The Stacks project

Proof. Let $Y' \subset Y$ be the reduction of $Y$. Set $X' = Y' \times _ Y X$ and $Z' = Y' \times _ Y Z$. As $Y' \to Y$ is a universal homeomorphism by Morphisms, Lemma 29.45.6 we see that it suffices to prove the lemma for $Z' \subset X' \to Y'$. Thus we may assume that $Y$ is integral, see Properties, Lemma 28.3.4. By Morphisms, Proposition 29.27.1 there exists a nonempty affine open $V \subset Y$ such that $X_ V \to V$ and $Z_ V \to V$ are flat and of finite presentation. We claim that $V$ works. Pick $y \in V$. If $Z_ y$ has a nonempty interior, then $Z_ y$ contains a generic point $\xi$ of an irreducible component of $X_ y$. Note that $\eta \leadsto f(\xi )$. Since $Z_ V \to V$ is flat we can choose a specialization $\xi ' \leadsto \xi$, $\xi ' \in Z$ with $f(\xi ') = \eta$, see Morphisms, Lemma 29.25.9. By Lemma 37.22.10 we see that

Hence some irreducible component of $Z_\eta$ passing through $\xi '$ has dimension $\dim _{\xi '}(X_\eta )$ which contradicts the assumption that $Z_\eta$ is nowhere dense in $X_\eta$ and we win. $\square$