Existence theorems for nonprojective complete algebraic varieties
Masayoshi Nagata · Illinois Journal of Mathematics · 1958
The purpose of the present paper is to prove the following two theorems" THEOREM 1.Let L be a function field over a ground field It.Assume that dim L is not less than 2. Assume furthermore that if dim L 2, then lc is sufficiently large.Then there exists a complete normal abstract variety of L which is not projective.THEOREM 2. If n is a natural number not less than 3, then there exists a complete nonsingular variety of dimension n which is not projective; more ex- plicitly, there exists a nonsingular complete variety of the rational function .field of dimension n, .which is defined over the prime field and which is not projective.We shall remark that, since Zariski [4] proved that a normal abstract sur- face can be imbedded in a projective suri'ace (as an open subset) if there exists an affine variety which carries all singular points of the given surface, our results give a complete answer for the imbedding problem in one sense.Therefore it will be an important problem to give some sufficient conditions for a given variety to be projective.It will be also an interesting problem to characterize function fields which have nonsingular complete nonprojective varieties.1. Two lemmas LEMMA 1.Let V and V' be varieties.If V is not projective, then V X V' is not prqective.Proof.V X V' contains a nonprojective subvariety V X P' (P'e V'), and therefore V X V' is not projective.LEMMA 2. Let V be a normal variety with function field L, and let L' be a finite algebraic extension of L. Let V' be the derived normal variety of V in L'.If V' can be imbedded in a projective variety V', then V can be imbedded in a projective variety.Proof.We may assume that V' is an open subset of V".Let P be a generic point of Y over a ground field It, and let Z(P) be P', where P' form