In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to
K ( U 1 , … , U d ) , {\displaystyle K(U_{1},\dots ,U_{d}),}
the field of all rational functions for some set { U 1 , … , U d } {\displaystyle \{U_{1},\dots ,U_{d}\}} of indeterminates, where d is the dimension of the variety.
Rationality and parameterization Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in K [ X 1 , … , X n ] {\displaystyle K[X_{1},\dots ,X_{n}]} . If V is rational, then there are n + 1 polynomials g0, ..., gn in K ( U 1 , … , U d ) {\displaystyle K(U_{1},\dots ,U_{d})} such that f i ( g 1 / g 0 , … , g n / g 0 ) = 0. {\displaystyle f_{i}(g_{1}/g_{0},\ldots ,g_{n}/g_{0})=0.} In other words, we have a rational parameterization x i = g i g 0 ( u 1 , … , u d ) {\displaystyle x_{i}={\frac {g_{i}}{g_{0}}}(u_{1},\ldots ,u_{d})} of the variety. Conversely, such a rational parameterization induces a field homomorphism of the field of functions of V into K ( U 1 , … , U d ) {\displaystyle K(U_{1},\dots ,U_{d})} . But this homomorphism is not necessarily onto. If such a parameterization exists, the variety is said to be unirational. Lüroth's theorem (see below) implies that unirational curves are rational. Castelnuovo's theorem implies also that, in characteristic zero, every unirational surface is rational.
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 {\displaystyle K\subset L} is this: is L {\displaystyle L} isomorphic to a rational function field over K {\displaystyle 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 {\displaystyle K} and L {\displaystyle L} are constructed. For example, let K {\displaystyle K} be a field, and let
{ y 1 , … , y n } {\displaystyle \{y_{1},\dots ,y_{n}\}}
be indeterminates over K and let L be the field generated over K by them. Consider a finite group G {\displaystyle G} permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of L {\displaystyle L} , typically denoted L G {\displaystyle L^{G}} . The rationality question for K ⊂ L G {\displaystyle K\subset 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 (Noether 1918) 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 (Noether 1913) where she attributed the problem to E. Fischer.) She showed this was true for n = 2, 3, or 4. R. G. Swan (1969) found a counter-example to the Noether's problem, with n = 47 and G a cyclic group of order 47.
Lüroth's theorem
A celebrated case is Lüroth's problem, which Jacob Lüroth solved in the nineteenth century. Lüroth's problem concerns subextensions L of K(X), the rational functions in the single indeterminate X. Any such field is either equal to K or is also rational, i.e. L = K(F) for some rational function F. In geometrical terms this states that a non-constant rational map 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.
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