Principalization of ideals on toroidal orbifolds

TL;DR

This paper develops a functorial method to transform ideals on toroidal orbifolds into monomial ideals via modifications, enabling functorial resolution of singularities for varieties with logarithmic structures.

Contribution

It adapts existing desingularization methods to toroidal orbifolds, providing a functorial approach to principalize ideals and resolve singularities in this setting.

Findings

Achieved functorial principalization of ideals on toroidal orbifolds.

Established functorial resolution of singularities for varieties with logarithmic structures.

Laid groundwork for functorial semistable reduction theorems.

Abstract

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 [W{\l}o05], 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.