Assume $M,N$ are smooth varieties over $\mathbb C$. Let $f,g$ be projective and flat morphisms over $\Delta:=\{z\in\mathbb C||z|<1\}$ and $h:M\to N$ is a dominant map and is a generically finite morphism over $\Delta$. So we have a commutative diagram $\require{AMScd}$ \begin{CD} M @>h>> N\\ @V{f}VV @V{g}VV\\ \Delta @= \Delta \end{CD} Let $Z$ be a component of $M_0=f^{-1}(0)$, so by our assumption $h(Z)$ is a component of $N_0=g^{-1}(0)$.
Take a general point $p\in Z$ and a disk $\Delta_p$ in $M$ which is normal to $M_0$ at $p$. Then the restriction $h|_{\Delta_p}$ is a map
$$h|_{\Delta_p}:\Delta_p\to N$$
Question 1: Is it true that $h|_{\Delta_p}$ can be identified with $z\mapsto z^k$ for some $k\ge 1$, so that we can call $k$ the $\textit{ramification index}$ of map $h$ along $Z$? Is this notion ever defined somewhere?
Question 2: Is it true that $k$ is constant on an open dense subset of $Z$? More precisely, (I guess) we can choose $U\subset Z$ to be the set such that $p\in U$ is a smooth point of $Z$, $h(p)$ a smooth point of $h(Z)$ and $p$ is not intersection of $Z$ with other components of $f^{-1}(0)$.
I'm sure these type of questions have been studied a long time ago, but I just can't find the relevant reference to the literature. I appreciate for any answer and suggestion on reference.