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The even master system and generalized Kummer surfaces

Published online by Cambridge University Press:  24 October 2008

Jose Bertin  and
Pol Vanhaecke
Jose Bertin
Affiliation:
Université de Grenoble I, Institut Fourier, Laboratoire de Mathématiques, Associé au CNRS (URA 188), BP 74, 38402 Saint Martin D'Héres, France
Pol Vanhaecke
Affiliation:
Université des Sciences et Technologies de Lille, U.F.R. de Mathématiques Pures et Appliquées, Associé au CNRS (URA 751), 59655 Villeneuve d'Ascq, France
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Abstract

In this paper we study a generalized Kummer surface associated to the Jacobian of those complex algebraic curves of genus two which admit an automorphism of order three. Such a curve can always be written as y2 = x6 + 2kx3 + 1 and k2 ╪ 1 is the modular parameter. The automorphism extends linearly to an automorphism of the Jacobian and we show that this extension has a 94 invariant configuration, i.e. it has 9 fixed points and 9 invariant theta curves, each of these curves contains 4 fixed points and 4 invariant curves pass through each fixed point. The quotient of the Jacobian by this automorphism has 9 singular points of type A2 and the 94 configuration descends to a 94 configuration of points and lines, reminiscent to the well-known 166 configuration on the Kummer surface. Our ‘generalized Kummer surface’ embeds in ℙ4 and is a complete intersection of a quadric and a cubic hypersurface. Equations for these hypersurfaces are computed and take a very symmetric form in well-chosen coordinates. This computation is done by using an integrable system, the ‘even master system’.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 131 - 142
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Adler, M. and Van Moerbeke, P.. Algebraic completely integrable systems: a systematic approach. Perspectives in Mathematics (Academic Press, to appear).Google Scholar
[2]Adler, M. and Van Moerbeke, P.. The complex geometry of the Kowalewski–Painlevé analysis. Invent. Math. 97 (1987), 3–51.CrossRefGoogle Scholar
[3]Beauville, A.. Complex Algebraic Surfaces. London Math. Soc. Lecture Note Series 68 (Cambridge University Press, 1982).Google Scholar
[4]Bertin, J. and Elencwajg, G.. Manuscript.Google Scholar
[5]Castelnuovo, G.. Sulle congruenze del 3° ordine del spazio a 4 dimensioni. Memoria Atti Istituto Venoto VI (1888).Google Scholar
[6]Griffiths, P. and Harris, J.. Principles of Algebraic Geometry. Pure and Applied Mathematics (Wiley-Interscience, 1978).Google Scholar
[7]Hudson, R. W. H.. Kummer's quartic surface. Cambridge Mathematical Library (Cambridge University Press, 1990; first published in 1905).Google Scholar
[8]Lange, H. and Birkenhake, C.. Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften (Springer-Verlag, 1992).Google Scholar
[9]Mumford, D.. On the equations defining Abelian varieties I. Invent. Math. 1 (1966), 287–354.CrossRefGoogle Scholar
[10]Mumford, D.. Tata Lectures on Theta 2. Progress in Mathematics (Birkhäuser, 1984).Google Scholar
[11]Roan, S.. A Characterization of ‘Rapidity’ Curve in the Chiral Potts Model. Commun. Math. Phys. 145 (1992), 605–634.Google Scholar
[12]Segre, . Sulle variata cubiche dello spazio a 4 dimensioni. Memorie Acad. Torino II 39 (1888).Google Scholar
[13]Vanhaecke, P.. Linearising two-dimensional integrable systems and the construction of action-angle variables. Math. Z. 211 (1992), 265–313.Google Scholar