Original sentence: A fundamental theorem of Birkar-Cascini-Hacon-McKernan from 2006 is that the canonical ring of a smooth projective algebraic variety is finitely generated.If the canonical class is an ample line bundle, then the canonical ring is the homogeneous coordinate ring of the image of the canonical map.In all cases the canonical ring of V is the direct sum and graded ring.Its dimension measured by transcendence degree is called the Kodaira dimension of V.The canonical model is the image of the canonical map, and therefore its homogeneous coordinate ring is the canonical ring.The canonical model is useful because it can sometimes be used to reduce the study of birational questions to biregular questions.The minimal model program sparked the question of just how generally this is true.To take in more examples, the Proj of the canonical ring is also called the canonical model; it is possible to take  even if R is not finitely generated, but the result would not be a variety
Edited sentence: A fundamental theorem of Birkar-Cascini-Hacon-McKernan from 2006 is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.Here the dimension of the canonical ring may be taken to mean Krull dimension or transcendence degree.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.If the canonical ring is not finitely generated, then  is not a variety, and so it cannot be birational to V; in particular, V admits no canonical model.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.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.The Kodaira dimension of V is the dimension of the canonical ring minus one.The canonical ring of V is the graded ring.The minimal model program proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated.This can be true even when the canonical class of V is not ample.When the canonical ring is finitely generated, the canonical model is Proj of the canonical ring