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数压缩的分解问题，得到了以下结果：(1)通过重复具有最小偏差的除数压缩，从只有指数为1的终端奇点的3重到指数m大于1的三维终端奇点，构造了二元态射。结果，我们确定了指数大于或等于2且偏差不大于1的三维终端奇点上的所有素因数。(2)我们证明了由于M.Reid的原始形式，不存在经济解，即如果存在三维终端奇点的部分解，且所有偏差不大于1的素因数为其例外因子。我们还证明了改变奇点条件后存在这样的经济解。(3)对于指数大于1的三维终端奇点，我们给出了收缩因子到偏差为1的点的除法压缩的显式刻画。这种除法压缩是通过在四维或五维空间中嵌入奇点并进行适当的加权爆破得到的。(4)我们给出了Cd和Ce型Gorenstein终端奇点与偏差奇点的除法压缩的显式刻画。

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.