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A figure in space of seven dimensions, and its sections

Published online by Cambridge University Press:  24 October 2008

W. L. Edge
W. L. Edge
Affiliation:
Trinity College
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Extract

Several papers have recently been published which include proofs of the following:

Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 1 , January 1934 , pp. 19 - 26
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

* Welchman, W. G., Proc. Camb. Phil. Soc. 28 (1932), 275CrossRefGoogle Scholar; H. G. Telling, Ibid. 403; D. W. Babbage, Ibid. 421; Semple, J. G., Journal London Math. Soc. 7 (1932), 266.CrossRefGoogle Scholar

† James, C. F. G., Proc. Camb. Phil. Soc. 21 (1923), 673.Google Scholar

* Cf. Edge, , Proc. London Math. Soc. (21), 33 (1931), 53.Google Scholar

* The locus is rational and is represented on [3] by means of cubic surfaces through three skew lines, and in other ways.

* A referee points out that these three systems of [5]'s are dual to the three systems of lines on . In this connection see also James, loc. cit. 674–675. Just as a trisecant plane of determines, by means of the three directrices through its intersections with , a secant [5], so a [4] which meets three generating solids of each in a line contains a directrix of , namely the transversal of the three lines. Viewed from this aspect Theorem 2 is the dual of the statement that any [4] which contains a directrix σ of lying on meets some generating solid of in a line and some generating solid of in a line. The truth of this statement is obvious; both the lines in question are the line σ itself, which is the dual of the space σ6 of Theorem 2.

* This configuration of three lines and three twisted cubics occurs in Welchman's paper: Proc. Camb. Phil. Soc. 28 (1932), 416–420 (418).Google Scholar

* Cf. Telling, loc. cit. 415.

† Segre, , Atti Torino, 19 (1884), 355.Google Scholar

‡ Cf. Baker, , Principles of Geometry, 4 (Cambridge, 1925)Google Scholar, chapter 5; Semple, loc. cit. 270.

* Segre, , Math. Annalen, 27 (1886), 296–314 (302).CrossRefGoogle Scholar