This involves only local discussion. By construction we get overlapping local holomorphic charts on $\mathbb{C}\mathbb{P}^{n+1}$ given by $\{v_{1},v_{2},u_{i}(i\geq 3)\}$ and $\{v_{1}^{\prime},v_{2}^{\prime},u_{i}^{\prime}(i\geq 3)\}$. Given a point in this overlap with coordinates $(v_{1},v_{2},u_{i})$ and $(v_{1}^{\prime},v_{2}^{\prime},u_{i}^{\prime})$ in these two coordinate charts respectively, then we have

(7.102)

$$\begin{cases}v_{1}^{\prime}=v_{1}\cdot Q_{1}(v_{1},v_{2},u_{i})\\ v_{2}^{\prime}=v_{2}\cdot Q_{2}(v_{1},v_{2},u_{i})\\ u_{i}^{\prime}=R_{i}^{\prime}(v_{1},v_{2},u_{i}).\end{cases}$$

where $Q_{1},Q_{2}$ are smooth and non-vanishing along $D$. More precisely, we have

(7.103)

$$Q_{i}=(\sigma_{\beta^{\prime}}^{1}/\sigma_{\beta}^{1})^{d_{i}}.$$

Correspondingly we obtain the transition maps on $U^{1}_{\beta}\cap U^{1}_{\beta^{\prime}}$ given by

(7.104)

$$\begin{cases}v_{2}^{\prime}=v_{2}\cdot Q_{2}(v_{1},v_{2},u_{i});\\ \zeta_{3}^{\prime}=\zeta_{3}\cdot Q_{2}^{-1}(v_{1},v_{2},u_{i});\\ u_{i}^{\prime}=R_{i}^{\prime}(v_{1},v_{2},u_{i});\end{cases}$$

where using (7.22) we can write $v_{1}$ implicitly as a function of $v_{2},\zeta_{3}^{\prime}$ and $u_{i}$. In particular, we obtain the transition function of $Y_{1}\cap U_{\beta}^{1}\cap U_{\beta^{\prime}}^{1}$ given by

(7.105)

$$\begin{cases}v_{2}^{\prime}=v_{2}\cdot Q_{2}(0,v_{2},u_{i});\\ u_{i}^{\prime}=R_{i}^{\prime}(0,v_{2},u_{i}),\end{cases}$$

and $\mathcal{N}\cap U_{\beta}^{1}\cap U_{\beta^{\prime}}^{1}$ given by

(7.106)

$$\begin{cases}\zeta_{3}^{\prime}=\zeta_{3}\cdot Q_{2}^{-1}(v_{1},0,u_{i});\\ u_{i}^{\prime}=R_{i}^{\prime}(v_{1},0,u_{i}).\end{cases}$$

Then the conclusion follows by a direct calculation. ∎