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A generalization of the concept of a [[Rational function|rational function]] on an [[Algebraic variety|algebraic variety]]. Namely, a rational mapping from an irreducible algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776101.png" /> to an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776102.png" /> (both defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776103.png" />) is an equivalent class of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776105.png" /> is a non-empty open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776107.png" /> is a morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776108.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r0776109.png" />. Two pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761011.png" /> are said to be equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761013.png" /> coincide on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761014.png" />. In particular, a rational mapping from a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761015.png" /> to an affine line is a rational function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761016.png" />. For every rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761017.png" /> there is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761019.png" /> for all equivalent pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761021.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761022.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761023.png" />. The open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761024.png" /> is called the domain of regularity of the rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761026.png" /> is the image of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761027.png" /> (written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761028.png" />) under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761029.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761030.png" /> is a rational mapping of algebraic varieties and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761031.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761033.png" /> determines an imbedding of fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761034.png" />. Conversely, an imbedding of the fields of rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761035.png" /> determines a rational mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761037.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761038.png" /> induces an isomorphism of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761040.png" /> of rational functions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761041.png" /> is called a birational mapping.
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The set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761042.png" /> at which the rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761043.png" /> is not regular has codimension 1, in general. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761044.png" /> is complete and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761045.png" /> is smooth and irreducible, then this set has codimension at least 2. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761047.png" /> are complete irreducible varieties over an algebraically closed field of characteristic 0, then the rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761048.png" /> can be included in a commutative diagram (see [[#References|[2]]]):
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A generalization of the concept of a [[Rational function|rational function]] on an [[Algebraic variety|algebraic variety]]. Namely, a rational mapping from an irreducible algebraic variety  $  X $
 +
to an algebraic variety  $  Y $(
 +
both defined over a field  $  k $)
 +
is an equivalent class of pairs  $  ( U , \phi _ {U} ) $,  
 +
where  $  U $
 +
is a non-empty open subset of  $  X $
 +
and  $  \phi _ {U} $
 +
is a morphism from  $  U $
 +
to  $  Y $.  
 +
Two pairs  $  ( U , \phi _ {U} ) $
 +
and  $  ( V , \psi _ {V} ) $
 +
are said to be equivalent if $  \phi _ {U} $
 +
and  $  \psi _ {V} $
 +
coincide on  $  U \cap V $.  
 +
In particular, a rational mapping from a variety  $  X $
 +
to an affine line is a rational function on  $  X $.  
 +
For every rational mapping  $  \phi : X \rightarrow Y $
 +
there is a pair  $  ( \widetilde{U}  , \phi _ {\widetilde{U}  }  ) $
 +
such that  $  U \subseteq \widetilde{U}  $
 +
for all equivalent pairs  $  ( U , \phi _ {U} ) $
 +
and $  \phi _ {U} $
 +
is the restriction of  $  \phi _ {\widetilde{U}  }  $
 +
to  $  U $.  
 +
The open subset  $  \widetilde{U}  $
 +
is called the domain of regularity of the rational mapping $  \phi $,
 +
and  $  \phi ( \widetilde{U}  ) $
 +
is the image of the variety  $  X $(
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written  $  \phi ( X) $)  
 +
under  $  \phi $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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If  $  \phi : X \rightarrow Y $
 +
is a rational mapping of algebraic varieties and  $  \phi ( X) $
 +
is dense in  $  Y $,
 +
then  $  \phi $
 +
determines an imbedding of fields,  $  \phi  ^ {*} : k ( Y) \rightarrow k ( Y) $.
 +
Conversely, an imbedding of the fields of rational functions  $  \phi  ^ {*} : k ( Y) \rightarrow k ( Y) $
 +
determines a rational mapping from  $  X $
 +
to  $  Y $.  
 +
If  $  \phi $
 +
induces an isomorphism of the fields  $  k ( X) $
 +
and  $  k ( Y) $
 +
of rational functions, then  $  \phi $
 +
is called a birational mapping.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761051.png" /> are morphisms of an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761053.png" /> is a composite of monoidal transformations (cf. [[Monoidal transformation|Monoidal transformation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761054.png" /> is a [[Birational mapping|birational mapping]] of complete non-singular surfaces, then there exists a diagram (*) in which both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761056.png" /> are composites of monoidal transformations with non-singular centres (Zariski's theorem), that is, every birational mapping of complete non-singular surface decomposes into monoidal transformations with non-singular centres and their inverses. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077610/r07761057.png" />, the question of whether every birational mapping can be decomposed in this way is open (1990).
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The set of points of  $  X $
 +
at which the rational mapping  $  \phi : X \rightarrow Y $
 +
is not regular has codimension 1, in general. But if  $  Y $
 +
is complete and  $  X $
 +
is smooth and irreducible, then this set has codimension at least 2. If  $  X $
 +
and  $  Y $
 +
are complete irreducible varieties over an algebraically closed field of characteristic 0, then the rational mapping  $  \phi :  X \rightarrow Y $
 +
can be included in a commutative diagram (see [[#References|[2]]]):
 +
 
