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

自15年前以来，我们一直在研究紧致复光滑曲线的退化与法面奇点之间的一些关系。大约10年前，我写了一篇论文《法面奇点的铅笔亏格》(J.Math.Soc.Japan，2007)。在这里，我们证明了给定一个正常曲面奇点(X，0)和奇点的极大理想的一个元素f，存在一个单参数退化曲线族，它自然地扩展了分解空间，纤维映射扩展了f。利用这个结果，我们定义了一个不变量，命名为(X，o)的铅笔亏格，并研究了它的几个性质。大约6年前，我们一直在研究C*-等变退化曲线族。另外，我们称它们为C*-曲线束。在这项研究中，我们研究了具有C*作用的曲线束与法向曲面奇点之间的几种关系。从这个角度出发，我们可以引入“对偶”C*-曲线束的概念。这些性质反映了一些不变量(例如，Milnor数和Goto-Watanabe a-不变量)的对偶性。为了证明这一点，我们还证明了关于循环商奇点的循环覆盖的一个基本公式。给出了从曲线上的全纯直线丛构造所有C*-曲线束的典范方法，完成了一篇题为[C*-等变曲线退化与具有C*作用的法线奇点]的论文，全文51页，已于今年5月4日发表在期刊上。

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.