On Polarized Varieties of Dimension 3 (I)

Abstract

Introduction. The problem we are interested in dealing with here is to find a reasonable upper bound for h2(V, ((rX)) for a nondegenerate divisor X on a non-singular projective 3-fold V, which satisfies l(rX) > 0, so that we can claim the following: l(rX) stays positive and maintains a reasonable value while a polarized (V, X) is being deformed globaly. The problem involves in part to find such a positive integer r too. Of course, a main purpose of doing this is to extend our result in [4] to polarized 3-folds over the universal domain of characteristic p. We first attempted this in [5], essentially under the assumption that either a suitable pluricanonical system or antipluricanonical system is free from fixed curves. But unfortunately our arguments contained some incompleteness in the discussion of the first case. In this paper, we shall extend the ideas and results in [5] to the case when a suitable pluricanonical system is not empty. Since the case when all pluri-genera vanish require additional discussions, this case will be treated in a paper which will follow this. After discussing some preliminary results, we discuss an elementary result on ruled surfaces U which states that l(Ku + 3X) > 0 when X is arithmetically effective and X(2) > 0 (X may not be positive), which is not self-evident for characteristic p. Then after expounding effects of monoidal transformations, we discuss a criterion of a subvariety D of codimension 1 of a non-singular projective 3-fold being a ruled surface. These are necessitated by a possibility that a fixed component of A(rX) may be a ruled surface, which requires a separate treatment. After these preparations, a choice of r is made in Section 1, Chap. II, and then we directly estimate an upper bound for h2(O(rX)), showing that our requirement is satisfied.