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A configuration of lines in three dimensions

Published online by Cambridge University Press:  20 January 2009

J. W. P. Hirschfeld
J. W. P. Hirschfeld
Affiliation:
University of Sussex, England
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In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew linesa1, …, a5with a single transversal b6such that no four of the ai lie in a regulus, the four ai excluding aj have a second transversal bj and the five lines b1, …, b5thus obtained have a transversal a6—the completing line of the double-six. The other fifteen lines of the cubic surface are then , where ai bj is the plane containing ai and bj.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 2 , December 1972 , pp. 105 - 123
Copyright
Copyright © Edinburgh Mathematical Society 1972

References

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