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Generalized adjunction and applications. (English) Zbl 0619.14004
The main purpose of the paper under review is to give a precise description of polarized pairs (X,H), where X is a complex projective manifold of dimension \(r\) and H is an ample divisor on it (not necessarily effective) such that \(K_ X+iH\) is not semiample (respectively ample) for \(1\leq i=r+1, r, r-1, r-2\) (respectively \(i=r+1, r, r-1)\). For example \(K_ X+(r+1)H\) is always semiample and ample except for the case where \(X\simeq {\mathbb{P}}^ r\) and \(H\in | {\mathcal O}(1)|\). Among the exceptions for smaller adjunctions the author obtains scrolls over curves and \(Del\quad Pezzo\quad manifolds\) [T. Fujita, ”On the structure of polarized manifolds with total deficiency one. I, II and III”, J. Math. Soc. Japan 32, 709-725 (1980; Zbl 0474.14017); 33, 415-434 (1981; Zbl 0474.14018); 36, 75-89 (1984; Zbl 0541.14036)].
The exact statement is contained in section \(1.\) Its proof is based on cone [S. Mori, Ann. Math., II. Ser. 116, 133-176 (1982; Zbl 0557.14021)] and contraction theorems [Y. Kawamata, Ann. Math., II. Ser. 119, 603-633 (1984; Zbl 0544.14009)]. Some applications are given in section \(2.\) Here you may find slightly improved results of A. J. Sommese [”The birational theory of hyperplane sections of projective threefolds” (preprint 1981)] and an alternative proof of the Bădescu theorem classifying smooth projective threefolds the support of which is a geometrically ruled surface as an ample divisor and so on.
Reviewer: V.V.Shokurov

MSC:
14C20 Divisors, linear systems, invertible sheaves
14J99 Surfaces and higher-dimensional varieties
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