Relative canonical model

In mathematics, the relative canonical model of a singular variety formula_1
is a particular canonical variety that maps to formula_1, which simplifies the structure. The precise definition is: 
If formula_3 is a resolution define the adjunction sequence to be the sequence of subsheaves formula_4 if formula_5 is invertible formula_6 where formula_7 is the higher adjunction ideal. Problem. Is formula_8 finitely generated? If this is true then formula_9 is called the "relative canonical model" of formula_10, or the "canonical blow-up" of formula_1.
Some basic properties were as follows:
The relative canonical model was independent of the choice of resolution. 
Some integer multiple formula_12 of the canonical divisor of the relative canonical model was Cartier and the number of exceptional components where this agrees with the same multiple of the canonical divisor of Y is also independent of the choice of Y. When it equals the number of components of Y it was called "crepant". It was not known whether relative canonical models were Cohen–Macaulay.
Because the relative canonical model is independent of formula_10, most authors simplify the terminology, referring to it as the relative canonical model "of " formula_1 rather than either the relative canonical model "of " formula_10 or the canonical blow-up of formula_1. The class of varieties that are relative canonical models have canonical singularities. Since that time in the 1970s other mathematicians solved affirmatively the problem of whether they are Cohen–Macaulay. The minimal model program started by Shigefumi Mori proved that the sheaf in the definition always is finitely generated and therefore that relative canonical models always exist.