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.今设Wp为(π ? f)-1(p)之例外除子,我们将证明Wp有唯一基本闭链分解Wp=2Z1或Wp=Σα=1l Zα使其满足若干性质.我们将定义π于p处的指标wp,并用上述分解求其值.特别地,可证(Y,P)为奇点当且仅当wp ≥ 1.作为Wp分解式的另一应用,我们将计算?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 covering π:(Y, P) → (X, p) with π-1(p)=P, and a Jung's resolution f:? → Y. Let Wp be the exceptional divisor of (π?f)-1(p). We will prove that Wp has a unique decomposition into fundamental cycles Wp=2Z1 or Wp=Σα=1l Zα satisfying some conditions. We will define a local index wp for π at p and compute it by the above decomposition of Wp. In particular, we will show that (Y, P) is singular iff wp ≥ 1. As another application of the decomposition of Wp, we also compute the number of blown-downs needed to get the minimal resolution from ?.
基本闭链 / 曲面奇点 / 有限覆盖 / Jung解消 / 典范解消 {{custom_keyword}} /
fundamental cycle / surface singularity / finite covering / Jung's resolution / canonical resolution {{custom_keyword}} /
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国家自然科学基金(11671140);上海市核心数学与实践重点实验室基金(18dz2271000)
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