Let $X$ and $Y$ be integral, finite-type $\mathbb{C}$-schemes with $Y$ nonsingular and $\dim X=\dim Y$. Let $f:X\rightarrow Y$ be a dominant morphism. Let $U\subseteq Y$ be the set of $y\in Y$ for which $f^{-1}\{y\}$ is a nonempty finite set of points. By generic flatness, $U$ contains a dense open subset of $Y$. What I would like to know is if $U$ itself is open.
Note 1: This will be the case when $f$ is a closed map by upper semicontinuity of fiber dimension on the target.
Note 2: David Speyer gives an example of $U$ failing to be open when $Y$ is not normal. See his answer here. So some kind of niceness re: singularities of $Y$ must be used if $U$ is to be proved open. Perhaps some form of Zariski's Main Theorem would be useful, but it's not clear to me how.
Note 3: One can reduce to the case $X$ and $Y$ affine, so I'm adding the commutative algebra tag. Let $A\subseteq B$ be an extension of finitely generated $\mathbb{C}$-domains of the same Krull dimension with $A$ regular. If $p$ is a prime ideal of $A$ with finitely many primes of $B$ lying over it, does there exist $f\in A-p$ such that every prime of $A_{f}$ has finitely many primes of $B$ lying over it?