View source for Rational variety

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An [[Algebraic variety|algebraic variety]]  $  X $, 
defined over an [[Algebraically closed field|algebraically closed field]]  $  k $, 
whose field of rational functions  $  k ( X) $
is isomorphic to a purely [[Transcendental extension|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|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|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|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|Lüroth problem]].

====References====
<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>
<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>
</table>