四重曲面奇点的一个注记

A Note on Surface Singularities of Multiplicity Four

令P为复曲面Y之四重孤立奇点.众所周知,存在局部不可约有限覆盖π:(Y,P)→(X,p)满足π^(-1)(p)=P,以及Jung氏解消f:Y^(~)→Y.今设W_(p)为(π■f)^(-1)(p)之例外除子,我们将证明W_(p)有唯一基本闭链分解W_(p)=2Z1或W_(p)=∑α=1l Zα使其满足若干性质.我们将定义π于p处的指标w_(p),并用上述分解求其值.特别地,可证(Y,P)为奇点当且仅当w_(p)≥1.作为W_(p)分解式的另一应用,我们将计算Y^(~)收缩到极小解消所需的步数.

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 coveringπ:(Y,P)→(X,p)withπ-1(p)=P,and a Jung’s resolution f:Y^(~)→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)=2 Z1 or W_(p)=∑α=1l Zαsatisfying some conditions.We will define a local index w_(p) forπat 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 fromY^(~).