algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a rational surface is a surface

birationally equivalent In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...

to the

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 (namely, parallel lines) that do ...

, or in other words a

rational variety In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), the ...

of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the

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. For most of the classes the modu ...

of complex surfaces, and were the first surfaces to be investigated.

Structure

Every non-singular rational surface can be obtained by repeatedly

blowing 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 th ...

a minimal rational surface. The minimal rational surfaces are the projective plane and the

Hirzebruch surface In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by . Definition The Hirzebruch surface \Sigma_n is the \mathbb^1-bundle, called a Projective bundle, over \mathbb^1 associated to the sheaf\mathca ...

s Σ_''r'' for ''r'' = 0 or ''r'' ≥ 2. Invariants: The

plurigenera In mathematics, the pluricanonical ring of an algebraic variety ''V'' (which is non-singular), or of a complex manifold, is the graded ring :R(V,K)=R(V,K_V) \, of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (for ...

are all 0 and the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

is trivial.

Hodge diamond Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address ...

: where ''n'' is 0 for the projective plane, and 1 for

s and greater than 1 for other rational surfaces. The

Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...

is the odd

unimodular lattice In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...

I_1,''n'', except for the

s Σ_2''m'' when it is the even unimodular lattice II_1,1.

Castelnuovo's theorem

Guido Castelnuovo Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also signi ...

proved that any complex surface such that ''q'' and ''P''₂ (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 In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), t ...

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 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. In particu ...

s) that are not rational. At one time it was unclear whether a complex surface such that ''q'' and ''P''₁ both vanish is rational, but a counterexample (an

Enriques surface In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity ''q'' = 0 and the canonical line bundle ''K'' is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex n ...

) was found by

Federigo Enriques Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) 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 ...

Examples of rational surfaces

* Bordiga surfaces: A degree 6 embedding of the projective plane into ''P''⁴ defined by the quartics through 10 points in general position. *

Châtelet surface In algebraic geometry, a Châtelet surface is a rational surface studied by given by an equation :y^2-az^2=P(x), \, where ''P'' has degree 3 or 4. They are conic bundle In algebraic geometry, a conic bundle is an algebraic variety that app ...

s *

Coble surface In 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 th ...

s *

Cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...

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 In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by : x^3 + y^3 + z^3 = 1. \ Methods of algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynom ...

, the Cayley cubic surface, and the

Clebsch diagonal surface In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral sur ...

. *

del Pezzo surface In 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. They are in some sense the opposite of surfaces of general ...

s (Fano surfaces) *

Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...

s Σ_''n'' * ''P''¹×''P''¹ The product of two projective lines is the Hirzebruch surface Σ₀. It is the only surface with two different rulings. * The

Segre surface In algebraic geometry, a Segre surface, studied by and , 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 Pez ...

An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points. *

Steiner surface In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; ...

A surface in ''P''⁴ with singularities which is birational to the projective plane. *

White surface In algebraic geometry, a White surface is one of the rational surfaces in ''P'n'' studied by , generalizing cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic s ...

s, a generalization of Bordiga surfaces. *

Veronese surface In 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 Giu ...

An embedding of the projective plane into ''P''⁵.

References

* * *{{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , title=On Castelnuovo's criterion of rationality p_a = P₂ = 0 of an algebraic surface , mr= 0099990 , year=1958 , journal=Illinois Journal of Mathematics , issn=0019-2082 , volume=2 , pages=303–315

External links

Le Superficie Algebriche
A tool to visually study the geography of (minimal) complex algebraic smooth surfaces Complex surfaces Birational geometry Algebraic surfaces

Structure

Castelnuovo's theorem

Examples of rational surfaces

See also

References

External links