##
**Threefolds whose canonical bundles are not numerically effective.**
*(English)*
Zbl 0557.14021

The exceptions to the numerical effectiveness of the canonical bundle play a fundamental role in the classification of algebraic surfaces. Actually, as is well known, if K is a canonical divisor, the inequality \(KC<0\) for a curve C can happen for two different reasons: (a) C is an exceptional curve (of the first kind), (b) no positive multiple of K is effective. Case (a) leads to Castelnuovo’s contraction theorem and to the concept of minimal model, while, case (b) leads, after contracting all exceptional curves, to a structure theorem for minimal models of ruled surfaces.

The paper under review is the most successful transplanting of the above dichotomy into the classification of algebraic threefolds. In this respect, the main results of the paper [already announced in Proc. Natl. Acad. Sci. USA 77, 3125-3126 (1980; Zbl 0434.14022)] are the following ones. Let X be a nonsingular projective 3-fold defined over an algebraically closed field of characteristic zero, whose canonical divisors are not numerically effective. Then two cases can occur. (A) (theorem 3.3:) There are a morphism \(\Phi:X\to Y\) to a projective 3-fold (singular, in some instances) and an irreducible divisor D on X (which can be of several types), such that \(\Phi\) is an isomorphism on \(X\setminus D\) and contracts D either to a smooth curve or to a point. - (B) (theorem 3.5:) There is a morphism \(\Phi:X\to Y\) onto a nonsingular projective variety Y which exhibits X as (i) a conic bundle, if dim Y\(=2\), (ii) a Del Pezzo fibration, if dim Y\(=1\), (iii) a Fano 3-fold with Picard number 1, if dim Y\(=0.\)

The heart of the paper is the author’s theory of extremal rays, which holds for nonsingular projective varieties defined over an algebraically closed field of any characteristic. Consider the real vector space N(X) of the numerical equivalence classes of 1-cycles on X, with real coefficients (endowed with any norm). Let \(\overline{NE}\) be the closure of the cone in N(X) generated by the classes of the effective 1-cycles and let \(\overline{NE}_-=\{Z\in \overline{NE}| ZK\geq 0\}.\) An extremal ray R is a half line \({\mathbb{R}}_+[Z]\) such that \((1)\quad KZ<0\) and \((2)\quad Z_ 1,Z_ 2\) in \(\overline{NE}\) satisfy \(Z_ 1,Z_ 2\in R,\) when \(Z_ 1+Z_ 2\in R.\) R can be numerically effective (n.e.) if \(ZD\geq 0\) for every irreducible divisor D, or not (n.n.e.). The author’s cone theorem (theorem 1.5) states the following. \(\overline{NE}\) is the smallest closed convex cone containing \(\overline{NE}_-\) and the extremal rays; every extremal ray is generated by the class of an extremal rational curve (i.e. a rational curve \(\ell \subset X\) such that \(-\ell K\leq \dim X+1).\) Moreover, in general the extremal rays are not in a finite number (e.g. there are surfaces containing infinitely many exceptional curves); however, any open convex cone in N(X) which contains \(\overline{NE}_-\) leaves out only finitely many extremal rays.

As an introduction to the more difficult case of 3-folds, the author classifies the extremal rational curves on surfaces (theorem 2.1). They are: exceptional curves (n.n.e. extremal rays), fibres of \({\mathbb{P}}^ 1\)- bundles and lines of \({\mathbb{P}}^ 2\) (n.e. extremal rays). This leads to the dichotomy (a), (b) for surfaces, which the author reviews in section 2. Coming to 3-folds, the author shows that for every extremal ray R there is a morphism \(\Phi =Cont_ R: X\to Y\) onto a projective variety Y, such that \(\Phi_*{\mathcal O}_ X={\mathcal O}_ Y\) and \(Cont_ R(C)=point\) for every irreducible curve C lying on R (theorem 3.1). Now, if X contains only n.e. extremal rays, one has \(\dim Y\leq 2\) and this leads to structure theorem (B). On the other hand, if R is n.n.e., then dim Y\(=3\) and the author classifies the exceptional divisors which can occur for \(Cont_ R\) as well as the singularities of Y. This gives reduction theorem (A).

