Discrete Painlevé equations from pencils of quadrics in $\mathbb P^3$ with branching generators
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| Format: | Preprint |
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2025
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| author | Alonso, Jaume Suris, Yuri B. |
| author_facet | Alonso, Jaume Suris, Yuri B. |
| contents | In this paper we extend the novel approach to discrete Painlevé equations initiated in our previous work [2]. A classification scheme for discrete Painlevé equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlevé equation is viewed as an autonomous transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial $Δ(λ)$ of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlevé map corresponds to a translation on the universal cover of the Riemann surface of $\sqrt{Δ(λ)}$, rather than to a Möbius transformation of the pencil parameter $λ$ as in [2]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_02275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Painlevé equations from pencils of quadrics in $\mathbb P^3$ with branching generators Alonso, Jaume Suris, Yuri B. Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems In this paper we extend the novel approach to discrete Painlevé equations initiated in our previous work [2]. A classification scheme for discrete Painlevé equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlevé equation is viewed as an autonomous transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial $Δ(λ)$ of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlevé map corresponds to a translation on the universal cover of the Riemann surface of $\sqrt{Δ(λ)}$, rather than to a Möbius transformation of the pencil parameter $λ$ as in [2]. |
| title | Discrete Painlevé equations from pencils of quadrics in $\mathbb P^3$ with branching generators |
| topic | Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2506.02275 |