en>Cherkash: simplified inline math

2013-04-01T14:35:27Z

simplified inline math

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	~~You could also~~ be ~~brought into use~~ the ~~pull down family gifts custom pinterest menu~~ on the ~~different programs~~.<br><br>~~Here~~ is ~~my webpage~~ - [http://www.~~pinterest~~.~~com~~/~~bottleyourbrand~~/~~food~~-~~labels~~/ ~~view this board~~ on ~~pinterest~~]		{{distinguish\|Picard modular group}}
			In [[mathematics]], the '''Picard group''' of a [[ringed space]] ''X'', denoted by Pic(''X''), is the group of [[isomorphism]] classes of [[invertible sheaves]] (or line bundles) on ''X'', with the [[group operation]] being [[tensor product]]. This construction is a global version of the construction of the divisor class group, or [[ideal class group]], and is much used in [[algebraic geometry]] and the theory of [[complex manifold]]s.

			Alternatively, the Picard group can be defined as the [[sheaf cohomology]] group

			:<math>H^1 (X, \mathcal{O}_X^{*}).\,</math>

			For integral [[Scheme (mathematics)\|schemes]] the Picard group can be shown to be isomorphic to the class group of [[Cartier divisor]]s. For complex manifolds the [[exponential sheaf sequence]] gives basic information on the Picard group.

			The name is in honour of [[Émile Picard]]'s theories, in particular of divisors on [[algebraic surface]]s.

			==Examples==
			* The Picard group of the spectrum of a [[Dedekind domain]] is its ''ideal class group''.

			* The invertible sheaves on [[projective space]] '''P'''<sup>''n''</sup>(''k'') for ''k'' a [[field (mathematics)\|field]], are the [[twisting sheaf\|twisting]] [[sheaf (mathematics)\|sheaves]] <math>\mathcal{O}(m),\,</math> so the Picard group of '''P'''<sup>''n''</sup>(''k'') is isomorphic to '''Z'''.

			*The Picard group of the affine line with two origins over ''k'' is isomorphic to '''Z'''.

			==Picard scheme==
			The construction of a scheme structure on ([[representable functor]] version of) the Picard group, the '''Picard scheme''', is an important step in algebraic geometry, in particular in the [[duality theory of abelian varieties]]. It was constructed by {{harvtxt\|Grothendieck\|1961/62}}, and also described by {{harvtxt\|Mumford\|1966}} and {{harvtxt\|Kleiman\|2005}}. The '''Picard variety''' is dual to the [[Albanese variety]] of classical algebraic geometry.

			In the cases of most importance to classical algebraic geometry, for a [[non-singular]] [[complete variety]] ''V'' over a [[field (mathematics)\|field]] of [[characteristic (algebra)\|characteristic]] zero, the [[connected space\|connected component]] of the identity in the Picard scheme is an [[abelian variety]] written Pic<sup>0</sup>(''V''). In the particular case where ''V'' is a curve, this neutral component is the [[Jacobian variety]] of ''V''. For fields of positive characteristic however, [[Jun-Ichi_Igusa\|Igusa]] constructed an example of a smooth projective surface ''S'' with Pic<sup>0</sup>(''S'') non-reduced, and hence not an [[abelian variety]].

			The quotient Pic(''V'')/Pic<sup>0</sup>(''V'') is a [[finitely-generated abelian group]] denoted NS(''V''), the '''[[Néron–Severi group]]''' of ''V''. In other words the Picard group fits into an [[exact sequence]]

			:<math>1\to \mathrm{Pic}^0(V)\to\mathrm{Pic}(V)\to \mathrm{NS}(V)\to 0.\,</math>

			The fact that the rank is finite is [[Francesco Severi]]'s '''theorem of the base'''; the rank is the '''Picard number''' of ''V'', often denoted ρ(''V''). Geometrically NS(''V'') describes the [[algebraic equivalence]] classes of [[divisor (algebraic geometry)\|divisors]] on ''V''; that is, using a stronger, non-linear equivalence relation in place of [[linear equivalence of divisors]], the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to [[numerical equivalence]], an essentially topological classification by [[intersection number]]s.

			==See also==
			*[[Sheaf cohomology]]
			*[[Cartier divisor]]
			*[[Holomorphic line bundle]]
			*[[Ideal class group]]
			*[[Arakelov class group]]

			==References==
			*{{citation\|first=A.\|last=Grothendieck\|authorlink=Alexander Grothendieck\|url=http://www.numdam.org/item?id=SB_1961-1962__7__143_0 \|title=V. Les schémas de Picard. Théorèmes d'existence\|series=Séminaire Bourbaki, t. 14, \|year=1961/62\|issue= 232}}
			*{{citation\|first=A.\|last=Grothendieck\|url=http://www.numdam.org/item?id=SB_1961-1962__7__221_0 \|title=VI. Les schémas de Picard. Propriétés générales\|series=Séminaire Bourbaki, t. 14, \|year=1961/62\|issue= 236}}
			* {{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 \| mr=0463157 \| year=1977}}
			*{{citation\|first=Jun-Ichi\|last=Igusa\|authorlink=Jun-Ichi_Igusa\|title=On some problems in abstract algebraic geometry\|journal=Proc. Nat. Acad. Sci. U.S.A.\| volume= 41\|
			pages= 964–967 \|year=1955}}
			*{{Citation \| last1=Kleiman \| first1=Steven L. \| author1-link=Steven Kleiman \| title=Fundamental algebraic geometry \| arxiv=math/0504020 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Math. Surveys Monogr. \| mr=2223410 \| year=2005 \| volume=123 \| chapter=The Picard scheme \| pages=235–321}}
			*{{Citation \| last1=Mumford \| first1=David \| author1-link=David Mumford \| title=Lectures on Curves on an Algebraic Surface \| publisher=[[Princeton University Press]] \| series=Annals of Mathematics Studies \| isbn=978-0-691-07993-6 \| mr=0209285 \| year=1966 \| volume=59\| oclc=171541070}}
			* {{Citation \| last1=Mumford \| first1=David \| author1-link= David Mumford \| title=Abelian varieties \| publisher=[[Oxford University Press]] \| location=Oxford \| isbn=978-0-19-560528-0 \| oclc=138290 \| year=1970}}

			[[Category:Geometry of divisors]]
			[[Category:Scheme theory]]
			[[Category:Abelian varieties]]

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