Rational Variety - Rationality and Parameterization

Rationality and Parameterization

Let V be an affine algebraic variety of dimension d defined by a prime ideal I=⟨f₁, ..., f_k⟩ in . If V is rational, then there are n+1 polynomials g₀, ..., g_n in such that In order words, we have a rational parameterization of the variety.

Conversely, such a rational parameterization induces a field homomorphism of the field of functions of V into . But this homomorphism is not necessarily onto. If such a parameterization exists, the variety is said 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.