Semistable 3-fold flips.

*(English)*Zbl 0860.14016
Russ. Acad. Sci., Izv., Math. 42, No. 2, 371-425 (1994); reprint from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 2, 165-222 (1993).

The author gives a notion of semistability generalizing the similar one for a semistable surface degeneration.

On a normal analytic 3-fold \(X\) with terminal singularities, a semistable divisor is defined to be a reduced divisor \(D\), where all irreducible components are normal \(\mathbb{Q}\)-Cartier-divisors such that locally there exists a resolution \(g:Y \to X\) with a divisor \(g^*D\) having nonsingular irreducible components with normal crossings and such that \(g\) is decomposable into a product of locally projective morphisms on a sequence of 3-folds with terminal singularities, where \(D\) induces divisors having irreducible components which are normal Cartier-divisors. The singularities on \(X\) (“semistable singularities”) are described in this paper in a “... half-inductive and half explicit” way. This includes results on “semistable blowing down” and “semistable flips”.

The applications in the final sections are devoted to more detailed studies and include the existence of a minimal semistable model. This has (among others) as a consequence the existence of the Brieskorn-Tjurina simultaneous resolution.

On a normal analytic 3-fold \(X\) with terminal singularities, a semistable divisor is defined to be a reduced divisor \(D\), where all irreducible components are normal \(\mathbb{Q}\)-Cartier-divisors such that locally there exists a resolution \(g:Y \to X\) with a divisor \(g^*D\) having nonsingular irreducible components with normal crossings and such that \(g\) is decomposable into a product of locally projective morphisms on a sequence of 3-folds with terminal singularities, where \(D\) induces divisors having irreducible components which are normal Cartier-divisors. The singularities on \(X\) (“semistable singularities”) are described in this paper in a “... half-inductive and half explicit” way. This includes results on “semistable blowing down” and “semistable flips”.

The applications in the final sections are devoted to more detailed studies and include the existence of a minimal semistable model. This has (among others) as a consequence the existence of the Brieskorn-Tjurina simultaneous resolution.

Reviewer: M.Roczen (Berlin)

##### MSC:

14E30 | Minimal model program (Mori theory, extremal rays) |

14J30 | \(3\)-folds |

14E05 | Rational and birational maps |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |