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

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 ${\text{CP}}^1\times {\text{CP}}^1$, 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.