Let $f: X \to Y$ be a morphism of (irreducible) varieties, where the dimension of every fiber dim$f^{-1}(y)=n$ is the same. Must it follow that dim$X=$ dim$Y+n$?
The reason I am asking this is that this is a theorem in some notes I am reading, but I am sure the proof presented is problematic, yet I cannot seem to patch the errors. However this does seem geometrically intuitive... Any help is appreciated!