On Essential Components of Singularities
Date Issued
2011
Date
2011
Author(s)
Su, Yu-Cheng
Abstract
The main purpose of this paper is to describe the correspondence between the irreducible components of arc space of singularities and the essential components.
In recent years, the development of motivic integration proposed by Kontsevich which was worked out by Denef and Loeser draws a lot of attention to the study of jet schemes and arc spaces. The study of arc spaces and jet schemes has become a very important and interesting tool in algebraic geometry, especially in the theory of singularities. Some important works are made by Denef-Loeser, Mustaţă, and Ein.
In the milestone work of Nash, he proved the injectivity of the map mapping from the set of irreducible components of the space of arcs through singular points to the set of essential component of a resolution of singularities. We call this map the Nash map. He also asked whether this map is always bijective.
In order to understand the Nash map explicitly, we consider many singularities in dimension two and three, and try to work out the correspondence explicitly. There are some potential difficulties. The first one is that in dimension three or higher, there is no “minimal resolution" in general. Therefore it is not easy to determine whether an exceptional divisor is essential or not. We can only see that those exceptional divisors with discrepancy not greater than one are essential. On the other hand, it is not clear how to determine irreducible components of arc space through singularities. We try to compute this explicitly in the straightforward manner.
In this paper, we first introduce some notations and definitions to help us dealing with the problem. After that in section four, we try to find those essential components over a 2-dimensional singularity via the minimal resolution of surface. We also make some discussion on discrepancy of exceptional divisors for 3-dimensional terminal cases to obtain the essential components. Next, we try to determine the irreducible components of the space of arcs through the singularities. At the end, we consider a 3-dimensional terminal singularity and use Hayakawa''s method to construct a resolution then try to find out the essential components.
We conclude that an exceptional divisor is essential if it appears in the minimal resolution for surface singularities or is of discrepancy less than or equal to one in the higher dimensional cases. And after enough many jet scheme computed without finding new components, we know the number of the components of arc space and how they looks like. Finally we know that if a 3-dimensional terminal singularity satisfies some extra condition, then it is enough to consider the vector in the toric language to decide whether a divisor is essential.
Subjects
algebraic geometry
Nash problem
arc spaces
essential components
singularities
discrepancy
Type
thesis
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