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**Classification of three-dimensional flips.**
*(English)*
Zbl 0773.14004

The celebrated Mori theory says that any smooth projective complex threefold \(X\) can be transformed, by two kinds of birational operations called divisorial contractions and flips, to a model \(X'\) which has only “very mild” singularities and satisfies one of the following conditions:

(1) The canonical bundle \(K'\) of \(X'\) is nef; in this case \(\kappa(X)\geq 0\).

(2) There is a fibration \(f:X'\to Y\) such that \(\dim Y<3\) and \(-K'\) is \(f\)-ample; in this case \(X\) is uniruled.

The main aim of this paper is to generalize this theory to the case of families of threefolds. The deformation invariance of plurigenera of projective threefolds with mild singularities follows from this result. As another important application, it is shown that the “birational moduli” functor of threefolds of general type with a given Hilbert function is coarsely represented by a separated algebraic space of finite type. This theory applies also to the study of nonprojective deformation families of a projective threefold.

For the proof of the main result, it is shown that the two operations, divisorial contractions and flips, can be performed continuously for families. The contraction morphisms of extremal rays can be done continuously, so the problem is reduced to the existence of flips for families. The following result serves as a key lemma at this step: Let \(f:X\to Y\) be a proper bimeromorphic morphism of complex threefolds such that (1) \(X\) has only terminal singularities, (2) there is a normal point \(Q\) on \(Y\) such that \(C=f^{-1}(Q)\) is an irreducible curve and \(X- C\cong Y-Q\), (3) \(K_ XC<0\) for the canonical bundle \(K_ X\) of \(X\). Let \(t\) be a general element of the ideal \(I_ Q\) of \(Q\) in \(Y\) and let \(H'=\{t=0\}\) be its zero divisor. Then the singularity of \(H'\) at \(Q\) is classified very precisely. The result seems too technical to be reviewed here, but by using deformation theory of such surface singularities the authors deduce the existence of flips for families, where a deformation family of a threefold \(Y\) as above is regarded as the total space of a deformation family of the surface \(H'\).

The proof of this “key lemma” consists of very precise case-by-case analysis and numerous computations, and occupies most pages of this article. But it is really worth the pages spent. In fact, it establishes a classification theory of three dimensional small contractions and flips, with many by-products including the solution of Reid’s conjecture about general elephants. Thus this paper deserves its title.

(1) The canonical bundle \(K'\) of \(X'\) is nef; in this case \(\kappa(X)\geq 0\).

(2) There is a fibration \(f:X'\to Y\) such that \(\dim Y<3\) and \(-K'\) is \(f\)-ample; in this case \(X\) is uniruled.

The main aim of this paper is to generalize this theory to the case of families of threefolds. The deformation invariance of plurigenera of projective threefolds with mild singularities follows from this result. As another important application, it is shown that the “birational moduli” functor of threefolds of general type with a given Hilbert function is coarsely represented by a separated algebraic space of finite type. This theory applies also to the study of nonprojective deformation families of a projective threefold.

For the proof of the main result, it is shown that the two operations, divisorial contractions and flips, can be performed continuously for families. The contraction morphisms of extremal rays can be done continuously, so the problem is reduced to the existence of flips for families. The following result serves as a key lemma at this step: Let \(f:X\to Y\) be a proper bimeromorphic morphism of complex threefolds such that (1) \(X\) has only terminal singularities, (2) there is a normal point \(Q\) on \(Y\) such that \(C=f^{-1}(Q)\) is an irreducible curve and \(X- C\cong Y-Q\), (3) \(K_ XC<0\) for the canonical bundle \(K_ X\) of \(X\). Let \(t\) be a general element of the ideal \(I_ Q\) of \(Q\) in \(Y\) and let \(H'=\{t=0\}\) be its zero divisor. Then the singularity of \(H'\) at \(Q\) is classified very precisely. The result seems too technical to be reviewed here, but by using deformation theory of such surface singularities the authors deduce the existence of flips for families, where a deformation family of a threefold \(Y\) as above is regarded as the total space of a deformation family of the surface \(H'\).

The proof of this “key lemma” consists of very precise case-by-case analysis and numerous computations, and occupies most pages of this article. But it is really worth the pages spent. In fact, it establishes a classification theory of three dimensional small contractions and flips, with many by-products including the solution of Reid’s conjecture about general elephants. Thus this paper deserves its title.

Reviewer: T.Fujita (Tokyo)

### MSC:

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

14D15 | Formal methods and deformations in algebraic geometry |

14J30 | \(3\)-folds |

14E05 | Rational and birational maps |

32J17 | Compact complex \(3\)-folds |

14D22 | Fine and coarse moduli spaces |