On ring-theoretical invariants of singular points in positive characteristic

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We have studied Hilbert-Kunz multiplicity as an invariant of singular points in positive char acteristic for three years. The most important result in our work is to give a characterization of regular local rings in terms of Hilbert-Kunz multiplicity. Actually, many researchers tried to gener alize our theorem. After this research, we have studied Hilbert-Kunz multiplicity of ideals defined by the dual graph of the resolution of singularities. Note that Hilbert-Kunz multiplicity for such an ideal is a ring-theoretical invariant associated to isolated singularity in positive characteristic. As one of our results, for integrally closed ideals in a rational double point, we obtained algorithm for calculating their Hilbert-Kunz multiplicities in terms of the dual graph. On the other hand, we have tried calculation of Hilbert-Kunz multiplicity for blow-up rings, but we could not get complete algorithm. As a partial result, we get some inequalities with respect to blow-up rings and the basering.Also, we introduced the notion of the minimal Hilbert-Kunz multiplicity and gave several method for calculation. This invariant can be described as the difference of the Hilbert-Kunz multiplicities of some pairs of ideals. Furthermore, we found that this invariant is equal to the invariant which is defined by other researchers. We gave a presentation of our results as above at Symposium on Commutative algebra and at Symposium on Algebra on Summer in 2001. Also, we have a project to study blow-up rings in positive characteristic.