Relative desingularization and principalization of ideals

A Simplified Proof of Desingularization and Applications

This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a…

Principalization of ideals on toroidal orbifolds

Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we produce a modification $X' \to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods of [Wło05], discarding steps which become redundant. We deduce functorial resolution of singularities for varieties with logarithmic structures. This is the first step in our program to apply logarithmic desingularization to a morphism $Z \to B$, aiming to prove functorial semistable reduction theorems.

函子单项式化与相对主元化中中心的唯一性

Functorial monomialization and uniqueness of centers for relative principalization

本文研究相对对数情形下理想主元化的函子性问题，证明本征映射中Kummer与加权对数中心均可函子式单项式化，从而得到群作用等变且与基局部化相容的特征0相对主元化与对数光滑约化。

对数叶状轨形上的主理想化

Principalization on logarithmically foliated orbifolds

该研究在特征零情形下构建了带正规交叉除子和叶状结构的光滑轨形的理想主理想化方法，并通过Darboux叶状结构奇点消解、一般横截截面转化两个应用验证其在叶状结构研究中的作用。

Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonic…

A strong desingularization theorem

Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $π: W_r\longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r\subset W_r$ denotes the strict transform of $X\subset W$ then: 1) The morphism $π:W_r\longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem); 2) The {\em total transform} of ${\mat…