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A two-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776501.png" />, whose field of rational functions is a purely [[Transcendental extension|transcendental extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776502.png" /> of degree 2. Every rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776503.png" /> is birationally isomorphic to the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776504.png" />.
 
A two-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776501.png" />, whose field of rational functions is a purely [[Transcendental extension|transcendental extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776502.png" /> of degree 2. Every rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776503.png" /> is birationally isomorphic to the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776504.png" />.
  
The [[Geometric genus|geometric genus]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776505.png" /> and the [[Irregularity|irregularity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776506.png" /> of a complete smooth rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776507.png" /> are equal to 0, that is, there are no regular differential 2- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776508.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776509.png" />. Every multiple genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765010.png" /> of a smooth complete rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765011.png" /> is also zero, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765012.png" /> is the canonical divisor of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765013.png" />. These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765014.png" /> is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765015.png" /> is a rational surface if and only if there is a non-singular rational curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765017.png" /> with index of self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765018.png" />.
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The [[Geometric genus|geometric genus]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776505.png" /> and the [[Irregularity|irregularity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776506.png" /> of a complete smooth rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776507.png" /> are equal to 0, that is, there are no regular differential 2- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776508.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r0776509.png" />. Every multiple genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765010.png" /> of a smooth complete rational surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765011.png" /> is also zero, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765012.png" /> is the [[canonical divisor]] of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765013.png" />. These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765014.png" /> is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765015.png" /> is a rational surface if and only if there is a non-singular rational curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765017.png" /> with index of self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765018.png" />.
  
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765019.png" /> and the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765020.png" /> (projectivization of two-dimensional vector bundles over the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765021.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765024.png" />. In other words, the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765025.png" /> is a fibration by rational curves over a rational curve with a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765026.png" /> which is a smooth rational curve with index of self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765027.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765028.png" /> is isomorphic to the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765029.png" />, and the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765030.png" /> are obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765031.png" /> by a sequence of elementary transformations (see [[#References|[1]]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765019.png" /> and the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765020.png" /> (projectivization of two-dimensional vector bundles over the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765021.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765024.png" />. In other words, the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765025.png" /> is a fibration by rational curves over a rational curve with a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765026.png" /> which is a smooth rational curve with index of self-intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765027.png" />. The surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765028.png" /> is isomorphic to the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765029.png" />, and the surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765030.png" /> are obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077650/r07765031.png" /> by a sequence of elementary transformations (see [[#References|[1]]]).

Revision as of 18:21, 19 October 2017

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


Comments

If is defined over a, not necessarily algebraically closed, field and is birationally equivalent to over , then is said to be a -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=42126
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article