Resolving toric varieties with Nash blow-ups

TL;DR

This paper investigates the resolution of singularities in toric varieties using Nash blow-ups, providing polyhedral interpretations, explicit solutions for surfaces, and computational evidence for higher dimensions.

Contribution

It offers a polyhedral framework for normalized Nash blow-ups, proves resolution for toric surfaces, and demonstrates computational success in resolving over a thousand higher-dimensional toric varieties.

Findings

Normalized Nash blow-ups resolve all tested toric varieties.

Continued fractions facilitate explicit resolutions for toric surfaces.

Computational methods effectively resolve complex higher-dimensional cases.

Abstract

It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blow-ups. We consider this question in the case of a toric variety. We interpret the normalized Nash blow-up in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.