Painlevé equations and deformations of rational surfaces with rational double points

Details Painlevé equations and deformations of rational surfaces with rational double points Painlevé equations and deformations of rational surfaces with rational double points

2000-07-26

arxiv.org/abs/math/0007156v1

Physics and combinatorics 1999 (Nagoya), 320--365, World Sci. Publishing, River Edge, NJ, 2001.

Mathematics

Algebraic Geometry

45 pages

Scientific paper

In this paper, we show that the B\"acklund transformations of Painlev\'e equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlev\'e equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlev\'e equations.

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