Rational surface

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.


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"n" for "n" = 0 or "n" ≥ 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 surfacesand greater than 1 for other rational surfaces.

The Picard group is the odd unimodular lattice I1,"n", except for the Hirzebruch surfaces Σ2"m" when it is the even unimodular lattice II1,1.

Hirzebruch surfaces

The Hirzebruch surface Σn is the "P"1 bundle over "P"1associated 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 "P"1×"P"1, and Σ1 is isomorphic to "P"2 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 "P"1). The Picard group is generated by the curve "C" and one of the fibers, and these generators have intersection matrix

:egin{bmatrix}0 & 1 \ 1 & -n end{bmatrix} ,

so the bilinear form is two dimensional unimodular, and is even or odd depending on whether "n" is even or odd.

The Hirzebruch surface Σ"n" ("n" > 1) blown up at a point on the special curve "C" is isomorphic to Σ"n" − 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 "P"2 (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 "P"1 both vanish is rational, but a counterexample (an Enriques surface) was found by Federigo Enriques.

*Zariski, Oscar "On Castelnuovo's criterion of rationality" "p""a" = "P"2 = 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 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 projective plane
* Bordiga surfaces: A degree 6 embedding of the projective plane into "P"4 defined by the quartics through 10 points in general position.
* Veronese surface An embedding of the projective plane into "P"5.
* Steiner surface A surface in "P"4 with singularities which is birational to the projective plane.

ee also

*rational variety
*list of algebraic surfaces


* "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

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • 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. This is a question on its function field: is it up to isomorphism the field of all… …   Wikipedia

  • Rational reconstruction — is a philosophical and linguistic method that systematically translates intuitive knowledge of rules into a logical form. [ * Habermas, Jurgen. (1979). Communication and the Evolution of Society. Toronto: Beacon Press.] In other words, it is an… …   Wikipedia

  • Surface of class VII — In mathematics, surfaces of class VII are non algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self… …   Wikipedia

  • Rational horizon — Horizon Ho*ri zon, n. [F., fr. L. horizon, fr. Gr. ? (sc. ?) the bounding line, horizon, fr. ? to bound, fr. ? boundary, limit.] 1. The line which bounds that part of the earth s surface visible to a spectator from a given point; the apparent… …   The Collaborative International Dictionary of English

  • rational horizon — Horizon Ho*ri zon, n. [F., fr. L. horizon, fr. Gr. ? (sc. ?) the bounding line, horizon, fr. ? to bound, fr. ? boundary, limit.] 1. The line which bounds that part of the earth s surface visible to a spectator from a given point; the apparent… …   The Collaborative International Dictionary of English

  • Rational singularity — In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map :f : Y ightarrow X from a… …   Wikipedia

  • Surface of general type — In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2.These are all algebraic, and in some sense most surfaces are in this class. ClassificationGieseker showed that there is a coarse moduli scheme for… …   Wikipedia

  • Zariski surface — In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p gt; 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface. In particular, all Zariski… …   Wikipedia

  • Cubic surface — A 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 (hence, cubic). Cubic surfaces are del Pezzo… …   Wikipedia

  • Châtelet surface — Some snapshots showing the real points of the Châtelet surface with P(x)=x^3 5*x^2 6*x. Axis: x=red, y=yellow, z=blu In algebraic geometry, a Châtelet surface is a rational surface studied by Châtelet (1959) given by an equation where …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”