论文标题
图像Milnor数字和出色的展开
The image Milnor number and excellent unfoldings
论文作者
论文摘要
我们在图像中显示了三个基本属性$μ_i（f）$ $ f \ colon（\ Mathbb {c}^{n}，s），s）\ rightArrow（\ Mathbb {c}^{n+1}，0）$具有孤立的不稳定。首先，我们展示了图像米尔诺数的保护，从中可以推断出家庭的上半持续点和家庭拓扑不变性。其次，我们证明了Mond的猜想，该猜想说$μ_i（f）= 0 $时，仅当$ f $稳定时。最后，我们表明休斯顿的一个猜想是，任何家庭$ f_t \ colon（\ mathbb {c}^{n}，s）\ rightArrow（\ mathbb {c}^{n+1}，0）$，带有$μ_i_i（f_t）$ constant的$。由于技术原因，在最后两个属性中，我们仅考虑Corank 1案例。
We show three basic properties on the image Milnor number $μ_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond's conjecture, which says that $μ_I(f)=0$ if and only if $f$ is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $μ_I(f_t)$ constant is excellent in Gaffney's sense. By technical reasons, in the two last properties we consider only the corank 1 case.
