奇異點上的重要部分

Abstract

本文主要探討對於一個代數流形(varieties)上的奇異點(singularities)進行解析(resolution)之後，所得到的異常因子(exceptional divisors)與通過該奇異點的弧空間(arc space)的對應關係。 弧空間的概念最早由Nash所提出，並由於Denef, Loeser, Mustata與Ein等人的研究並推廣，在代數幾何的領域上蓬勃發展。然而早在Nash的文章裡，他已經給出了一個映射從弧空間中的最簡化部分(irreducible components)到異常因子，並且證明這個映射是一對一的。 為了更明確的了解這個映射，我們嘗試了許多二維度與三維度的例子。但是要明白的寫出對應關係的話，我們碰到了許多困難。首先，在三維度或是更高維度的空間中，我們並沒有基本且標準的「最小解析」(minimal resolution)，這使得我們沒辦法有效的觀察異常因子並檢驗它們是否重要(essential)。再者，我們並不清楚如何檢查一個弧空間上的一個部分是否可以再簡化，只好直接計算並套用數學歸納法。 在文章中，我們首先定義問題並且介紹所需要的工具。接著，我們使用最小解析的手法並確定所有二維度的重要部分。同一章節中我們也使用差異性(discrepancy)的概念來處理高維度的情況，讓我們對於重要部分有基本的了解。再來我們直接從代數流形的定義式去計算通過奇異點的弧空間上的最簡化部分，並得知當我們的計算程序滿足特定條件之後，歸納就可以保證弧空間中最簡化部分的個數與長相。最後，我們考慮三維度中大家比較熟悉的終端奇異點(terminal singularities)，並嘗試去決定哪些異常因子是重要的。在使用了Hayakawa的解析手法之後，我們最後的結論是當這個代數流形滿足一些特定條件之後，可以把問題化簡到解析時使用的向量，並配合之前的差異性來判斷異常因子的重要性。

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.