Threefold Divisorial Contractions to Singularities of cE Type

• Main Authors: Ip Fai Ng, 伍業輝
• Other Authors: Jungkai Alfred Chen
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/73705437856070119166

碩士 === 國立臺灣大學 === 數學研究所 === 102 === 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.