Equisingularidades de funções definidas em ICIS e IDS

Abstract

We study the equisingularity of a family of function germs $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$, where $\{(X_t,0)\}$ is a family of $d$-dimensional isolated determinantal singularity. We define the $(d-1)$th polar multiplicity of the fibers $X_t\cap f_t^{-1}(0)$, $m_{d-1}(X_t\cap f_t^{-1}(0),0)$, and we present results relating the constancy of $m_{k}(X_t\cap f_t^{-1}(0),0)$ for $k=0,\ldots,d-1$ and $m_i(X_t,0)$ for $i=0,\ldots,d$ with the constancy of the Milnor number of $f_t$ and the Whitney equisingularity of the families $\{(X_t\cap f_t^{-1}(0),0)\}$ and $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$.

In the particular case where $\{(X_t,0)\}$ is a family of isolated complete intersection singularity we provide a condition to ensure the Whitney conditions in terms of the integral closure of the ideal defining the singular set of each member of family $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$. We also relate the constancy of the Milnor number of $f_t$ with the strict integral closure of the module formed by the partial derivatives of the application that defines $X_t\cap f_t^{-1}(0)$.