The smoothness of $Z$ and the assumption that the components of $\mathscr{X}_{k}$ are Cartier divisors imply that $\mathscr{X}$ is regular at any point of $Z$. Indeed, for any point $p\in Z$ and $j\in J$, let $z_{j}\in\mathcal{O}_{\mathscr{X},p}$ be a local equation of $D_{j}$ at $p$. As $\mathcal{O}_{Z,p}\simeq\mathcal{O}_{\mathscr{X},p}/(z_{0},\ldots,z_{n-r})$ is a regular local ring of dimension $r$, $(z_{0},\ldots,z_{n-r})$ can be extended to form a regular system of parameters for $\mathcal{O}_{\mathscr{X},p}$.