TITLE: When is the flatness locus non-empty
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: Let $f:X\to S$ be a morphism of schemes.
Assume that $S$ is integral, and let $K$ be its function field. As "everything is flat over a field", the generic fibre $f_K:X_K\to \mathrm{Spec} \ K$ is a flat morphism.
In particular, the locus of flatness is non-empty.
As Laurent Moret-Bailly points out below, if $f$ is of finite presentation (and $S$ still integral), then $f$ is flat over a dense open of $S$.