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

1.关于Hilbert-Kunz重数的下界：在开始这项研究之前，我们已经证明了任何Hilbert-Kunz重数为1的非混合局部环是正则局部环。实际上，这个定理是Nagata经典定理在正特征方面的推广。在这项研究中，我们考虑了一个问题，找到一个下界的Hilbert-Kunz重数的非正则局部环。结果，我们发现了一个猜想，即二次超曲面可以达到这样的下界，并证明了对于至多4维的局部环，这个猜想是成立的。我们给出的下界在代数几何中很有趣，但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该猜想.最小Hilbert-Kunz重数我们引入了最小Hilbert-Kunz重数的概念来估计F-正则局部环的劣性。不变量是介于0和1之间的真实的数。最近Aberbach等人证明了局部环是F-正则的当且仅当它的最小Hilbert-Kunz重数是正的。确定了仿射环面奇异环和商奇异环的最小Hilbert-Kunz重数，这是典型的F-正则环.具有最小重数的Buchsbaum Stanley-Reisner环的刻画我们与佐贺大学的寺井直树一起研究了Buchsbaum Stanley-Reisner环的最小自由分解、重数、h-向量。特别地，我们给出了这类环的重数下界，并刻画了这类环。

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.