Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-30T17:23:15.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic tetrads and harmonic elliptic quartic curves

Published online by Cambridge University Press:  14 February 2012

R. H. Dye
R. H. Dye
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne
Get access Rights & Permissions [Opens in a new window]

Synopsis

The group of a harmonic elliptic quartic has exceptional action on the edges of the self-polar tetrahedron, singling out one pair of opposite edges. Geometrical explanations are given. One concerns the possession by each of these two edges of a certain harmonic tetrad constructible in three ways. The same constructions yield, for a general quartic, three distinct tetrads on each edge of its tetrahedron, with a related fourth. The cases of exceptional overlap of these tetrads are examined: they occur for quartics of moduli 0, ∞ and — 32/49.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 1-2 , 1977 , pp. 91 - 96
Copyright
Copyright © Royal Society of Edinburgh 1977

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

1Dye, R. H.Pencils of elliptic quartic curves and an identification of Todd's quartic combinant. Proc. London Math. Soc, 34 (1977), 459–478.CrossRefGoogle Scholar
2Edge, W. L.The principal chords of an elliptic quartic. Proc. Roy. Soc. Edinburgh Sect. A 71 (1973), 43–50.Google Scholar
3Enriques, F. and Chisini, O.Teoria geometrica delle equazioni e dellefunzioni algebriche I (Bologna: Zanichelli, 1929).Google Scholar
4Enriques, F. and Chisini, O.Teoria geometrica delle equazioni e delle funzioni algebriche III (Bologna: Zanichelli, 1924).Google Scholar
5Salmon, G.A treatise on the analytic geometry of three dimensions (Dublin: Hodges and Figgis, 1882).Google Scholar
6Todd, J. A.Combinants of a pencil of quadric surfaces, I. Proc. Cambridge Philos. Soc. 43 (1947), 475–487.CrossRefGoogle Scholar
7Todd, J. A.Projective and analytical geometry (London: Pitman, 1947).Google Scholar