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

Adler, V. E.; Bobenko, A. I.; Suris, Yu. B.

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

Authors

Adler, V. E.

Bobenko, A. I.

Suris, Yu. B.

Description

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.
LaTeX, 40pp, 3 Figs