alp$, $\forall\alp\in A$.

We leave the converse to the reader.
\end{proof}

\begin{defi}\label{def:4}
Let $\Gcal$ be a germ of foliation at $(\C^2,0)$ with an isolated singularity at $0$. We say that a separatrix $\g$ of $\Gcal$ is \emph{distinguished} if for any $f\in \Diff(\C^2,0)$ such that $f^*(\Gcal)=\Gcal$ then $f(\g)=\g$.
\end{defi}

\begin{lemm}\label{l:25}
Let $\Gcal$ be a germ of foliation at $(\C^2,0)$ with an isolated singularity at $0$ which is not equivalent to the radial foliation. Then $\Gcal$ has a distinguished separatrix.
\end{lemm}

\begin{proof}
In the proof we use Seidenberg's resolution theorem~\cite{sd}. Let $S$ be a smooth complex surface and $\Gcal$ be a foliation by curves on $S$. Given $p\in Sing(\Gcal)\sub S$ we denote $\Diff(S,p)$ the set of germs at $p\in S$ of biholomorphisms $f\colon(S,p)\to S$ with a fixed point at $p$. Assume that the germ of $\Gcal$ at $p$ is defined by a germ of holomorphic vector field $X$ with an isolated singularity at $p$. We use also the notations
\begin{align*}
\Diff_{\Gcal}(S,p)&=\left\{f\in \Diff(S,p)\,\middle|\,f^*(\Gcal)=\Gcal\right\}\,.
\intertext{and}
\Diff^0_\Gcal(S,p)&=\left\{f\in \Diff_\Gcal(S,p)\,\middle|\,f\,\,\text{preserves the leaves of }\Gcal\right\}\,.\qedhere
\end{align*}
\end{proof}

\begin{rema}\label{r:23}
Note that:
\begin{enumerate}
\item \label{rem2.11.1}Given $f\in \Diff_\Gcal(S,p)$, then $f^*(X)=h_X.\,X$, where $h_X\in\Ocal^*_p$.
\item \label{rem2.11.2}$\Diff_\Gcal(S,p)$ is a sub-group of $\Diff(S,p)$.
\item \label{rem2.11.3}Given $f\in \Diff_\Gcal(S,p)$ and an irreducible separatrix $\g$ of $\Gcal$ through $p$ then $f(\g)$ is also a separatrix of $\Gcal$ through $p$.
\end{enumerate}
\end{rema}

Let $Sep(\Gcal)$ be the set of irreducible separatrices of $\Gcal$ through $p$. By (3) of Remark~\ref{r:23}, $\Diff_\Gcal(S,p)$ acts in $Sep(\Gcal)$ as $(f,\del)\in \Diff_\Gcal(S,p)\times Sep(\Gcal)\to f(\del)\in Sep(\Gcal)$. The idea of the proof is to find a finite subset $G_o:=\{\g_1,\,\dots,\,\g_k\}\sub Sep(\Gcal)$ such that $f(G_o)\sub G_o$ for all $f\in \Diff_\Gcal(S,p)$. In this case, the set $\g:=\{f(\g_1)\,|\,f\in \Diff_\Gcal(S,p)\}\sub G_o$ contains finitely many irreducible separatrices of $\Gcal$ through $p$ and can be considered as a germ of curve through $p$ such that $f(\g)=\g$ for all $f\in
\Diff_\Gcal(S,p)$, and so $\g$ is a distinguished separatrix of $\Gcal$ through $p$. Let us prove the existence of the finite set $G_o$.

%\vskip.1in

First of all, we observe that there are two possibilities for the foliation $\Gcal$:
\begin{enumerate}\Romanenumi
\item \label{rem2.11.I}$\Gcal$ has finitely many irreducible separatrices through $p$. This case is trivial and the details are left to the reader.
\item \label{rem2.11.II}$\Gcal$ has infinitely many irreducible separatrices through $p$. Let us prove Lemma \ref{l:25} in this case.
\end{enumerate}

We will consider a blowing-up process used to resolve the foliation $\Gcal$ (see~\cite{cs}). The first case, is when $\Gcal$ has a simple singularity at $p$ and no blowing-ups are needed in the process. Let $\la_1$ and $\la_2$ be the eigenvalues of $DX(p)$. The singularity is simple if:
\begin{enumerate}\alphenumi
\item \label{rem2.11.a}$\la_1.\,\la_2\ne0$ and $\frac{\la_2}{\la_1}\notin\Q_+$.
\item \label{rem2.11.b}$\la_1\ne0$ and $\la_2=0$ (or vice-versa). In this case, $p$ is a saddle-node.
\end{enumerate}
In both cases $\Gcal$ has one or two separatrices through $p$ and so Lemma~\ref{l:25} is true.

When the singularity is not simple, Seidenberg's theorem says that after a finite process of blowing-ups $\Pi\colon(\td{S},E)\to (S,p)$ then all the singularities of the strict transform $\Pi^*(\Gcal)$ in the exceptional divisor $E$ are simple. The blowing-up process $\Pi$ can be considered as a composition blowing-ups of points
\begin{multline}\label{eq:3}
\left(\td{S},E\right)\\
:=\left(\td{S}_k,E_k\right)\overset{\Pi_k}\longrightarrow(\td{S}_{k-1},E_{k-1})\overset{\Pi_{k-1}}\longrightarrow\,\dots\,\overset{\Pi_{2}}\longrightarrow\left(\td{S}_1,E_1\right)
\overset{\Pi_{1}}\longrightarrow\left(\td{S}_0,E_0\right)=(S,p)
\end{multline}
where in the $j^{\rm th}$ step $\Pi_j\colon(\td{S}_j,E_j)\to(\td{S}_{j-1},E_{j-1})$, $j\ge2$, we blow-up in a point $p_{j-1}\in E_{j-1}$. The exceptional divisor obtained in this step will be denoted as $\p^1\simeq\td{E}_j\sub E_j$, so that $\Pi_j(\td{E}_j)=p_{j-1}$. We use also the notation $\td\Pi_j:=\Pi_1\circ\,\dots\,\circ\Pi_j$. We will denote also $\td\Gcal_j:=\td\Pi^*(\Gcal)$. The point $p_{j-1}\in E_{j-1}$ is chosen between the non simple singularities of $\td{\Gcal}_{j-1}$ on $E_{j-1}$. Seidenberg's theorem can be stated as follows

