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On the canonical curve of genus five

Published online by Cambridge University Press:  24 October 2008

H. W. Richmond
H. W. Richmond
Affiliation:
King's College
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Extract

This paper contains an attempt, begun several years ago and only partially successful, to do for the canonical curve of genus five something similar to what W. P. Milne had done for that of genus four. A fundamental feature of Milne's work was the use of a rational normal curve of order three drawn through the six points of contact of a quadric with the sextic curve; here it is not in general possible to pass a rational curve of order four through the eight points of contact. (Exceptionally it may be possible to do so and then the development follows much the same lines as in Milne's paper: only the general case is discussed in what follows.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 2 , April 1932 , pp. 174 - 189
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Proc. Lond. Math. Soc. (2), 21 (1923), 373–380Google Scholar; 26 (1927), 119–134; 28 (1928), 484–492.

† In four dimensions the terms used are line, plane, prime for flat loci, curve, surface, primal for curved loci. Quadric means a primal of order two, a locus defined by a single equation of the second degree in the coordinates of points.

‡ An exception occurs in connection with the plane quintic curve with one node, and the curves birationally transformable into it. The three quadrics are then found to have in common a cubic scroll, and so do not define the curve.

* Should the prime touch the cone at all points of a line through A, the root is found to be triple. The next section indicates the line of proof of these statements sufficiently.

* This is not the only possible configuration of the base-points. It can for example be arranged that the cubics should all pass through three definite points and have definite tangents at two of them.

* We are here associating any one of the 120 bitangent planes of the curve with any one of the 1023 families of contact-quadrics; wherefore it appears that the loci and equations that we discover present themselves 122,760 times in connection with the curve.

† Another form for D, apparently simpler but actually less convenient, is obtained by adding IΔ or I (Aa+Hh+Gg) to the value of D 0 that has been found

With this value of D it is remarkable that, of the ten sextic functions of x, y, z which appear as coefficients of the squares and products of ξ, η, ζ, ω in the equation of the general contact-sextic, all except one (the coefficient of ξ, ω) are divisible by xy.