# 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