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.
We give a characteristic free proof of the main result of [L. Ghezzi, H.T. Hà, O. Kashcheyeva, Toroidalization of generating sequences in dimension two function fields, J. Algebra 301 (2) (2006) 838-866. ArXiv:math.AC/0509697.] concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of twodimensional algebraic regular local rings R ⊂ S satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in R and S has a toroidal structure.
Let (R, m R ) be a local domain, with quotient field K. Suppose that ν is a valuation of K with valuation ring (V, m V ), and that ν dominates R; that is, R ⊂ V and m V ∩R = m R . The possible value groups Γ of ν have been extensively studied and classified, including in the papers MacLane [8], MacLane and Schilling [9], Zariski and Samuel [12], and Kuhlmann [7]. Γ can be any ordered abelian group of finite rational rank (Theorem 1.1 [7]). The semigroupis however not well understood, although it is known to encode important information about the topology and resolution of singularities of Spec(R) and the ideal theory of R.In Zariski and Samuel's classic book on Commutative Algebra [12], two general facts about the semigroup S R (ν) are proven (in Appendix 3 to Volume II).1. S R (ν) is a well ordered subset of the positive part of the value group Γ of ν of ordinal type at most ω h , where ω is the ordinal type of the well ordered set N, and h is the rank of ν. 2. The rational rank of ν plus the transcendence degree of V /m V over R/m R is less than or equal to the dimension of R.The second condition is the Abhyankar inequality [1]. The only semigroups which are realized by a valuation on a one dimensional regular local ring are isomorphic to the natural numbers. The semigroups which are realized by a valuation on a regular local ring of dimension 2 with algebraically closed residue field are much more complicated, but are completely classified by Spivakovsky in [10]. A different proof is given by Favre and Jonsson in [5], and the theorem is formulated in the context of semigroups by Cutkosky and Teissier [3].In [3], Teissier and the first author give some examples showing that some surprising semigroups of rank > 1 can occur as semigroups of valuations on noetherian domains, and raise the general questions of finding new constraints on value semigroups and classifying semigroups which occur as value semigroups.In this paper, we consider semigroups of rank 1 valuations. We show in Theorem 2.1 that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S R (ν). This allows us to give (in Corollary 2.4) a very simple example of a well ordered subsemigroup of Q + of ordinal type ω, which is not a value semigroup of a local domain. This shows that the above conditions 1 and 2 do not characterize value semigroups on local domains.The simple bound of Theorem 2.1 of this paper is extended in the article [4] of Teissier and the first author to give a very general bound on the growth of a value semigroup of arbitrary rank, from which a rough description of the (extremely bizzare) shape of a higher rank valuation semigroup is derived.
Abstract. We work with rational rank 1 valuations centered in regular local rings. Given an algebraic function field K of transcendence degree 3 over k, a regular local ring R with QF (R) = K and a k-valuation ν of K, we provide an algorithm for constructing a generating sequences for ν in R. We then develop a method for determining a valuation ν on k(x, y, z) through the sequence of defining values. Using the above results we construct examples of valuations centered in k[x, y, z] (x,y,z) and investigate their semigroups of values.
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