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| In [[mathematics]], an '''algebraic surface''' is an [[algebraic variety]] of [[dimension of an algebraic variety|dimension]] two. In the case of geometry over the field of [[complex number]]s, an algebraic surface has complex dimension two (as a [[complex manifold]], when it is [[non-singular]]) and so of dimension four as a [[smooth manifold]].
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| The theory of algebraic surfaces is much more complicated than that of [[algebraic curve]]s (including the [[compact space|compact]] [[Riemann surface]]s, which are genuine [[surface]]s of (real) dimension two). Many results were obtained, however, in the [[Italian school of algebraic geometry]], and are up to 100 years old.
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| == classification by the Kodira dimension ==
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| In the case of dimension one varieties are classified by only the [[genus|topological genus]], but dimension two, the difference between the [[arithmetic genus]] <math>p_a</math> and the geometric genus <math>p_g</math> turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the [[Irregularity of a surface|irregularity]] for the classification of them. Let's summerize the results. (in detail, for each kind of surfaces refer to each redirections)
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| Examples of algebraic surfaces include (κ is the [[Kodaira dimension]]):
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| * κ=−∞: the [[complex projective plane|projective plane]], [[quadric]]s in '''P'''<sup>3</sup>, [[cubic surface]]s, [[Veronese surface]], [[del Pezzo surface]]s, [[ruled surface]]s
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| * κ=0 : [[K3 surface]]s, [[abelian surface]]s, [[Enriques surface]]s, [[hyperelliptic surface]]s
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| * κ=1: [[Elliptic surface]]s
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| * κ=2: [[surface of general type|surfaces of general type]].
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| For more examples see the [[list of algebraic surfaces]].
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| The first five examples are in fact [[birationally equivalent]]. That is, for example, a cubic surface has a [[function field of an algebraic variety|function field]] isomorphic to that of the [[projective plane]], being the [[rational function]]s in two indeterminates. The cartesian product of two curves also provides examples.
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| == birational geometry of surfaces ==
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| The [[birational geometry]] of algebraic surfaces is rich, because of [[blowing up]] (also known as a [[monoidal transformation]]); under which a point is replaced by the ''curve'' of all limiting tangent directions coming into it (a [[projective line]]). Certain curves may also be blown ''down'', but there is a restriction (self-intersection number must be −1).
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| == properties ==
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| [[ample line bundle#Intersection theorem|'''Nakai criterion''']] says that:
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| :A Divisor ''D'' on a surface ''S'' is ample if and only if ''D<sup>2</sup> > 0'' and for all irreducible curve ''C'' on ''S'' ''D•C > 0.
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| Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let <math>\mathcal{D}(S)</math> be the abelian group consisting of all the divisors on ''S''. Then due to the [[intersection number|intersection theorem]]
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| :<math>\mathcal{D}(S)\times\mathcal{D}(S)\rightarrow\mathbb{Z}:(X,Y)\mapsto X\cdot Y</math> | |
| is viewed as a [[quadratic form]]. Let
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| :<math>\mathcal{D}_0(S):=\{D\in\mathcal{D}(S)|D\cdot X=0,\text{for all } X\in\mathcal{D}(S)\}</math>
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| then <math>\mathcal{D}/\mathcal{D}_0(S):=Num(S)</math> becomes to be a '''numerical equivalent class group''' of ''S'' and
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| :<math>Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E</math>
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| also becomes to be a quadratic form on <math>Num(S)</math>, where <math>\bar{D}</math> is the image of a divisor ''D'' on ''S''. (In the bellow the image <math>\bar{D}</math> is abbreviated with ''D''.)
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| For an ample bundle ''H'' on ''S'' the definition
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| :<math>\{H\}^\perp:=\{D\in Num(S)|D\cdot H=0\}.</math>
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| leads the '''Hodge index theorem''' of the surface version.
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| :for <math>D\in\{\{H\}^\perp|D\ne0\}, D\cdot D < 0</math>, i.e. <math>\{H\}^\perp</math> is a negative definite quadratic form.
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| This theorem is proved by using the Nakai criterion and the Riemann-Roch theorem for surfaces. For all the divisor in <math>\{H\}^\perp</math> this theorem is true. This theorem is not only the tool for the research of surfaces but also used for the proof of the [[Weil conjecture]] by Deligne because it is true on the algebraically closed field.
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| Basic results on algebraic surfaces include the [[Hodge index theorem]], and the division into five groups of birational equivalence classes called the [[classification of algebraic surfaces]]. The ''general type'' class, of [[Kodaira dimension]] 2, is very large (degree 5 or larger for a non-singular surface in '''P'''<sup>3</sup> lies in it, for example).
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| There are essential three [[Hodge number]] invariants of a surface. Of those, ''h''<sup>1,0</sup> was classically called the '''irregularity''' and denoted by ''q''; and ''h''<sup>2,0</sup> was called the '''geometric genus''' ''p''<sub>''g''</sub>. The third, ''h''<sup>1,1</sup>, is not a [[birational invariant]], because [[blowing up]] can add whole curves, with classes in ''H''<sup>1,1</sup>. It is known that [[Hodge cycle]]s are algebraic, and that [[algebraic equivalence]] coincides with [[homological equivalence]], so that ''h''<sup>1,1</sup> is an upper bound for ρ, the rank of the [[Néron-Severi group]]. The [[arithmetic genus]] ''p''<sub>''a''</sub> is the difference
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| :geometric genus − irregularity.
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| In fact this explains why the irregularity got its name, as a kind of 'error term'.
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| == Riemann-Roch theorem for surfaces == | |
| {{main|Riemann-Roch theorem for surfaces}}
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| The [[Riemann-Roch theorem for surfaces]] was first formulated by [[Max Noether]]. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.
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| ==References==
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| *{{eom|id=A/a011640|first=I.V.|last= Dolgachev}}
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| *{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58658-6 | mr=1336146 | year=1995}}
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| == External links ==
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| * [http://imaginary.org/program/surfer Free program SURFER] to visualize algebraic surfaces in real-time, including a user gallery.
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| * [http://www.singsurf.org/singsurf/SingSurf.html SingSurf] an interactive 3D viewer for algebraic surfaces.
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| * [http://www.mathematik.uni-kl.de/%7Ehunt/drawings.html Some beautiful algebraic surfaces]
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| * [http://www1-c703.uibk.ac.at/mathematik/project/bildergalerie/gallery.html Some more, with their respective equations]
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| * [http://www.bru.hlphys.jku.at/surf/index.html Page on Algebraic Surfaces started in 2008]
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| * [http://maxwelldemon.com/2009/03/29/surfaces-2-algebraic-surfaces/ Overview and thoughts on designing Algebraic surfaces]
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| [[Category:Algebraic surfaces]]
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