By Leonard Roth
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Extra info for Algebraic Threefolds: With Special Regard to Problems of Rationality
SEVERI has also established the duality law that the base number for varieties ~ is equal to that of varieties Va- k • In the case where Va is either a Grassmannian or a SEGRE variety, the problem has been solved from first principles, independently of the above result. Thus, on a Grassmannian, a base for subvarieties of any given dimension is provided by the SCHUBERT varieties of that dimension (see SEVERI, c; HODGE and PEDOE, a). The various bases for submanifolds on a SEGRE variety have been determined by BENEDICTY .
The adjoint curves of order n - 2 cut on W~ a linear series g~=~, determinable in K, which, if n > 3, can be mapped on the prime sections of a non-singular normal curve V~-2. By repetition of this process, we can transform V~ birationally in K either to a twisted cubic or a conic, according as n is odd or even; and in the former case we can further transform V~ birationally to a line, for the twisted cubic projects into a monoid. Hence, any nonsingular rational curve of order n can be transformed birationally, in K, to a line or a conic, according as n is odd or even (N OETHER [1 J).
E. not reflected in automorphisms of t(x, y, z, ... ). Examples of these phenomena are given in ROTH [I5J, where the corresponding theory for fourfolds is also illustrated. The problem of the base for subvarieties Vk of any dimension k on Va has been solved by SEVERI [7J, using topological methods, subject to the hypothesis that, on ~, arithmetical equivalence of subvarieties implies algebraic equivalence. SEVERI has also established the duality law that the base number for varieties ~ is equal to that of varieties Va- k • In the case where Va is either a Grassmannian or a SEGRE variety, the problem has been solved from first principles, independently of the above result.