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''birational isomorphism''
 
''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165001.png" /> is said to be a birational mapping if it satisfies one of the following equivalent conditions: 1) there exist dense open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165003.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165004.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165005.png" /> and realizes an isomorphism of subschemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165006.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165008.png" /> are the sets of generic points of the irreducible components of the schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b0165009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650010.png" /> respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650011.png" /> induces a bijective correspondence between the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650012.png" /> and an isomorphism of local rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650013.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650014.png" />.
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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 }  $,  
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$  \{ y _ {j} \} _ {j \in J }  $
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650016.png" /> are irreducible and reduced, the local rings of their generic points become identical with the fields of rational functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650018.png" />, respectively. In such a case the birational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650019.png" /> induces, in accordance with condition 2), an isomorphism of the fields of rational functions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650020.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650022.png" /> are said to be birationally equivalent or birationally isomorphic if a birational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650023.png" /> exists. A [[Birational morphism|birational morphism]] is a special case of a birational mapping.
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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|birational morphism]] is a special case of a birational mapping.
  
The simplest birational mapping is a [[Monoidal transformation|monoidal transformation]] with a non-singular centre. For smooth complete varieties of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016500/b01650024.png" /> 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.
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The simplest birational mapping is a [[Monoidal transformation|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====
 
====References====

Latest revision as of 10:59, 29 May 2020


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
How to Cite This Entry:
Birational mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birational_mapping&oldid=34205
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