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 $X$. 



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Image of curve along a finite etale Galois map
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 $p$.


