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**Zariski decomposition of divisors on algebraic varieties.**
*(English)*
Zbl 0604.14002

Let V be a smooth projective variety over \({\mathbb{C}}\) and let D be an effective divisor on V. If V is a surface the Zariski decomposition of D is the datum of the largest numerically effective \({\mathbb{Q}}\)-divisor P such that \(N=D-P\) is an effective \({\mathbb{Q}}\)-divisor [see O. Zariski, Ann. Math., II. Ser. 76, 560-615 (1962; Zbl 0124.370)]. Different extensions of this notion to higher dimensional varieties, in view of the solution of the problem of the finite generation of the canonical ring of varieties of general type, have been proposed by various authors: X. Benveniste [Invent. Math. 73, 157-164 (1983; Zbl 0539.14025)], T. Fujita in Classification of algebraic and analytic manifolds, Proc. Symp., Katata/Jap. 1982, Prog. Math. 39, 65-70 (1983; Zbl 0525.14004)], A. Morikawi [”Semi-ampleness if the numerically effective part of Zariski decomposition” and ”Several properties of Zariski decomposition” (both preprints)], and the author in the paper under review. The definition proposed by the author is the following: \(D=D'+F\) is a Zariski decomposition if D’ is an effective and numerically effective \({\mathbb{Q}}\)-divisor and \(h^ 0(nD)=h^ 0([nD'])\) for any \(n>0\), [nD’] denoting the integral part of nD’.

Then two results are proved. First it is shown, by giving an example, that there are divisors D on some threefold V such that for any birational morphism \(f: V'\to V\) the pull-back \(f^*(D)\) does not have a Zariski decomposition (This phenomenon can only happen, as the author shows, if the D-dimension of the divisor is 3). - The second main result is that the canonical ring of a threefold V of general type is finitely generated if there is a canonical divisor K of V and a birational morphism \(f: V'\to V\) such that \(f^*(K)\) has a Zariski decomposition. It should however be mentioned that the recent conclusion of Mori’s minimal model program implies the finite generation of the canonical ring of any threefold of general type [S. Mori, ”Flip theorem and the existence of minimal models for threefolds” (preprint)]. An excellent account on the subject of minimal models and canonical rings may be found in S. Mori’s expository paper ”Classification of higher dimensional varieties” (preprint) to appear in the proceedings of the Summer Institute in Algebraic Geometry of the Am. Math. Soc., Bowdoin 1985.

Then two results are proved. First it is shown, by giving an example, that there are divisors D on some threefold V such that for any birational morphism \(f: V'\to V\) the pull-back \(f^*(D)\) does not have a Zariski decomposition (This phenomenon can only happen, as the author shows, if the D-dimension of the divisor is 3). - The second main result is that the canonical ring of a threefold V of general type is finitely generated if there is a canonical divisor K of V and a birational morphism \(f: V'\to V\) such that \(f^*(K)\) has a Zariski decomposition. It should however be mentioned that the recent conclusion of Mori’s minimal model program implies the finite generation of the canonical ring of any threefold of general type [S. Mori, ”Flip theorem and the existence of minimal models for threefolds” (preprint)]. An excellent account on the subject of minimal models and canonical rings may be found in S. Mori’s expository paper ”Classification of higher dimensional varieties” (preprint) to appear in the proceedings of the Summer Institute in Algebraic Geometry of the Am. Math. Soc., Bowdoin 1985.

Reviewer: C.Ciliberto

### Keywords:

effective divisor; finite generation of canonical ring of a threefold; Zariski decomposition
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### References:

[1] | X. Benveniste, Sur l’anneau canonique de certaines variétés de dimension \(3\) , Invent. Math. 73 (1983), no. 1, 157-164. · Zbl 0539.14025 |

[2] | T. Fujita, Semipositive line bundles , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 353-378. · Zbl 0561.32012 |

[3] | T. Fujita, Canonical rings of algebraic varieties , Classification of algebraic and analytic manifolds (Katata, 1982) ed. K. Ueno, Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 65-70. · Zbl 0525.14004 |

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[6] | Y. Kawamata, Pluricanonical systems on minimal algebraic varieties , Invent. Math. 79 (1985), no. 3, 567-588. · Zbl 0593.14010 |

[7] | S. Kleiman, Toward a numerical theory of ampleness , Ann. of Math. (2) 84 (1966), 293-344. JSTOR: · Zbl 0146.17001 |

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[9] | P. M. H. Wilson, On the canonical ring of algebraic varieties , Compositio Math. 43 (1981), no. 3, 365-385. · Zbl 0459.14006 |

[10] | O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface , Ann. of Math. (2) 76 (1962), 560-615. JSTOR: · Zbl 0124.37001 |

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