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Three related quartic curves in four dimensions

Published online by Cambridge University Press:  24 October 2008

H. G. Telling
H. G. Telling
Affiliation:
Newnham College
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Extract

1. It has been shown by Pieri, and independently by James, that the trisecant planes of a quartic curve in [4] which meet a line meet also another quartic curve intersecting the former in six points. James generalises the theorem and shows that trisecant planes of a quartic curve C, which meet a quartic curve C1 having six points in common with C, also meet another quartic curve C2, and that the relation between the three curves is symmetrical. The object of this note is to give a simple proof of this theorem and to discuss the representation by which this proof is effected. The method used is analogous to that of Pieri; an interesting differential method is adopted by C. Segre who shows that the foci of the first and second orders, of any linear system of ∞2 planes in [4] of which two planes pass through a point, determine a conic and five points respectively, in any plane of the system: the trisecant planes of C meeting C1 give a particular case of such a system.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 4 , October 1932 , pp. 403 - 415
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Gior. di Mat., 28 (1890), 209.Google Scholar

† Proc. Camb. Phil. Soc., 21 (1923), 664.Google Scholar

‡ Rendiconti Lincei, 30 (1921), 67.Google Scholar

* This locus is not that of the vertices of all quadric coues containing C and C 1: there are ∞1 points, which are intersections of chords of C 1 and C 2, from which these curves project into nodal quartics which are (2, 2) upon a quadric.

* Baker, H. F., Principles of Geometry, 4 (1925), 114.Google Scholar

† This figure arises significantly by projection from three points of itself of the discussed by James (loc. cit.).

* Gordan, , Invariantentheorie, 2 (1887), 298Google Scholar; Meyer, , Apolarität (1883), 273.Google Scholar

† Cf. Wood, , Twisted Cubic (1913), 33.Google Scholar

* Cf. Jessop, , Line Complex (1903), 280Google Scholar, for order of the singularities.

* This is the primal which is obtained by James (loc. cit.) as the section of a sextic locus in [7].