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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]]
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| :<math>R(V,K)=R(V,K_V) \,</math>
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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:
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| :<math>R_n := H^0(V, K^n),\ </math>
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| that is, the space of [[Section (fiber bundle)|sections]] of the ''n''-th [[tensor product]] ''K''<sup>''n''</sup> of the [[canonical bundle]] ''K''.
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| 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''.
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| 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.
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| ==Properties==
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| ===Birational invariance===
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| 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.
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| ===Fundamental conjecture of birational geometry===
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| A basic conjecture is that the pluricanonical ring is [[Finitely generated algebra|finitely generated]]. This is considered a major step in the [[Mori program]].
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| {{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.
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| ==The plurigenera==
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| The dimension
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| :<math>P_n = h^0(V, K^n) = \operatorname{dim}\ H^0(V, K^n)</math> | |
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| 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.
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| The size of ''R'' is a basic invariant of ''V'', and is called the [[Kodaira dimension]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{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}}
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| * {{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 }}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| [[Category:Algebraic geometry]]
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| [[Category:Birational geometry]]
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| [[Category:Structures on manifolds]]
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I'm a 35 years old, married and work at the high school (Nutritional Sciences).
In my spare time I try to learn Norwegian. I've been there and look forward to go there anytime soon. I love to read, preferably on my ipad. I like to watch Breaking Bad and The Big Bang Theory as well as documentaries about anything scientific. I enjoy Knapping.
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