Research for multiplicities and tight closures on singular points of positive characteristic

1. 关于Hilbert-Kunz多重度的下界：在开始本研究之前，我们已经证明了任何Hilbert-Kunz多重度为1的非混合局部环是正则局部环。实际上，这个定理是对永田经典定理在正性质上的推广。在本研究中，我们考虑了一个寻找非正则局部环的Hilbert-Kunz多重度下界的问题。结果，我们得到了二次超曲面可以得到这样一个下界的一个猜想，并证明了它对于不超过4维的局部环是成立的。我们给出的下界在代数几何中是很有趣的，但是我们不能得到任何充分的理论。此外，Enescu-Shimamoto在完全相交的情况下证明了这个猜想。极小Hilbert-Kunz复数我们引入极小Hilbert-Kunz复数的概念来估计f正则局部环的坏性。不变量是0到1区间内的实数。最近，Aberbach等证明了局部环是f正则的当且仅当其最小Hilbert-Kunz多重为正。我们确定了典型的f正则环仿射环奇点和商奇点的最小Hilbert-Kunz复数。具有最小多重性的Buchsbaum Stanley-Reisner环的性质。我们与佐贺大学的Naoki Terai一起研究了Buchsbaum Stanley-Reisner环的最小自由分辨率，多重度，h向量。特别地，我们给出了这些环的多重度的下界，并对这些环进行了表征。

1. On Lower bounds for Hilbert-Kunz multiplicities:We have proved that any unmixed local ring with Hilbert-Kunz multiplicity one is a regular local ring before starting this research. Indeed, this theorem is a generalization of Nagata's classical theorem in positive characteristic. In this research, we considered a problem of finding a lower bound on Hilbert-Kunz multiplicities for non-regular local rings. As a result, we found a conjecture that such a lower bound is attained by quadratic hypersurface, and proved that it is true for local rings of at most dimension 4. The lower bound given by us is interesting in algebraic geometry, but we cannot obtain any sufficient theory. Moreover, the conjecture was proved by Enescu-Shimamoto in the case of complete intersections.2. Minimal Hilbert-Kunz multiplicityThe notion of minimal Hilbert-Kunz multiplicities was introduced by us to estimate of badness of F-regular local rings. The invariant is a real number in the interval between 0 and 1. Recently, Aberbach etc. proved that a local ring is F-regular if and only if its minimal Hilbert-Kunz multiplicity is positive. We determined the minimal Hilbert-Kunz multiplicities for affine toric singularities and quotient singularities, which are typical F-regular rings.3. Characterization of Buchsbaum Stanley-Reisner rings with minimal multiplicity.We studied minimal free resolutions, multiplicities, h-vectors for Buchsbaum Stanley-Reisner rings together with Naoki Terai at Saga University. In particular, we gave a lower bound for multiplicities for those rings, and characterized such rings.