Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T01:09:37.339Z Has data issue: false hasContentIssue false
  • English
  • Français

C-Nodal Surfaces of Order Three

Published online by Cambridge University Press:  20 November 2018

Tibor Bisztriczky
Tibor Bisztriczky*
Affiliation:
University of Calgary, Calgary, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of synthetic surfaces of order three in [9]. While his theory resulted in a partial classification of a now larger class of surfaces, it was too general to permit a global description. In [1], we added a differentiability condition to Marchaud's definition. This resulted in a partial classification and description of surfaces of order three with exactly one singular point in [2]-[5]. In the present paper, we examine C-nodal surfaces and thus complete this survey.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 1 , 01 February 1983 , pp. 68 - 100
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Bisztriczky, T., Surfaces of order three with a peak. I, J. of Geometry 11 (1978), 55–83.Google Scholar
2. Bisztriczky, T., Surfaces of order three with a peak. II, J. of Geometry 11 (1978), 110–138.Google Scholar
3. Bisztriczky, T., Uniplanar surfaces of order three, Geometriae Dedicata 8 (1979), 259–277.Google Scholar
4. Bisztriczky, T., Biplanar surfaces of order three, Can. J. Math. 31 (1979), 396–418.Google Scholar
5. Bisztriczky, T., Biplanar surfaces of order three. II, Can. J. Math. 32 (1980), 839–866.Google Scholar
6. Bisztriczky, T., On surfaces of order three, Can. Math. Bull. 22 (1979), 351–355.Google Scholar
7. Bisztriczky, T., On the lines of a surface of order three, Math. Ann. 243 (1979), 191–195.Google Scholar
8. Haupt, O. and Kiinneth, H., Geometrische Ordnungen (Springer, Berlin, Heidelberg, New York, 1967).Google Scholar
9. Marchaud, A., Sur les surfaces du troisième ordre delà géométrie finie, J. Math. Pure Appi. 18 (1939), 323–362.Google Scholar
10. Marchaud, A., Sur les propriétés différentielles du premier ordre des surfaces simples de Jordan et quelques applications, Ann. Ec. Norm. Sup. 63 (1947), 81–108.Google Scholar
11. Marchaud, A., Sur les courbes et surfaces du troisième ordre en géométrie finie, Bull. Cl. Sci. Acad. Roy. Belgique 49 (1963), 555–575.Google Scholar