Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties.
If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism $V\simeq f(V)$ ? I am interested in the case of a double cover (i.e. $d=2$), but I expect the answer is general.
Added: Let $\iota$ denote the sheet-interchange involution induced on $X$. I expect the answer to the above question to be NO whenever $\iota(V)=V$, of course. But can this actually happen (does it always happen)? Secondly, is the answer automatically YES whenever $\iota(V)\neq V$?
Ok, here is a sketch. Let $f:X\to Y$ be an etale cover of smooth projective surfaces of degree 2 (higher degrees will work too). Let $p\neq q$ be points in $X$ such that $f(p)=f(q)$. I claim that there is an irreducible curve $C$ passing through $p,q$ and then clearly $f:C\to f(C)$ is not injective.
For this, I use a lovely argument by C. P. Ramanujam (see his collected works, if you have access). Let $g:Z\to X$ be the blow up of $p,q$, then $Z$ is projective and thus a general hyperplane section $H\subset Z$ is irreducible, by the usual Bertini. But, being a hyperplane, $H$ has to meet both the exceptional curves over $p,q$ and thus $g(H)=C$ is an irreducible curve on $X$ passing through both $p,q$.