Rational mapping

Kind of partial function between algebraic varieties In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal definition Formally, a rational map f : V → W {\displaystyle f\colon V\to W} between two varieties is an equivalence class of pairs ( f U , U ) {\displaystyle (f_{U},U)} in which f U {\displaystyle f_{U}} is a morphism of varieties from a non-empty open set U ⊂ V {\displaystyle U\subset V} to W {\displaystyle W} , and two such pairs ( f U , U ) {\displaystyle (f_{U},U)} and ( f ′ U ′ , U ′ ) {\displaystyle ({f'}_{U'},U')} are considered equivalent if f U {\displaystyle f_{U}} and f ′ U ′ {\displaystyle {f'}_{U'}} coincide on the intersection U ∩ U ′ {\displaystyle U\cap U'} (this is, in particular, vacuously true if the intersection is empty, but since V {\displaystyle V} is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: If two morphisms of varieties are equal on some non-empty open set, then they