\begin{theo}
It is possible to choose a blowing-up process as above in such a way that all singularities of the strict transform $\td\Gcal_k=\td\Pi_k^*(\Gcal)$ are simple.
\end{theo}

\begin{rema}
%{\rm
There are two possibilities in each step
\[
\Pi_j\colon\left(\td{S}_j,\td{E}_j\right)\to\left(\td{S}_{j-1},p_{j-1}\right).
\]
We assume that $p_{j-1}$ is a non simple singularity of $\td\Gcal_{j-1}$. Let $X_{j-1}$ be a germ at $p_{j-1}$ of holomorphic vector field that represents the germ of $\td\Gcal_{j-1}$ at $p_{j-1}$. Let $X_\nu\linebreak=P_\nu(x,y)\tfrac{\pa}{\pa x}+Q_\nu(x,y)\tfrac{\pa}{\pa y}$ be the first non-zero jet of $X_{j-1}$ at $p_{j-1}$, where $P_\nu$ and $Q_\nu$ are homogeneous polynomials of degree $\nu\ge1$. Set $F_{\nu+1}(x,y)=x.\,Q_\nu(x,y)-y\,P_\nu(x,y)$.
\begin{enumerate}\romanenumi
\item \label{rem2.13.i}If $F_{\nu+1}\not\equiv0$ then $F_{\nu+1}$ is homogeneous of degree $\nu+1$ and the blowing-up is called \emph{non-dicritical}. The divisor $\td{E}_j$ is invariant for the foliation $\td\Gcal_j$ and the singularities of $\td\Gcal_j$ on $\td{E}_j$ are the directions correspondent to the directions defined by $F_{\nu+1}(x,y)=0$.
\item \label{rem2.13.ii}If $F_{\nu+1}\equiv0$ then $X_\nu=F_{\nu-1}(x,y)\,R$, where $R=x\tfrac{\pa}{\pa x}+y\tfrac{\pa}{\pa y}$ is the radial vector field in $\C^2$ and $F_{\nu-1}$ is homogeneous of degree $\nu-1$. In this case, the blowing-up is called \emph{dicritical}. The divisor $\td{E}_j$ is non-invariant for $\td\Gcal_j$ and it is transverse to $\td{E}_j$ outside the set $V_j\sub\td{E}_j$ corresponding to the directions defined by the equation $F_{\nu-1}(x,y)=0$.

If $\nu=1$ then $\td\Gcal_{j-1}$ is equivalent to the radial foliation at $p_{j-1}$. We will say that $p_{j-1}$ is a \emph{radial} singularity of $\td\Gcal_{j-1}$. If $p_{j-1}$ is not radial for $\td\Gcal_{j-1}$ then $V_j\ne\emp$ and we can divide it into two disjoint subsets $V_j=\tau_j\cup\si_j$, where
\begin{itemize}
\item $\si_j=Sing(\td\Gcal_j)\cap\td{E}_j$.
\item $\tau_j=V_j\setminus Sing(\td\Gcal_j)$. We call $\tau_j$ the set of tangencies of $\td\Gcal_j$ with $\td{E}_j$.
\end{itemize}
\end{enumerate}
Remark also that $Sep(\Gcal)$ is finite if, and only if, all blowing-ups in the process are non-dicritical.

Since in the blowing-up process, in each step, $1\le j\le k$, we blow-up in some non-simple singularity of $\td\Gcal_{j-1}$, if $\td{E}_j$ is dicritical, at the end the tangencies $\tau_j$ ``survive'', in the sense that there exists a set $\tau\sub E_k$ such that for any $1\le j<k$ such that $\tau_j\ne\emp$ then
\[
\tau_j\sub\Pi_k\circ\dots\circ\Pi_{j+1}(\tau)
\]
\end{rema}

%\vskip.1i
For each $1\le j\le k$ denote by $\Diff(\td{S}_j,E_j)$ the set of germs of biholomorphisms $f\colon(\td{S}_j,E_j)\to(\td{S}_j,E_j)$.

\begin{defi}\label{def:5}
We say that $f\in \Diff(S,p)$ can be lifted to $\Diff(\td{S}_j,E_j)$ if there exists a germ of biholomorphism $\td{f}_j\in \Diff(\td{S}_j,E_j)$ such that the diagram below commutes:
%\[
%\begin{matrix}
%\left(\td{S}_j,E_j\right)&\overset{\td{f}_j}\longrightarrow&\left(\td{S}_j,E_j\right)\\
%\td\Pi_j\downarrow&\,&\downarrow\td\Pi_j\\
%(S,p)&\overset{f}\longrightarrow&(S,p)\\
%\end{matrix}
%\]
\[
\xymatrix{
\left(\td{S}_j,E_j\right) \ar[r]^{\td{f}_j} \ar[d]_{\displaystyle{\td\Pi_j}} & \left(\td{S}_j,E_j\right)\ar[d]^{\displaystyle{\td\Pi_j}}\\
(S,p) \ar[r]^f & (S,p)
}
\]

