Jet schemes and minimal toric embedded resolutions of rational double point singularities

TL;DR

This paper constructs minimal embedded toric resolutions for rational double point singularities using jet schemes, establishing a bijective correspondence between certain jet scheme components and divisors, except for the E8 case.

Contribution

It introduces a novel method linking jet scheme components to divisors in minimal embedded toric resolutions of rational double points, extending Nash correspondence concepts.

Findings

Established a bijective correspondence between jet scheme components and divisors for most singularities.

Constructed minimal embedded toric resolutions using jet scheme structures.

Identified exceptions in the E8 singularity case.

Abstract

Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.