On polarized normal varieties, I

Abstract

In this paper, we deal with the first part of an application of our Riemann-Roch type inequalities (cf. [13], [23]) toward deformations of polarized normal varieties. In Chapter I, we discuss the problem of eliminating fixed components from complete linear systems defined by multiples of a given divisor. Let Un be a complete normal variety and Y0 an ample Cartier divisor on U. The main result of the chapter is that there is a positive integer c0, predicted by the first two leading coefficients of the polynomial X(U, O(m Y0)), such that the complete linear system Λ(rc0Y0) has no fixed component whenever r is a positive integer (which is essentially contained in Theorem 1.1 (cf. [13], Lemma 5.2)). An easy consequence of this result is that when n = 2, we can find another positive integer c1 predicted by the same coefficients as above, such that rc1c0Y0 is very ample on U whenever r is a positive integer. Even though this has been generalized to n = 3 by J. Kollár (cf. [12]), we have included this in Section 3 since it is very simple.