Electronic filter - Revision history

en>Gadget850: refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB

2014-12-30T22:21:40Z

refbegin, replaced: </div> → {{refend}}, <div class="references-small"> → {{refbegin}} using AWB

← Older revision		Revision as of 00:21, 31 December 2014
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en>Spinningspark: Reverted 1 edit by 123.63.112.145 (talk): An L filter requires two elements to form the L. (TW)

2014-02-25T13:07:56Z

Reverted 1 edit by 123.63.112.145 (talk): An L filter requires two elements to form the L. (TW)

← Older revision		Revision as of 15:07, 25 February 2014
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	~~In [[mathematics]], the '''pluricanonical ring''' of an [[algebraic variety]] ''V'' (which is [[non-singular]]), or of a [[complex manifold]], is the [[graded commutative ring\|graded ring]]~~		I'm Marina and I live with my husband and our two children in Salez, in the south part. My hobbies are Record collecting, Australian Football League and Auto audiophilia.<br><br>Review my website :: [http://Www.mikomak.com/?p=76793 fifa Coin Generator]
	~~:<math>R(V,K)=R(V,K_V) \,</math>~~

	~~of sections of powers of the [[canonical bundle]] ''K''. Its ''n''th graded component (for <math>n\geq 0</math>) is:~~
	~~:<math>R_n := H^0(V, K^n),\ </math>~~

	~~that is, the space of [[Section (fiber bundle)\|sections]] of the '~~'~~n''-th [[tensor product]] ''K''<sup>''n''</sup> of the [[canonical bundle]] ''K''.~~

	~~The 0th graded component <math>R_0</math> is sections of the trivial bundle,~~ and ~~is one dimensional as ''V'' is projective. The projective variety defined by this graded ring is called the '''canonical model''' of ''V'',~~ and ~~the dimension of the '''canonical model'''~~, ~~is called~~ the ~~[[Kodaira dimension]] of ''V''.~~

	One can define an analogous ring for any [[line bundle]] ''L'' over ''V''; the analogous dimension is called the '''Iitaka dimension'''. A line bundle is called '''big''' if the Iitaka dimension equals the dimension of the variety.

	~~==Properties==~~
	~~===Birational invariance===~~
	The canonical ring and therefore likewise the Kodaira dimension is a [[birational invariant]]: Any birational map between smooth compact complex manifolds induces an isomorphism between the respective canonical rings. As a consequence one can define the Kodaira dimension of a singular space as the Kodaira dimension of a [[desingularization]]. Due to the birational invariance this is well defined, i.e., ~~independent of the choice of the desingularization.~~

	~~===Fundamental conjecture of birational geometry===~~
	~~A basic conjecture is that the pluricanonical ring is [[Finitely generated algebra\|finitely generated]]. This is considered a major step in the [[Mori program]].~~
	~~{{harvs\|txt\|first=Caucher \|last=Birkar\|first2= Paolo \|last2=Cascini\|first3= Christopher D. \|last3=Hacon\|first4= James\|last4= McKernan\|year=2010}}~~ and ~~{{harvs\|txt\|last1=Siu \| first1=Yum-Tong \|year=2006}} have announced proofs of this conjecture~~.

	~~==The plurigenera==~~
	~~The dimension~~

	:<~~math~~>~~P_n = h^0(V, K^n) = \operatorname{dim}\ H^0(V, K^n)~~<~~/math~~>

	~~is the classically-defined ''n''-th '''''plurigenus''''' of ''V''. The pluricanonical divisor <math>K^n</math>, via the corresponding~~ [~~[linear system of divisors]], gives a map to projective space <math>\mathbf{P}(H^0(V, K^n)) = \mathbf{P}^{P_n - 1}<~~/~~math>, called the ''n''-canonical map.~~

