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Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$.

Is $f(C)$ a smooth curve?

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No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

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    $\begingroup$ +1 for the frightening anecdote $\endgroup$ – Piotr Achinger Mar 7 '18 at 21:30
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    $\begingroup$ @PiotrAchinger: ironically my anecdote had to be withdrawn, or at least substantially revised. $\endgroup$ – Pop Mar 7 '18 at 22:41
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If you want an explicit counterexample, take a smooth genus $2$ curve $C$ and embed it in its Jacobian $X:=J(C)$. Choose a point $\alpha$ of order $2$ in $C$ and let $f \colon X \to Y$ the quotient map by the translation $x \mapsto x+ \alpha$.

Then $f$ is a degree $2$ isogeny of abelian surfaces and, setting $C':= C + \alpha$, the two curves $C$, $C'$ intersect transversally at two points.

Since $f(C)=f(C \cup C')$, it follows that the curve $f(C)$ has a node, image of the two nodes of $C \cup C'$.

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