Rational surfaces associated with affine root systems and geometry of the Painlevé equations

摘要: We present a geometric approach to the theory of Painleve equations based on rational surfaces. Our starting point is compact smooth surface X which has unique anti-canonical divisor D canonical type. classify all such surfaces X. To each X, there corresponds root subsystem E (1) 8 inside Picard lattice realize action corresponding affine Weyl group as Cremona family these show that translation part gives rise discrete equations, and above constitutes their symmetries by Backlund transformations. The six differential appear degenerate cases this construction. In latter context, Okamoto's space initial conditions pole symplectic form defining Hamiltonian structure.