Let $f : X \dashrightarrow S$ be a rational map of smooth projective varieties. Is it true that, after a birational modification of $S$, every fiber intersects the domain of definition? Explicitly, is there a birational map $S \dashrightarrow S'$ of projective varieties with mild singularities, such that the map $\mathrm{dom}(f) \to S \to S'$ is surjective?
I think the above is true when $\dim{S} \le 2$ where it can be done explicitly by contracting certain rational curves on $S$.
