The Stacks project

Definition 29.49.11. Let $X$ and $Y$ be irreducible schemes.

  1. We say $X$ and $Y$ are birational if $X$ and $Y$ are isomorphic in the category of irreducible schemes and dominant rational maps.

  2. Assume $X$ and $Y$ are schemes over a base scheme $S$. We say $X$ and $Y$ are $S$-birational if $X$ and $Y$ are isomorphic in the category of irreducible schemes over $S$ and dominant $S$-rational maps.

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