Unirational

Unirationality

A unirational varietyV over a field K is one dominated by a rational variety, so that its function field K(V) lies in a pure transcendental field of finite type (which can be chosen to be of finite degree over K(V) if K is infinite). The solution of Lüroth's problem shows that for algebraic curves, rational and unirational are the same, and Castelnuovo's theorem implies that for complex surfaces unirational implies rational, because both are characterized by the vanishing of both the arithmetic genus and the second plurigenus. Zariski found some examples (Zariski surfaces) in characteristic p > 0 that are unirational but not rational. Clemens & Griffiths (1972) showed that a cubic three-fold is in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. Their work used an intermediate Jacobian. Iskovskih & Manin (1971) showed that all non-singular quartic threefolds are irrational, though some of them are unirational. Artin & Mumford (1972) found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational.