Universal Characteristic-free Resolution of Singularities, I

TL;DR

This paper introduces a universal, characteristic-free method for resolving singularities of integral affine varieties over perfect fields, applicable to all singularities without characteristic restrictions.

Contribution

It presents a novel universal blowup process that resolves all singularities simultaneously, independent of the variety's specific type or characteristic.

Findings

Universal blowup process resolves all singularities

Method is characteristic-free and applies broadly

Provides a smooth resolution for any singular integral affine variety

Abstract

We prove that for any singular integral affine variety $X$ of finite presentation over a perfect field defined over $\mathbb Z$, there exists a smooth morphism from $Y$ onto $X$ such that $Y$ admits a resolution. That is, there exists a smooth scheme $\widetilde{Y}$ and a projective birational morphism from $\widetilde{Y}$ onto $Y$, followed by a smooth morphism from $Y$ onto $X$. Our approach differs fundamentally from existing methods, as we neither restrict to any specific singular variety nor fix the characteristic. Instead, we design a {\it universal} blowup process that {\it simultaneously} resolves all possible singularities, and, our method is entirely characteristic-free.