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**Flip theorem and the existence of minimal models for 3-folds.**
*(English)*
Zbl 0649.14023

From the introduction: “One of the main achievements of the 19th century Italian school of algebraic geometry was a complete understanding of the birational geometry of surfaces, including the construction of minimal models. During the past several years, a program has emerged to construct minimal models in higher dimensions using something called extremal rays. This paper completes the final step of that minimal model program (MMP) in dimension 3.” - The main theorem is the following. Two main points of the proof are a result due to Y. Kawamata [Ann. Math., II. Ser. 127, No.1, 93-163 (1988; Zbl 0651.14005)] and a detailed analysis of 3- dimensional terminal singularities.

(Flip theorem) Let \(f: X\to Y\) be a birational morphism between complex normal projective 3-folds such that X has only \({\mathbb{Q}}\)-factorial terminal singularities, \(\rho (X)=\rho (Y)+1\), \(-K_ X\) is f-ample and f contracts no divisors (to curves or points). Then there is a birational morphism \(f': X'\to Y\) from a projective 3-fold X’ with only \({\mathbb{Q}}\)- factorial terminal singularities such that f’ contracts no divisors and \(K_{X'}\) is f’-ample.

The map \(X\to X'\) is isomorphic in codimension 1, \(\rho(X)=\rho(X')\) (=the Picard numbers) and f’ (or X’) is called the directed flip (or flip) of f. If \(\dim(X)\geq 4\) the existence of the directed flip is unknown. Let us point out three important corollaries which follow from MMP by virtue also of the works of Fujita, Benveniste, Kawamata (corollary 1) and Miyaoka (corollary 2).

1. For every nonsingular projective threefold X of general type, the graded canonical ring R(X) is finitely generated.

2. A nonsingular projective threefold X is uniruled if and only if the Kodaira dimension \(\kappa(X)\) is negative.

3. Every birational morphism \(f: X\to Y\) between nonsingular projective 3-folds is a composition of divisorial contractions and directed flips.

One should mention that, even in dimension 3, there are several things yet to be done in classification theory. Indeed the results above can be seen as the beginning of a detailed structure theory.

(Flip theorem) Let \(f: X\to Y\) be a birational morphism between complex normal projective 3-folds such that X has only \({\mathbb{Q}}\)-factorial terminal singularities, \(\rho (X)=\rho (Y)+1\), \(-K_ X\) is f-ample and f contracts no divisors (to curves or points). Then there is a birational morphism \(f': X'\to Y\) from a projective 3-fold X’ with only \({\mathbb{Q}}\)- factorial terminal singularities such that f’ contracts no divisors and \(K_{X'}\) is f’-ample.

The map \(X\to X'\) is isomorphic in codimension 1, \(\rho(X)=\rho(X')\) (=the Picard numbers) and f’ (or X’) is called the directed flip (or flip) of f. If \(\dim(X)\geq 4\) the existence of the directed flip is unknown. Let us point out three important corollaries which follow from MMP by virtue also of the works of Fujita, Benveniste, Kawamata (corollary 1) and Miyaoka (corollary 2).

1. For every nonsingular projective threefold X of general type, the graded canonical ring R(X) is finitely generated.

2. A nonsingular projective threefold X is uniruled if and only if the Kodaira dimension \(\kappa(X)\) is negative.

3. Every birational morphism \(f: X\to Y\) between nonsingular projective 3-folds is a composition of divisorial contractions and directed flips.

One should mention that, even in dimension 3, there are several things yet to be done in classification theory. Indeed the results above can be seen as the beginning of a detailed structure theory.

Reviewer: M.Beltrametti

### MSC:

14J30 | \(3\)-folds |

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

14J10 | Families, moduli, classification: algebraic theory |

14E05 | Rational and birational maps |

14C20 | Divisors, linear systems, invertible sheaves |