Geometry of autonomous discrete Painlevé equations related to the Weyl group $W(E_8^{(1)})$
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908725040644096 |
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| author | Alonso, Jaume Suris, Yuri B. |
| author_facet | Alonso, Jaume Suris, Yuri B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometry of autonomous discrete Painlevé equations related to the Weyl group $W(E_8^{(1)})$ Alonso, Jaume Suris, Yuri B. Exactly Solvable and Integrable Systems Mathematical Physics 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. |
| title | Geometry of autonomous discrete Painlevé equations related to the Weyl group $W(E_8^{(1)})$ |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2512.18288 |