Rational variety

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