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On certain nets of plane curves

Published online by Cambridge University Press:  24 October 2008

F. P. White
F. P. White
Affiliation:
St John's College
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Extract

1. The plane quartic curves which pass through twelve fixed points g, of which no three lie on a straight line, no six on a conic and no ten on a cubic, form a net of quartics represented by the equation

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 1 , February 1924 , pp. 1 - 10
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* See Küpper, C., “Über geometrische Netze” [Abh. d. k. bōhm. Gesell. d. Wiss., Prag, (7) I (1886)].Google Scholar The existence of such nets affords an example of the exceptions pointed out by Bacharach, [Math. Ann. XXVI (1886), 275]CrossRefGoogle Scholar to a theorem of Cayley's [Papers, I, 25] that a curve of order p=m+n-γ passing through

of the points of intersection of two curves of orders m and n passes also through the remaining

. This no longer holds if these

points lie on a curve of order γ -3. In our case m=n=γ=4. This example has been given for many years by Mr Richmond in his lectures on Plane Algebraic Geometry at Cambridge.

† See White, F. P., Proc. Camb. Phil. Soc. XXI (1922), 216.Google Scholar

* Projecting from P on to a hyperplane, we get a quintic surface in ordinary space with a double twisted cubic curve; this surface was investigated years ago by Clebsch, [Math. Ann. 1 (1869), 284]Google Scholar; the hyperelliptic curve is the representation of the double curve.