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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 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''=⟨''f''1, ..., ''f''''k''⟩ in . If ''V'' is rational, then there are ''n''+1 polynomials ''g''0, ..., ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「rational variety」の詳細全文を読む スポンサード リンク
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