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

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 E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.

KEYWORDS:

• Embedded nash problem
• resolution of singularities
• toric geometry

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14E15
• 14E18
• 14M25

Acknowledgment

We would like to thank Shihoko Ishii, Monique Lejeune-Jalabert, Patrick Popescu-Pampu and Bernard Teissier for several discussions about this article.