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.

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 |