On pencil genus of 2-dimensional singularities

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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).