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XVII.—The Tacnodal Form of Humbert's Sextic*

Published online by Cambridge University Press:  14 February 2012

W. L. Edge
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Synopsis

Humbert's 5-nodal plane sextic first appeared in his 1894 paper. Its canonical curve C was identified in 1951, when it was shown that the sextic is the outcome of projecting C from one of its own chords on to a plane.

In this present paper it is remarked that there are 60 chords of C such that the projection has two tacnodes, each a confluence of two of Humbert's 5 nodes, and an equation is found for this tacnodal curve.

A certain specialization permits C to be invariant for a group of 32, not merely 16, projectivities. Further specializations, described in the proper place, permit groups of orders 64, 96, 160. The resulting tacnodal sextics have groups of birational self-transformations isomorphic to these.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 68 , Issue 3 , 1970 , pp. 257 - 269
Copyright
Copyright © Royal Society of Edinburgh 1970

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References

References to Literature

Baker, H. F., 1933. Principles of Geometry, V. Cambridge University Press.Google Scholar
Edge, W. L., 1951. “Humbert's plane sextics of genus 5”, Proc. Camb. Phil. Soc. Math. Phys., 47, 483–495.CrossRefGoogle Scholar
Edge, W. L., 1969. “Three plane sextics and their automorphisms”, Can. J. Math. 21, 1263–1278.CrossRefGoogle Scholar
Humbert, G., 1894. “Sur un complex remarquable de coniques et sur la surface du troisième ordre”, J. Ec. Poly tech., 64, 123–149.Google Scholar
Wiman, A., 1895. “Uber die algebraischen Curven von den Geschlectern p = 4, 5 und 6, welche eindeutigen Transformationen in sich besitzen”, Bih. K. Svenska Vetensk Akad. Handl., 21, (1), 3.Google Scholar