Let $f: X \to Y$ be an arbitrary dominant morphism of locally noetherian schemes over any field $k$. Assume that $X,Y$ are also irreducible with dimensions $\operatorname{X}=n $ and $ \operatorname{Y}=m$. We don't make any additional assumptions (like smoothness, flatness etc. )
Is there any result known generically estimating the dimension of fibers of $f$. Generically means here that I'm seeking for a result like 'there exist an open subscheme $U \subset Y$' such that for all $y \in U$ the the dimension of the fiber $f^{-1}(y)$ is $\le n-m$?
I know there are some known results if we impose additional assumptions on (generical) flatness but here I would like to discuss the case if we not use such assumpions. Is there moreover any dependence on the base field $k$ known?