Geometry of autonomous discrete Painlev\'e equations related to the Weyl group $W(E_8^{(1)})$

TL;DR

This paper explores the geometric structure of autonomous discrete Painlevé equations linked to the Weyl group $W(E_8^{(1)})$, introducing new birational involutions to construct commuting maps on rational elliptic surfaces.

Contribution

It provides a geometric construction of commuting maps for autonomous discrete Painlevé equations using novel birational involutions related to pencils of curves.

Findings

Construction of commuting maps via birational involutions

Connection between Painlevé equations and Weyl group symmetries

Use of rational elliptic surfaces in the geometric framework

Abstract

Discrete Painlev\'e equations are integrable two-dimensional birational maps associated to a family of generalized Halphen surfaces. The latter can be seen either as $\mathbb P^2$ blown up at nine points or as $\mathbb P^1\times\mathbb P^1$ blown up at eight points. These maps become autonomous if the blow-up points are in a special position (support a pencil of cubic curves in $\mathbb P^2$, respectively a pencil of biquadratic curves in $\mathbb P^1\times\mathbb P^1$), so that the generalized Halphen surfaces become rational elliptic surfaces. In the generic case, the symmetry of a discrete Painlev\'e equation is the Weyl group $W(E_8^{(1)})$. One has a system of commuting maps which correspond to translational elements of $W(E_8^{(1)})$ associated to the roots of the lattice $E_8^{(1)}$. In the present note, we give a geometric construction of these commuting maps. For this, we use some novel birational involutions based on the above mentioned pencils of curves.