Nash blowup fails to resolve singularities in dimensions four and higher

TL;DR

This paper demonstrates that Nash blowups, even when iterated or normalized, do not effectively resolve singularities in algebraic varieties of dimension four or higher over any algebraically closed field.

Contribution

It proves that Nash blowups fail to resolve singularities in higher dimensions, highlighting limitations of this approach in algebraic geometry.

Findings

Nash blowups do not resolve singularities in dimension four or higher.

Iterating Nash blowups does not improve resolution in higher dimensions.

The failure occurs over fields of arbitrary characteristic.

Abstract

In this paper we show that iterating Nash blowups or normalized Nash blowups does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic.