Q-universal desingularization

TL;DR

This paper proves that a desingularization algorithm for algebraic varieties in characteristic zero is functorial and Q-universal, meaning it can be extended to all varieties over Q and related structures through a specific factorization of morphisms.

Contribution

It establishes the functoriality and Q-universality of the desingularization algorithm in characteristic zero, enabling broader applicability and extension to localizations and Henselizations.

Findings

Algorithm is functorial with respect to regular morphisms.

Every variety admits a regular morphism to a Q-variety.

Desingularization extends functorially to localizations and Henselizations.

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.