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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
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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.

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

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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