Lemma 37.22.1. Let $f : X \to Y$ be a finite type morphism of schemes. Assume $Y$ irreducible with generic point $\eta $. If $X_\eta = \emptyset $ then there exists a nonempty open $V \subset Y$ such that $X_ V = V \times _ Y X = \emptyset $.
Proof. Follows immediately from the more general Morphisms, Lemma 29.8.4. $\square$
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