# Birational mapping

birational isomorphism

A rational mapping between algebraic varieties inducing an isomorphism of their fields of rational functions. In a more general setting, a rational mapping of schemes $f: X \rightarrow Y$ is said to be a birational mapping if it satisfies one of the following equivalent conditions: 1) there exist dense open sets $U \subset X$ and $V \subset Y$ such that $f$ is defined on $U$ and realizes an isomorphism of subschemes $f\mid _ {U} : U \rightarrow V$; 2) if $\{ x _ {i} \} _ {i \in I }$, $\{ y _ {j} \} _ {j \in J }$ are the sets of generic points of the irreducible components of the schemes $X$ and $Y$ respectively, $f$ induces a bijective correspondence between the sets $\alpha : I \rightarrow J$ and an isomorphism of local rings ${\mathcal O} _ {X, x _ {i} } \rightarrow {\mathcal O} _ {Y, y _ {\alpha (i) } }$ for each $i \in I$.

If the schemes $X$ and $Y$ are irreducible and reduced, the local rings of their generic points become identical with the fields of rational functions on $X$ and $Y$, respectively. In such a case the birational mapping $f: X \rightarrow Y$ induces, in accordance with condition 2), an isomorphism of the fields of rational functions: $R(Y) \simeq R(X)$.

Two schemes $X$ and $Y$ are said to be birationally equivalent or birationally isomorphic if a birational mapping $f: X \rightarrow Y$ exists. A birational morphism is a special case of a birational mapping.

The simplest birational mapping is a monoidal transformation with a non-singular centre. For smooth complete varieties of dimension $\leq 2$ any birational mapping may be represented as the composite of such transformations and their inverses. At the time of writing (1986) this question remains open in the general case.

How to Cite This Entry:
Birational mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_mapping&oldid=46070
This article was adapted from an original article by I.V. DolgachevV.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article