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Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-...

I will start the story by the end: Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...