Birational transformations possessing fundamental curves

Abstract

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.