\end{defi}

\begin{rema}
Observe that, if the lift $\td{f}_j$ of $f$ exists then it is unique. When $j=1$ (just one blowing-up) the lifting exists for any $f\in
\Diff(S,p)$, but if $j\ge2$ then there are germs $f\in
\Diff(S,p)$ that cannot be lifted to $\Diff(\td{S}_j,E_j)$. However, we have the following:
\end{rema}

\begin{enonce}{Claim}\label{cl:21}
The blowing-up process can be done in such a way that any $f\in
\Diff_\Gcal(S,p)$ can be lifted to the last step in an unique $\td{f}=\td{f}_k\in \Diff(\td{S}_k,E_k)$. Moreover, $\td{f}$ preserves $\td\Gcal_k$ in the sense that $\td{f}^*(\td\Gcal_k)=\td\Gcal_k$.
\end{enonce}

\begin{proof}
We say that the $j^{\rm th}$ step of the blowing-up process is admissible if any $f\in \Diff_\Gcal(S,p)$ has a lifting $\td{f}_j\in
\Diff(\td{S}_j,E_j)$. We will obtain by induction a blowing-up process, as in~\eqref{eq:3}, for which there are steps $1=\ell_1<\ell_2<\dots<\ell_r=k$ such that the $\ell_j^{\rm th}$ step is admissible, for any $1\le j\le r$, and $\td\Pi_k\colon(\td{S}_k,E_k)\to(S,p)$ is a resolution of the foliation $\Gcal$.

First of all, the first step is admissible, because any $f\in
\Diff(S,p)$ admits a lifting $\td{f}_1\in \Diff(\td{S}_1,E_1)$.

Assume that we have found some process for which the $\ell:=\ell_s$ step is admissible, $\ell\ge1$, so that any $f\in \Diff_\Gcal(S,p)$ admits a lifting $\td{f}=\td{f}_\ell\in Diff(\td{S}_\ell,E_\ell)$ satisfying $\td{f}^*(\td\Gcal_\ell)=\td\Gcal_\ell$. Given $f\in \Diff_\Gcal(S,p)$, with lifting $\td{f}$, and $q\in E_\ell$ then $\td{f}$ is an equivalence between the two germs of $\td\Gcal_\ell$ at $q$ and at $\td{f}(q)$. In particular, $\td{f}$ preserves the set of non simple singularities of $\td\Gcal_\ell$.

If $\td\Gcal_\ell$ is not a resolution of $\Gcal$ then it has at least one non simple singularity $q_1$. Let $Sat(q_1)=\{\td{f}(q_1)\,|\,f\in
\Diff_\Gcal(S,p)\}=\{q_1,\,\dots,\,q_m\}$. We then blow-up once at all points $q_j\in Sat(q_1)$, passing from the $\ell=\ell_s$ step to the $\ell_{s+1}:=\ell_s+m$ step directly. Let $\wh{E}_j$ be the divisor obtained by the blowing-up at $q_j$.

Given $f\in \Diff_\Gcal(S,p)$ and its lifting $\td{f}_s$, let $\td{f}_s(q_j)=q_{i(j)}$, $1\le j\le m$. Then we can obtain a lifting $\td{f}_{s+1}$ of $\td{f}_s$ such that $\td{f}_{s+1}(\wh{E}_j)=\wh{E}_{i(j)}$, $1\le j\le m$. By Seidenberg's theorem this process must end at some step, when the final foliation $\td\Gcal_k=\td\Pi_k^*(\Gcal)$ has all singularities simple.
\end{proof}
%\vskip.1inT


\begin{proof}[{Proof of Lemma~\ref{l:25}}]
Let us finish the proof of the lemma. We will consider two cases:
\begin{enumerate}
\item \label{prooflem2.10.1}There is $q_1\in Sing(\td\Gcal_k)\cap E_k$ that has some separatrix $\td\g$ not contained in $E_k$.
\item \label{prooflem2.10.2}All the separatrices of the singularities of $\td\Gcal_k$ are contained in $E_k$.
\end{enumerate}

In the first case, let $Sat(q_1)=\{\td{f}(q_1)\,|\,f\in
\Diff_\Gcal(S,p)\}=\{q_1,\,\dots,\,q_m\}$. Given $f\in \Diff_\Gcal(S,p)$ and $q_j=\td{f}(q_1)$ then $\td{f}(\td\g):=\td\g_f$ is a separatrix of $\td\Gcal_k$ not contained in $E_k$. Since $\td\g_f$ is not contained in $E_k$, its image $\g_f:=\td\Pi_k(\td\g_f)$ is a separatrix of $\Gcal$ through $p$. Moreover, since $q_1,\,\dots,\,q_m$ are simple singularities of $\td\Gcal_k$ the set $\{\td\g_f\,|\,f\in \Diff_\Gcal(S,p)\}$ is finite. Therefore, if we set
\begin{equation}\label{eq:4}
\g=\bigcup_{f\,\in\,\Diff_\Gcal\,(S,\,p)}\,\g_f
\end{equation}
then $f(\g)=\g$, $\forall f\in \Diff_\Gcal(S,p)$, and $\g$ is a distinguished separatrix of $\Gcal$.

In the second case necessarily there are dicritical irreducible divisors of $\td\Gcal_k$, say $\td{E}_1,\,\dots,\,\td{E}_m$, contained in $E_k$ (by Camacho--Sad theorem). This case will be divided into two sub-cases:
\begin{itemize}
\item\label{prooflem2.10.2.1}(2.1) The set of tangencies $\tau$ is not empty.
\item\label{prooflem2.10.2.2}(2.2) $\tau=\emp$.
\end{itemize}

In case (2.1) let $q_o\in \tau$ and $\td\g_{\,id}$ be the leaf of $\td\Gcal_k$ through $q_o$. Then, for any $f\in \Diff_\Gcal(S,p)$ we have $q_f:=\td{f}(q_o)\in\tau$ and $\td\g_f:=\td{f}(\td\g_{\,id})$ is the leaf of $\td\Gcal_k$ through $\td{f}(q_o)$. Since $q_f\in E_k$, but $\td\g_f$ is not contained in $E_k$, $\forall f\in \Diff_\Gcal(S,p)$, the image $\td\Pi_k(\td\g_f):=\g_f$ is an irreducible separatrix of $\Gcal$ through $p$. Therefore, if we define $\g$ as in~\eqref{eq:4} then $\g$ is a distinguished separatrix of $\Gcal$.