Much progress has been made on this subject, after the publication of the paper under review. An updated list of references can be found in Reid’s translation of Shokurov’s paper ”On the closed cone of curves on algebraic 3-folds” [cf. V. V. Shokurov, Math. USSR, Izv. 24, 193- 198 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, 203-208 (1984)].

The paper under review is the most successful transplanting of the above dichotomy into the classification of algebraic threefolds. In this respect, the main results of the paper [already announced in Proc. Natl. Acad. Sci. USA 77, 3125-3126 (1980; Zbl 0434.14022)] are the following ones. Let X be a nonsingular projective 3-fold defined over an algebraically closed field of characteristic zero, whose canonical divisors are not numerically effective. Then two cases can occur. (A) (theorem 3.3:) There are a morphism \(\Phi:X\to Y\) to a projective 3-fold (singular, in some instances) and an irreducible divisor D on X (which can be of several types), such that \(\Phi\) is an isomorphism on \(X\setminus D\) and contracts D either to a smooth curve or to a point. - (B) (theorem 3.5:) There is a morphism \(\Phi:X\to Y\) onto a nonsingular projective variety Y which exhibits X as (i) a conic bundle, if dim Y\(=2\), (ii) a Del Pezzo fibration, if dim Y\(=1\), (iii) a Fano 3-fold with Picard number 1, if dim Y\(=0.\)

The heart of the paper is the author’s theory of extremal rays, which holds for nonsingular projective varieties defined over an algebraically closed field of any characteristic. Consider the real vector space N(X) of the numerical equivalence classes of 1-cycles on X, with real coefficients (endowed with any norm). Let \(\overline{NE}\) be the closure of the cone in N(X) generated by the classes of the effective 1-cycles and let \(\overline{NE}_-=\{Z\in \overline{NE}| ZK\geq 0\}.\) An extremal ray R is a half line \({\mathbb{R}}_+[Z]\) such that \((1)\quad KZ<0\) and \((2)\quad Z_ 1,Z_ 2\) in \(\overline{NE}\) satisfy \(Z_ 1,Z_ 2\in R,\) when \(Z_ 1+Z_ 2\in R.\) R can be numerically effective (n.e.) if \(ZD\geq 0\) for every irreducible divisor D, or not (n.n.e.). The author’s cone theorem (theorem 1.5) states the following. \(\overline{NE}\) is the smallest closed convex cone containing \(\overline{NE}_-\) and the extremal rays; every extremal ray is generated by the class of an extremal rational curve (i.e. a rational curve \(\ell \subset X\) such that \(-\ell K\leq \dim X+1).\) Moreover, in general the extremal rays are not in a finite number (e.g. there are surfaces containing infinitely many exceptional curves); however, any open convex cone in N(X) which contains \(\overline{NE}_-\) leaves out only finitely many extremal rays.

As an introduction to the more difficult case of 3-folds, the author classifies the extremal rational curves on surfaces (theorem 2.1). They are: exceptional curves (n.n.e. extremal rays), fibres of \({\mathbb{P}}^ 1\)- bundles and lines of \({\mathbb{P}}^ 2\) (n.e. extremal rays). This leads to the dichotomy (a), (b) for surfaces, which the author reviews in section 2. Coming to 3-folds, the author shows that for every extremal ray R there is a morphism \(\Phi =Cont_ R: X\to Y\) onto a projective variety Y, such that \(\Phi_*{\mathcal O}_ X={\mathcal O}_ Y\) and \(Cont_ R(C)=point\) for every irreducible curve C lying on R (theorem 3.1). Now, if X contains only n.e. extremal rays, one has \(\dim Y\leq 2\) and this leads to structure theorem (B). On the other hand, if R is n.n.e., then dim Y\(=3\) and the author classifies the exceptional divisors which can occur for \(Cont_ R\) as well as the singularities of Y. This gives reduction theorem (A).

Much progress has been made on this subject, after the publication of the paper under review. An updated list of references can be found in Reid’s translation of Shokurov’s paper ”On the closed cone of curves on algebraic 3-folds” [cf. V. V. Shokurov, Math. USSR, Izv. 24, 193- 198 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, 203-208 (1984)].

Reviewer: A.Lanteri

### MSC:

14J30 | \(3\)-folds |

14C20 | Divisors, linear systems, invertible sheaves |

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