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Geometry of Yang--Baxter maps: pencils of conics and quadrirational mappings

V. E. Adler·A. I. Bobenko·Yu. B. Suris
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Abstract

Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map $(x,y)\mapsto(u,v)$ is called quadrirational, if its graph is also a graph of a birational map $(x,v)\mapsto(u,y)$. We obtain a classification of quadrirational maps on $\CP^1\times\CP^1$, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.

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BibTeXRIS

V. E. Adler, A. I. Bobenko, Yu. B. Suris. 2003-07-01. Geometry of Yang--Baxter maps: pencils of conics and quadrirational mappings. https://arxiv.org/abs/math/0307009

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