Logarithmic resolution via weighted toroidal blow-ups

Abstract

Let $X$ be a fs logarithmic scheme that is generically logarithmically smooth, and that admits a strict closed embedding into a logarithmically smooth scheme $Y$ over a field $\kk$ of characteristic zero. We construct a simple and fast procedure to functorial logarithmic resolution of $X$, where the end result is in particular a stack-theoretic modification $X' \rightarrow X$ such that $X'$ is logarithmically smooth over $k$. In particular, if $X$ is a closed subscheme of a smooth $k$-scheme $Y$, the procedure not only shares the same desirable features as the 'dream resolution algorithm' of Abramovich-Temkin-Wlodarczyk (arXiv:1906.07106), but also accounts for a key feature of Hironaka's Main Theorem I, which was not addressed in arXiv:1906.07106. As a consequence, we recover a different and simpler approach to Hironaka's resolution of singularities in characteristic zero.