Generalized adjunction and applications

Abstract

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 ℙn 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 Kx + iH is not semiample (respectively ample) for 1 ≤ i = r + 1, r, r − 1, r − 2 (respectively i = r + 1, r, r − 1).