Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces.

*(English)*Zbl 0651.14005This long paper under review seems to be a by-product of the author’s trial to show the so-called “flip conjecture” (existence of a flip) and to verify the existence of so-called “minimal models” for 3-dimensional algebraic varieties [see e.g. S. Mori, J. Am. Math. Soc. 1, No.1, 117-253 (1988; Zbl 0649.14023)]. - It falls into two parts.

The first part consists of sections 1 through 7: Let X be a 3-dimensional normal algebraic variety over the complex number field with only so- called canonical singularities. We fix this notation in the following. Let D be a Weil divisor on X. In the first part, as the main result, it is shown that the sheaf of graded \({\mathcal O}_ X\)-algebras \({\mathcal R}_ X(D)=\oplus_{m\geq 0}{\mathcal O}_ X(mD)\) is finitely generated. - Applying the functor Proj and setting \(X'=Proj({\mathcal R}_ X(D))\), this claim is equivalent to that there exists a projective morphism \(f: X'\to X\) satisfying the following three conditions: (1) The strict transform D’ of D on X’ is f-ample; (2) Some multiple nD’ of D’ is a locally principal divisor on X’ (note that since X is singular, D is not necessarily locally principal); (3) The morphism f is an isomorphism in codimension one, i.e., for some finite set \(A\subset X\), the inverse image \(f^{- 1}(A)\) is one-dimensional and the restriction \(X'-f^{-1}(A)\to X-A\) of f is an isomorphism. (For understanding, consider the hypersurface in \({\mathbb{C}}^ 4 \)defined by \(xz-yw=0\) as X, and the 2-dimensional linear space defined by \(x=y=0\) as D. In this case X’ is the blowing-up of X along D. X’ is smooth.)

The three main results of the author’s previous paper - vanishing theorem, base point free theorem, cone theorem - are regarded as the basis of this paper. In addition to these, in section 2 he introduces the cone denoted by Mov in the space of numerical equivalence class of locally principal divisors with coefficients in real numbers and develops the theory of sectional decompositions for divisors. The sectional decomposition is a generalization of the so-called Zariski decomposition in dimension 2. The analysis of the cone Mov and the notion of sectional decompositions are the keys of this paper. - As an important corollary, the author shows the existence of an isomorphism \(g: Y\to X\) of codimension one such that Y has only canonical singularities and such that for every irreducible subvariety E of Y of codimension one, some multiple nE of E is locally defined by a single function. (Y can be called the \({\mathbb{Q}}\)-factorization of X.)

Moreover, there are two by-products of the proof. First it is shown that any birational map between 3-dimensional \({\mathbb{Q}}\)-factorial terminal good minimal models is a composition of a finite number of terminal special log-flips. Here a log-flip is a certain birational map which can be described relatively explicitly. The complement of its defining domain is one-dimensional and isomorphic to the projective space. [Obviously the notion of log-flips is an analogy of that of flips in the important paper “Flip conjecture. I” by S. Mori cited above. However, the roles played by the canonical sheaf in the notions are essentially different.] - Secondly a global version of the main result is shown for a relative big divisor under the assumption that the relative numerical class of the canonical sheaf of X with respect to the projective surjective morphism \(h: X\to S\) to a normal variety S is zero.

Sections 8, 9 and 10 are the second part. The author shows the following theorem. If we apply the resolution of singularities and the stable reduction beforehand, we can regard it as the solution of the minimal model conjecture for a one-dimensional family of surfaces:

Theorem 10.1. Let \(f: Z\to S\) be a projective surjective morphism of smooth varieties with connected fibers such that \(\dim (Z)=3\) and \(\dim (S)=1\). Assume that singular fibers of f are reduced and have simple normal crossings. Then there exists a projective surjective morphism \(g: X\to S\) such that X has at most \({\mathbb{Q}}\)-factorial terminal singularities and such that one of the following holds: (1) the canonical sheaf \(K_ X\) of X is numerically effective relative to g, i.e., g is a minimal model of f; (2) There are a two-dimensional normal variety Y and a projective surjective morphism \(h: X\to Y\) and \(k: Y\to S\) such that \(g=kh\), \(\rho (X/Y)=1\) (\(\rho\) denotes the relative Picard number) and \(- K_ X\) is h-ample. (3) \(\rho (X/S)=1\) and \(-K_ X\) is g-ample.