	~~The size of ''R'' is a basic invariant of ''V'', and is called the [[Kodaira dimension]].~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	*{{Citation \| last1=Birkar \| first1=Caucher \| last2=Cascini \| first2=Paolo \| last3=Hacon \| first3=Christopher D. \| last4=McKernan \| first4=James \| title=Existence of minimal models for varieties of log general type \| arxiv=math.AG/~~0610203 \| doi=10~~.~~1090/S0894-0347-09-00649-3 \| mr=2601039 \| year=2010 \| journal=[[Journal of the American Mathematical Society]] \| volume=23 \| issue=2 \| pages=405–468}}~~
	* {{Citation \| author=P. Griffiths \| authorlink=Phillip Griffiths \| coauthors=[[Joe Harris (mathematician)\|J. Harris]] \| title=Principles of Algebraic Geometry \| series=Wiley Classics Library \| publisher=Wiley Interscience \| year=1994 \| isbn=0-471-05059-8 \| page=573 }}
	*{{Citation \| last1=Siu \| first1=Yum-Tong \| title=Invariance of plurigenera \| doi=10.1007/~~s002220050276 \| mr=1660941 \| year=1998 \| journal=[[Inventiones Mathematicae]] \| volume=134 \| issue=3 \| pages=661–673}}~~
	*{{citation\|arxiv=math.AG/0610740\|title=A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring\|last1=Siu \| first1=Yum-Tong \|year=2006}}
	*{{citation\|arxiv=0704.1940\|title=Additional Explanatory Notes on the Analytic Proof of the Finite Generation of the Canonical Ring\|last1=~~Siu \| first1=Yum-Tong \|year=2007}}~~

	~~[[Category:Algebraic geometry]]~~
	~~[[Category:Birational geometry]]~~
	~~[[Category:Structures on manifolds]~~]

en>Monkbot: /* SAW filters */Fix CS1 deprecated date parameter errors

2014-01-17T21:38:15Z

SAW filters: Fix CS1 deprecated date parameter errors

@@ Line 1: / Line 1: @@
-== who died still hard to say.' Luo Feng eyes firmly. ==
+In [[mathematics]], the '''pluricanonical ring''' of an [[algebraic variety]] ''V'' (which is [[non-singular]]), or of a [[complex manifold]], is the [[graded commutative ring|graded ring]]
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+of sections of powers of the [[canonical bundle]] ''K''. Its ''n''th graded component (for <math>n\geq 0</math>) is:
-相关的主题文章：
+:<math>R_n := H^0(V, K^n),\ </math>
- <ul>
+that is, the space of [[Section (fiber bundle)|sections]] of the ''n''-th [[tensor product]] ''K''<sup>''n''</sup> of the [[canonical bundle]] ''K''.
-   <li>[http://www.fooyso.com/zhidao/question.php?qid=399710 http://www.fooyso.com/zhidao/question.php?qid=399710]</li>
+The 0th graded component <math>R_0</math> is sections of the trivial bundle, and is one dimensional as ''V'' is projective. The projective variety defined by this graded ring is called the '''canonical model''' of ''V'', and the dimension of the '''canonical model''', is called the [[Kodaira dimension]] of ''V''.
-   <li>[http://www.elixirpublishers.com/index.php?item/create_form/214 http://www.elixirpublishers.com/index.php?item/create_form/214]</li>
+One can define an analogous ring for any [[line bundle]] ''L'' over ''V''; the analogous dimension is called the '''Iitaka dimension'''. A line bundle is called '''big''' if the Iitaka dimension equals the dimension of the variety.
-   <li>[http://the-shadow-aitp.com/cgi-bin/ybs/ybs.cgi http://the-shadow-aitp.com/cgi-bin/ybs/ybs.cgi]</li>
+==Properties==
- </ul>
+===Birational invariance===

en>ClueBot NG: Reverting possible vandalism by 83.67.67.165 to version by 83.65.26.208. False positive? Report it. Thanks, ClueBot NG. (1154201) (Bot)

2012-07-26T15:13:36Z

Reverting possible vandalism by 83.67.67.165 to version by 83.65.26.208. False positive? Report it. Thanks, ClueBot NG. (1154201) (Bot)

New page

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Electronic filter - Revision history

en>Gadget850: refbegin, replaced: → {{refend}}, → {{refbegin}} using AWB

en>Spinningspark: Reverted 1 edit by 123.63.112.145 (talk): An L filter requires two elements to form the L. (TW)

en>Monkbot: /* SAW filters */Fix CS1 deprecated date parameter errors

en>ClueBot NG: Reverting possible vandalism by 83.67.67.165 to version by 83.65.26.208. False positive? Report it. Thanks, ClueBot NG. (1154201) (Bot)

en>Gadget850: refbegin, replaced: → {{refend}},
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