Generalized adjunction and applications

The linear system |K + C| ‘adjoint’ to a curve C on a projective surface was studied by the classical Italian geometers. The adjoint system to a hyperplane section H of smooth projective surface was investigated systematically, in modern terms, by Sommese [22] and Van de Ven [26]. The map associated to the linear system | K + ( r −1) H |, where H is a hyperplane section of a smooth variety of arbitrary dimension r , was used to classify submanifolds of ℙ n with ‘small invariants’ (e.g. degree, sectional genus, etc.); see [10]. On the other hand, Sommese [ 23, 24, 25 ] studied adjoint systems to a smooth ample divisor H on a smooth threefold X and obtained, as applications, many interesting results about the pair ( X, H ). As noticed independently by several authors (see e.g. [17], [4], [11]) the appearance of Mori's deep contribution [20] (see also [21]) put the subject of adjunction in a new perspective. Accordingly, the present paper–which relies heavily on Mori's results and on the contraction theorem due to Kawamata-Shokurov (see [14])–contains a systematical study of various adjoint systems to an ample (possibly non-effective) divisor on a manifold of arbitrary dimension. More precisely, the main result (which is contained in Section 1) gives the precise description of polarized pairs ( X, H ), where X is a complex projective mani–fold of dimension r and H an ample divisor on it (not necessarily effective), such that K x + iH is not semiample (respectively ample) for 1 ≤ i = r + 1, r , r − 1, r − 2 (respectively i = r + 1, r , r − 1).