Birational isomorphism

In mathematics, in particular the subfield of algebraic geometry, a rational map is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.

Formally, a rational map ${\displaystyle f\colon V\to W}$ between two varieties is an equivalence class of pairs ${\displaystyle (f_{U},U)}$ in which ${\displaystyle f_{U}}$ is a morphism of varieties from an open set ${\displaystyle U\subset V}$ to ${\displaystyle W}$, and two such pairs ${\displaystyle (f_{U},U)}$ and ${\displaystyle (f_{U}',U')}$ are considered equivalent if ${\displaystyle f_{U}}$ and ${\displaystyle f_{U}'}$ coincide on the intersection ${\displaystyle U\cap U'}$ (this is, in particular, vacuously true if the intersection is empty, but since ${\displaystyle V}$ is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma:

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