Toroidalization of generating sequences in dimension two function fields

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

We give a characteristic free proof of the main result of our previous paper (math.AC/0509697) concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two dimensional algebraic regular local rings $R\subset S$ 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.

Strong Toroidalization of Dominant Morphisms of 3-folds

Suppose that $f:X\to Y$ is a dominant morphism of 3-folds over an algebraically closed field of characteristic zero. We prove that there exist sequences of blow ups of points and nonsingular curves $Φ:X_1\to X$ and $Ψ:Y_1\to Y$ such that the induced map $f_1:Y_1\to X_1$ is a toroidal morphism. This extends an earlier proof of the author of this theorem with the extra assumption that $f$ is birational.

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ o…

Generating sequences and key polynomials

The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.

双图与二维函数域上非除数估值的生成序列

Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields

本文基于Spivakovsky的双图理论，证明了二维函数域上非除数估值的最小生成序列，且定义方式与原用法不同，通过双图数据明确生成序列值。

生成秩1赋值的序列

Generating sequences of rank 1 valuations

本文研究三维正则局部环上的秩1赋值，提出构造其生成序列的算法，推广了此前仅适用于有理秩1赋值的算法，并给出有理秩3赋值的示例完成显式计算。