Minimal Models of Canonical 3-Folds

TL;DRAbstract

<!-- *** Custom HTML *** --> This paper introduces a temporary definition of <i>minimal models</i> of 3-folds (0.7), and studies these under extra hypotheses. The main result is Theorem (0.6), in which I characterise the singularities which necessarily appear on a minimal model, and prove the existence of a minimal model $S$ of a 3-fold of f.g. general type, by blowing up the canonical model $X$ studied in [C3-f], imitating closely the minimal resolution of Du Val surface singularities. Apart from techniques familiar from [C3-f] (computations of the valuations of differentials; cyclic covers; crepant blow-ups of index 1 points which are not cDV), the main new element (Theorem (2.6)) is a method of blowing up the 1-dimensional singular locus, based on the Brieskorn–Tyurina result on the existence of simultaneous resolutions of a family of Du Val surface singularities, together with the elementary transformations in $(-2)$-curves of Burns and Rapoport. Part II is devoted to an exposition

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