Algebraic Geometry the Fundamental Ideas of Abstract Algebraic Geometry

ALGEBRAIC GEOMETRY 79 It may be advisable to give a special name to those varieties which admit every (allowable) ground field as field of definition. Obviously, these are the varieties which are defined over the prime field of the given characteristic p. I propose to call them universal varieties. The projective space and the Grassmannian varieties are examples of universal varieties. Another important class of universal varieties is obtained by considering the set of all algebraic varieties, of a given order and dimension, in the w-dimensional universal projective space and introducing in that set an algebro-geometric structure based on the Chow coordinates of a cycle. The study of these varieties (of which the Grassmannian varieties are special cases) is closely connected with the outstanding problem of developing a theory of algebraic equivalence of cycles on a given variety, and will no doubt be a fundamental object of future research. The definition of a variety as a set of points having coordinates in the universal domain has some startling, and perhaps unpleasant, set-theoretic implications. We have populated our varieties with points having coordinates which are transcendental over k. Thus, if x and y are independent variables the pair (x, y) is a legitimate point of the plane; and—what is worse—if a/and y are other independent variables, then (x, y') is another point of the plane, quite distinct from the point (x, y). This is shocking, especially if we recall that our universal domain has infinite transcendence degree and that consequently we have created infinitely many replica of that ghostlike point (x, y). However, we are dealing here with a methodological fiction which is extremely useful in proving very real theorems. For instance, the entire theory of specializations is based on this set-theoretic conception of a variety, and the entire elementary theory of algebraic correspondences can be developed on that basis in the most effortless and simple fashion. Furthermore, most results concerning irreducible subvarieties of a given variety can be best expressed and derived as results concerning the general points of these subvarieties. Nevertheless fiction remains fiction even if it is useful, and I feel that perhaps our varieties have altogether too many points to be good geometric objects. As the theory progresses beyond its foundational stage, some cuts and reductions may become necessary. Thus, one may begin first of all by eliminating isomorphic replica of points, by identifying points which are isomorphic over the given ground field k. Or one may restrict the coordinate domain to the algebraic closure of k. Or one may do both of these things at the same time. I have no strong convictions on these issues, and I am quite content in leaving their settlement to the future development of the theory of algebraic varieties. But to round up this discussion, let me indicate briefly some topological aspects of these issues. Given a variety V and given any field k (not necessarily a field of definition of V), there is a natural topology on V, relative to k: it is the topology in which the closed sets arc intersections of V with varieties which are defined over A. In particular, if V itself is defined over k, then the closed sets on V are the subvarieties of V which are also defined over k, and in terms of this topology