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

TLDR: In this article, a geometric approach to the theory of Painleve equations based on rational surfaces is presented, where a compact smooth rational surface X has a unique anti-canonical divisor D of canonical type.

Abstract: We present a geometric approach to the theory of Painleve 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 subsystem 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 Painleve equations, and that the above action constitutes their group of symmetries by Backlund transformations The six Painleve 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