Let $f: X \to Y$ be a finite, surjective morphism of smooth, projective, irreducible varieties over $\mathbb{C}$ and let $y \in Y$. 


1. Can I find a smooth curve $C \subseteq Y$ with $y \in C$ such that $f^{-1}(C)$ is again a smooth curve?
2. If not, can I find a curve $C \subseteq Y$ such that $y$ is a smooth point of $C$ and such that every $x \in f^{-1}(\{y\})$ is a smooth point of $f^{-1}(C)$?
3. Can I find a curve $C \subseteq Y$ with the property of 1. or 2. with the additional property that $f$ is unramified along a Zariski dense subset of $f^{-1}(C)$?


Edit: As Yusuf recommended, I'd like to point out that I am especially interested in the case where $y$ lies in the branch locus. Also $y$ is not necessarily a smooth point of the underlying reduced scheme of the branch divisor.



Preimage of smooth curves under morphism of smooth varieties
Consider the map $\mathbb P^2 \to \mathbb P^2$ with equations $(xy,x^2-y^2,z^2)$. Any smooth curve passing through the point $(0,0,1)$ has an equation that looks locally linear, and its inverse image has an equation that looks locally quadratic.


