the morphism $f_i$ is ramified at $p_i$ on $C_i$ if $k_i > 1$ and we say it is unramified at $C_i$ if $k_i = 1$.

Let $f_i : C_i \rightarrow X$, $i \in \{ 1, 2 \}$ be two non-constant morphisms from two smooth curves to a smooth surface $X$ and let $p_1 \in C_1$ and $p_2 \in C_2$ be points such that $f_1 (p_1) = f_2 (p_2) = q$. Denote by $\tilde f _1$ and $\tilde f _2$ the morphisms induced by $f_1$ and $f_2$ from each curve to the blow-up of $X$ at $q$, and assume $\tilde f _1 (p_1) = \tilde f _2 (p_2) = \tilde q$. Then the following conditions are equivalent:

1.  $\tilde f _1$ and $\tilde f _2$ are unramified at $\tilde q$ and $C_1$ and $C_2$ are transverse at $\tilde q$;

2.  after possibly renumbering the curves $C_1$ and $C_2$, there are coordinates $u,v$ on $X$ near $q$ and $x_i$ on $C_i$ near $p_i$ such that $$\label{bln2}
    \begin{array} {ll}
    f_1 ^* : 
    \left\{ \begin{tabular}{c@{ $\longmapsto$ }c}
    $u$ & $x_1 U_1 (x_1)$ \\[5pt]
    $v$ & $x_1 ^3 V_1 (x_1)$ \\[5pt]
    \end{tabular} \right. & U_1 (0) \neq 0 \vspace{10pt} \\ 
    f_2 ^* : \left\{ 
    \begin{tabular}{c@{ 