Principalization of ideals on toroidal orbifolds

Abstract

Given an ideal $\mathcal{I}$ on a variety $X$ with toroidal singularities, we produce a modification $X^{\prime }\to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X^{\prime }$. 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.