Is the set $\{y\in Y\mid f^{-1}(y)\subset H\}$ closed?

Question

Suppose $f\colon X\to Y$ is a surjective morphism of projective varieties, $H\subseteq X$ is a closed subset in $X$, does this hold: $\{y\in Y \mid f^{-1}(y)\subset H\}$ is a closed subset of $Y$?

Answer 1

This looks like it should be true, but it isn't. Here's an example.

Let $Y=\mathbf P^2$, let $X$ be the blowup of $\mathbf P^2$ in a point $p$, and let $f: X \rightarrow Y$ be the blowdown morphism.

Now take $H$ to the proper transform on $X$ of a smooth curve $C$ through $p$. Then the set you are asking about will be exactly $C - \{p\}$.