We only need to consider around a point $(x,t,s)$ on the exceptional set $\mathcal{N}$, so $(x,t)\in D\times\{0\}$. Without loss of generality may assume $x_{0}\neq 0$. Since $D$ is a complete intersection by assumption (iii), 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 $\mathbb{C}\mathbb{P}^{n+1}$. So we can write

(7.29)

$$\Gamma=-\frac{J^{-1}}{(n+2)t^{n+1}f(u)}dv_{1}\wedge dv_{2}\wedge du_{3}\cdots\wedge du_{n+1},$$

where $J$ is the Jacobian given by

(7.30)

$$J=\frac{\partial f_{1}}{\partial u_{1}}\frac{\partial f_{2}}{\partial u_{2}}-\frac{\partial f_{1}}{\partial u_{2}}\frac{\partial f_{2}}{\partial u_{1}}.$$

Suppose first we work on the affine chart $\{s_{1}\neq 0\}$. Then we get the local equations for $\widehat{\mathcal{X}}$ given by (7.22). Since we are away from $D_{1}$, we must have $\zeta_{3}\neq 0$. Then we can use $\zeta_{3},t,u_{3},\cdots,u_{n+1}$ as local holomorphic coordinates on $\widehat{\mathcal{X}}$. We have

(7.31)

$$dv_{1}=-t^{d_{2}}fd\zeta_{3}-d_{2}t^{d_{2}-1}\zeta_{3}fdt-t^{n+2}\zeta_{3}df,$$

(7.32)

$$dv_{2}=d_{1}t^{d_{1}-1}\zeta_{3}^{-1}dt-\zeta_{3}^{-2}t^{d_{1}}d\zeta_{3}$$

and

(7.33)

$$df=\frac{\partial f}{\partial v_{1}}dv_{1}+\frac{\partial f}{\partial v_{2}}dv_{2}+\sum_{j\geq 3}\frac{\partial f}{\partial u_{j}}du_{j}.$$

So we get

		$\displaystyle(1+t^{d_{2}}\zeta_{3}\frac{\partial f}{\partial v_{1}})dv_{1}$
(7.34)		$\displaystyle=$	$\displaystyle(-t^{d_{2}}f+t^{n+2}\zeta_{3}^{-1}\frac{\partial f}{\partial v_{2}})d\zeta_{3}-(d_{2}t^{d_{2}-1}\zeta_{3}f+d_{1}t^{n+1}\frac{\partial f}{\partial v_{2}})dt$	$\displaystyle\mod(du_{3},\cdots,du_{n+1}).$

Hence we get

(7.35)

$$\Gamma=\frac{\zeta_{3}^{-1}}{(1+t^{d_{2}}\zeta_{3}\frac{\partial f}{\partial v_{1}})}J^{-1}d\zeta_{3}\wedge dt\wedge du_{3}\wedge\cdots\wedge du_{n+1}.$$

Near $t=0$ we see $\Gamma$ is smooth around such a point. Similarly we can deal with the chart $\{s_{2}\neq 0\}$.

Now on $\{s_{3}\neq 0\}$, we only need to consider a point on $D$ where $f=0$, then by our assumption (iv) we may use $v_{3}=f$ as a local holomorphic coordinate to replace $u_{3}$ for instance. Then we can write

(7.36)

$$\Gamma=-\frac{1}{(n+2)t^{n+1}f}K^{-1}dv_{1}\wedge dv_{2}\wedge dv_{3}\wedge du_{4}\cdots\wedge du_{n+1},$$

where $K$ is the Jacobian for the change of coordinates. We have

(7.37)

$$dv_{3}=-(\zeta_{1}d\zeta_{2}+\zeta_{2}d\zeta_{1}),$$

(7.38)

$$dv_{1}=t^{d_{2}}d\zeta_{2}+d_{2}\zeta_{2}t^{d_{2}-1}dt,$$

(7.39)

$$dv_{2}=t^{d_{1}}d\zeta_{1}+d_{1}\zeta_{1}t^{d_{1}-1}dt.$$

Then we get

(7.40)

$$\Gamma=K^{-1}dt\wedge d\zeta_{1}\wedge d\zeta_{2}\wedge du_{4}\cdots\wedge du_{n+1},$$

which is smooth. ∎