將因子收縮為cE型奇異點的因子收縮映射之研究

Abstract

極小模型理論是雙有理幾何的一個有力工具。為了更清楚暸解三維多樣體的幾何學，我們希望利用明確極小模型理論對三維多樣體進行明確分類，其中「明確」（explicit）指的是透過對具體方程式的研究以得到更細緻的結論。而divisorial contractions、flips和flops被視為極小模型理論中的初等雙有理映射，對它們的認識將有助加深我們對三維幾何學的瞭解。 在這裡我們會著眼於divisorial contractions，亦即因子收縮映射。由[Kwk05]我們知道把因子收縮到指標大於1的點的因子收縮映射皆可以寫成一個加權blowup。我們猜想把因子收縮到指標是1的點的因子收縮映射也都可以寫成一個加權blowup。 本文探討把因子收縮到cE型奇異點而且discrepancy為1的因子收縮映射。我們將會整理並介紹早川貴之（Takayuki Hayakawa）的工作[HayP2]。特別地，在discrepancy 1的假設下，我們可以對cE型奇異點引入適當的結構，使得在建構因子收縮映射時對cE型奇異點有較好的分類。最後，我們按照前述cE型奇異點的分類建構出一些因子收縮映射，從而部分地驗證我們的猜想。

The minimal model program (MMP) has long been a powerful tool in birational geometry. In order to know more about the geometry in dimension 3, we hope to develop an explicit classification of threefolds by using the MMP explicitly. By explicit we mean the study concerns concrete equations so as to gain more details. Note that divisorial contractions, flips and flops are considered elementary birational maps in the MMP. Having some explicit awareness about these birational maps allows us to have a better understanding of threefolds. Here we intend to study divisorial contractions. It is known that a divisorial contraction to a point of index greater than 1 can be realized as some weighted blowup ([Kwk05]). We conjecture that the statement is also true for points of index 1; that is, every divisorial contraction to a point of index 1 can also be realized as a weighted blowup. This thesis considers divisorial contractions to cE points with discrepancy 1. We will survey the work [HayP2] of Hayakawa. In particular, certain structure will be introduced to cE singularities so that we would have a better classification for constructing or studying the divisorial contractions. Finally, we construct some divisorial contractions according to that classification of cE points in order to partially examine our conjecture.