en>David Eppstein: Undo — this is the wrong article for this level of technical detail; it is a survey of random graph models in general, not specifically about Erdos-Renyi-Gilbert graphs

2015-01-12T08:49:54Z

Undo — this is the wrong article for this level of technical detail; it is a survey of random graph models in general, not specifically about Erdos-Renyi-Gilbert graphs

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en>Jérôme: /* See also */

2013-12-06T10:38:49Z

See also

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	~~Jerrie Swoboda~~ is ~~what you can call me~~ and ~~I totally dig whom name~~. ~~Managing people~~ is ~~my time job~~ now. As a ~~girl what I really like~~ is to ~~take up croquet~~ but ~~I are unable make~~ it ~~my position really~~. ~~My husband~~'s ~~comments~~ and ~~I chose~~ to ~~maintain in Massachusetts~~. ~~Go to my web site to find out more~~: http://~~prometeu~~.~~net<br><br>my homepage~~ ... [http://~~prometeu~~.~~net clash~~ of ~~clans hack apk~~]		In [[algebraic geometry]], a branch of [[mathematics]], '''Serre duality''' is a [[duality (mathematics)\|duality]] present on [[non-singular]] [[projective variety\|projective]] [[algebraic variety\|algebraic varieties]] ''V'' of dimension ''n'' (and in greater generality for [[vector bundle]]s and further, for [[coherent sheaf\|coherent sheaves]]). It shows that a [[cohomology]] group ''H''<sup>''i''</sup> is the [[dual space]] of another one, ''H''<sup>''n''−''i''</sup>. If the variety is defined over the [[complex number]]s, this yields different information from [[Poincaré duality]], which relates ''H''<sup>''i''</sup> to ''H''<sup>2''n''−''i''</sup>, considering ''V'' as a real manifold of dimension 2''n''.

			In the case for holomorphic vector bundle ''E'' over a smooth compact complex manifold ''V'', the statement is in the form:

			:::<math>H^q(V,E)\cong H^{n-q}(V,K\otimes E^{\ast})^{\ast},</math>

			in which ''V'' is not necessarily projective.

			==Algebraic curve==
			The case of [[algebraic curve]]s was already implicit in the [[Riemann-Roch theorem]]. For a curve ''C'' the coherent groups ''H''<sup>''i''</sup> vanish for ''i'' > 1; but ''H''<sup>1</sup> does enter implicitly. In fact, the basic relation of the theorem involves ''l''(''D'') and ''l''(''K''−''D''), where ''D'' is a [[divisor]] and ''K'' is a divisor of the [[canonical class]]. After [[Jean-Pierre Serre\|Serre]] we recognise ''l''(''K''−''D'') as the dimension of ''H''<sup>1</sup>(''D''), where now ''D'' means the [[line bundle]] determined by the divisor ''D''. That is, Serre duality in this case relates groups ''H''<sup>1</sup>(''D'') and ''H''<sup>0</sup>(''KD''), and we are reading off dimensions (notation: ''K'' is the canonical line bundle, ''D'' is the dual line bundle, and juxtaposition is the [[tensor product]] of line bundles).

			In this formulation the Riemann-Roch theorem can be viewed as a computation of the [[Euler characteristic of a sheaf]]

			:''h''<sup>0</sup>(''D'') − ''h''<sup>1</sup>(''D''),

			in terms of the [[genus (mathematics)\|genus]] of the curve, which is

			:''h''<sup>1</sup>(''C'',''O''<sub>''C''</sub>),

			and the degree of ''D''. It is this expression that can be generalised to higher dimensions.

			Serre duality of curves is therefore something very classical; but it has an interesting light to cast. For example, in [[Riemann surface]] theory, the [[deformation theory]] of complex structures is studied classically by means of [[quadratic differential]]s (namely sections of ''L''(''K''<sup>2</sup>)). The [[deformation theory]] of [[Kunihiko Kodaira]] and [[D. C. Spencer]] identifies deformations via ''H''<sup>1</sup>(''T''), where ''T'' is the [[tangent bundle]] sheaf ''K''*. The duality shows why these approaches coincide.

			==Origin and generalisations==
			The origin of the theory lies in Serre's earlier work on [[several complex variables]]. In the generalisation of [[Alexander Grothendieck]], Serre duality becomes a part of [[coherent duality]] in a much broader setting. While the role of ''K'' above in general Serre duality is played by the [[determinant line bundle]] of the [[cotangent bundle]], when ''V'' is a manifold, in full generality ''K'' cannot merely be a ''single'' sheaf in the absence of some hypothesis of [[non-singular]]ity on ''V''. The formulation in full generality uses a [[derived category]] and [[Ext functor]]s, to allow for the fact that ''K'' is now represented by a [[chain complex]] of sheaves, namely, the dualizing complex. Nevertheless, the statement of the theorem is recognisably Serre's.

			==References==
			* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link=Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90244-9 \| oclc=13348052 \| id={{MathSciNet \| id = 0463157}} \| year=1977}}, see Ch. III.7
			*{{Springer\|id=D/d034120\|title=Duality}}
			* {{Citation \| last1=Huybrechts \| first1=Daniel \| title=Complex geometry\| publisher=[[Springer-Verlag]] \| location=Berlin \| year=2005}}, see p. 171.
			* {{Citation \| last1=Tate \| first1=John \| author1-link=John Tate \| title=Residues of differentials on curves \| url=http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1968_4_1_1/ASENS_1968_4_1_1_149_0/ASENS_1968_4_1_1_149_0.pdf \| year=1968 \| journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série \| issn=0012-9593 \| volume=1 \| pages=149–159}} contains a proof for Serre duality for curves
			*[http://rigtriv.wordpress.com/2008/05/29/serre-duality/ Serre duality] at the [[weblog]] [http://rigtriv.wordpress.com/ Rigorous trivialities]
			*[http://math.stackexchange.com/questions/2403/precise-connection-between-poincare-duality-and-serre-duality A link between Poincaré and Serre dualities via Hodge theory] on [[Stack exchange]]

			[[Category:Topological methods of algebraic geometry]]
			[[Category:Complex manifolds]]
			[[Category:Duality theories]]

en>VolkovBot: r2.7.2) (Robot: Adding et:Juhuslik graaf

2012-08-14T17:24:38Z

r2.7.2) (Robot: Adding et:Juhuslik graaf

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Jerrie Swoboda is what you can call me and I totally dig whom name. Managing people is my time job now. As a girl what I really like is to take up croquet but I are unable make it my position really. My husband's comments and I chose to maintain in Massachusetts. Go to my web site to find out more: http://prometeu.net<br><br>my homepage ... [http://prometeu.net clash of clans hack apk]

Random graph - Revision history

en>David Eppstein: Undo — this is the wrong article for this level of technical detail; it is a survey of random graph models in general, not specifically about Erdos-Renyi-Gilbert graphs

en>Jérôme: /* See also */

en>VolkovBot: r2.7.2) (Robot: Adding et:Juhuslik graaf