Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T22:10:23.143Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Topology of Certain Algebraic Threefold Loci

Published online by Cambridge University Press:  20 January 2009

J. A. Todd
J. A. Todd
Affiliation:
University of Manchester.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The study of the topological properties of algebraic surfaces, considered as continua of four real dimensions, has thrown much light on the theory of the birational invariants of such loci. The results obtained for surfaces have been generalised to varieties of higher dimension by Hodge, and, particularly, by Lefschetz. Apart from this, little seems to be known about the general topological properties of algebraic loci of three (or more) dimensions, the detailed study of which seems to present considerable difficulty. In particular, apart from the general theorems of Lefschetz, nothing seems to be known about the cycles of three dimensions of an algebraic V3. The object of the present paper is to study these cycles on certain quite special V3, in the hope that some insight may be gained into the general theory.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 4 , Issue 3 , August 1935 , pp. 175 - 184
Copyright
Copyright © Edinburgh Mathematical Society 1935

References

REFERENCES

1.Baker, H. F.Principles of Geometry (Cambridge), III (1923), VI (1933).Google Scholar
2.Coxeter, H. S. M. “The polytopes with regular-prismatic vertex figures,” Part 2, Proc. London Math. Soc. (2) 34 (1932) 126–189.Google Scholar
3.Coxeter, H. S. M., “Finite groups generated by reflections, and their subgroups generated by reflections.” Proc. Comb. Phil. Soc, 30 (1934) 466–482.CrossRefGoogle Scholar
4.Fano, G., “Sulle superficie algebriche contenute in una varieta cubica dello spazio a quattro dimensioni.” Atti Ace. Torino, 39 (1904) 597–615.Google Scholar
5.Fano, G., “Sul sistema ∞2 di rette contenute in una varietà cubica generale dello spazio a quattro dimensioni.” Atti Ace. Torino, 39 (1904) 778–792.Google Scholar
6.Hodge, W. V. D., “On multiple integrals attached to an algebraic variety.” Jour. London Math. Soc., 5 (1930) 283–290.CrossRefGoogle Scholar
7.Hodge, W. V. D., ” Proc. Camb. Phil. Soc., 31 (1935) 18–25.CrossRefGoogle Scholar
8.Lefschetz, S., L'analysis situs et la géométrie algébrique (Paris, 1924).Google Scholar
9.Lefschetz, S., Géométrie sur lea surfaces et les variétés algébriques (Mémorial des Sciences Mathématiques, 40, 1929).Google Scholar
10.Lefschetz, S., Topology (New York, 1930).Google Scholar
11.Lefschetz, S., “Invariance absolueet invariance relative en géométrie algébrique.” Recueil Math. Soc. Moscou, 39 (1932) 98–103.Google Scholar
12.Schoute, P. H., “On the relation between the vertices of a definite six-dimensional polytope and the lines of a cubic surface.” Proc Roy. Acad. Sci. (Amsterdam) 13 (1910) 375–383.Google Scholar
13.Segre, C., Mehrdimensionale Räume (Encyk. Math. Wiss., IIIC7).Google Scholar
14.Todd, J. A., “Polytopes associated with the general cubic surface.” Jour. London Math. Soc. 7 (1932) 200–205.CrossRefGoogle Scholar
15.Zariski, O., Algebraic Surfaces (Ergebnisse der Math. III 5, 1935).Google Scholar