Relative desingularization and principalization of ideals

TL;DR

This paper develops a canonical method for desingularizing morphisms of logarithmic schemes in characteristic zero, leading to a semistable reduction theorem applicable over arbitrary valuation rings.

Contribution

It introduces a relative, functorial principalization process for ideals in logarithmic schemes, advancing the theory of desingularization and semistable reduction.

Findings

Constructed relative principalization of ideals for logarithmic schemes.

Achieved functorial, canonical desingularization of morphisms.

Deduced semistable reduction theorem over arbitrary valuation rings.

Abstract

In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are relatively canonical and even functorial with respect to logarithmically regular morphisms and arbitrary base changes. Relative canonicity means, that the principalization requires a fine enough non-canonical modification of the base, and once it is chosen the process is canonical. As a consequence we deduce the semistable reduction theorem over arbitrary valuation rings. In another our work in progress, the same problems will be solved canonically in the case of proper morphisms.