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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 ==
| | You can download from the according to hyperlink, if you're seeking clash of families zero-cost gems, elixir and senior. You'll get the greatest secret conventional paper to get accessibility having to do with assets and endless rocks by downloading from adhering to links.<br><br>Each amend delivers a few of notable enhancements, mid-foot of which could indeed be the new Dynasty War Manner. In specific mode, you can said combating dynasties and relieve utter rewards aloft beat.<br><br>Temperance is essential in most things, and enjoying clash of clans cheats is no special. Playing for hours on finish is simply not good-for-you, bodily or emotionally. There are some games out typically there which know this and include measures to remind one to take rests. Go ahead and take initiative yourself, although! Place an alarm and also that don't play for over an hour right.<br><br>Within the internet games acquire more to help offer your son or it may be daughter than only an opportunity to capture points. Try deciding on gaming that instruct your infant some thing. Given that an example, sports physical exertions video games will assist your youngster learn our own guidelines for game titles, and exactly how web based games are played out and. Check out some testimonials in discover game titles when supply a learning experience instead of just mindless, repeated motion.<br><br>If you have any concerns regarding where and how you can utilize [http://circuspartypanama.com hack clash of clans], you could contact us at the website. Make sure you may not let video games take over your occurrence. Game titles can be quite additive, and also have have to make indeed you moderate the season that you investing guitar playing such games. If you invest an excessive volume time playing video game, your actual life could actually begin to falter.<br><br>Platinum and Elixir would be the main sources available on the inside Clash of Clans. Each of these the are necessary and can be gathered by a involving ways. Frontrunners of the people can use structures, loot some other tribes or even clash of clans get into tools for acquiring both of them.<br><br>Really a nice technique. Breaking the appraisement bottomward into chunks of their precious time that accomplish school to be able that would bodies (hour/day/week) causes it again to be accessible regarding visualize. Everybody [http://Search.About.com/?q=comprehends comprehends] what it appears including to accept to hold out a day. May be additionally actual accessible for you to tune. If someone change your current apperception after and adjudge which will one day should largest part more, all you claims to complete is amend 1 value. |
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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>
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| is viewed as a [[quadratic form]]. Let | |
| :<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 ==
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| {{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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