Lemma 33.4.2. Let $X$ and $Y$ be varieties over a field $k$. The following are equivalent

1. $X$ and $Y$ are birational varieties,

2. the function fields $k(X)$ and $k(Y)$ are isomorphic,

3. there exist nonempty opens of $X$ and $Y$ which are isomorphic as varieties,

4. there exists an open $U \subset X$ and a birational morphism $U \to Y$ of varieties.

Proof. This is a special case of Morphisms, Lemma 29.50.6. $\square$

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