# Image of curve along a finite etale Galois map

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?

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 $$p$$.
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 $C \cup C'$.