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Symbolical algebra and the quadrics containing a rational curve

Published online by Cambridge University Press:  20 January 2009

H. F. Baker
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In regard to the algebra of binary forms and the theory of rational curves there exists a wide literature, with which I am not well acquainted. Very possibly the simple remark made in this note is found elsewhere. But the note is a grateful echo of recent delightful colloquy with persons of good will.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 44 , February 1925 , pp. 131 - 143
Copyright
Copyright © Edinburgh Mathematical Society 1925

References

page 131 note * Cf. , Segre, Enzykl. Math. III, Mehrdimensionale Räume, No. 27 (1912; published 1921)Google Scholar. He refers to Clifford, Classification of Loci, 1878 (Papers, p. 311); Schoute, P. H., Proc. Akad. Amsterdam I (1899), p. 313Google Scholar; Brusotti, , Ann. di. Mat. IX (1904), p. 311CrossRefGoogle Scholar, and Rend. 1st. Lomb., XLII (1909), p. 144Google Scholar.

page 131 note † Skeat Concise Etymol. Dict, gives, under Colloquy, “see Loquacious”; under which however are given Soliloquy, and Obloquy. Floreat Congressus Mathematicus.

page 133 note * It is remarked by Clifford (loc. cit.) that for this rational quartic curve, the four points whose osculating threefolds pass through an arbitrary point of the fourfold space lie on the polar threefold of the point in regard to the quadric; with a similar result for the rational curve of even order in its own normal space of the same order. That the tangents of the quartic curve lie on the quadric is recognised by Brusotti (loc. cit.) The tangents also lie on the cubic locus expressed by the vanishing of the determinant of which the three rows consist respectively of the elements x 0, x 1, x 2; x 1, x 2, x 3; x 2, x 3, x 1. Segre (loc. cit.), following Schoute (loc. cit.), remarks that if we take the locus of the (l–1) folds joining l points of the rational curve of order n in n – fold space, where , this locus, which is of dimension 2l – 1, and of order satisfies the equations,

For n = 3, this gives the equations of the cubic curve; for n = 4, beside the equations of the curve, it gives the equation of the cubic threefold, just referred to, containing all the chords of the curve and in particular its tangents; and so on.

page 134 note * See Wood, P. W., Cambridge Math. Tracts, No. 14, p. 21Google Scholar. Professor Dixon's discovery is equivalent to saying tliat four tangents of a cubic curve in space cut, upon their two transversals, ranges whose cross ratios are – p 3q, and – pq 3, where p, q are two numbers which are harmonic conjugates with respect to the two imaginary cube roots of unity.

page 134 note † I should like to mention that this relation was remarked to me by Professor Turnbull.

page 143 note * In general we have

where , as we may see by considering