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Rational surfaces associated with affine root systems and geometry of the Painlevé equations (1999)
| Citations: | 159 - 6 self |
BibTeX
@MISC{Sakai99rationalsurfaces,
author = {Hidetaka Sakai},
title = {Rational surfaces associated with affine root systems and geometry of the Painlevé equations},
year = {1999}
}
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Abstract
We present a geometric approach to the theory of Painlev'e equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystems of E (1) 8 inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlev'e equations, and that the above action constitutes their group of symmetries by Backlund transformations. The six Painlev'e differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure.
Keyphrases
rational surface painlev equation affine root system latter context hamiltonian structure symplectic form differential equation pole divisor cremona action picard lattice rational surfa geometric approach unique anti-canonical divisor discrete equation initial condition degenerate case corresponding affine weyl group backlund transformation root subsystem translation part affine weyl group canonical type compact smooth rational surface
