Difference between revisions of "Rational mapping"

Revision as of 14:53, 7 June 2020

A generalization of the concept of a rational function on an algebraic variety. Namely, a rational mapping from an irreducible algebraic variety to an algebraic variety (both defined over a field ) is an equivalent class of pairs , where is a non-empty open subset of and is a morphism from to . Two pairs and are said to be equivalent if and coincide on . In particular, a rational mapping from a variety to an affine line is a rational function on . For every rational mapping there is a pair such that for all equivalent pairs and is the restriction of to . The open subset is called the domain of regularity of the rational mapping , and is the image of the variety (written ) under .

If is a rational mapping of algebraic varieties and is dense in , then determines an imbedding of fields, . Conversely, an imbedding of the fields of rational functions determines a rational mapping from to . If induces an isomorphism of the fields and of rational functions, then is called a birational mapping.

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

(*)

where , are morphisms of an algebraic variety and is a composite of monoidal transformations (cf. Monoidal transformation). If is a birational mapping of complete non-singular surfaces, then there exists a diagram (*) in which both and 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 , 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=49390

This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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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" />.
-r0776101.png
-$#A+1 = 57 n = 0
-$#C+1 = 57 : ~/encyclopedia/old_files/data/R077/R.0707610 Rational mapping
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
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-{{TEX|auto}}
+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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-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 $
+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]]]):
-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 $
+<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>
-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 $
+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).
-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).
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Difference between revisions of "Rational mapping"

Revision as of 14:53, 7 June 2020

References