Geometry of two- and three-dimensional integrable systems related to affine Weyl groups $W(E_8^{(1)})$ and $W(E_7^{(1)})$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910032605478912 |
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| author | Alonso, Jaume Suris, Yuri B. |
| author_facet | Alonso, Jaume Suris, Yuri B. |
| contents | We find a general framework for the construction of birational involutions on two- and three-dimensional varieties obtained from $\mathbb P^2$, $\mathbb P^1\times \mathbb P^1$, and $\mathbb P^3$ by blow-up at nine, respectively eight points. Each such involution is based on a divisor class with a one-dimensional linear system with a generic element of genus zero. Classical Manin involutions represent the simplest particular case. Novel, more sophisticated cases identified here include birational involutions of $\mathbb P^2$ along conics and along nodal cubic curves, as well as birational involutions of $\mathbb P^3$ along quadratic cones and along Cayley nodal cubic surfaces. We prove a general formula for the induced action of geometric birational involutions on the respective Picard group, and give a general result about decomposition of translational elements of the respective affine Weyl group of symmetries into a product of two geometric birational involutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21391 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry of two- and three-dimensional integrable systems related to affine Weyl groups $W(E_8^{(1)})$ and $W(E_7^{(1)})$ Alonso, Jaume Suris, Yuri B. Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry We find a general framework for the construction of birational involutions on two- and three-dimensional varieties obtained from $\mathbb P^2$, $\mathbb P^1\times \mathbb P^1$, and $\mathbb P^3$ by blow-up at nine, respectively eight points. Each such involution is based on a divisor class with a one-dimensional linear system with a generic element of genus zero. Classical Manin involutions represent the simplest particular case. Novel, more sophisticated cases identified here include birational involutions of $\mathbb P^2$ along conics and along nodal cubic curves, as well as birational involutions of $\mathbb P^3$ along quadratic cones and along Cayley nodal cubic surfaces. We prove a general formula for the induced action of geometric birational involutions on the respective Picard group, and give a general result about decomposition of translational elements of the respective affine Weyl group of symmetries into a product of two geometric birational involutions. |
| title | Geometry of two- and three-dimensional integrable systems related to affine Weyl groups $W(E_8^{(1)})$ and $W(E_7^{(1)})$ |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Algebraic Geometry |
| url | https://arxiv.org/abs/2602.21391 |