Let $C, D$ be curves (so $1$-dimensional, proper $k$-schemes).
Assume that $C,D$ are irreducible and $f: C \to D$ is a surjective morphism of schemes. Let $c \in C$ be a closed point.
My question is if the fiber $f^{-1}(c)$ must be in this case a finite set? What is the argument? Or conterexample?
If generally not what are the weakest conditions which garantee this?
Sure it would be a boring game if we assume that $C$ is Noetherian since the fiber as a closed subset inherits aslo this property.