四重曲面奇点的一个注记

洪杰, 陆俊

数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 455-462.

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数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 455-462. DOI: 10.12386/A2021sxxb0038
论文

四重曲面奇点的一个注记

    洪杰, 陆俊
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A Note on Surface Singularities of Multiplicity Four

    Jie HONG, Jun LU
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文章历史 +

摘要

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-1p)=P,以及Jung氏解消f?Y.今设Wp为(π ? f-1p)之例外除子,我们将证明Wp有唯一基本闭链分解Wp=2Z1Wpα=1l Zα使其满足若干性质.我们将定义πp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.

Abstract

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 Wpα=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 ?.

关键词

基本闭链 / 曲面奇点 / 有限覆盖 / Jung解消 / 典范解消

Key words

fundamental cycle / surface singularity / finite covering / Jung's resolution / canonical resolution

引用本文

导出引用
洪杰, 陆俊. 四重曲面奇点的一个注记. 数学学报, 2021, 64(3): 455-462 https://doi.org/10.12386/A2021sxxb0038
Jie HONG, Jun LU. A Note on Surface Singularities of Multiplicity Four. Acta Mathematica Sinica, Chinese Series, 2021, 64(3): 455-462 https://doi.org/10.12386/A2021sxxb0038

参考文献

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基金

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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