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Minimal models of canonical 3-folds. (English) Zbl 0558.14028
Algebraic varieties and analytic varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983).
[For the entire collection see Zbl 0504.00008.]
This is a continuation of a previous paper [Journées de géométrie algebrique, Angers/France 1979, 273-310 (1980; Zbl 0451.14014)] here quoted as C.3. These two papers together give new, very interesting, insight in the difficult problems regarding the classification of 3-folds of ”general type”. In the case of surfaces (of general type) it is well known that the canonical model X may have only very simple singularities, the minimal model S is non singular, $$K_ S$$ is ”numerically effective and free” $$(=:nef)$$ and the morphism $$f: S\to X$$ is such that $$f^*\omega_ X=\omega_ S$$ (f is ”crepant”). In this paper a definition of minimal model S for a 3-fold X is proposed (X of f. g. general type (C.3.) and $$\kappa_{num}\geq 0)$$ which preserves the maximum of the previous properties for surfaces. S may have singularities of a specified simple type called ”quick” and is obtained by blowing up the canonical models X. Before stating the main result we need some more definitions. $$f: Y\to X$$ is a partial resolution if it is a proper birational morphism in which Y is always assumed normal. If f is a partial resolution an exceptional prime divisor of f is any prime divisor $$\Gamma$$ $$\subset Y$$ such that $$co\dim f(\Gamma)\geq 2.$$ Let X be a variety of dimension 3 with canonical singularities (C.3.), $$P\in X$$ is a terminal singularity if it has a resolution $$f: Y\to X$$ such that (i) f has at least one exceptional prime divisor and (ii) if $$K_ Y=f^*K_ X+\Delta$$ every exceptional prime divisor of f appears in $$\Delta$$ with strictly positive coefficient.
The main theorem is the following: 1. Let $$P\in X$$ a 3-fold point then P is terminal if and only if it is quick. - 2. Let X be a 3-fold with canonical singularities. Then there exists a partial resolution $$f: S\to X$$ such that (a) f is crepant, and (b) S has quick singularities. Furthermore this f can be chosen as the composite of certain elementary steps (blow-ups) which are intrinsic to X and is then uniquely determined and projective. - The paper contains many other results of interest in themselves and many appealing conjectures and open problems.
Reviewer: F.Gherardelli

##### MSC:
 14J30 $$3$$-folds 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14E30 Minimal model program (Mori theory, extremal rays) 14C20 Divisors, linear systems, invertible sheaves 14B05 Singularities in algebraic geometry