Let $k$ be an algebraically closed field, $f:X \to Y$ be a surjective proper $k$-morphism locally of finite presentation between irreducible noetherian schemes. Assume that $Y$ is reduced. Under what additional condition on $f$ (other than flatness/ generic flatness) does there exist a *non-empty* open set $U$ of $X$ such that $f|_U$ is flat?

If I further assmume that $X$ is reduced, then does there exists a non-empty open subset $U$ of $Y$ such that $f|_{f^{-1}(U)}$ is flat?

notassume that $Y$ is reduced... $\endgroup$ – Matthieu Romagny Oct 21 '15 at 19:34