We prove that a map germ \(f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)\) with isolated instability is stable if and only if \(\mu _I(f)=0\), where \(\mu _I(f)\) 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 \(\mu _I(f)\) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the \(\mathscr {A}_e\)-codimension of f is \(\le \mu _I(f)\), 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.

我们证明具有孤立不稳定性的映射胚\(f:(\mathbb {C}^n,S)\rightarrow (\mathbb {C}^{n+1},0)\) 是稳定的当且仅当\ (\mu _I(f)=0\)，其中\(\mu _I(f)\)是 Mond 定义的图像 Milnor 数。在之前的一篇论文中，我们通过附加假设f具有 corank 1 来证明了这个结果。这里的证明对于 corank \(\ge 2\)也有效，前提是\((n,n+1)\)在马瑟意义上是很好的维度（所以\(\mu _I(f)\)是明确定义的）。我们的结果可以看作 Mond 猜想的弱版本，它说f的\(\mathscr {A}_e\)余维数为\(\le \mu _I(f)\)，如果f是加权齐次的，则相等。作为一个应用，我们推论f的范数展开的分叉集是一个超曲面。