is, the fibers of $\mathcal F$ and $\mathcal G$ are transversal at $p$.

An important consequence for us is that the pull-back by $\Phi_{F,G}$ of a holomorphic foliation $\mathcal L$ on $\Pp^2$ is also a holomorphic foliation in $S$. In the next Section we describe the singularities of ${\Phi^*_{F,G}}(\mathcal L)$.

New foliations on S {#sec-new-foliation}
===================

Let us take a foliation $\mathcal L$ on $\mathbb P^2$ defined by $\omega= LdP-d.PdL=0$, where $P(X,Y)=\sum_{i+j \leq d} a_{ij}X^iY^j$ is a polynomial of degree $d$ such that $C_0=\{P=0\}$ (we may assume $a_{0d}\neq 0$) and $L$ is a linear polynomial such that $L=0$ is transverse to $C_0$ . The singularities of $\mathcal L$ contained in $C_0$ are supposed to be disjoint of $A_0\cup \phi(A)$.

We proceed to compute the multiplicity $Z(\mathcal L^{*},C,p)$ along $C$ of $p$ as a singularity of $\mathcal L^{*}=\Phi_{F,G}^{*}(\mathcal L)$ at the points where $\Phi_{F,G}$ maybe fails to be a biholomorphism. In order to make the computation easier, we take $L(X,Y)=X+b$.

\[Indices-Z\] With notation of the proof of Proposition \[extension\], we have

-   Case A1: $Z(\mathcal L^{*},C,p)= d+mult_0(A_1(x,0))$.

-   Case A2: $Z(\mathcal L^{*},C,p)= mult_0(A_2(x,0))$.

-   Case B: $Z(\mathcal L^{*},C,p)=mult_0(B(x,0))$.

[**Case A1**]{}: $x$ and $\xi$ are relatively prime. It follows that $$P(\Phi_{F,G}(x,y))=\dfrac{yv(x,y)}{\xi(x,y)^d}$$ In 