Relative canonical model

In mathematics, the term relative canonical model was coined in the 1970's

If $f:Y\to X$ is a resolution of a normal variety define the adjunction sequence to be the sequence of subsheaves $f_{*}\omega _{Y}^{\otimes n};$ if $\omega _{X}$ is invertible $f_{*}\omega _{Y}^{\otimes n}=I_{n}\omega _{X}^{\otimes n}$ where $I_{n}$ is the higher adjunction ideal. Problem. Is $\oplus _{n}f_{*}\omega _{Y}^{\otimes n}$ finitely generated? If this is true then $\oplus _{n}f_{*}\omega _{Y}^{\otimes n}\to X$ is called the relative canonical model of Y, or the canonical blow-up of X.

The relative canonical model is independent of the choice of resolution.

Some basic properties were as follows: some integer multiple $r$ of the canonical divisor of the relative canonical model is Cartier and the number of irreducible components (including multiplicity) by which this differs from the same multiple of the canonical divisor of Y is a natural number also independent of the choice of Y. If this number is zero the same article coined the phrase that $Y$ is a crepant resolution. It also asked whether the relative canonical model is Cohen-Macaulay.

Because the relative canonical model is independent of $Y$, most authors simplify the terminology, referring to it as the relative canonical model of $X$ rather than the relative canonical model of $Y$ and the term canonical blow-up of $X$ is no longer used. Since that time 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. Also other mathematicians solved affirmatively the second stated problem of whether they are Cohen-Macaulay. The class of varieties which are relative canonical models are those which have canonical singularities.

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