We will divide case (2.2) into two subcases:
\begin{subequations}
\begin{equation}\label{prooflem2.10.2.2.1}
E_k\setminus\bigcup_j\td{E}_j\ne\emp.\tag{$2.2.1$} 
\end{equation}
\begin{equation}\label{prooflem2.10.2.2.2}
 E_k=\bigcup_j\td{E}_j.\tag{$2.2.2$}
\end{equation}
\end{subequations}


In case~\eqref{prooflem2.10.2.2.1}, let $\wh{E}$ be a connected component of $E_k\setminus\bigcup_j\td{E}_j$. Let $\bigcup_{i=1}^rD_i$ be decomposition of $\wh{E}$ into irreducible components, $D_i\simeq\p^1$. Note that:
\begin{enumerate}\romanenumi
\item \label{prooflem2.10.i}The graph formed by the divisors $D_i$ is a tree.
\item \label{prooflem2.10.ii}The intersection matrix $(D_i\,.\,D_j)_{1\,\le\:i,\,j\:\le\,r}$ is negative.
\item \label{prooflem2.10.iii}If $D$ is an irreducible divisor of $E_k$ such that $D\not\sub\wh{E}$ but $D\cap\wh{E}\ne\emp$ then $D=\td{E}_j$ for some $j$. In particular, $D$ is dicritical.
\end{enumerate}
In this case, $Sing(\td\Gcal_k)\cap\wh{E}\ne\emp$ and contains a singularity $q$ with a separatrix $\td\g$ not contained in $\wh{E}$. This is a consequence of Sebastiani's version of Camacho--Sad theorem (see~\cite{sb}). In fact, $\td\g$ is not contained in $E_k$, for otherwise it would be contained in some irreducible divisor $D$ of $E_k$ not contained in $\wh{E}$, and $D$ is non dicritical, which contradicts (iii). Therefore, we reduce the problem to case~\eqref{prooflem2.10.1}.

In case~\eqref{prooflem2.10.2.2.2} all irreducible divisors $\td{E}_j$ of $E_k$ are dicritical. We can assume that $Sing(\td\Gcal_k)=\emp$. In fact, if $q_o\in Sing(\td\Gcal_k)$ then $q_o$ is simple and any of their separatrices cannot be contained in $E_k$, for otherwise some of the divisors $\td{E}_j$ would be non-dicritical. Therefore, we are again in case~\eqref{prooflem2.10.1}. In particular, we can assume that all divisors $\td{E}_j$ are radial, in the sense that for any $q\in \td{E}_j$ the leaf of $\td\Gcal_k$ through $q$ is transverse to $\td{E}_j$. Moreover, $m\ge2$ because otherwise $p$ would be a radial singularity of $\Gcal$. In particular, we can assume that $\td{E}_1\cap\td{E}_2=\{q_o\}\ne\emp$. Let $\td\g_{\,id}$ be the leaf of $\td\Gcal_k$ through $q_o$. Note that, for any $f\in
\Diff_\Gcal(S,p)$ then
\[
q_f:=\td{f}(q_o)=\td{f}(\td{E}_1)\cap\td{f}(\td{E}_2)\in E_k\,,
\]
so that $\td\g_f:=\td{f}(\td\g_{\,id})$ is the leaf of $\td\Gcal_k$ through $q_f$. For each $f\in \Diff_\Gcal(S,p)$ the projection $\g_f:=\td\Pi_k(\td\g_f)$ is an irreducible separatrix of $\Gcal$. Since
\[
A:=\left\{q_f\,\middle|\,f\in \Diff_\Gcal(S,p)\right\}\sub\bigcup_{i\,\ne\,j}\,\td{E}_i\cap\td{E}_j
\]
then $A$ is finite. Therefore, we can construct a distinguished separatrix $\g$ of $\Gcal$ as in~\eqref{eq:4}.
\end{proof}

%\vskip.1in

Finally, note that Lemma~\ref{l:21} is a consequence of Lemmas~\ref{l:24} and~\ref{l:25}.

%\vskip.1in

\subsection{Proof of Lemma~\ref{l:22}.}
Since $U$ is a tubular neighborhood of $\Ga$ the map $\Te\in H^2_{Dr}(U)\mapsto\Te|_\Ga\in H^2_{Dr}(\Ga)$ is an isomorphism. Therefore, it is sufficient to prove that $c_1(N_\Si)|_\Ga=0$.

Recall that the germ of $\fa$ at any $q\in\Ga$ is equivalent to a product of a singular foliation by curves on $(\C^2,0)$ by a regular foliation of dimension $n-2$. This implies that there exist a local coordinate system around $q$, $z=(x,y)\colon U\to\C^2\times\C^{n-2}$, $x=(x_1,x_2)$, $y=(y_1,\,\dots,\,y_{n-2})$, and a holomorphic vector field $X=P(x)\tfrac{\pa}{\pa x_1}+Q(x)\tfrac{\pa}{\pa x_2}$, with an isolated singularity at $0\in\C^2$, such that
\begin{itemize}
\item $\fa|_U$ is generated by the $n-1$ commuting vector fields $X$, $Y_1:=\tfrac{\pa\,}{\pa y_1},\,\dots,\, Y_{n-2}\linebreak:=\tfrac{\pa}{\pa y_{n-2}}$.
\end{itemize}
Moreover, the separatrix $\Si$ of $\fa$ along $\Ga$ is induced by a separatrix $\g=(f(x_1,x_2)=0)$ of $X$, such that $X(f)=h.\,f$, where we have assumed $h(0)=0$.
%\vskip.1in 
It follows that we can find a Leray covering $\Ucal=(U_\alp)_{\alp\,\in\,A}$ of $\Ga$ by open sets with the following properties:
\begin{enumerate}\alphenumi
\item \label{sect.2.3.a}For each $\alp \in A$, there exists a coordinate system $z_\alp=(x_\alp,y_\alp)\colon U_\alp\to\C^2\times\C^{n-2}$, where $\Si\cap U_\alp=(x_\alp=0)$, $x_\alp=(x_{\alp 1},x_{\alp 2})$ and $y_\alp=(y_{\alp 1},\,\dots,\,y_{\alp\,n-2})$.
\item \label{sect.2.3.b}For each $\alp\in A$, $\fa|_{U_\alp}$ is generated by the $n-1$ holomorphic vector fields
\[
X_\alp=P(x_\alp)\frac{\pa}{\pa x_{\alp 1}}+Q(x_\alp)\frac{\pa}{\pa x_{\alp 2}}\,,\,Y_{\alp j}=\frac{\pa}{\pa y_{\alp j}},\,\,j=1,\,\dots,\,n-2\,\,.
\]
\item \label{sect.2.3.c}$\Si\cap U_\alp$ has the reduced equation $f_\alp=0$, where $f_\alp=f(x_\alp)$. In particular, if we set $h_\alp=h(x_\alp)$ then
\[
X_{\alp}(f_{\alp})=h_\alp\cdot\,f_a\,,\,Y_{\alp j}(f_{\alp})=0,\quad\forall\,1\le j\le n-2\,.
\]
\end{enumerate}
Consider the multiplicative cocycle $g=(g_{\alp\,\be})_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$ such that $f_\alp=g_{\alp\,\be}.\,f_\be$ on $U_\alp\,\cap\,U_\be\,\ne\,\emp$. 
%\vskip.1in

