Hostname: page-component-54dcc4c588-9xpg2 Total loading time: 0 Render date: 2025-10-07T01:45:20.447Z Has data issue: false hasContentIssue false
  • English
  • Français

Quartic curves in characteristic 2

Published online by Cambridge University Press:  24 October 2008

C. T. C. Wall
C. T. C. Wall
Affiliation:
Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX
Get access Rights & Permissions [Opens in a new window]

Extract

Simple singularities in positive characteristic

Simple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 3 , May 1995 , pp. 393 - 414
Copyright
Copyright © Cambridge Philosophical Society 1995

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Artin, M.. Coverings of the rational double points in characteristic p, pp. 11–21 in Complex Analysis and Algebraic Geometry, (ed. Bailey, W. L. and Shioda, T.). (Cambridge University Press, 1977).CrossRefGoogle Scholar
[2]Brieskorn, E.. Singular elements of semisimple algebraic groups, pp. 279–284. In Actes Congrès Int. Math. (Nice, 1970) (Gauthier-Villars, 1971).Google Scholar
[3]Demazure, M.. Surfaces de del Pezzo III: positions presque générales, pp. 36–49 in Séminaire sur les Singularités des Surfaces, (ed. Demazure, M., Pinkham, H. and Teissier, B.). (Springer Lecture Notes in Math. 777, 1980).CrossRefGoogle Scholar
[4]Val, P. du. On isolated singularities which do not affect the conditions of adjunction III. Proc. Camb. Phil. Soc. 30 (1934), 483–491.Google Scholar
[5]Greuel, G.-M. and Kroning, H.. Simple singularities in positive characteristic. Math. Zeits. 203 (1990), 339–354.CrossRefGoogle Scholar
[6]Hefez, M.. Non-reflexive curves. Compositio Math. 69 (1989), 3–36.Google Scholar
[7]Hirschfeld, J. W. P.. Protective Geometries over Finite Fields. (Oxford University Press, 1979).Google Scholar
[8]Husemoller, D.. Elliptic curves. (Springer graduate texts in math. 111 1989).Google Scholar
[9]Kiyek, K. and Steinke, G.. Einfache Kurvensingularitäten in beliebiger Charakteristik Arch. Math. 45 (1985), 565–573.Google Scholar
[10]Knop, H.. Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. Invent. Math. 90 (1987), 579–604.Google Scholar
[11]Mérindol, J.-Y.. Les singularités simples elliptiques, leurs déformations, les surfaces de del Pezzo et les transformations quadratiques. Ann. Sci. Ec. Norm. Sup. 15 (1982), 17–44.CrossRefGoogle Scholar
[12]Naruki, I. and Urabe, T.. On singularities on degenerate del Pezzo surfaces of degree 1,2, preprint (57pp) RIMS, Kyoto (1970).Google Scholar
[13]De Resmini, M. J.. Sulle quartiche piane sopra un campo di caratteristica due. Richerche Math. 19 (1970), 133–160.Google Scholar
[14]De Resmini, M. J.. On quartics in a plane over a field of characteristic 2, pp. 187–197 in Atti del Convegno di Geometria Combinatoria e sue Applicazione. (Università di Perugia, 1971).Google Scholar
[15]Roczen, M.. Cubic surfaces with double points in positive characteristic, preprint MPI92–18 (8pp) Max-Planck-Institut, Bonn, 1992.Google Scholar
[16]Wall, C. T. C.. Is every quartic a conic of conies? Math. Proc. Camb. Phil. Soc. 109 (1991), 419–424.Google Scholar