On the configuration of the singular fibers of jet schemes of rational double points

Abstract

To each variety X and a nonnegative integer m, there is a space X_m over X, called the jet scheme of X of order m, parametrizing m-th jets on X. Its fiber over a singular point of X is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of X_m and the exceptional curves of the minimal resolution of X for $m\gg 0.$ In this article, for a surface X over $\mathbb{C}$ with a singularity of A_n or D₄-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of A_n or D₄-type singularity, we show that this graph is isomorphic to the resolution graph for $m\gg 0.$

Keywords:

• Jet scheme
• rational double point singularities

2020 Mathematics Subject Classification:

• Primary: 14J17

Acknowledgement

The author would like to thank Nobuyoshi Takahashi for valuable advice.