$Q$-universal desingularization

Abstract

We prove that the algorithm for desingularization of algebraic varieties in characteristic zero of the first two authors is functorial with respect to regular morphisms. For this purpose, we show that, in characteristic zero, a regular morphism with connected affine source can be factored into a smooth morphism, a ground-field extension and a generic-fibre embedding. Every variety of characteristic zero admits a regular morphism to a $Q$-variety. The desingularization algorithm is therefore $Q$-universal or absolute in the sense that it is induced from its restriction to varieties over $Q$. As a consequence, for example, the algorithm extends functorially to localizations and Henselizations of varieties.