\begin{enonce}{Claim }\label{cl:22}
If $U_\alp\cap U_\be\ne\emp$ then $g_{\alp\,\be}$ is locally constant on $U_\alp\cap U_\be\cap\Ga: dg_{\alp\,\be}|_{U_\alp\,\cap\,U_\be\,\cap\,\Si}\equiv0$. In particular $c_1(N_\Si)=0$.
\end{enonce}

\begin{proof}
Let $U_\alp\cap U_\be\cap \Si\ne\emp$. We assert that there exists a $(n-1)\times(n-1)$ matrix $A_{\alp\be}$, with entries in
\[
\Ocal(U_\alp\cap U_\be), A_{\alp\be}=\left(a_{\alp\be}^{ij}\right)_{0\,\le\:i,\,j\:\le\,n-2}\,,
\]
such that
\begin{equation}\label{eq:5}
\begin{cases}
X_\alp &=a_{\alp\,\be}^{00}.\,X_\be+\displaystyle{\sum_{j=1}^{n-2}}a_{\alp\,\be}^{0j}.\,Y_{\be\,j}\,\\
 Y_{\alp i} &=a_{\alp\,\be}^{i0}.\,X_\be+\displaystyle{\sum_{j=1}^{n-2}} a_{\alp\,\be}^{ij}.\,Y_{\be\,j},\,1\le i\le n-2\\
\end{cases}
\end{equation}

In fact, since $\fa|_{U_\alp\,\cap\,U_\be}$ is generated by both systems $\left<X_\alp,Y_{\alp\,i}\,|\,1\le i\le n-2\right>$ and $\left<X_\be,Y_{\be\,i}\,|\,1\le i\le n-2\right>$, we can find a matrix $A_{\alp\,\be}$ with entries in $\Ocal(U_\alp\,\cap\,U_\be\,\setminus\,\Si)$ as in~\eqref{eq:5}. But since $cod(\Si)=2$ the entries of $A_{\alp\,\be}$ can be extended to $U_\alp\,\cap \,U_\be$ by Hartog's theorem.

Now, from (c) we get
\[
0=Y_{\alp i}(f_a)=Y_{\alp i}(g_{\alp\be}.\,f_\be)=Y_{\alp i}(g_{\alp\be}).\,f_\be+g_{\alp\be}.\,Y_{\alp i}(f_\be)
\]
and from (c) and~\eqref{eq:5}
\begin{multline*}
Y_{\alp\,i}(f_\be)=a_{\alp\,\be}^{i0}.\,X_\be(f_\be)+\sum_{j=1}^{n-2}a_{\alp\be}^{ij}.\,Y_{\be\,j}(f_\be)=a_{\alp\,\be}^{i0}.\,h_\be.\,f_\be
\\
\implies\left(Y_{\alp\,i}(g_{\alp\,\be})+a_{\alp\,\be}^{i0}.\,h_\be\right)\,f_\be=0\,\implies\,Y_{\alp\,i}(g_{\alp\,\be})=-a_{\alp\,\be}^{i0}.\,h_\be\,.
\end{multline*}
Now, $h_\be|_{U_\alp\,\cap\,U_\be\,\cap\,\Si}=h(0)=0$ and so
\[
Y_{\alp\,i}(g_{\alp\,\be})|_{U_\alp\,\cap\,U_\be\,\cap\Si}=\frac{\pa g_{\alp\,\be}}{\pa y_{\alp i}}(0,y_\alp)=0\,,\,1\le i\le n-2\,\,\implies\,\,dg_{\alp\,\be}|_{U_\alp\,\cap\,U_\be\,\cap\,\Si}=0\,.
\]
This finishes the proof of Lemma~\ref{l:22}. 
\end{proof}

%\vskip.1inT


\subsection{Proof of Lemma~\ref{l:23}.}\label{ss:23}
The case in which $\Si$ is smooth was proved in~\cite{lc2}. Here we give a more general proof (suggested by J. V. Pereira). Let us consider first the case $n=3$: $M=\p^3$. In this case, $\Ga$ is a compact algebraic curve so that $H^2_{DR}(\Ga)\simeq\R$ and the map
\[
\Te\in H^2_{DR}(\Ga)\mapsto\int_\Ga\Te\in\R
\]
is an isomorphism. In fact, we will prove that
\[
\int_\Ga c_1(N_\Si)\in\N\;\implies\;c_1(N_\Si)\ne0\,\,.
\]
We will see that $\int_\Ga c_1(N_\Si)$ represents the intersection number of a small deformation $\Ga_t$ of $\Ga$ with $\Si$.

