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Point configurations, Cremona transformations and the elliptic difference Painlevé equation

K. Kajiwara·T. Masuda·M. Noumi·Y. Ohta·Y. Yamada
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Abstract

A theoretical foundation for a generalization of the elliptic difference Painlevé equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $τ$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $τ$-functions on the lattice.

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K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada. 2004-11-02. Point configurations, Cremona transformations and the elliptic difference Painlevé equation. https://arxiv.org/abs/nlin/0411003

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