The embedded Nash problem of birational models of rational triple singularities

TL;DR

This paper investigates whether jet space data can be used to construct embedded resolutions of singularities, finding it generally impossible but feasible for certain rational triple surface singularities.

Contribution

It demonstrates the limitations of jet space data for resolution construction and identifies specific cases where it is effective.

Findings

Jet space data cannot generally produce embedded resolutions.

It is possible to construct resolutions for some rational triple surface singularities.

The study clarifies the scope of jet-based resolution methods.

Abstract

We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset \textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X.$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.