Let $\Xcal(\p^3)$ be the vector space of holomorphic vector fields on $\p^3$: $dim(\Xcal(\p^3))=15$. Given $Z\in\Xcal(\p^3)$ we will denote by $(t,q)\in\C\times\p^3\mapsto Z_t(q)\in\p^3$ its flow and $\Ga_t:=Z_t(\Ga)$. Let $U$ be a tubular neighborhood $U$ of $\Ga$ with $U\sub\bigcup_\alp U_\alp$.
\begin{rema}\label{r:26}
There exist $Z\in\Xcal(\p^3)$ and $\ep>0$ with the following properties:
\begin{enumerate}\alphenumi
\item \label{rem.2.18.a}If $t\in D_\ep\sub\C$ then $\Ga_t\sub U$, where $D_\ep=\{t\,|\,|t|<\ep\}$.
\item \label{rem.2.18.b}If $t\in D_\ep^*:=D_\ep\setminus\{0\}$ then $\Ga_t\cap\Ga=\emp$.
\item \label{rem.2.18.c}The set $B:=\{t\in D_\ep\,|\,\Ga_t\,$ is not transverse to $\Si\}$ is discrete in $D^*$.
\item \label{rem.2.18.d}There exists $t_o\in D^*\setminus B$ such that $\Ga_{t_o}\cap\Si\ne\emp$.
\end{enumerate}
\end{rema}


We leave the proof of Remark~\ref{r:26} for the reader. Let us finish the proof of Lemma~\ref{l:23} in the case of $\p^3$.

\begin{proof}[{Proof of Lemma~\ref{l:23}}]
The idea is to prove that, if $t\in D_\ep^*\setminus B$ then $\int_\Ga c_1(N_\Si)=\#(\Ga_t\cap\Si)$, the intersection number of $\Ga_t$ with $\Si$. By (d) of Remark~\ref{r:26} $\#(\Ga_t\cap\Si)>0$ and so $c_1(N_\Si)\ne0$.

%\vskip.1in

First of all, note that
\[
\Ga_t\cap\Si=Z_t\left(\Ga\cap Z_{-t}(\Si)\right)\,\implies\,\#\,\left[\Ga_t\cap\Si\right]=\#\,\left[\Ga\cap Z_{-t}(\Si)\right]\,.
\]
On the other hand, $Z_{-t}(\Si)$ can be defined in the covering $\Ucal_t:=(Z_{-t}(U_\alp))_{\alp\,\in\,A}$ by the divisor $(f_\alp\circ Z_t)_{\alp\,\in\,A}$, with associated cocycle $g_t:=(g_{\alp\,\be}\circ Z_t)_{U_\alp\,\cap\,U_\be\,\ne\,\emp}$. Since $t\in D^*_\ep\setminus B$, $\Ga$ is transverse to $Z_{-t}(\Si)$ and so $\Ga\cap Z_{-t}(\Si)$ is finite and is defined by the divisor
\begin{multline*}
\left(f_\alp\circ Z_t|_{\Ga\,\cap\,Z_{-t}(U_\alp)}\right)_{\alp\,\in\,A}\\
\text{ with associated cocycle}\quad g_t|_\Ga=\left(g_{\alp\,\be}\circ Z_t|_{\Ga\,\cap Z_{-t}(U_\alp\,\cap\,U_\be)}\right)_{U_\alp\,\cap\,U_\be\,\ne\,\emp}\,.
\end{multline*}
This divisor can be interpreted as a holomorphic section of the line bundle induced by $g_t|_\Ga$ on $Pic(\Ga)$. In particular, if $c_1(g_t|_\Ga)$ is its first Chern class then its degree is given by
\[
\int_\Ga c_1(g_t|_\Si)=\#\,[\Ga\cap Z_{-t}(\Si)]=\#\,[\Ga_t\cap \Si]\,\,.
\]
Since the map
\[
t\in D_\ep\mapsto \int_\Ga c_1(g_t|_\Ga)
\]
is continuous and constant in $D_\ep^*\setminus B$, we get $\int_\Ga c_1(g_0|_\Si)>0 \implies$ $c_1(g_0|_\Si)\linebreak=c_1(N_\Si)\ne0$. This finishes the proof of Lemma~\ref{l:23} in the case of $\p^3$.

The case of $\p^n$, $n\ge4$, can be reduced to the previous by taking sections by generic 3-planes linearly embedded in $\p^n$. We leave the details to the reader.
\end{proof}

\subsection{Proof of Corollary~\ref{c:21}.}
\begin{proof}
Recall that $X(f)=h.\,f$, where $X$ represents the normal type $\Gcal$ of $\fa$ along $\Ga$ and $f\in\Ocal_2$ is reduced. By lemma~\ref{l:23} we have $h(0)\ne0$. Let $f_\mu$ and $X_\nu$ be the first non-zero jets of $f$ and $X$ at $0\in\C^2$, respectively. Then
\[
X(f)=h.\,f\quad\implies\quad X_\nu(f_\mu)=h(0).\,f_\mu\quad\implies\quad\nu=1
\]
and $X_\nu=X_1$ is not nilpotent; has at least one non-zero eigenvalue. On the other hand, we have seen that $\Ga$ is a Kupka component of $\fa$ if, and only if, $tr(X_1)\ne0$. If $tr(X_1)=0$ and $X_1$ has a non-zero eigenvalue, then we can assume that $X_1\linebreak=\la\,(x_1\tfrac{\pa}{\pa x_1}-x_2\tfrac{\pa}{\pa x_2}), \la\ne0$. In this case, $X$ has exactly two separatrices through $0\in\C^2$ which are smooth and tangent to $x_1=0$ and $x_2=0$. We can assume that these separatrices have equations $f_1(x_1,x_2)=x_1+h.o.t$ and $f_2(x_1,x_2)=x_2+h.o.t$. Consider the separatrix $\g=(f_1.\,f_2=0)$ of $X$. Note that $f(\g)=\g,\,\forall f\in
\Diff_\Gcal(\C^2,0)$. By Lemma~\ref{l:24} $\g$ generates a separatrix $\Si$ of $\fa$ along $\Ga$. However $X(f_1.\,f_2)=h.\,f_1.\,f_2$ where $h(0)=0$, because $X_1(x_1.\,x_2)=0$. Therefore, we must have $tr(X_1)\ne0$ and $\Ga$ is a Kupka component of $\fa$. \end{proof}

