On a Computational Approach to the Nash Blowup Problem

TL;DR

This paper presents computational evidence and new examples related to the Nash blowup conjectures, showing both counterexamples and cases where the conjecture holds, thus advancing understanding of singularity resolution.

Contribution

The authors implement computational methods that produce counterexamples and new resolved cases for the Nash blowup conjectures in toric varieties.

Findings

Counterexamples to the Nash blowup conjectures are constructed.

Many toric varieties are resolved by iterating the Nash blowup.

Extensive computational data supports remaining open cases.

Abstract

In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not resolved by the normalized Nash blowup, including cyclic quotient singularities, toric hypersurfaces, and Q-factorial Gorenstein singularities. In addition, we report extensive computational evidence: tens of thousands of two-dimensional toric varieties that are resolved by iterating the Nash blowup, and millions of three-dimensional toric varieties that are resolved by iterating the normalized Nash blowup. This provides positive evidence for the remaining open cases of the conjectures.