Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces

A simpler proof of toroidalization of morphisms from 3-folds to surfaces

We give a simpler and more conceptual proof that a morphism from a 3-fold to a surface, over an algebraically closed field of characteristic 0, can be made into a toroidal morphism by sequences of blow ups of nonsingular subvarieties above the domain and range. Our original proof, which is much longer and more complicated, appeared in Springer Lecture Notes in 2002. The final version of this paper is much clearer, thanks to the encouragement of the referee. In particular, Section 5 on global reduction has been completely rewritten. The paper will appear in the Annals of the Fourier Institute.

环面态射的饱和基变换

Saturated base change of toroidal morphisms

本文证明了支配性环面态射的饱和基变换保持环面性，并系统梳理了环面嵌入、环面态射与对数正则对数簇、对数光滑态射之间的定义等价关系。

Some remarks on toroidal morphisms

This note contains some results related to the definitions of toroidal embeddings and toroidal morphisms over non-closed fields of characteristic zero.

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.

Toroidal orbifolds, destackification, and Kummer blowings up

We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to X_{tcs}$ is logarithmically smooth. Further, we use torification results of [AT17] to construct a destackification functor, a variant of the main result of Bergh [Ber17], on the category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of blowings up of toroidal stacks $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$ such that $Y_{tc}$ coincides with the usual coarse moduli space $Y_{cs}$. In particular, this provides a toroi…

Monomialization of Morphisms From 3 Folds to Surfaces

Suppose that $Φ:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that the morphism $Φ_1:X_1\to Y_1$ is a ``Monomial Morphism''. As a corollary, we show that we can make $Φ_1$ a toroidal morphism.

Weak semistable reduction in characteristic 0

Let X->B be a morphism of varieties in characteristic zero. Semistable reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat), dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar, Shepherd-Barron). In this paper we consider the general case. First we define what we mean by a semistable morphism in terms of toroidal embeddings. Then we reduce the varieties to toroidal embeddings and solve a slightly weaker version of semistable reduction. We also state the full semistable reduction problem in terms of combinatorics of the associated polyhedral complexes.

Universal Weak Semistable Reduction

We show that a toroidal morphism can be reduced to a weakly semistable one in a universal way if we allow families to be modified to Deligne-Mumford stacks instead of schemes.

骨架的调和覆盖

Harmonic covers of skeleta

本文证明环状概形的有限覆盖诱导对偶复形间的调和态射，并利用平衡条件推导出任意维度骨架覆盖的黎曼-胡尔维茨公式。