Definitions

# Rational variety

In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K. This is a question on its function field: is it up to isomorphism

$K\left(x_1, dots , x_n\right),$

the field of all rational functions for some set $\left\{x_1, dots, x_n\right\}$ of indeterminates?

## Rationality questions

A rationality question asks whether a given field extension is rational, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as purely transcendental. More precisely, the rationality question for the field extension $K < L$ is this: is $L$ isomorphic to a rational function field over $K$ in the number of indeterminates given by the transcendence degree?

There are several different variations of this question, arising from the way in which the fields $K$ and $L$ are constructed.

For example, let $K$ be a field, and let

$\left\{y_1, dots, y_n \right\}$

be indeterminates over K and let L be the field generated over K by them. Consider a finite group $G$ permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of $L$, typically denoted $L^G$. The rationality question for $K < L^G$ is called Noether's problem and asks if this field of fixed points is or is not a purely transcendental extension of K. In the paper on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to "Noether's problem". (She first mentioned this problem in where she attributed the problem to E. Fischer.) She showed this was true for n=2, 3, or 4. found a counter-example to the Noether's problem, with n=47 and G a cyclic group of order 47.

## Classical results

A celebrated case is Lüroth's problem, which was solved in the nineteenth century, and its generalisations to higher dimensions which lie much deeper. Lüroth's problem concerns subfields L of K(X), the rational functions in the single indeterminate X, for which the degree

[K(X):L]

is finite, and with K algebraically closed. Any such field is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant birational mapping from the projective line to a curve C can only occur when C also has genus 0. That fact can be read off geometrically from the Riemann-Hurwitz formula.

## Unirationality

A unirational variety is one covered by a rational variety, so that on the function field level it has a function field that lies in a pure transcendental field that has finite degree over it. The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo's theorem implies that for complex surfaces unirational implies rational. Zariski found some examples (Zariski surfaces) in characteristic p > 0 that are unirational but not rational. It was first shown by Clemens and Griffiths that a cubic threefold is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an intermediate Jacobian. For the field of complex numbers, Guido Castelnuovo had characterised the unirational and the rational varieties amongst algebraic surfaces by the same criterion: the vanishing of both the arithmetic genus and the second plurigenus.

János Kollár proved in 2000 that a smooth cubic hypersurface is unirational over any field K for which it has a point defined. This is an improvement of many classical results, beginning with the case of cubic surfaces (which are rational varieties over an algebraic closure). Other examples of varieties that are shown to be unirational are many cases of the moduli space of curves .