# Rational surface

A two-dimensional algebraic variety, defined over an algebraically closed field , whose field of rational functions is a purely transcendental extension of of degree 2. Every rational surface is birationally isomorphic to the projective space .

The geometric genus and the irregularity of a complete smooth rational surface are equal to 0, that is, there are no regular differential 2- or -forms on . Every multiple genus of a smooth complete rational surface is also zero, where is the canonical divisor of the surface . These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface is a rational surface if and only if there is a non-singular rational curve on with index of self-intersection .

With the exception of rational surfaces and ruled surfaces, every algebraic surface is birationally isomorphic to a unique minimal model. In the class of rational surfaces there is a countable set of minimal models. It consists of the projective space and the surfaces (projectivization of two-dimensional vector bundles over the projective line ), , where and . In other words, the surface is a fibration by rational curves over a rational curve with a section which is a smooth rational curve with index of self-intersection . The surface is isomorphic to the direct product , and the surfaces are obtained from by a sequence of elementary transformations (see [1]).

Rational surfaces have a large group of birational transformations (called the group of Cremona transformations).

If the anti-canonical sheaf on a smooth complete rational surface is ample (cf. Ample sheaf), then is called a Del Pezzo surface. The greatest integer such that for some divisor on is called the index of the Del Pezzo surface. The index is equal to 1, 2 or 3 (see [2]). A Del Pezzo surface of index 3 is isomorphic to . For a Del Pezzo surface of index 2, the rational mapping defined by the sheaf gives a birational isomorphism onto a quadric in . Del Pezzo surfaces of index 1 can be obtained by monoidal transformations (cf. Monoidal transformation) of the plane with centres at points in general position, where (see [2]).

#### References

 [1] I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001 [2] V.A. Iskovskii, "Anticanonical models of three-dimensional algebraic varieties" , Current problems in mathematics , 12 , Moscow (1979) pp. 59–157; 239 (In Russian) [3] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001