Geometry of the Yang-Baxter Maps: pencils of conics and quadrirational mappings

Abstract

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