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On pencil genus of 2-dimensional singularities

关于二维奇点的铅笔亏格

基本信息

批准号：
12640060
负责人：
TOMARU Tadashi
金额：
$ 0.77万
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：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640060/
关键词：
complex analytic space normal singularity degenerate family of algebraic curves pencil genus fundamental cycle cyclic covering Kodaira singularity Kulikov singularity 二次元特異点巡回被覆特異点基本種数

项目摘要

On any singularity (X,o) of a complex analytic space, we consider a good resolution space. Then we prove that it is embedded into a total space a pencil of algebraic curves. Let consider the minimal value of the genus of such pencil. We call the value "pencil genus of (X,o)" and write it p_e(X,o). Also, let f be an element of the maximal ideal of (X,o) which satisfies some properties. Let Φ : S → Δ be a pencil of algebraic curves and π : (Y,E)→(X,o) a good resolution such that S contains Y and satisfies the composition of f and π is equal to the restriction of Φ to Y. For such pair (X,o) and f, we consider such pencils and resolutions. Let consider the minimal value of the genus of such pencil. We call the value "pencil genus of a pair of (X,o) and f" and write it p_e(X,o,f). In this paper we researched the fundamental properties of p_e(X,o) and p_e(X,o,f). Our most important results are as follows :Theorem 1. Let (X,o) be a normal surface singularity and let h an element of the maximal ideal of (X,o). Let π : (Y,E)→(X,o) be a good resolution such that (h o π) is a simple normal crossing divisor on Y. Then there exists a pencil of curves Φ : S → Δ of genus p_e(X,o,h) which satisfies the above property and all connected components of supp(S_o)＼ E are minimal P^1-chains started from E.Theorem 2. Let (X,o) be a normal surface singularity. Suppose p_f(X,o)≧ 2. Then (X,o) is a Kodaira singularity if and only if the w.d.graph for the minimal good resolution of (X,o) is a Kodaira graph and p_e(X,o)=p_f(X,o)Theorem 3. (X,o) is a Kulikov singularity if and only if there is a reduced element h in the maximal ideal of (X,o) with p_e (X,o,h) =p_f(X,o).

在复解析空间的任何奇点（X，o）上，我们考虑一个好的分解空间.然后证明它是嵌入到一个总空间的一束代数曲线。让我们考虑这种铅笔的亏格的最小值。我们称这个值为“（X，o）的铅笔亏格”并将其记为p_e（X，o）。另外，设f是（X，o）的极大理想中满足某些性质的元素.令Φ：S → Δ是一束代数曲线，π：（Y，E）→（X，o）是一个好的分解，使得S包含Y，满足f的复合，π等于Φ对Y的限制.对于这样的对（X，o）和f，我们考虑这样的铅笔和决议。让我们考虑这种铅笔的亏格的最小值。我们称该值为“一对（X，o）和f的铅笔亏格”，并将其写为p_e（X，o，f）。本文研究了p_e（X，o）和p_e（X，o，f）的基本性质。我们最重要的结果如下：定理1。设（X，o）是法曲面奇点，h是（X，o）的极大理想的元素.设π：（Y，E）→（X，o）是一个好的分解，使得（ho π）是Y上的一个简单的正规交叉因子.则存在亏格p_e（X，o，h）的曲线束Φ：S → Δ满足上述性质，且supp（S_o）\ E的所有连通分支都是从E出发的极小P^1-链。设（X，o）为法向曲面奇点。设p_f（X，o）<$2.则（X，o）是科代拉奇点当且仅当（X，o）的最小好分解的w.d.图是科代拉图且p_e（X，o）=p_f（X，o）定理3.（X，o）是库利科夫奇性当且仅当（X，o）的极大理想中存在约化元h，其中p_e（X，o，h）=p_f（X，o）.