 +
$$ \tag{* }
 +
 
 +
where  $  \eta $,
 +
$  f $
 +
are morphisms of an algebraic variety $  Z $
 +
and $  \eta $
 +
is a composite of monoidal transformations (cf. [[Monoidal transformation|Monoidal transformation]]). If $  \phi : X \rightarrow Y $
 +
is a [[Birational mapping|birational mapping]] of complete non-singular surfaces, then there exists a diagram (*) in which both $  f $
 +
and $  \eta $
 +
are composites of monoidal transformations with non-singular centres (Zariski's theorem), that is, every birational mapping of complete non-singular surface decomposes into monoidal transformations with non-singular centres and their inverses. In the case when $  \mathop{\rm dim} X \geq  3 $,  
 +
the question of whether every birational mapping can be decomposed in this way is open (1990).
  
 
====References====
 
====References====

Revision as of 08:09, 6 June 2020


A generalization of the concept of a rational function on an algebraic variety. Namely, a rational mapping from an irreducible algebraic variety $ X $ to an algebraic variety $ Y $( both defined over a field $ k $) is an equivalent class of pairs $ ( U , \phi _ {U} ) $, where $ U $ is a non-empty open subset of $ X $ and $ \phi _ {U} $ is a morphism from $ U $ to $ Y $. Two pairs $ ( U , \phi _ {U} ) $ and $ ( V , \psi _ {V} ) $ are said to be equivalent if $ \phi _ {U} $ and $ \psi _ {V} $ coincide on $ U \cap V $. In particular, a rational mapping from a variety $ X $ to an affine line is a rational function on $ X $. For every rational mapping $ \phi : X \rightarrow Y $ there is a pair $ ( \widetilde{U} , \phi _ {\widetilde{U} } ) $ such that $ U \subseteq \widetilde{U} $ for all equivalent pairs $ ( U , \phi _ {U} ) $ and $ \phi _ {U} $ is the restriction of $ \phi _ {\widetilde{U} } $ to $ U $. The open subset $ \widetilde{U} $ is called the domain of regularity of the rational mapping $ \phi $, and $ \phi ( \widetilde{U} ) $ is the image of the variety $ X $( written $ \phi ( X) $) under $ \phi $.

If $ \phi : X \rightarrow Y $ is a rational mapping of algebraic varieties and $ \phi ( X) $ is dense in $ Y $, then $ \phi $ determines an imbedding of fields, $ \phi ^ {*} : k ( Y) \rightarrow k ( Y) $. Conversely, an imbedding of the fields of rational functions $ \phi ^ {*} : k ( Y) \rightarrow k ( Y) $ determines a rational mapping from $ X $ to $ Y $. If $ \phi $ induces an isomorphism of the fields $ k ( X) $ and $ k ( Y) $ of rational functions, then $ \phi $ is called a birational mapping.

The set of points of $ X $ at which the rational mapping $ \phi : X \rightarrow Y $ is not regular has codimension 1, in general. But if $ Y $ is complete and $ X $ is smooth and irreducible, then this set has codimension at least 2. If $ X $ and $ Y $ are complete irreducible varieties over an algebraically closed field of characteristic 0, then the rational mapping $ \phi : X \rightarrow Y $ can be included in a commutative diagram (see [2]):

$$ \tag{* }

where $ \eta $, $ f $ are morphisms of an algebraic variety $ Z $ and $ \eta $ is a composite of monoidal transformations (cf. Monoidal transformation). If $ \phi : X \rightarrow Y $ is a birational mapping of complete non-singular surfaces, then there exists a diagram (*) in which both $ f $ and $ \eta $ are composites of monoidal transformations with non-singular centres (Zariski's theorem), that is, every birational mapping of complete non-singular surface decomposes into monoidal transformations with non-singular centres and their inverses. In the case when $ \mathop{\rm dim} X \geq 3 $, the question of whether every birational mapping can be decomposed in this way is open (1990).

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I" Ann. of Math. , 79 : 1–2 (1964) pp. 109–326 MR0199184 Zbl 0122.38603
How to Cite This Entry:
Rational mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_mapping&oldid=34204
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article