An algebraic variety , defined over an algebraically closed field , whose field of rational functions is isomorphic to a purely transcendental extension of of finite degree. In other words, a rational variety is an algebraic variety that is birationally isomorphic to a projective space .
A complete smooth rational variety possesses the following birational invariants. The dimensions of all spaces of regular differential -forms on are equal to 0. In addition, the multiple genus
In low dimension the above invariants uniquely distinguish the class of rational varieties among all algebraic varieties. Thus, if and the genus of is equal to 0, then is a rational curve. If , the arithmetic genus
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Rational variety. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rational_variety&oldid=42127