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 thatS dominatesR 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.

Ramification of local rings along valuations

“…We obtain the following theorem (Theorem 1.5 from Section 1 of this paper). Theorem 4.2 is proven by Ghezzi, Hà and Kashcheyeva in [7] when k is algebraically closed of characteristic zero. Proof.…”

“…We have the following theorem, which generalizes Proposition 1.3 to this case. This surprising theorem was proven when k is algebraically closed of characteristic zero and dim K = 2 by Ghezzi, Hà and Kashcheyeva in [7]. If n = 2, ν is rational and R → S is stable, then R has regular parameters u, v, S has regular parameters x, y and there exist a unit γ in S such that…”

Irrational behavior of algebraic discrete valuations

“…The algorithm in [7] is valid in arbitrary two dimensional regular local rings. This technique of generating sequences is also used in [8] and [9] to find stable toric forms of extensions of associated graded rings along a valuation in finite defectless extensions of algebraic function fields of dimension two.…”

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

“…A similar analysis was done in [5] (Section 3), but we outline it below for completeness. We refer to [3] (Section 7.2) for the background needed in this section.…”

“…We refer to Section 2 of this paper or to Section 2 of [5] for the precise definition of toroidal structure. Theorem 1.1 ([5, 8.1]).…”