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Non-normalized Nash blowup fails to resolve singularities in dimension three

Federico Castillo, Daniel Duarte, Maximiliano Leyton-Álvarez, Alvaro Liendo

TL;DR

The paper proves that iterating the non-normalized Nash blowup does not necessarily resolve singularities for 3-dimensional varieties over characteristic zero fields. It constructs an explicit 3D non-normal affine toric variety $X(S)$ and uses the toric combinatorial description of Nash charts to show the second Nash blowup contains a chart isomorphic to $X(S)$, yielding a counterexample. A field-independent monomial-map argument with explicit defining equations reinforces the result across characteristic zero fields. This establishes a concrete limitation of Nash blowup-based resolution in dimension three and provides a computational framework for exploring toric Nash blowups in both theoretical and experimental settings.

Abstract

In this paper we show that iterating (non-normalized) Nash blowups does not necessarily resolve the singularities of algebraic varieties of dimension three over fields of characteristic zero.

Non-normalized Nash blowup fails to resolve singularities in dimension three

TL;DR

The paper proves that iterating the non-normalized Nash blowup does not necessarily resolve singularities for 3-dimensional varieties over characteristic zero fields. It constructs an explicit 3D non-normal affine toric variety and uses the toric combinatorial description of Nash charts to show the second Nash blowup contains a chart isomorphic to , yielding a counterexample. A field-independent monomial-map argument with explicit defining equations reinforces the result across characteristic zero fields. This establishes a concrete limitation of Nash blowup-based resolution in dimension three and provides a computational framework for exploring toric Nash blowups in both theoretical and experimental settings.

Abstract

In this paper we show that iterating (non-normalized) Nash blowups does not necessarily resolve the singularities of algebraic varieties of dimension three over fields of characteristic zero.

Paper Structure

This paper contains 3 sections, 3 theorems, 18 equations.

Table of Contents

  1. The Nash blowup of an affine toric variety
  2. Proof of the Theorem
  3. Remarks on computer experimentation

Key Result

Theorem 1

For any field of characteristic zero, there exists a non-normal affine algebraic variety $X$ of dimension $3$ such that the second iteration of the Nash blowup of $X$ contains an open affine subset isomorphic to $X$.

Theorems & Definitions (7)

  • Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Remark 4