_i : C_i \rightarrow \tilde X$ are given by $$\begin{aligned}
\tilde f_1 ^* : & \left\{ \begin{tabular}{c@{ $\longmapsto $ }c}
$\tilde u$&$x_1 U_1 (x_1)$ \vphantom{$\frac{\frac{1}{1}}{\frac{1}{1}}$} \\
$\tilde v$&$x_1 ^2 \frac {V_1 (x_1)} {U_1 (x_1)}$
\end{tabular} \right. & U_1 (0) \neq 0 \\[5pt]
\tilde f_2 ^* : & \left\{ \begin{tabular}{c@{ $\longmapsto $ }c}
$\tilde u$&$x_2 ^{m} U_2 (x_2)$ \vphantom{$\frac{\frac{1}{1}}{\frac{1}{1}}$} \\
$\tilde v$&$x_2 \frac {V_2 (x_2)} {U_2 (x_2)}$ 
\end{tabular} \right. & U_2 (0), V_2 (0) \neq 0\end{aligned}$$

Clearly these maps are unramified at $x_i = 0$ and since $(1,0)$ and $(\star ,1)$ are tangent vectors at $\tilde q$ to $C_1$ and $C_2$ respectively, the maps are also transverse at $\tilde q$. This simple computation proves the first half of the lemma.

Suppose conversely that in the blow-up $\tilde X$ of $X$ at $q$, the curves $C_1$ and $C_2$ meet transversely at the point $\tilde q = \tilde f_i (p_i) \in \tilde X$. Fix coordinates $x_1$ on $C_1$ at $p_1$ and $x_2$ on $C_2$ at $p_2$, and choose coordinates $u,v$ near $q$ and $\tilde u , \tilde v$ near $\tilde q$ such that (\[bloco\]) are the equations of the blow-up morphism. We have $$ f\_1 \^\* : {

  ------------- -------------------------
   $\tilde u $   