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Fricke's octavic curve

Published online by Cambridge University Press:  20 January 2009

W. L. Edge
W. L. Edge
Affiliation:
Inveresk HouseMusselburghEH21 7UA
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The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 1 , February 1984 , pp. 91 - 101
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

REFERENCES

1.Cayley, A., On the theory of the curve and torse, Quarterly J. of Pure and Applied Maths. 11 (1871), 294–317:Google Scholar
Mathematical Papers 8 (Cambridge Univ. Press, 1895), 72–91.Google Scholar
2.Cayley, A., On the curves situate on a surface of the second order, Phil. Mag. 22 (1861), 35–38:CrossRefGoogle Scholar
Mthematical Papers 5 (Cambridge Univ. Press, 1892), 70–72.Google Scholar
3.Edge, W. L., Bring's curve, J. London Math. Soc. (2) 18 (1978), 539–545.CrossRefGoogle Scholar
4.Edge, W. L., Tritangent planes of Bring's curve, J. London Math. Soc. (2) 23 (1981), 215–222.CrossRefGoogle Scholar
5.Fricke, R., Entwicklungen zur Transformationen fiinfter und siebenter Ordnung einiger spezieller automorpher Funktionen, Ada Mathematica 17 (1893).Google Scholar
6.Gordan, P., Ueber die Auflosung der Gleichungen von funften Grade, Math. Annalen 13 1878), 375–404.CrossRefGoogle Scholar
7.Klein, F., Vorlesungen über das Ikosaeder (Teubner, Leipzig, 1884).Google Scholar
8.Segre, C.Introduzione alia geometna sopra un ente algebrico semplicemente infinito, Annali di mat. (2) 22 (1894), 41–142:CrossRefGoogle Scholar
Opere I (Rome, 1957), 198–304.Google Scholar
9.Severi, F., Vorlesüngen tiber algebraische Geometrie (Teubner, Leipzig, 1921).CrossRefGoogle Scholar
10.Wima, A., Ueber die algebraischen Curven von den Geschlechtern p = 4, 5, 6 welche eindeutige Transformationen in sich besitzen, Svenska Vet. Akad. Handlingar: Bihang till Handlingar 21 (1895), afd. 1, No. 3, 41 pp.Google Scholar