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Two Formulae for Space Curves

Published online by Cambridge University Press:  24 October 2008

J. W. Archbold
J. W. Archbold
Affiliation:
St John's College
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Extract

We consider in space [3] a curve C of order n and genus p without multiple points. If we represent the lines of [3] by the points of a quadric Ω in [5], the chords of C will be represented by the points of a surface F of order (n−1)2−p lying on Ω. This surface has a triple curve M (with multiple points) corresponding to the ruled surface of trisecants of C (and the quadrisecants) of order ⅓(n−1)(n−2)(n−3)−p(n−2). It is the object of this note to find the genera of M and of a prime section ϑ of F; these being also the genera of the ruled surface of trisecants of C and of the ruled surface of chords of C which belong to a linear complex.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 3 , July 1930 , pp. 358 - 360
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* Baker, H. F., Principles of Geometry, IV, 50.Google Scholar

† Enriques-Chisini, , Teoria Geometrica delle Equazioni, III, 471.Google Scholar

‡ This number is equal to the number of tangents of C meeting x, which is the rank r.

* Zeuthen, , “Sur les singularités des courbes gauches,” Annali di Matematiche, (2) 3 (1869) 186.Google Scholar