四重曲面奇点的一个注记

Let P be an isolated singularity of multiplicity 4 of a complex surface Y. It is well-known that there is a locally irreducible finite covering π:(Y, P) → (X, p) with π^-1(p)=P, and a Jung's resolution f:Ỹ → Y. Let W_p be the exceptional divisor of (π◦f)^-1(p). We will prove that W_p has a unique decomposition into fundamental cycles W_p=2Z₁ or W_p=Σ_α=1^l Z_α satisfying some conditions. We will define a local index w_p for π at p and compute it by the above decomposition of W_p. In particular, we will show that (Y, P) is singular iff w_p ≥ 1. As another application of the decomposition of W_p, we also compute the number of blown-downs needed to get the minimal resolution from Ỹ.