Rational Variety

A rational variety over a field is an integral algebraic variety birationally equivalent over to projective space. Equivalently, dense open subsets of the two varieties are isomorphic. A smooth projective variety over an algebraically closed field that is rational is also a stably rational variety and a retract rational variety (Hotchkiss 2026).

See also

Algebraic Variety, Birational Transformation, Projective Space, Retract Rational Variety, Stably Rational Variety

Explore with Wolfram|Alpha

More things to try:

References

Hotchkiss, J. "The Rationality Problem for Hypersurfaces of Degree at Least Five." 7 Oct 2026. https://arxiv.org/abs/2610.10509.

Cite this as:

Weisstein, Eric W. "Rational Variety." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RationalVariety.html

Subject classifications