Abstract
Let V be a nonsingular variety defined over an algebraically closed field. Call a divisor D on V arithmetically positive if the intersection number (DiY) is strictly positive for all i-dimensional subvarieties Y of V, i , r, where r is the dimension of V. Call D ample if some multiple nD of i is linearly equivalent to a hyperplane section for some projective embedding of V, i. e., if the rational map of V defined by i nD I is biregular. Clearly if D is ample, it is arithmetically positive. Conversely, the NakaiMoisezon test states, if D is arithmetically positive, it is ample. This test was first discovered by Nakai [4] for nonsingular surfaces. Then Moisezon outlined a proof for higher dimensional nonsingular varieties in [2], and in [3] he suggested a definition of the intersection number (Di Y) on singular varieties and remarked that with that definition the test continues to be valid. Independently, Nakai [5] extended the test to projective algebraic schemes. In Section 1 below, we give a proof of the test, which, in our opinion, is simpler and more self-contained although based on that of Nakai [5]. Since we have stated the test for nonsingular varieties, we cannot immediately apply induction on the dimension r of V. Therefore, we prove the following slightly stronger theorem, in substance stated by Moisezon [2], from which the test follows immediately by taking U V:
Published Version
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