Rational Surface - Castelnuovo's Theorem

Castelnuovo's Theorem

Guido Castelnuovo proved that any complex surface such that q and P₂ (the irregularity and second plurigenus) both vanish is rational. This is used in the Enriques-Kodaira classification to identify the rational surfaces. Zariski (1958) proved that Castelnuovo's theorem also holds over fields of positive characteristic.

Castelnuovo's theorem also implies that any unirational complex surface is rational, because if a complex surface is unirational then its irregularity and plurigenera are bounded by those of a rational surface and are therefore all 0, so the surface is rational. Most unirational complex varieties of dimension 3 or larger are not rational. In characteristic p > 0 Zariski (1958) found examples of unirational surfaces (Zariski surfaces) that 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.