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

我们研究了三维端点奇点和三维除数收缩的分辨率，得到了如下结果：(1)通过重复差值最小的除法收缩，构造了从只有索引为1的端点奇点的3折到索引为m大于1的3维端点奇点的双分态射。结果，我们确定了指数大于或等于2且差异不大于1的三维终端奇点上的所有质因数。(2)我们证明了M.Reid不存在其原始形式的经济解决方案，即如果存在以差异不大于1的所有素数作为其例外除数的三维终端奇点的部分解决方案。我们还证明了在改变奇点上的条件后存在这样的经济决议。(3)对于指数大于1的三维端点奇点，给出了将除数压缩到差值为1的点的除数压缩的显式描述。通过将奇异点嵌入四维或五维空间，并通过适当的加权膨胀，可以获得这种除法压缩。(4)给出了cD型和cE型的Gorenstein终端奇点的分缩，其差值为1。

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.