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

Abstract Discrete Painlevé equations are integrable two-dimensional birational maps associated to a family of generalized Halphen surfaces. A generalized Halphen surface can be realized either as $${\mathbb {P}}^2$$ P 2 blown up at nine points or as $${\mathbb {P}}^1\times {\mathbb {P}}^1$$ P 1 × P 1 blown up at eight points. These maps become autonomous if the blow-up points are in a special position (nine points in $${\mathbb {P}}^2$$ P 2 supporting a pencil of cubic curves, resp. eight points in $${\mathbb {P}}^1\times {\mathbb {P}}^1$$ P 1 × P 1 supporting a pencil of biquadratic curves), so that a generalized Halphen surface becomes a rational elliptic surface. In the generic case, the symmetry of a discrete Painlevé equation is the Weyl group $$W(E_8^{(1)})$$ W ( E 8 ( 1 ) ) . One has a system of commuting maps which correspond to translational elements of $$W(E_8^{(1)})$$ W ( E 8 ( 1 ) ) associated to the roots of the lattice $$E_8^{(1)}$$ 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.