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 Σ_r for r = 0 or r ≥ 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 I_1,n, except for the Hirzebruch surfaces Σ_2m when it is the even unimodular lattice II_1,1.

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.

Examples of rational surfaces

Bordiga surfaces: A degree 6 embedding of the projective plane into P⁴ defined by the quartics through 10 points in general position.
Châtelet surfaces
Coble 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 surface, and the Clebsch diagonal surface.
del Pezzo surfaces (Fano surfaces)
Enneper surface
Hirzebruch surfaces Σ_n
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
Segre surface An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points.
Steiner surface A surface in P⁴ with singularities which is birational to the projective plane.
White surfaces, a generalization of Bordiga surfaces.
Veronese surface An embedding of the projective plane into P⁵.

References

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314
Zariski, Oscar (1958), "On Castelnuovo's criterion of rationality p_a = P₂ = 0 of an algebraic surface", Illinois Journal of Mathematics, 2: 303–315, ISSN 0019-2082, MR 0099990

External links

Le Superficie Algebriche: A tool to visually study the geography of (minimal) complex algebraic smooth surfaces