We know $\widehat{\mathcal{X}}$ is isomorphic to $\mathcal{X}$ away from $D\times\{0\}$, so it suffices to consider around a point $(x,0)$ where $f_{1}(x)=f_{2}(x)=0$. Locally in an affine chart $\{s_{1}\neq 0\}$, $\widehat{\mathcal{X}}$ is then cut out by the equations

(7.21)

$$\begin{cases}f_{2}(u)\zeta_{3}=t^{d_{1}};\\ f_{1}(u)\zeta_{3}=t^{d_{2}}\zeta_{2};\\ \zeta_{2}+\zeta_{3}^{2}f(u)=0;\\ f_{2}(u)\zeta_{2}=t^{d_{1}-d_{2}}f_{1}(u);\\ t^{d_{2}}\zeta_{3}f(u)+f_{1}(u)=0.\end{cases}$$

These can be reduced to two equations on the coordinates $u$, $t$ and $\zeta_{3}$, given by

(7.22)

$$\begin{cases}f_{2}(u)\zeta_{3}-t^{d_{1}}=0\\ t^{d_{2}}\zeta_{3}f(u)+f_{1}(u)=0.\end{cases}$$

By our assumption (iii) locally we may use $v_{1}=f_{1}(u)$ and $v_{2}=f_{2}(u)$ to replace $u_{1},u_{2}$ (say) as local holomorphic coordinates on a neighborhood of $x$ in $\mathbb{C}\mathbb{P}^{n+1}$. Then it is easy to see the corresponding subvariety is smooth if $\zeta_{3}\neq 0$, and has transversal $A_{d_{1}-1}$ singularities along $D_{1}$. So this gives the local description of $\widehat{\mathcal{X}}$ in a neighborhood of $D_{1}$. Similarly on $\{s_{2}\neq 0\}$ we also know the space is smooth except with transversal $A_{d_{2}-1}$ singularities along $D_{2}$.

On $\{s_{3}\neq 0\}$, we use $u,t,\zeta_{1},\zeta_{2}$ as coordinates, and we get the constraint equations

(7.23)

$$\begin{cases}\zeta_{1}\zeta_{2}+f(u)=0,\\ f_{2}(u)-t^{d_{1}}\zeta_{1}=0,\\ f_{1}(u)-t^{d_{2}}\zeta_{2}=0.\end{cases}$$

We only need to consider the points where $\zeta_{1}=\zeta_{2}=t=0$, so in particular we also have $f(u)=0$. At such a point, the differentials of these three equations are $(\nabla f(u),\nabla f_{2}(u),\nabla f_{1}(u))$. This is non-zero by our assumption (iv). ∎