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.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001