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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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