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?



When is the flatness locus non-empty
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 $S$.


