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Additional note on plane congruences and fifth incidence theorems

Published online by Cambridge University Press:  24 October 2008

W. G. Welchman
W. G. Welchman
Affiliation:
Sidney Sussex College
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Extract

In a recent paper I discussed plane congruences of order two in [4] and obtained congruences of types (2, 6)1, (2, 6)2, (2, 5), (2, 4) and (2, 3). The method employed was due to Segre, who showed that a plane congruence of order two in [4] has in general a curve locus of singular points which is met by each plane in five points. Then, if we can find a curve in [4], composite or not, with an ∞2 system of quadrisecant planes of which two pass through an arbitrary point, the planes must all meet a residual curve, and we shall have obtained a congruence of the second order and a fifth incidence theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 4 , October 1932 , pp. 416 - 420
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

† Welchman, , Proc. Camb. Phil. Soc., 28 (1932), 275–284CrossRefGoogle Scholar; I shall refer to this paper as W.

‡ Todd, , Proc. Lond. Math. Soc. (2), 33 (1932), 328–52.CrossRefGoogle Scholar

§ The general ones should have been obtained in W, but I overlooked a valid degeneration.

|| This process is of course equivalent to dualising in the normal space of each threefold locus and taking a section by a [4].

* See the table given in W, p. 276.

* In the configuration obtained in W, § 6·3, one of the cubics has degenerated into three lines.

† Brown, L. M., Journal Lond. Math. Soc., 5 (1930), 168–176.CrossRefGoogle Scholar

‡ Telling, H. G., Proc. Camb. Phil. Soc., 28 (1932), 403–415.CrossRefGoogle Scholar

§ Babbage, D. W., Proc. Camb. Phil. Soc., 28 (1932), 421–426.CrossRefGoogle Scholar

|| See, for instance, Todd, loc. cit., 343.