In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two.
Guido Castelnuovo proved that any complex surface such that q and P2 (the irregularity and second plurigenus) both vanish is rational.
Zariski (1958) proved that Castelnuovo's theorem also holds over fields of positive characteristic.
Most unirational complex varieties of dimension 3 or larger are not rational.
At one time it was unclear whether a complex surface such that q and P1 both vanish is rational, but a counterexample (an Enriques surface) was found by Federigo Enriques.