On the finiteness of generators of a pluricanonical ring for a 3-fold of general type. (English) Zbl 0587.14027

Let X be a complete normal algebraic 3-fold of general type defined over \({\mathbb{C}}\) and let \(K_ X\) be the canonical (Weil) divisor of X. Assume that: \((a)\quad X\quad has\) only canonical singularities (i.e. there exists a positive integer \(\rho\) such that \(\rho K_ X\) is a Cartier divisor and there is a proper birational morphism \(f:\quad X'\to X\) from a non-singular 3-fold inducing a natural homomorphism \(f^*{\mathcal O}_ X(\rho K_ X)\to {\mathcal O}_{X'}(\rho K_{X'}))\); and \((b)\quad K_ X\quad is\) numerically effective. The main result of the paper states that under the above assumptions the canonical ring R(X) of X is a finitely generated \({\mathbb{C}}\)-algebra, where, by definition, \(R(X)=R(X')=\otimes_{m\geq 0}H^ 0(X',{\mathcal O}_{X'},(mK_{X'})).\) The corresponding result for surfaces, defined over an arbitrary algebraically closed field goes back to D. Mumford [Ann. Math., II. Ser. 76, 612-615 (1962); appendix to a paper of O. Zariski, ibid. 560-615 (1962; Zbl 0124.370)]. Due to Hironaka and to results of Fujita the author reduces to work with the desingularization \(f:\quad X'\to X\) and to show that the stable base locus of \(f^*\rho K_ X\) is empty. This is achieved by proving the following general result. Let D be a Cartier divisor on X such that \(D^ 3>0\) and assume that both D and \(D- K_ X\) are numerically effective; then the complete linear system \(| mD|\) is base point-free for some positive integer m. Recent progress on the subject, including generalizations, can be found in the recent paper of the author in Complex analysis and algebraic geometry, Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 41-55 (1986).
Reviewer: A.Lanteri


14J30 \(3\)-folds
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C20 Divisors, linear systems, invertible sheaves
14B05 Singularities in algebraic geometry


Zbl 0124.370
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