\section{Corollaries~\ref{c:2} and~\ref{c:3}}\label{ss:3}

\subsection{Proof of Corollary~\ref{c:2}.}
A codimension one foliation $\Gcal$ on $\p^n$ of degree zero has a rational first integral of degree one. It is defined in some coordinate system $(x_1,\,\dots,\,x_{n+1})\in\C^{n+1}$ by a the form $\om=x_1\,dx_2-x_2\,dx_1$. In particular, $\Pi^{-1}(Sing(\Gcal))=(x_1=x_2=0)$, which is a l.t.p component.

%\vskip.1in

Conversely, let $\fa$ be a codimension one foliation on $\p^n$, $n\ge3$. It is known that $Sing(\fa)$ has at least one irreducible component of codimension two~\cite{ln}. Assume that all components of $Sing(\fa)$ are l.t.p. Let $\Om$ be a 1-form on $\C^{n+1}$ that represents $\fa$ in homogeneous coordinates: $\fa_\Om=\Pi^*(\fa)$. Then
\begin{enumerate}\alphenumi
\item \label{sect.3.1.a}$i_R\Om=0$, where $R$ is the radial vector field on $\C^{n+1}$.
\item \label{sect.3.1.b}The coefficients of $\Om$ are homogeneous of degree $d+1$, where $d=deg(\fa)$.
\item \label{sect.3.1.c}$i_Rd\Om=(d+2)\,\Om$ (see~\cite{lc}). In particular, $Sing(d\Om)\sub Sing(\Om)$.
\end{enumerate}

\begin{enonce}{Claim}\label{cl:31}
Let $q\in Sing(\fa)$ and $p\in\Pi^{-1}(q)\sub\C^{n+1}\setminus\{0\}$. Then $d\Om_p\ne0$. In particular, $Sing(d\Om)=\emp$ and $deg(\fa)=0$.
\end{enonce}

\begin{proof}
Let $\om$ be a holomorphic 1-form that represents $\fa$ in a neighborhood of $q$. The hypothesis and Theorem~\ref{t:1} imply that $d\om(q)\ne0$.

On the other hand, $\Pi^*(\om)$ represents $\fa_\Om$ in a neighborhood, say $U$, of $p$. It follows that $\Pi^*(\om)=\var.\,\Om$ on $U$, where $\var\in\Ocal^*(U)$. Therefore,
\begin{multline*}
\Pi^*(d\om)=d\,\Pi^*(\om)=d\var\wedge\Om+\var.\,d\Om
\\
\begin{aligned}
\implies\forall\,u,v\in T_p\C^{n+1}\quad\text{we get}\,\,\var(p).\,d\Om_p(u,v)
&=\Pi^*(d\om)_p\,(u,v)\\
&=d\om_q\,\left(d\Pi(p).u,d\Pi(p).v\right)\,.
\end{aligned}
\end{multline*}
Since $\Pi$ is a submersion, it follows that $d\Om_p\ne0$. Therefore, the coefficients of $\Om$ must be of degree one and $\fa$ has degree zero, as asserted in Corollary~\ref{c:2}. 
\end{proof}

%\vskip.1inR

\subsection{Proof of corollary~\ref{c:3}.}
The idea is to use Corollary~\ref{c:2}. Assume that there exists an integrable 2-form $\eta$ on $\C^n$, $n\ge4$, with homogeneous coefficients of degree $d\ge1$ and such that $Sing(\eta)=\{0\}$. Denote by $\fa_\eta$ the holomorphic codimension two foliation of $\C^n$ generated by $\eta$. By assumption $Sing(\fa_\eta)=\{0\}$. Note also that the codimension two distribution of $\C^n\setminus\{0\}$ tangent to $\fa_\eta$ is given by
\[
ker(\eta)(p)=\left\{v\in T_p\C^n\,\middle|\,i_v\,\eta(p)=0\right\}\,,\,\forall p\ne0\,,
\]
where $i_v$ denotes the interior product. The fact that $ker(\eta)$ has codimension two is equivalent to
\begin{equation}\label{eq:e2}
\eta\wedge\eta=0\,.
\end{equation}

Let $\om=i_R\eta$, where $R=\sum_{j=1}^nz_j\tfrac{\pa}{\pa z_j}$ is the radial vector field on $\C^n$. We have two possibilities: either $\om\equiv0$, or $\om\not\equiv0$.

In the first case, $\eta$ generates a codimension two foliation on $\p^{n-1}$: there exists a codimension two foliation $\fa$ on $\p^{n-1}$ such that $\Pi^*(\fa)=\fa_\eta$, where $\Pi\colon\C^n\setminus\{0\}\linebreak\to\p^{n-1}$ denotes the canonical projection. However, any codimension two foliation on $\p^{n-1}, n\ge4$, has at least one singularity: if $q\in Sing(\fa)$ then the line $\ov{\Pi^{-1}(q)}\sub\C^n$ is contained in the singular set of $\eta$.

