Toroidalization of generating sequences in dimension two function fields

Abstract

Let k be an algebraically closed field of characteristic 0, and let K ∗ / K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let ν ∗ be a k-valuation of K ∗ with valuation ring V ∗ , and let ν be the restriction of ν ∗ to K. Suppose that R → S is an extension of algebraic regular local rings with quotient fields K and K ∗ respectively, such that V ∗ dominates S and S dominates R. We prove that there exist sequences of quadratic transforms R → R ¯ and S → S ¯ along ν ∗ such that S ¯ dominates R ¯ and the map between generating sequences of ν and ν ∗ has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.