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An [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776601.png" />, defined over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776602.png" />, whose field of rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776603.png" /> is isomorphic to a purely [[Transcendental extension|transcendental extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776604.png" /> of finite degree. In other words, a rational variety is an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776605.png" /> that is birationally isomorphic to a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776606.png" />.
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A complete smooth rational variety possesses the following birational invariants. The dimensions of all spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776607.png" /> of regular differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776608.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r0776609.png" /> are equal to 0. In addition, the multiple genus
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<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/r077660/r07766010.png" /></td> </tr></table>
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An [[Algebraic variety|algebraic variety]]  $  X $,
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defined over an [[Algebraically closed field|algebraically closed field]]  $  k $,
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whose field of rational functions  $  k ( X) $
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is isomorphic to a purely [[Transcendental extension|transcendental extension]] of  $  k $
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of finite degree. In other words, a rational variety is an algebraic variety  $  X $
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that is birationally isomorphic to a projective space  $  \mathbf P  ^ {n} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766011.png" /> is the canonical divisor of the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766012.png" />, that is, the [[Kodaira dimension|Kodaira dimension]] of the rational variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766013.png" /> is equal to 0.
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A complete smooth rational variety possesses the following birational invariants. The dimensions of all spaces  $  H  ^ {0} ( X , \Omega _ {X}  ^ {k} ) $
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of regular differential  $  k $-
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forms on  $  X $
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are equal to 0. In addition, the multiple genus
  
In low dimension the above invariants uniquely distinguish the class of rational varieties among all algebraic varieties. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766014.png" /> and the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766015.png" /> is equal to 0, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766016.png" /> is a [[Rational curve|rational curve]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766017.png" />, the arithmetic genus
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$$
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P _ {n}  =   \mathop{\rm dim} _ {k} \
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H  ^ {0} ( X , {\mathcal O} _ {X} ( n K _ {X} ) )  = 0 \ \
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\textrm{ for }  n > 0 ,
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$$
  
<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/r077660/r07766018.png" /></td> </tr></table>
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where  $  K _ {X} $
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is the [[canonical divisor]] of the algebraic variety  $  X $,
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that is, the [[Kodaira dimension|Kodaira dimension]] of the rational variety  $  X $
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is equal to 0.
  
and the multiple genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766020.png" /> is a [[Rational surface|rational surface]]. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077660/r07766021.png" />, there is no good criterion of rationality, due to the negative solution of the [[Lüroth problem|Lüroth problem]].
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In low dimension the above invariants uniquely distinguish the class of rational varieties among all algebraic varieties. Thus, if  $  \mathop{\rm dim} _ {k}  X = 1 $
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and the genus of  $  X $
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is equal to 0, then $  X $
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is a [[Rational curve|rational curve]]. If  $  \mathop{\rm dim} _ {k}  X = 2 $,  
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the arithmetic genus
  
====References====
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$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
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p _ {a}  =   \mathop{\rm dim} _ {k} \
 
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H  ^ {0} ( X , \Omega _ {X}  ^ {2} ) -
 
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\mathop{\rm dim} _ {k} H  ^ {0} ( X , \Omega _ {X}  ^ {1} )  =  0
 
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$$
====Comments====
 
  
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and the multiple genus  $  P _ {2} = 0 $,
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then  $  X $
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is a [[Rational surface|rational surface]]. However, if  $  \mathop{\rm dim} _ {k}  X \geq  3 $,
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there is no good criterion of rationality, due to the negative solution of the [[Lüroth problem|Lüroth problem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Beauville, J.-L. Colliot-Hélène, J.J. Sansuc, P. Swinnerton-Dyer, "Variétés stablement rationelles non-rationelles" ''Ann. of Math.'' , '''121''' (1985) pp. 283–318</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Beauville, J.-L. Colliot-Thélène, J.J. Sansuc, P. Swinnerton-Dyer, "Variétés stablement rationelles non-rationelles" ''Ann. of Math.'' , '''121''' (1985) pp. 283–318</TD></TR>
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</table>

Latest revision as of 07:33, 13 November 2023


An algebraic variety $ X $, defined over an algebraically closed field $ k $, whose field of rational functions $ k ( X) $ is isomorphic to a purely transcendental extension of $ k $ of finite degree. In other words, a rational variety is an algebraic variety $ X $ that is birationally isomorphic to a projective space $ \mathbf P ^ {n} $.

A complete smooth rational variety possesses the following birational invariants. The dimensions of all spaces $ H ^ {0} ( X , \Omega _ {X} ^ {k} ) $ of regular differential $ k $- forms on $ X $ are equal to 0. In addition, the multiple genus

$$ P _ {n} = \mathop{\rm dim} _ {k} \ H ^ {0} ( X , {\mathcal O} _ {X} ( n K _ {X} ) ) = 0 \ \ \textrm{ for } n > 0 , $$

where $ K _ {X} $ is the canonical divisor of the algebraic variety $ X $, that is, the Kodaira dimension of the rational variety $ X $ is equal to 0.

In low dimension the above invariants uniquely distinguish the class of rational varieties among all algebraic varieties. Thus, if $ \mathop{\rm dim} _ {k} X = 1 $ and the genus of $ X $ is equal to 0, then $ X $ is a rational curve. If $ \mathop{\rm dim} _ {k} X = 2 $, the arithmetic genus

$$ p _ {a} = \mathop{\rm dim} _ {k} \ H ^ {0} ( X , \Omega _ {X} ^ {2} ) - \mathop{\rm dim} _ {k} H ^ {0} ( X , \Omega _ {X} ^ {1} ) = 0 $$

and the multiple genus $ P _ {2} = 0 $, then $ X $ is a rational surface. However, if $ \mathop{\rm dim} _ {k} X \geq 3 $, there is no good criterion of rationality, due to the negative solution of the Lüroth problem.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1] A. Beauville, J.-L. Colliot-Thélène, J.J. Sansuc, P. Swinnerton-Dyer, "Variétés stablement rationelles non-rationelles" Ann. of Math. , 121 (1985) pp. 283–318
How to Cite This Entry:
Rational variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_variety&oldid=23952
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article