Computing Versal Deformations of Singularities with Hauser's Algorithm
Computing Versal Deformations of Singularities with Hauser's Algorithm
复制标题
用 Hauser 算法计算奇点的横向变形
DOI：
10.1007/978-3-642-39131-6_6
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
J. Stevens
中科院分区：
文献类型：
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作者：
J. Stevens
在用方程局部描述复杂解析空间时，一般需要比余维更多的方程。虽然这些方程是线性独立的，但它们在代数上不是独立的，而是通过所谓的合子联系在一起的。因此，不能独立地选择方程的摄动。这使得对所有可能的扰动的描述变成了一个不平凡的问题。通常情况下，一个人一步一步地扩展无限小的变形。这可能导致永无止境的计算，在有限数量的步骤后必须停止计算。大多数有记录的计算通过只考虑拟齐次奇点的负度变形来避免这个问题，其中基空间也具有多项式拟齐次方程。另一种方法是基于Teissier的思想(见[20])，由Herwig Hauser在他的论文[10]中发展出来的，另见[12，11]。给定由方程组f=(f1，…，fk)定义的奇点(X0，0)C(Cn，0)，首先确定映射f：(Cn，0)→(Ck，0)的横向开折，然后计算出其上有原始奇点变形的地层。一般地，映射f不是有限奇点类型，所以它的顶点展开是无限维的，并且
In describing complex analytic spaces locally by equations one needs in general more equations than the codimension. Although linearly independent, these equations will not be algebraically independent, but are related by so called syzygies. Therefore the perturbations of the equations cannot be chosen independently. This makes the description of all possible perturbations into a nontrivial problem.Normally one extends infinitesimal deformations step for step. This can lead to a never ending computation, which has to be cut off after a finite number of steps. Most calculations on record avoid this problem by only considering deformations of negative degree of quasi-homogeneous singularities, where the base space also has polynomial quasi-homogeneous equations. An alternative method, based on an idea of Teissier’s (see [20]), was developed by Herwig Hauser in his thesis [10], see also [12, 11]. Given a singularity (X0, 0) C (Cn, 0) defined by a system of equations f=(f1,..., fk), one first determines the versal unfolding of the map f:(Cn, 0)→(Ck, 0) without bothering about syzygies and then computes the stratum over which one has a deformation of the original singularity. In general the map f is not of finite singularity type, so its versal unfolding is infinite dimensional and