Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T02:43:02.568Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Edge, W. L.. Humbert's plane sextics of genus 5. Proc. Camb. Philos. Soc. 47 (1951), 483–495.CrossRefGoogle Scholar
2Edge, W. L.. Three plane sextics and their automorphisms. Canad. J. Math. 21 (1969), 1263–1278.CrossRefGoogle Scholar
3Humbert, G.. Sur un complex remarquable de coniques et sur la surface du troisième ordre. Journal de I'école polytechnique (1894), 123–149.Google Scholar
4Klein, F.. Zur theorie der Linienkomplexe des ersten und zweiten Grades. Math. Annalen 2 (1870), 198–226; Gesammelte Math. Werke 1 (Berlin 1921), 53–80.CrossRefGoogle Scholar
5Salmon, G.. Lessons introductory to the Modern Higher Algebra (Dublin: Hodges, Foster & Co., 1876).Google Scholar