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