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Some enumerative results for curves

Published online by Cambridge University Press:  24 October 2008

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

1. The problem of finding the number of space cubic curves which pass through p given points and have 6 – p given lines as chords has been solved by several different methods. The similar problem, in space of four dimensions, of the number of rational quartic curves which pass through p given points and have 7 – p given trisecant planes has been solved for p > 1 by F. P. White and for p = 1 by J. A. Todd. In a recent paper I discussed a similar problem for elliptic quartic curves. My present object is to apply the method of that paper to the problems mentioned above and to obtain two other sets of numbers which I believe to be new. They are (1) the number of rational normal quartic curves which have three assigned chords, pass through p assigned points, and have 3 – p assigned trisecant planes; (2) the number of curves of intersection of three quadrics in [4] which have a suitable number of assigned points and chords. The evaluation of Schubert symbols which occur in the work is done by means of a formula due to Giambelli‖. This formula is stated in a more simple form in a note at the end of this paper (7).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 1 , January 1932 , pp. 18 - 22
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* See, for instance, Baker, , Principles of Geometry, 3 (1923), 139–142.Google Scholar

† White, , Journ. Lond. Math. Soc., 4 (1929), 11.CrossRefGoogle Scholar

‡ Todd, , Proc. Camb. Phil. Soc., 26 (1930), 323.CrossRefGoogle Scholar

§ Welchman, , Proc. Camb. Phil. Soc., 27 (1931), 20.CrossRefGoogle Scholar

|| See Segre, , “Mehrdimensionale Räume”, Encykl. Math. Wiss., iii c 7, 818Google Scholar.

* See Segre, , “Mehrdimensionale Räume”, Encykl. Math. Wiss., iii c 7, 815Google Scholar, and compare § 7, below.

† This method is due to Prof. J. G. Semple.

* The Schubert symbols are explained by Segre, , Encykl. Math. Wiss., iii c 7, 795Google Scholar. For the calculation of the numbers see § 7, below.

* See Segre, loc. cit., 815.