Section 8 begins with explanations of the notion of flips, because the author knows that it is essential for the proof. He assumes a certain condition and by considering a branched double covering, he reduces the problem to the situation where the result of the first part can be applied. As the result, he shows that if the anti-canonical linear system of a 3-dimensional flipping singularity contains a normal member with at most rational singularities, then there exists a certain diagram of birational morphisms which he calls a “flip”. - Section 9 is devoted to the classification of log-canonical surface singularities. - In section 10 the author verifies the above mentioned theorem after complicated arguments using the classification in section 9.

After reading this paper, we notice that the condition that the relative numerical class of the canonical sheaf is zero is stressed in several parts. Sometimes this condition is induced from the assumption that singularities are only canonical singularities. However, under this condition we can treat only special log-flips and cannot treat original important flips. This is the reason why this paper is divided into two independent parts, and we have to start again in the second part with the results of the first part. This seems to be also the reason why the author has left the most difficult part of the minimal model conjecture to Mori.

Lastly the meaning of the word “minimal model” used by him and his cooperators should be remarked. First their minimal model is not a relatively minimal model whose existence is trivial. Secondly they think that the most important property to be satisfied by minimal models is for their canonical sheaf K to be numerically effective, i.e. for every curve C on the model the intersection \(K\cdot C\) is non-negative. Thirdly their minimal models are not necessarily smooth, because otherwise in dimension 3 we can construct a variety such that any candidate for the minimal model does not satisfy the above numerical effectiveness condition. They claim that terminal singularities or canonical singularities can be admitted to sit on minimal models. Fourthly they use this word even for families of varieties. If the total space X of a family of varieties has only terminal singularities and if X has the relatively numerically effective canonical sheaf, then they call X a minimal model of the family.

The first part consists of sections 1 through 7: Let X be a 3-dimensional normal algebraic variety over the complex number field with only so- called canonical singularities. We fix this notation in the following. Let D be a Weil divisor on X. In the first part, as the main result, it is shown that the sheaf of graded \({\mathcal O}_ X\)-algebras \({\mathcal R}_ X(D)=\oplus_{m\geq 0}{\mathcal O}_ X(mD)\) is finitely generated. - Applying the functor Proj and setting \(X'=Proj({\mathcal R}_ X(D))\), this claim is equivalent to that there exists a projective morphism \(f: X'\to X\) satisfying the following three conditions: (1) The strict transform D’ of D on X’ is f-ample; (2) Some multiple nD’ of D’ is a locally principal divisor on X’ (note that since X is singular, D is not necessarily locally principal); (3) The morphism f is an isomorphism in codimension one, i.e., for some finite set \(A\subset X\), the inverse image \(f^{- 1}(A)\) is one-dimensional and the restriction \(X'-f^{-1}(A)\to X-A\) of f is an isomorphism. (For understanding, consider the hypersurface in \({\mathbb{C}}^ 4 \)defined by \(xz-yw=0\) as X, and the 2-dimensional linear space defined by \(x=y=0\) as D. In this case X’ is the blowing-up of X along D. X’ is smooth.)

The three main results of the author’s previous paper - vanishing theorem, base point free theorem, cone theorem - are regarded as the basis of this paper. In addition to these, in section 2 he introduces the cone denoted by Mov in the space of numerical equivalence class of locally principal divisors with coefficients in real numbers and develops the theory of sectional decompositions for divisors. The sectional decomposition is a generalization of the so-called Zariski decomposition in dimension 2. The analysis of the cone Mov and the notion of sectional decompositions are the keys of this paper. - As an important corollary, the author shows the existence of an isomorphism \(g: Y\to X\) of codimension one such that Y has only canonical singularities and such that for every irreducible subvariety E of Y of codimension one, some multiple nE of E is locally defined by a single function. (Y can be called the \({\mathbb{Q}}\)-factorization of X.)

