Rational surface
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.
Structure
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 Σn for n = 0 or n ≥ 2.
Invariants: The plurigenera are all 0 and the fundamental group is trivial.
Hodge diamond:
| 1 | ||||
|---|---|---|---|---|
| 0 | 0 | |||
| 0 | 1+n | 0 | ||
| 0 | 0 | |||
| 1 |
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.
Hirzebruch surfaces
The Hirzebruch surface Σn is the P1 bundle over P1 associated to the sheaf
- O(0) + O(n).
The notation here means: O(n) is the n-th tensor power of the Serre twist sheaf O(1), the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface Σ0 is isomorphic to P1×P1, and Σ1 is isomorphic to P2 blown up at a point so is not minimal.
Hirzebruch surfaces for n>0 have a special curve C on them given by the projective bundle of O(n). This curve has self intersection number −n, and is the only irreducible curve with negative self intersection number. The only irreducible curves with zero self intersection number are the fibers of the Hirzebruch surface (considered as a fiber bundle over P1). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix
so the bilinear form is two dimensional unimodular, and is even or odd depending on whether n is even or odd.
The Hirzebruch surface Σn (n > 1) blown up at a point on the special curve C is isomorphic to Σn − 1 blown up at a point not on the special curve.
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 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.
This application of Castenuovo's theorem in charactersitic p > 0 is false: Zariski 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.
- Zariski, Oscar On Castelnuovo's criterion of rationality pa = P2 = 0 of an algebraic surface. Illinois J. Math. 2 1958 303--315.
Examples of rational surfaces
- del Pezzo surfaces (Fano surfaces)
- Cubic surfaces Nonsingular cubic surfaces are isomorphic to the projective plane blown up in 6 points, and are Fano surfaces. Named examples include the Fermat cubic, the Cayley cubic, and the Clebsch cubic.
- Hirzebruch surfaces Σn
- P1×P1 The product of two projective lines is the Hirzebruch surface Σ0. It is the only surface with two different rulings.
- The projective plane
- Bordiga surfaces: A degree 6 embedding of the projective plane into P4 defined by the quartics through 10 points in general position.
- Veronese surface An embedding of the projective plane into P5.
- Steiner surface A surface in P4 with singularities which is birational to the projective plane.
See also
References
- Compact Complex Surfaces by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven ISBN 3-540-00832-2
- Complex algebraic surfaces by Arnaud Beauville, ISBN 0-521-49510-5