Point configurations, Cremona transformations and the elliptic difference Painlevé equation

Details Point configurations, Cremona transformations and the elliptic difference Painlevé equation Point configurations, Cremona transformations and the elliptic difference Painlevé equation

2004-11-02

arxiv.org/abs/nlin/0411003v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

29 pages

Scientific paper

A theoretical foundation for a generalization of the elliptic difference Painlev\'e 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 $\tau$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $\tau$-functions on the lattice.

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