Birational morphisms of the plane

Abstract

Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings \phi of the polynomial algebra K[x,y] such that for the quotient fields, one has K(\phi(x), \phi(y)) = K(x,y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping \tau_x given by x \to x, y \to xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of \tau_x. This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.