Multiplicities and equisingularity of ICIS germs
Let \(F : (\mathbb{C} \times \mathbb{C}^n, 0) \to \mathbb{C}^{n - d}\) define a complete intersection \(X\) with reduced structure such that for fixed \(t\) \(F(t) : (\mathbb{C}^n, 0) \to \mathbb{C}^{n - d}\) defines a reduced complete intersection \(X_t\) with isolated singularity at 0. It is proved that the following conditions are equivalent: (i) \(X \backslash T\) is smooth and Whitney regular over \(T = \mathbb{C} \times 0\); (ii) the polar multiplicities \(m_k (X_t)\) are independent of \(t\); (iii) \({\partial F \over \partial t} \in {\mathfrak m} ({\partial F \over \partial z_1}, \dots, {\partial F \over \partial z_n})\) and \(X \backslash T\) is smooth; (iv) for the submodules \({\mathfrak m} ({\partial F(t) \over \partial z_1}, \dots, {\partial F(t) \over \partial z_n}) \subseteq 0^{n - d}_{X_t, 0}\) the Buchsbaum-Rim multiplicity \(e ({\mathfrak m} ({\partial F(t) \over \partial z_1}, \dots, {\partial F(t) \over \partial z_n}))\) is constant. A formula is given to express the multiplicities of (iv) in terms of polar multiplicities.
scientific article; zbMATH DE number 856585