Lemma 37.23.3. Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. If $Z \subset X$ is a closed subset with $Z_\eta$ nowhere dense in $X_\eta$, then there exists a nonempty open $V \subset Y$ such that $Z_ y$ is nowhere dense in $X_ y$ for all $y \in V$.

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 37.21.10 we see that

$\dim _{\xi '}(Z_\eta ) = \dim _{\xi }(Z_ y) = \dim _{\xi }(X_ y) = \dim _{\xi '}(X_\eta ).$

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. $\square$

Comment #5566 by Yatir on

In the fourth line from the bottom, should $Z_V\to Z$ be $Z_V\to V$?

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