Geometry of 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) → (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 ℂℙ 1 × 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.