Unirational variety

An algebraic variety $X$ over a field $k$ into which there exists a rational mapping from a projective space, $\phi : \mathbf{P}^n \rightarrow X$, such that $\phi(\mathbf{P}^n)$ is dense in $X$ and the extension of the field of rational functions $k(\mathbf{P}^n)/k(X)$ is separable. In other words, $k(X)$ has a separable extension which is purely transcendental (cf. Transcendental extension).

Unirational varieties are close to rational varieties (cf. Rational variety), e.g. on a unirational variety there are no regular differential forms, $H^0(X,\Omega_X^p) = 0$ for $p \ge 1$. The problem of the coincidence of rational and unirational varieties is called the Lüroth problem; the answer is negative in general, although all unirational algebraic curves are rational.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1]	János Kollár, Karen E. Smith, Alessio Corti, "Rational and Nearly Rational Varieties" , Cambridge Studies in Advanced Mathematics 92, Cambridge University Press (2004) ISBN 0-521-83207-1