Rational surface
Encyclopedia
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.
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.
Hodge diamond:
where n is 0 for the projective plane, and 1 for Hirzebruch surface
s
and greater than 1 for other rational surfaces.
The Picard group is the odd unimodular lattice
I1,n, except for the Hirzebruch surface
s Σ2m when it is the even unimodular lattice II1,1.
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. 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 found examples of unirational surfaces (Zariski surface
s) 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
.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a rational surface is a surface birationally equivalent
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
to the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, or in other words a rational variety
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K...
of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques-Kodaira classification
Enriques-Kodaira classification
In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space...
of complex surfaces,
and were the first surfaces to be investigated.
Structure
Every non-singular rational surface can be obtained by repeatedly blowing upBlowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point...
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
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
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 surface
Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .-Definition:The Hirzebruch surfaceΣn is the P1 bundle over P1associated to the sheafO+O.\...
s
and greater than 1 for other rational surfaces.
The Picard group is the odd unimodular lattice
Unimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
I1,n, except for the Hirzebruch surface
Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .-Definition:The Hirzebruch surfaceΣn is the P1 bundle over P1associated to the sheafO+O.\...
s Σ2m when it is the even unimodular lattice II1,1.
Castelnuovo's theorem
Guido CastelnuovoGuido Castelnuovo
Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
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. 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 found examples of unirational surfaces (Zariski surface
Zariski surface
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface...
s) 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
Enriques surface
In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
) was found by Federigo Enriques
Federigo Enriques
Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
.
Examples of rational surfaces
- Bordiga surfaceBordiga surfaceIn algebraic geometry, a Bordiga surface is a certain sort of rational surface of degree 6 in P4.A Bordiga surface is isomorphic to the projective plane blown up in 10 points, the embedding into P4 is given by the 5-dimensional space of quartics passing through the 10 points. White surfaces...
s: A degree 6 embedding of the projective plane into P4 defined by the quartics through 10 points in general position. - Châtelet surfaceChâtelet surfaceIn algebraic geometry, a Châtelet surface is a rational surface studied by given by an equationy^2-az^2=P \, where P has degree 3 or 4. They are conic bundles....
s - Coble surfaceCoble surfaceIn algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system |−K| and non-empty anti-bicanonical linear system |−2K|. An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of...
s - Cubic surfaceCubic surfaceA cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
s 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 surfaceDel Pezzo surfaceIn mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class...
s (Fano surfaces) - Enneper surfaceEnneper surfaceIn mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a surface that can be described parametrically by:It was introduced by Alfred Enneper in connection with minimal surface theory....
- Hirzebruch surfaceHirzebruch surfaceIn mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .-Definition:The Hirzebruch surfaceΣn is the P1 bundle over P1associated to the sheafO+O.\...
s 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 planeProjective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
- Segre surfaceSegre surfaceIn algebraic geometry, a Segre surface, named for Beniamino Segre, is an intersection of two quadrics in 4-dimensional projective space.They are rational surfaces isomorphic to a projective plane blown up in 5 points with no 3 on a line, and are del Pezzo surfaces of degree 4, and have 16 rational...
An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points. - Steiner surfaceSteiner surfaceIn geometry, a branch of mathematics, the Steiner surfaces, discovered by Jakob Steiner, are mappings of the real projective plane into three-dimensional real projective space...
A surface in P4 with singularities which is birational to the projective plane. - White surfaceWhite surfaceIn algebraic geometry, a White surface is one of the rational surfaces in Pn studied by , generalizing cubic surfaces and Bordiga surfaces, which are the cases n = 3 or 4....
s, a generalization of Bordiga surfaces. - Veronese surfaceVeronese surfaceIn mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
An embedding of the projective plane into P5.