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Rational surface

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A two-dimensional algebraic variety, defined over an algebraically closed field , 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
How to Cite This Entry:
Rational surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_surface&oldid=52156
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article