Lemma 37.23.2. Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta$. If $X_\eta \not= \emptyset$ then there exists a nonempty open $V \subset Y$ such that $X_ V = V \times _ Y X \to V$ is surjective.

Proof. This follows, upon taking affine opens, from Algebra, Lemma 10.30.2. (Of course it also follows from generic flatness.) $\square$

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