Rational surface

A two-dimensional algebraic variety, defined over an algebraically closed field $ k $, whose field of rational functions is a purely transcendental extension of $ k $ of degree 2. Every rational surface $ X $ is birationally isomorphic to the projective space $ \mathbf P ^ {2} $.

The geometric genus $ p _ {g} $ and the irregularity $ q $ of a complete smooth rational surface $ X $ are equal to 0, that is, there are no regular differential 2- or 1-forms on $ X $. Every multiple genus $ P _ {n} = \mathop{\rm dim} H ^ {0} ( X , {\mathcal O} _ {X} ( n K _ {X} ) ) $ of a smooth complete rational surface $ X $ is also zero, where $ K _ {X} $ is the canonical divisor of the surface $ X $. These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants $ p _ {g} = q = P _ {2} = 0 $ is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface $ X $ is a rational surface if and only if there is a non-singular rational curve $ C $ on $ X $ with index of self-intersection $ ( C ^ {2} ) _ {X} > 0 $.

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 $ \mathbf P ^ {2} $ and the surfaces $ F _ {n} \simeq P ( {\mathcal L} _ {n} ) $ (projectivization of two-dimensional vector bundles over the projective line $ \mathbf P ^ {1} $), $ {\mathcal L} _ {n} \simeq {\mathcal O} _ {\mathbf P ^ {1} } \oplus {\mathcal O} _ {\mathbf P ^ {1} } ( - n ) $, where $ n \geq 0 $ and $ n \neq 1 $. In other words, the surface $ F _ {n} $ is a fibration by rational curves over a rational curve with a section $ S _ {n} $ which is a smooth rational curve with index of self-intersection $ ( S _ {n} ^ {2} ) _ {F} = - n $. The surface $ F _ {0} $ is isomorphic to the direct product $ \mathbf P ^ {1} \times \mathbf P ^ {1} $, and the surfaces $ F _ {n} $ are obtained from $ F _ {0} $ 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 $ {\mathcal O} _ {X} ( - K _ {X} ) $ on a smooth complete rational surface is ample (cf. Ample sheaf), then $ X $ is called a Del Pezzo surface. The greatest integer $ r > 0 $ such that $ - K _ {X} \sim r D $ for some divisor $ D $ on $ X $ is called the index of the Del Pezzo surface. The index $ r $ is equal to 1, 2 or 3 (see [2]). A Del Pezzo surface of index 3 is isomorphic to $ \mathbf P ^ {2} $. For a Del Pezzo surface $ X $ of index 2, the rational mapping $ {\mathcal O} _ {X} : X \rightarrow \mathbf P ^ {3} $ defined by the sheaf $ {\mathcal O} _ {X} ( D) $ gives a birational isomorphism onto a quadric in $ \mathbf P ^ {3} $. Del Pezzo surfaces of index 1 can be obtained by $ n $ monoidal transformations (cf. Monoidal transformation) of the plane $ \mathbf P ^ {2} $ with centres at points in general position, where $ 1 \leq n \leq 8 $ (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

Comments

If $ X $ is defined over a, not necessarily algebraically closed, field and $ X $ is birationally equivalent to $ \mathbf P _ {k} ^ {2} $ over $ k $, then $ X $ is said to be a $ k $-rational surface.

References

[a1]	A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) MR0485887 Zbl 0394.14014
[a2]	J. Semple, L. Roth, "Introduction to algebraic geometry" , Oxford Univ. Press (1985) MR0814690 Zbl 0576.14001