A Criterion of an Ample Sheaf on a Projective Scheme

Abstract

In his previous paper [2],2 the author proved a criterion of an ample divisor 3 on a non-singular surface, i. e., a divisor X on a non-singular surface F is ample if and only if (X2) > 0 and X is arithmetically positive.4 In this paper he will prove a generalization of this result to a projective scheme over a field. Originally the author intended to prove this generalization only for a non-singular variety of any dimension, say n. However, in the course of the proof it became necessary to treat the problem on a variety of dimension n-1 with singularities, and then on a scheme of dimension n2. This is the reason why he finally decided to treat the problem on a general projective scheme right from the beginning. The author wishes to express his heartfelt thanks to Professor 0. Zariski and to D. Mumford. Without their encouragement and suggestions this work would not have been done. In this paper we shall make extensive use of notations, terminologies, and the results in Grothendieck's book, flements de G-4ometrie Algebrique, which will be cited as [G]. Most of them will be used freely without any further explanations, except some less fundamental notions. We hope the readers will not find much inconvenience in this way.