On some relations between complex surface singularities of some types and degeneration families of compact Riemann surfaces.

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Since 15 years ago, we have been researching some relations between normal surface singularities and degenerations of compact complex smooth curves.Around ten years ago, I wrote a paper 「Pencil genus for normal surface singularities」(J.Math.Soc.Japan, 2007). There, we prove that given a normal surface singularity (X,0) and an element f of the maximal ideal of the singularity, there exists a one parameter degeneration family of of curves which naturally extends the resolution space and the fiber map extends f. Using this result, we defined an invariant whose name is pencil genus of (X,o), and also studied the several properties.Now, let C* be the complex multiplicative group. Around 6 years ago, we have been studying the C*-equivariant degenerations family of curves. Also, we call them C*-pencil of curves. In this research, we studied the several relations between C*-pencil of curves and normal surface singularities with C*-action. From this point of view, we can introduced the notion of "dual" C*-pencil of curves. This properties reflect dualities of some invariants (i.e., for example, Milnor numbers and Goto-Watanabe a-invariant). To prove this, we also prove a fundamental formula on cyclic covers of cyclic quotient singularities. Also, we gave a canonical method to construct all C*-pencil of curves from holomorphic line bundle on curves.We complete a paper whose title is [C*-equivariant degenerations of curves and normal surface singularities with C*-action], which contains 51 pages and was submitted a journal in May 4 in this year.