论文标题
蒙德猜想的薄弱版本
A weak version of Mond's conjecture
论文作者
论文摘要
我们证明了一个地图细菌$ f：（\ mathbb {c}^n，s）\ to（\ mathbb {c}^{n+1}，0）$隔离不稳定的$稳定，并且仅当$μ_i（f）= 0 $时，且仅当$μ_i（f）= 0 $时，$μ_i（f）$是$μ_i（f）$是图像MILNOR编号。在上一篇论文中，我们证明了这一结果，并以$ f $具有Corank One的另一种假设。这里的证明也对Corank $ \ ge 2 $有效，前提是$（n，n+1）$在马瑟的意义上是不错的维度（因此$μ_i（f）$定义得很好）。我们的结果可以看作是Mond的猜想的弱版本，该版本说$ f $的$ \ MATHCAL {a} _e $ -sodimension是$ f $的$ \leμ_i（f）$，如果$ f $加权同质，则具有均等。作为一个应用程序，我们推断出$ f $的广泛展开的分叉集是一个超出表面。
We prove that a map germ $f:(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instability is stable if and only if $μ_I(f)=0$, where $μ_I(f)$ is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that $f$ has corank one. The proof here is also valid for corank $\ge 2$, provided that $(n,n+1)$ are nice dimensions in Mather's sense (so $μ_I(f)$ is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $\mathcal{A}_e$-codimension of $f$ is $\le μ_I(f)$, with equality if $f$ is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of $f$ is a hypersurface.
