Weak toroidalization over non-closed fields

TL;DR

This paper proves that any dominant morphism between algebraic varieties over a characteristic zero field can be modified into a toroidal morphism through projective birational transformations, extending previous results to non-closed fields.

Contribution

It generalizes the weak toroidalization theorem to non-closed fields, allowing for broader applicability of toroidalization techniques.

Findings

Dominant morphisms can be transformed into toroidal morphisms over non-closed fields.

The transformations preserve certain additional structural requirements.

The results extend previous algebraically closed field cases.

Abstract

We prove that any dominant morphism of algebraic varieties over a field k of characteristic zero can be transformed into a toroidal (hence monomial) morphism by projective birational modifications of source and target. This was previously proved by the first and third author when k is algebraically closed. Moreover we show that certain additional requirements can be satisfied.