Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T14:16:46.531Z Has data issue: false hasContentIssue false

Analytical degeneration of complete twisted cubics

Published online by Cambridge University Press:  24 October 2008

A. R. Alguneid
Affiliation:
Faculty of ScienceCairo University

Extract

In Schubert's book Abzählende Geometrie(3) there are listed, with no formal verifications, eleven first-order degenerations of the twisted cubic curve in S3. The curve, in this connexion, was seen by Schubert as a union C = (CL, CE, CT) of three simply infinite element systems, the locus CL of its points, the envelope CE of its osculating planes and the system CT of its tangent lines, the complete twisted cubic so envisaged being regarded as a single geometric variable of freedom 12, and any degeneration (projective specialization) of it, = (L, E, T), being said to be of the first order if it has freedom 11. Schubert used the symbols λ, λ′, k, kw, w′, θ, θ′, δ, δ′, η to denote the eleven first-order degenerations in question, the dashed symbol denoting always the dual of that denoted by the undashed. In this paper we exhibit analytically the existence of the above degenerations and indicate further how degenerations of higher order can be systematically investigated. We use for this purpose the analytical theory of complete collineations which has just recently been developed (1,4), and also some of the ideas contained in van der Waerden's earlier development of his general method of degenerate collineations (5).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1956

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Alguneid, A. R.Degeneration of space collineations. Proc. Egypt. Acad. Sci. 7 (1951), 117.Google Scholar
(2)Dixon, A. C.On twisted cubics which fulfil certain given conditions. Quart. J. pure appl. Math. 23 (1889), 343–57.Google Scholar
(3)Schubert, H.Kalkul der abzählenden Geometrie (Leipzig, 1879).Google Scholar
(4)Semple, J. G.The variety whose points represent complete collineations of S r on . E.C. Math. R. Univ. Roma (5), 10 (1951), 18.Google Scholar
(5)Van Der Waerden, B. L.Zur Algebraische Geometrie, XIV, XV. Math. Ann. 115 (1938), 619–42 and 645–55.CrossRefGoogle Scholar
(6)Wood, P. W.The twisted cubic (Camb. Tracts Math. no. 14, 1913).Google Scholar