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

TL;DR

This paper investigates the structure of singular fibers in jet schemes of surfaces with rational double points, establishing a correspondence between their irreducible components and the resolution graphs for specific singularity types.

Contribution

It extends the understanding of jet schemes by explicitly describing the intersection patterns of singular fiber components for $A_n$ and $D_4$-type singularities, linking them to resolution graphs.

Findings

The intersection graph of singular fiber components matches the resolution graph for large m.

The study provides a combinatorial description of singular fibers for specific rational double points.

The results connect jet scheme geometry with classical resolution theory.

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 paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-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 $D_4$-type singularity, we show that this graph is isomorphic to the resolution graph for $m \gg 0$.