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Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in ℙ3(ℂ)

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Abstract

A 3 dimensional analogue of Sakai’s theory concerning the relation between rational surfaces and discrete Painlevé equations is studied. For a family of rational varieties obtained by blow-ups at 8 points in general position in ℙ3, we define its symmetry group using the inner product that is associated with the intersection numbers and show that the group is isomorphic to the Weyl group of type E 7 (1). By parametrizing the configuration space by means of elliptic curves, the action of the Weyl group and the dynamical system associated with a translation are explicitly described. As a result, it is found that the action of the Weyl group on ℙ3 preserves a one parameter family of quadratic surfaces and that it can therefore be reduced to the action on ℙ1×ℙ1.

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  1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku Tokyo, 153-8914, Japan

    Tomoyuki Takenawa

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  1. Tomoyuki Takenawa
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Communicated by L. Takhtajan

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Takenawa, T. Discrete Dynamical Systems Associated with the Configuration Space of 8 Points in ℙ3(ℂ). Commun. Math. Phys. 246, 19–42 (2004). https://doi.org/10.1007/s00220-004-1043-5

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  • DOI: https://doi.org/10.1007/s00220-004-1043-5

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