Toroidalization of generating sequences in dimension two function fields of positive characteristic

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 → S̄ 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. 1