Castelnuovo's theorem
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. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to be investigated.
Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σr for r = 0 or r ≥ 2.
Invariants: The plurigenera are all 0 and the fundamental group is trivial.
where n is 0 for the projective plane, and 1 for Hirzebruch surfaces and greater than 1 for other rational surfaces.
The Picard group is the odd unimodular lattice I1,n, except for the Hirzebruch surfaces Σ2m when it is the even unimodular lattice II1,1.
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.
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