On pencil genus of 2-dimensional singularities

在复解析空间的任意奇点(X，o)上，我们考虑一个好的分解空间。然后，我们证明了它是嵌入到一个全空间中的一束代数曲线。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”，记为p_e(X，o)。同时，设f是(X，o)的极大理想的一个元素，它满足某些性质。设Φ：S→Δ是一支代数曲线束，π：(Y，E)→(X，o)是一个好的分解，使得S包含Y且满足f的合成，且π等于Φ对Y的限制。让我们考虑这种铅笔的亏格的最小值。我们称这个值为“一对(X，o)和f的铅笔亏格”，记为p_e(X，o，f)。本文研究了Pe(X，o)和Pe(X，o，f)的基本性质。我们最重要的结果如下：定理1.设(X，o)是正规曲面奇点，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(h oπ)是Y上的一个简单的法交因子，则存在一束满足上述性质的曲线Φ：亏格p_e(X，o，h)的S→Δ，且Supp(S_O)\E的所有连通分支都是从E.定理2开始的极小P^1-链。设p_f(X，o)≧2.则(X，o)是Kodaira奇点当且仅当(X，o)的极小好分解的W.D.图是Kodaira图，且p_e(X，o)=p_f(X，o)定理3.(X，o)是Kulikov奇点当且仅当(X，o)的极大理想中存在约化元素h且p_e(X，o，h)=p_f(X，o).

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).