Relative canonical model

In mathematics, the relative canonical model of a singular variety ${\displaystyle X}$ is a particular canonical variety that maps to ${\displaystyle X}$, which simplifies the structure. The precise definition is:

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

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