Editing Canonical bundle (section)

===Canonical rings===
{{main|Canonical ring}}

The '''canonical ring''' of ''V'' is the [[graded ring]]
:<math>R = \bigoplus_{d = 0}^\infty H^0(V, K_V^d).</math>
If the canonical class of ''V'' is an [[ample line bundle]], then the canonical ring is the [[homogeneous coordinate ring]] of the image of the canonical map. This can be true even when the canonical class of ''V'' is not ample. For instance, if ''V'' is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a ''k''-canonical map, where ''k'' is any sufficiently divisible positive integer.

The [[minimal model program]] proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated. In particular, this was known to imply the existence of a '''canonical model''', a particular birational model of ''V'' with mild singularities that could be constructed by blowing down ''V''. When the canonical ring is finitely generated, the canonical model is [[Proj construction|Proj]] of the canonical ring. If the canonical ring is not finitely generated, then {{nowrap|Proj ''R''}} is not a variety, and so it cannot be birational to ''V''; in particular, ''V'' admits no canonical model. One can show that if the canonical divisor ''K'' of ''V'' is a [[Nef line bundle|nef]] divisor and the [[Intersection theory|self intersection]] of ''K'' is greater than zero, then ''V'' will admit a canonical model (more generally, this is true for normal complete Gorenstein algebraic spaces<ref>{{cite book |last=Badescu |first=Lucian |author-link=Lucian Badescu |date=2001 |title=Algebraic Surfaces |publisher=Springer Science & Business Media  |page=242 |isbn= 9780387986685}}</ref>).<ref>{{cite book |last=Badescu |first=Lucian |author-link=Lucian Badescu |date=2001 |title=Algebraic Surfaces |publisher=Springer Science & Business Media  |page=123 |isbn= 9780387986685}}</ref>

A fundamental theorem of Birkar–Cascini–Hacon–McKernan from 2006<ref>{{cite web | url=http://www.birs.ca/birspages.php?task=displayevent&event_id=09w5033 | title=09w5033: Complex Analysis and Complex Geometry &#124; Banff International Research Station }}</ref> is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.

The [[Kodaira dimension]] of ''V'' is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean [[Krull dimension]] or [[transcendence degree]].