Characteristic p method in singularity theory

我们将“特征p”方法应用于奇异性理论，得到了一些有意义的结果：1利用Frobenius自同态刻画特征0中的奇异性。我们发现当环是Q-Gorenstein环时，对数-终端奇点与F-正则环是等价的。对于有理奇点和F-有理环也是如此。此外，我们发现，同样是真实的“奇点对”（KLT，PLT等）。后一个结果是与N.Hara.2的联合工作。“Hilbert-Kunz多重性”的概念，这是局部环的一个新的“多重性”。我们刻画了正则局部环，某些有理奇点在2维的这个多重性。我们还发现了二维有理双点中的整闭理想的一个非常美丽而神秘的公式。（一个联合工作与吉田）3我们调查链的整体封闭的理想，并发现存在的一个组成系列只由整体封闭的理想。我们发现，族的整闭理想的色长为1对应的封闭点的纤维锥。同时，我们还得到了2维正则局部环中单整闭理想的一个新的刻画。(The最后的成果是与S. Noh的合作。

We applied "characteristic p" methods to singularity theory and obtained the fokkowing results.1 Characterization of Singularities in Characteristic 0 via Frobenius endomorphism. We found that log-terminal singularity and F-regular rings are equivalent notions in the case the ring is Q- Gorenstein. The same is true for rational singularities and F-rational rings. Also, we found that the same is true for "singularity of pairs" (KLT, PLT etc.). The latter result is a joint work with N.Hara.2 The notion of "Hilbert-Kunz multiplicity", which is a new "multiplicity" for local rings. We characterized regular local rings, certain rational singularities in dimension 2 by this multiplicity. Also we found a very beautiful and mysterious formula for integrally closed ideals in 2-dimensional rational double points. (a joint work with K.Yoshida)3 We investigated chains of integrally closed ideals and found the existence of a composition series only by integrally closed ideals. We found that the family of integrally closed ideals of colength 1 corresponds to the closed points of the fiber cone. Also, we found a new characterization of simple integrally closed ideals in 2-dimensional regular local rings. (The last result is a joint work with S.Noh.)