Moreover, there are two by-products of the proof. First it is shown that any birational map between 3-dimensional \({\mathbb{Q}}\)-factorial terminal good minimal models is a composition of a finite number of terminal special log-flips. Here a log-flip is a certain birational map which can be described relatively explicitly. The complement of its defining domain is one-dimensional and isomorphic to the projective space. [Obviously the notion of log-flips is an analogy of that of flips in the important paper “Flip conjecture. I” by S. Mori cited above. However, the roles played by the canonical sheaf in the notions are essentially different.] - Secondly a global version of the main result is shown for a relative big divisor under the assumption that the relative numerical class of the canonical sheaf of X with respect to the projective surjective morphism \(h: X\to S\) to a normal variety S is zero.

Sections 8, 9 and 10 are the second part. The author shows the following theorem. If we apply the resolution of singularities and the stable reduction beforehand, we can regard it as the solution of the minimal model conjecture for a one-dimensional family of surfaces:

Theorem 10.1. Let \(f: Z\to S\) be a projective surjective morphism of smooth varieties with connected fibers such that \(\dim (Z)=3\) and \(\dim (S)=1\). Assume that singular fibers of f are reduced and have simple normal crossings. Then there exists a projective surjective morphism \(g: X\to S\) such that X has at most \({\mathbb{Q}}\)-factorial terminal singularities and such that one of the following holds: (1) the canonical sheaf \(K_ X\) of X is numerically effective relative to g, i.e., g is a minimal model of f; (2) There are a two-dimensional normal variety Y and a projective surjective morphism \(h: X\to Y\) and \(k: Y\to S\) such that \(g=kh\), \(\rho (X/Y)=1\) (\(\rho\) denotes the relative Picard number) and \(- K_ X\) is h-ample. (3) \(\rho (X/S)=1\) and \(-K_ X\) is g-ample.

Section 8 begins with explanations of the notion of flips, because the author knows that it is essential for the proof. He assumes a certain condition and by considering a branched double covering, he reduces the problem to the situation where the result of the first part can be applied. As the result, he shows that if the anti-canonical linear system of a 3-dimensional flipping singularity contains a normal member with at most rational singularities, then there exists a certain diagram of birational morphisms which he calls a “flip”. - Section 9 is devoted to the classification of log-canonical surface singularities. - In section 10 the author verifies the above mentioned theorem after complicated arguments using the classification in section 9.

After reading this paper, we notice that the condition that the relative numerical class of the canonical sheaf is zero is stressed in several parts. Sometimes this condition is induced from the assumption that singularities are only canonical singularities. However, under this condition we can treat only special log-flips and cannot treat original important flips. This is the reason why this paper is divided into two independent parts, and we have to start again in the second part with the results of the first part. This seems to be also the reason why the author has left the most difficult part of the minimal model conjecture to Mori.

Lastly the meaning of the word “minimal model” used by him and his cooperators should be remarked. First their minimal model is not a relatively minimal model whose existence is trivial. Secondly they think that the most important property to be satisfied by minimal models is for their canonical sheaf K to be numerically effective, i.e. for every curve C on the model the intersection \(K\cdot C\) is non-negative. Thirdly their minimal models are not necessarily smooth, because otherwise in dimension 3 we can construct a variety such that any candidate for the minimal model does not satisfy the above numerical effectiveness condition. They claim that terminal singularities or canonical singularities can be admitted to sit on minimal models. Fourthly they use this word even for families of varieties. If the total space X of a family of varieties has only terminal singularities and if X has the relatively numerically effective canonical sheaf, then they call X a minimal model of the family.

Reviewer: T.Urabe

##### MSC:

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

14J30 | \(3\)-folds |

14J17 | Singularities of surfaces or higher-dimensional varieties |

14E05 | Rational and birational maps |

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