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Point-curve correspondences

II. Induced and extended correspondences

Published online by Cambridge University Press:  24 October 2008

D. B. Scott
D. B. Scott
Affiliation:
Queen Mary CollegeUniversity of London
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Extract

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 3 , October 1946 , pp. 229 - 239
Copyright
Copyright © Cambridge Philosophical Society 1946

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References

REFERENCES

(1)Lefschetz, S.Correspondences between algebraic curves. Ann. Math. (2), 28 (1972), 342–54.Google Scholar
(2)Scott, D. B.Invariant groups associated with an algebraic surface. Proc. Cambridge Phil. Soc. 36 (1940), 414–23.CrossRefGoogle Scholar
(3)Scott, D. B.Point-curve correspondences. I. Proc. Cambridge Phil. Soc. 41 (1945), 135–45.Google Scholar
(4)Severi, F. Le corrispondenze fra i punti di una curva variable in un sistema lineare sopra una superficie algebriche. Math. Ann. 74 (1913), 515–44.Google Scholar