Imbedding of an abstract variety in a complete variety

Abstract

The purpose of the present paper is to prove that an arbitrary abstract variety can be imbedded in a com plete variety as an open set.As for the terminology, we shall employ the one in the sequence o f papers of ours in the American Journal of Mathematics ([2] (I, II, III)).We note that we need not assume that a ground ring is a Dedekind domain.Namely, our proof is valid without any modification in the case of m odels over a Noetherian integral domain, models being adapted to the case.T h e re fo re the ground rin g ca n b e rep la ced a lso b y a so-called Noetherian scheme, provided that every localities are integral domains.In § 1 , we state some o f known theorems on birational correspondences.In 2 , we discuss a special kind of birational transformation, called dilatation.In 3 , we give some auxiliary results and in § 4 w e give the proof o f our main theorem.The writer likes to add here that there has been one contribution by J. O h m [4 ] to this problem saying that i f V i s an abstract variety, C is a curve o n V a n d if th e re is a quasi- projective open covering { Ui l o f V such that C meets all the U1 , then there is an abstract variety V ' containing V as an open subset in such a way that the closure o f C in V ' is a complete variety. . B ir a t io n a l correspondences.We consider from now on only models whose function fields 1 ) The work was supported by NSF grant G14736. Masayoshi Nagataare contained in a field, hence correspondences between models are well defined as follows.Let M and M ' be models.W hen M dominates M ', th en the map (f) such that P > clo(P )E M ' for every PE M is a well defined map.This (/) is called the projection or geometric projection or morphism, from M into M ' .(4 )(M ) is not necessarily an open set but contains a non-empty open set of M '.) Th e projection GP is denoted by proj m , m , or proj m , or proj.N ow , in th e general case, M "-J(M , M ') dominates both M and M ' .The correspondence T between M and M ' is defined to be (proj m , , , m , )•(proj m i , , m ) ' .This T is denoted by T m ÷ m i. T m + m t gives in general a many to many correspondence of spots in M and M ' .W e say that a model M of a function field L is complete with respect to a spot P , if, fo r a given function field K containing L and P , the following is true :Every place of K dominating P has a center on M.This property is obviously independent o f th e choice o f K. W e say that a model M is complete with respect to a set M ' of spots if M is complete with respect to every spots of M'. W e sa y th a t2 ) The name of Riemann is added because Zariski [5] called this space "Riemann manifold" in the case of a projective variety, though this is not a Riemann manifold in th e usual sense in differential geometry.The writer believes that the motivation of Zariski for the terminology came from the case of a c u rv e .A n y way, the notion has nearly nothing to do with Riemann, hence the name "Zariski space" is seemingly preferable.But, unfortunately, the term "Zariski space" has been usen in a different meaning.Therefore we are proposing name "Zariski-Riemann space".