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Kodaira dimension: Difference between revisions

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The following integers are equal if they are non-negative. A good reference is {{harvtxt|Lazarsfeld|2004}}, Theorem 2.1.33.
 
* If the canonical ring is finitely generated, which is true in [[characteristic (algebra)|characteristic]] zero and conjectured in general: theThe dimension of the [[Proj construction]] <math>\operatorname{Proj} R (K_X)</math>, (thisa projective variety is called the '''canonical model''' of ''X''; itdepending only depends on the birational equivalence class of ''X.'' (This is defined only if the canonical ring <math>R = R(K_X)</math>is finitely generated, which is true in [[characteristic (algebra)|characteristic]] zero and conjectured in general.)
* The dimension of the image of the ''d''-canonical mapping for all positive multiples ''d'' of some positive integer <math>d_0</math>.
* The [[transcendence degree]] of the fraction field of ''R'', minus one,; i.e., <math>t-1</math>, where ''t'' is the number of [[algebraically independent]] generators one can find.
* The rate of growth of the plurigenera: that is, the smallest number ''κ'' such that <math>P_d/d^{\kappa}</math> is bounded. In [[Big O notation]], it is the minimal ''κ'' such that <math>P_d = O(d^{\kappa})</math>.
 
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