Is it true that for any proper dominant morphism $\pi:X \rightarrow Y$, there is an open set $U\subset Y$ such that $\pi|_U$ is a composition of smooth morphisms and radicial morphisms (assuming $X$ is regular, also can assume that $X$ and $Y$ are of finite type over a finite field)?

dense nonempty open set $U \subseteq X$instead of a subset of $Y$? – Karl Schwede Jul 1 '12 at 0:45