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 → A 2 given by polynomial mappings φ of the polynomial algebra K[x, y] such that for the quotient fields, one has K(φ(x), φ(y)) = K(x, y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ x given by x → x, y → xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ 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.