Classical Algebraic Geometry and Discrete Integrable Systems
Gessica Alecci, Michele Graffeo, Alexander Stokes
TL;DR
These notes survey how classical algebraic geometry—surface theory, resolution of indeterminacies, and linear systems—illuminate the study of discrete integrable systems, particularly Painlevé-type equations. They develop blow-up constructions to build spaces of initial conditions and relate degree growth to algebraic entropy via the action on $H^2(S,\mathbb{Z})$, illustrating this with a three-dimensional Cremona-based example that exhibits either quadratic or exponential growth depending on the orbit structure. They then present Sakai's framework of generalized Halphen surfaces and affine root systems, detailing root variables, Dynkin data, and symmetry groups that govern non-autonomous discrete Painlevé dynamics, including Okamoto-style spaces for differential Painlevé equations. Collectively, the work demonstrates how birational dynamics on rational surfaces underpin integrability criteria and provide a geometric classification framework for discrete and differential Painlevé equations.
Abstract
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry side, which is applied to differential and discrete Painlevé equations on the integrable systems side. Along the way we also discuss the theory of resolution of indeterminacies, which is applied to the cohomological computation of algebraic entropy of birational transformations of projective spaces, which is closely related to the integrability of the discrete systems they define.