DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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Birational maps and birational equivalence of classical affine varieties

Definition

Let $X$ and $Y$ be classical affine varieties.

A rational map $η : X ⇢ Y$ is birational when there exists a rational map $θ : Y ⇢ X$ such that $θ \circ η = {id}_{X} and η \circ θ = {id}_{Y}$ as rational maps.

The varieties $X$ and $Y$ are birationally equivalent when such a birational map exists.

Depends on

Rational maps between irreducible classical affine varieties

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, paragraph after Proposition 5.38 (standard reference, not scraped)
Donu Arapura, Notes on Basic Algebraic Geometry, §3.2 (standard reference, not scraped)