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

摘要

Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map (x, y) --> (u, v) is called quadrirational, if its graph is also a graph of a birational map (x, v) --> (u, y). We obtain a classification of quadrirational maps on CP1 x CP1, 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.