Rational varieties

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

the field of all rational functions for some set ${\displaystyle \{U_{1},\dots ,U_{d}\}}$ of indeterminates, where d is the dimension of the variety.

Let V be an affine algebraic variety of dimension d defined by a prime ideal I=⟨f₁, ..., f_k⟩ in ${\displaystyle K[X_{1},\dots ,X_{n}]}$. If V is rational, then there are n+1 polynomials g₀, ..., g_n in ${\displaystyle K(U_{1},\dots ,U_{d})}$ such that ${\displaystyle f_{i}(g_{1}/g_{0},\ldots ,g_{n}/g_{0})=0.}$ In order words, we have a rational parameterization ${\displaystyle x_{i}={\frac {g_{i}}{g_{0}}}(u_{1},\ldots ,u_{d})}$ of the variety.

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