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Birational transformations possessing fundamental curves

Published online by Cambridge University Press:  24 October 2008

J. A. Todd
J. A. Todd
Affiliation:
Trinity College
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 2 , April 1938 , pp. 144 - 155
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

REFERENCES

(1)Bassi, A.Rend. di Palermo, 60 (1937), 107–23.CrossRefGoogle Scholar
(2)Segre, B.Mém. Acad. Royale de Belgique (2), 14 (1936), 1–99.Google Scholar
(3)Todd, J. A.Proc. London Math. Soc. (2), 43 (1937), 127–38.Google Scholar
(4)Todd, J. A.Proc. London Math. Soc. (2), 43 (1937), 190–225.Google Scholar
(5)Todd, J. A.Proc. Cambridge Phil. Soc. 33 (1937), 425–37.CrossRefGoogle Scholar
(6)Todd, J. A.Proc. Edinburgh Math. Soc. (2), 5 (1938), 117–24.Google Scholar