Explicit studies on singularities which appear on 3-dimensional algebraic varieties

我们研究了三维终端奇点和三维除收缩的分解，得到了以下结果。(1)我们构造的双有理态射从3倍，只有终端奇点的指数为1的3维终端奇点的指数m大于1，通过重复的除法压缩最小的差异。结果确定了指数大于或等于2的三维终端奇点上的所有素因子，它们的偏差不大于1。(2)我们证明了不存在由M.Reid提出的原始形式的经济归结，即如果存在三维终端奇点的部分归结，其中所有的素因子的偏差不大于1，作为其例外因子。我们还证明了在改变奇点的条件后，存在这样的经济解决方案。(3)对于指数大于1的三维终端奇点，我们给出了将因子收缩到偏差为1的点的因子收缩的明确描述。这种除法压缩是通过将奇点嵌入到四维或五维空间中并进行适当的加权爆破而获得的。(4)我们给出了具有偏差1的Gorenstein型cD和cE终端奇点的整除收缩的一个明确的描述。

We studied on resolutions of 3-dimensional terminal singularities and 3-dimensional divisorial contractions and get the results below.(1) We construct birational morphisms from 3-folds with only terminal singularities with index 1 to 3-dimensional terminal singularities with index m greater than one, by repeating divisorial contractions with minimal discrepancy. As a result we determined all prime divisors over 3-dimensional terminal singularities with index greater than or equal to 2 which have discrepancy not greater than 1.(2) We showed that there does not exist economic resolutions due to M.Reid in its original form, which states that if there are partial resolutions of 3-dimensional terminal singularities with all prime divisors with discrepancy not greater than one as its exceptional divisors. We also showed that there are such economic resolutions after changing conditions on singularities.(3) We gave an explicit description of divisorial contractions which contract divisors to points with discrepancy one, for 3-dimensional terminal singularities with index greater than one. Such divisorial contractions are obtained by embedding singularities into four or five dimensional spaces and by making suitable weighted blowing ups.(4) We gave an explicit description of divisorial contractions of Gorenstein terminal singularities of type cD and cE with discrepancy one.