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A plane sextic and its five cusps

Published online by Cambridge University Press:  14 November 2011

W. L. Edge
W. L. Edge
Affiliation:
Nazareth House, Hillhead, Bonnyrigg, Midlothian EH 19 2JF, U.K.
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Synopsis

A certain plane sextic of genus 5 was encountered by Humbert and publicised by him [3] in 1894. Its striking geometrical properties clamour for elucidation; this was eventually supplied in 1951. For the canonical curve of genus 5 is the base curve C of a net N of quadrics in projective space [4], and C models a Humbert curve when all the quadrics of N have a common self-polar simplex [1]. The projection of C from one of its chords onto a plane is a 5-nodal sextic, the nodes all becoming cusps when the chord of C becomes a tangent. The properties to be elucidated become clear visually in the projection.

The sextic H described here is a specialisation of the cusped curve; it emerges as linearly dependent on a pair of reducible plane sextics concocted ad hoc.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 3-4 , 1991 , pp. 209 - 223
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Edge, W. L.. Humbert's plane sextics of genus 5. Proc. Camb. Philos. Soc. 47 (1951), 483–495.CrossRefGoogle Scholar
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