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Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is 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 $f$, I stronly feel that $f_{|C}$ is an immersion. Is it true ?

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  • $\begingroup$ I know this is true in the case where $f$ is also open, proper and surjective ($f$ is a branched cover). in that case, $C$ is in the critical set and there are local coordinates such that $f$ takes the form $(z_1,z_2) \mapsto (z_1^k,z_2)$ where $C=\{ z_1=0\}$ locally $\endgroup$
    – Albert
    Apr 10, 2017 at 9:46

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