ADE classification - Revision history

en>Cesium 133: fixed the ambig link...

2014-11-21T09:52:26Z

fixed the ambig link...

← Older revision		Revision as of 11:52, 21 November 2014
Line 1:		Line 1:
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en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (9930)

2014-02-10T12:06:09Z

WP:CHECKWIKI error fixes + other fixes using AWB (9930)

← Older revision		Revision as of 14:06, 10 February 2014
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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]]~~.		They contact me Emilia. Since she was 18 she's been working as a receptionist but her marketing never comes. My family members lives in Minnesota and my family loves it. Playing baseball is the pastime he will never stop doing.<br><br>my website :: at home std testing ([http://speedtax.com.au/bbs/index.php?mid=Speedtax_qna&page=1&listStyle=webzine&document_srl=2693885 click the up coming internet site])

	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.~~

	~~== classification by the Kodira dimension ==~~

	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)

	~~Examples of algebraic surfaces include (κ is the [[Kodaira dimension]]):~~

	* 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
	* κ=0 : [[K3 surface]]s, [[abelian surface]]s, [[Enriques surface]]s, [[hyperelliptic surface]]s
	* κ=1: [[Elliptic surface]]s
	* κ=2: [[surface of general type\|surfaces of general type]].

	~~For more examples see the [[list of algebraic surfaces]].~~

	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.

	~~== birational geometry of surfaces ==~~
	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).~~

	~~== properties ==~~

	~~[[ample line bundle#Intersection theorem\|'''Nakai criterion''']] says that:~~
	~~: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.~~

	~~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]]~~
	:~~<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~~
	~~:<math>\mathcal{D}_0(S):=\{D\in\mathcal{D}(S)\|D\cdot X=0,\text{for all } X\in\mathcal{D}(S)\}</math>~~
	~~then <math>\mathcal{D}/\mathcal{D}_0(S):=Num(S)</math> becomes to be a '''numerical equivalent class group''' of ''S'' and~~
	~~:<math>Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E</math>~~
	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''.)

	~~For an ample bundle ''H'' on ''S'' the definition~~
	~~:<math>\{H\}^\perp~~:~~=\{D\in Num(S)\|D\cdot H=0\}.<~~/~~math>~~
	~~leads the '''Hodge index theorem''' of the surface version.~~
	~~: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.~~
	~~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.

	~~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).~~

	~~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

	~~:geometric genus~~ &~~minus; irregularity.~~

	~~In fact this explains why the irregularity got its name, as a kind of 'error term'.~~

	== ~~Riemann-Roch theorem for surfaces ==~~
	~~{{main\|Riemann-Roch theorem for surfaces}}~~
	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.

	~~==References==~~
	*{{eom\|id=A/a011640\|first=I.V.\|last= Dolgachev}}
	*{{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}}

	~~== External links ==~~
	* [http://imaginary.org/program/surfer Free program SURFER] to visualize algebraic surfaces in real-time, including a user gallery.
	* [http://www.singsurf.org/singsurf/SingSurf.html SingSurf] an interactive 3D viewer for algebraic surfaces.
	* [http://www.mathematik.uni-kl.de/%7Ehunt/drawings.html Some beautiful algebraic surfaces]
	* [http://www1-c703.uibk.ac.at/mathematik/project/bildergalerie/gallery.html Some more, with their respective equations]
	* [http://www.bru.hlphys.jku.at/surf/index.html Page on Algebraic Surfaces started in 2008]
	* [http://maxwelldemon.com/2009/03/29/surfaces-2-algebraic-surfaces/ Overview and thoughts on designing Algebraic surfaces]

	~~[[Category:Algebraic surfaces]~~]

en>Mdd: Correction(s)

2013-09-10T13:36:28Z

Correction(s)

← Older revision		Revision as of 15:36, 10 September 2013
Line 1:		Line 1:
	~~Jerrie Swoboda~~ is ~~what~~ the ~~individual can call me along with I totally dig~~ that ~~name~~. ~~What me~~ and ~~my family completely love~~ is ~~acting~~ but ~~I can~~'~~t make~~ it ~~my profession really~~. ~~The job I~~'~~ve been occupying~~ for ~~years~~ is ~~an actual people manager~~. ~~Guam~~ is ~~where I~~'~~ve always been having~~. ~~You does find my website here~~: ~~http~~://~~circuspartypanama~~.~~com~~<br><br>~~My web blog~~; [http://~~circuspartypanama~~.com ~~hack clash of clans~~]		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]].

			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.

			== classification by the Kodira dimension ==

			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)

			Examples of algebraic surfaces include (κ is the [[Kodaira dimension]]):

			* 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
			* κ=0 : [[K3 surface]]s, [[abelian surface]]s, [[Enriques surface]]s, [[hyperelliptic surface]]s
			* κ=1: [[Elliptic surface]]s
			* κ=2: [[surface of general type\|surfaces of general type]].

			For more examples see the [[list of algebraic surfaces]].

			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.

			== birational geometry of surfaces ==
			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).

			== properties ==

			[[ample line bundle#Intersection theorem\|'''Nakai criterion''']] says that:
			: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.

			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]]
			:<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
			:<math>\mathcal{D}_0(S):=\{D\in\mathcal{D}(S)\|D\cdot X=0,\text{for all } X\in\mathcal{D}(S)\}</math>
			then <math>\mathcal{D}/\mathcal{D}_0(S):=Num(S)</math> becomes to be a '''numerical equivalent class group''' of ''S'' and
			:<math>Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E</math>
			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''.)

			For an ample bundle ''H'' on ''S'' the definition
			:<math>\{H\}^\perp:=\{D\in Num(S)\|D\cdot H=0\}.</math>
			leads the '''Hodge index theorem''' of the surface version.
			: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.
			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.

			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).

			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

			:geometric genus − irregularity.

			In fact this explains why the irregularity got its name, as a kind of 'error term'.

			== Riemann-Roch theorem for surfaces ==
			{{main\|Riemann-Roch theorem for surfaces}}
			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.

			==References==
			*{{eom\|id=A/a011640\|first=I.V.\|last= Dolgachev}}
			*{{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}}

			== External links ==
			* [http://imaginary.org/program/surfer Free program SURFER] to visualize algebraic surfaces in real-time, including a user gallery.
			* [http://www.singsurf.org/singsurf/SingSurf.html SingSurf] an interactive 3D viewer for algebraic surfaces.
			* [http://www.mathematik.uni-kl.de/%7Ehunt/drawings.html Some beautiful algebraic surfaces]
			* [http://www1-c703.uibk.ac.at/mathematik/project/bildergalerie/gallery.html Some more, with their respective equations]
			* [http://www.bru.hlphys.jku.at/surf/index.html Page on Algebraic Surfaces started in 2008]
			* [http://maxwelldemon.com/2009/03/29/surfaces-2-algebraic-surfaces/ Overview and thoughts on designing Algebraic surfaces]

			[[Category:Algebraic surfaces]]

en>Tomruen: /* Binary polyhedral groups */

2012-07-23T04:01:20Z

Binary polyhedral groups

New page

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