In the second case $\om$ is a 1-form on $\C^n$ with homogeneous coefficients of degree $d+1$.

\begin{lemm}\label{int}
The form $\om$ is integrable: $\om\wedge d\om=0$.
\end{lemm}

\begin{proof} 
The following is equivalent to the integrability of the distribution $ker(\eta)$:
\begin{enumerate}\Romanenumi
\item \label{proof.3.2.I}for any $p\in \C^n\setminus\{0\}$ there exists a germ coordinate system $(x,y)\colon(\C^n,p)\to(\C^2,0)\times(\C^{n-2},0)$, with $x=(x_1,x_2)$, such that $\eta_p=\var(x,y)\,dx_1\wedge dx_2$, where $\eta_p$ is the germ of $\eta$ at $p$ and $\var\in\Ocal_p^*$.
\end{enumerate}
Since the coefficients of $\eta$ are homogeneous of degree $d$ we have $L_R\eta=(d+2)\,\eta$, where $L_R$ denotes the Lie derivative in the direction of $R$. From this we get
\begin{multline*}
(d+2)\,\eta=L_R\eta=i_Rd\eta+d\,i_R\eta=i_Rd\eta+d\om
\\
\implies\om\wedge d\om=i_R\eta\wedge d\om=(d+2)\,i_R\eta\wedge \eta-i_R\eta\wedge i_Rd\eta\,.
\end{multline*}
Now, from~\eqref{eq:e2} we get
\[
0=i_R(\eta\wedge\eta)=2\,i_R\eta\wedge\eta=2\,\om\wedge\eta\,\implies\,\om\wedge d\om=-i_R\eta\wedge i_Rd\eta\,.
\]
If we consider a coordinate system as in (I) we have $\eta_p=\var\,dx_1\wedge dx_2$ and $d\eta_p\linebreak=d\var\wedge dx_1\wedge dx_2$ and this implies that $i_R\eta_p\wedge i_Rd\eta_p=0$, as the reader can check. 
\end{proof}
% \vskip.1in
 Write $\om=\phi.\,\om_1$, where $\phi$ is homogeneous and $cod(Sing(\om_1))\ge2$.
\begin{rema}\label{r:31.}
%{\rm 
Note that:
\begin{enumerate}\alphenumi
\item $\om_1\wedge\eta=0$. This is a consequence of $\om\wedge\eta=0$.
\item $\om_1\wedge d\om_1=0$ and $i_R\om_1=0$. This is a consequence of $\om\wedge d\om=0$ and $i_R\om=0$.
\end{enumerate}
%}
\end{rema}
Denote by $\fa_{\om_1}$ the foliation generated by $\om_1$. It follows from (b) of Remark~\ref{r:31.} that there exists a codimension one foliation $\fa$ on $\p^{n-1}$ such that $\Pi^*(\fa)=\fa_{\om_1}$.
\goodbreak
\begin{lemm}\label{l:32}
All irreducible components of $Sing(\fa)$ are l.t.p.
\end{lemm}

\begin{proof}
Fix $q\in Sing(\fa)$ and $p\in\C^n\setminus\{0\}$ with $\Pi(p)=q$. Note that $p\in Sing(\fa_{\om_1})$, the foliation generated by $\om_1$. Let $(x,y)\colon(\C^n,p)\to(\C^2,0)\times(\C^{n-2},0)$ be as in (I), so that $\eta=\var.\,dx_1\wedge dx_2$, $\var\in\Ocal_p^*$. It follows from $\om_1\wedge\eta=0$ that in these coordinates we have $\om_1=A(x,y)\,dx_1+B(x,y)\,dx_2$ and from $\om_1\wedge d\om_1=0$ that
\[
(A\,dB-B\,dA)\wedge dx_1\wedge dx_2=0\,\,\implies\,\,\om_1=h(x,y).\,\left(C(x_1,x_2)\,dx_1+D(x_1,x_2)\,dx_2\right)\,.
\]


Since $cod(Sing(\om_1))\ge2$ we get $h\in\Ocal_p^*$ and the germ of $Sing(\om_1)$ at $p$ is defined by $(x_1=x_2=0)$. Moreover, the germ of $\fa_{\om_1}$ at $p$ is defined by the form $C(x_1,x_2)\,dx_1+D(x_1,x_2)\,dx_2$ and so $\fa_{\om_1}$ is a transversely product at $p$. Since $p\in\Pi^{-1}(q)$ and $\Pi$ is a submersion at $p$, $\fa$ is a transversely product at $q$.
\end{proof}

%\vskip.1in

Corollary~\ref{c:2} implies that $\om_1$ has a linear rational first integral that we can assume to be $x_2/x_1$, so that $\om_1=x_1\,dx_2-x_2\,dx_1=i_R(dx_1\wedge dx_2)$. Let $\eta=\sum_{i<j}\eta_{ij}\,dx_i\wedge dx_j$, where $\eta_{ij}$ is homogeneous of degree $d$, $\forall\,i<j$. From $\om_1\wedge\eta=0$ we get $\eta_{ij}=0$, $\forall\,j>i\ge3$. Therefore, we can write $\eta=dx_1\wedge\alp +dx_2\wedge\be+\g\,dx_1\wedge dx_2$, where $\alp=\sum_{j\ge3}\eta_{1j}\,dx_j$, $\be=\sum_{j\ge3}\eta_{2j}\,dx_j$ and $\g=\eta_{12}$. Hence,
\begin{gather*}
0=\om_1\wedge\eta=\left(x_1\,dx_2-x_2\,dx_1\right)\wedge\left(\alp\wedge dx_1+\be\wedge dx_2+\g\,dx_1\wedge dx_2\right)\,\implies
\\
\left(x_1\,\alp+x_2\,\be\right)\wedge dx_1\wedge dx_2=0\,\,\implies\,\,x_1\,\alp=-x_2\,\be\,\,\implies
\end{gather*}
there exists 1-form $\mu$ with homogeneous coefficients of degree $d-1$ such that $\alp\alp=-x_2\,\mu$ and $\be=x_1\,\mu$. In particular, we get
\[
\eta=\om_1\wedge\mu+\g\,dx_1\wedge dx_2=\left(x_1\,dx_2-x_2\,dx_1\right)\wedge\mu+\g\,dx_1\wedge dx_2\,\implies\,
\]
\[
Sing(\eta)\sup(x_1=x_2=\g=0)\,\implies
\]
$d=0$ and $\g$ is a constant, for otherwise $cod(Sing(\eta))\le3$ and $Sing(\eta)\supsetneq\{0\}$. This finishes the proof of Corollary~\ref{c:3}.
\qed
%\vskip.2in


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