Geometry of autonomous discrete Painlevé equations related to the Weyl group $W(E_8^{(1)})$

TL;DR

This work provides a geometric realization of autonomous discrete Painlevé equations associated with the affine Weyl group $W(E_8^{(1)})$ by expressing translations as compositions of birational involutions tied to families of curves. It constructs new involutions from pencils of cubics in ${\mathbb P}^2$ to realize translations like $T_{\mathcal E_0-\mathcal E_i-\mathcal E_j-\mathcal E_k}$ on the Picard lattice of the rational elliptic surface, and extends the framework to pencils of biquadratic curves on ${\mathbb P}^1\times{\mathbb P}^1$ to realize additional translations via horizontal/vertical switches, $(1,1)$-curves, and $(2,1)$- or $(1,2)$-curves. The results connect the Cremona-action on base-point configurations with the $E_8^{(1)}$ translation lattice, offering explicit geometric mechanisms for commuting autonomous maps and suggesting avenues for non-autonomous and higher-dimensional generalizations. Overall, the paper deepens the geometric understanding of autonomous discrete Painlevé dynamics through concrete involutive constructions on rational surfaces.

Abstract

Discrete Painlevé 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 Painlevé 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.

Theorems & Definitions (14)

• Proposition 1
• Definition 1
• Theorem 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Definition 2
• Theorem 2