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

关于某些类型的复杂曲面奇点与紧致黎曼曲面简并族之间的某些关系。

基本信息

批准号：
20540062
负责人：
TOMARU Tadashi
金额：
$ 1.33万
依托单位：
Gunma University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2008
资助国家：
日本
起止时间：
2008 至 2010
项目状态：
已结题

项目摘要

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.

近15年来，我们一直在研究法向曲面奇异性与紧致复光滑曲线退化之间的关系，大约10年前，我写了一篇论文“法向曲面奇异性的亏格”（J.Math.Soc.Japan，2007）。在这里，我们证明了：给定一个法曲面奇点（X，0）和该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，它自然地扩展了分辨空间，纤维映射扩展了f.利用这个结果，我们定义了一个不变量，它的名字叫（X，o）的笔亏格，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。我们也称之为C*-曲线束。本文研究了C ~*-作用下C ~*-曲线束与法曲面奇点之间的若干关系。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的对偶性（即，例如，Milnor数和Goto-Watanabe a-不变量）。为了证明这一点，我们还证明了一个基本公式的循环覆盖的循环商奇点。我们完成了一篇题为[C ~*-equivariant degenerations of curves and normal surface singularities with C ~*-action]的论文，共51页，已于今年5月4日投稿。