The research of 2-dimensional complex singularities associated to degenerations of closed Riemann surfaces

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In this research, we investigated the following and obtained following results.(1) We have been investigated the structure of degenerations of closed Riemann surfaces with C^*-action. Four years ago, Tomaru proved that there is a very natural construction of degenerations of closed Riemann surfaces from complex surface singularities and holomorphic functions on the singularities. We prove similar result for normal surface singularities with C^*-action.(2) Let (X,o) be a normal surface singularity obtained by the contraction of the zero-section of a line bundle on a curve. We gave a necessary and sufficient condition for (X,o) to be Kodaira (or Kulikov) singularity. Using this, we gave an example which is a Kodaira singularity but not a Kulikov singularity.(3) We determined the value of pencil genus of rational triple points by using Artin's classification of rational triple points and Kodaira's classification of elliptic degenerations.(4) We prove some results on some relation between quasi-rational singularities and cyclic coverings.