Let $f : S \to X$ be a dominant morphism of smooth complex compact surfaces. Let $C \subset S$ be a smooth curve such that $df$, seen as a map from $T_S \to T_X$, is generically of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an isomorphism outside $C$. Suppose that $C$ is not contracted by $f$, is it true that $f_{|C}$ is an immersion?

One might want to decompose it into two sub-questions:

- Can one show that the rank of $df$ is exactly $1$ along $C$?
- Suppose the rank of $df$ is exactly $1$ along $C$, can we show that $f_{|C}$ is an immersion?