Let $\sigma$ and $\sigma^{\prime}$ be two maximal cones of $\Sigma$ intersecting along a face of codimension one. Setting $L_{\sigma\sigma^{\prime}}=L_{\sigma}\cap L_{\sigma^{\prime}}$, we may write $L_{\sigma}=L_{\sigma\sigma^{\prime}}\cup\{i_{0}\}$ and $L_{\sigma^{\prime}}=L_{\sigma\sigma^{\prime}}\cup\{i_{\infty}\}$. The sets $\mathcal{B}=((v_{i})_{i\in L_{\sigma\sigma^{\prime}}},v_{i_{0}},(v_{j})_{j\in J})$ and $\mathcal{B}^{\prime}=((v_{i})_{i\in L_{\sigma\sigma^{\prime}}},v_{i_{\infty}},(v_{j})_{j\in J})$ are bases of $\hat{N}=N\oplus\mathbb{Z}^{J}$. They induce isomorphisms $\beta,\beta^{\prime}:\mathbb{Z}^{r}\oplus\mathbb{Z}^{J}\rightarrow\hat{N}$ such that the change of basis from $\mathcal{B}$ to $\mathcal{B}^{\prime}$ is

$$M_{\mathcal{B}^{\prime}\mathcal{B}}=\beta^{\prime}\circ\beta^{-1}=\begin{array}[]{cccc}L_{\sigma\sigma^{\prime}}&i_{0}&J\\ \Id&(-C\cdot D_{i})_{i\in L_{\sigma\sigma^{\prime}}}&0&L_{\sigma\sigma^{\prime}}\\ 0&-1&0&i_{\infty}\\ 0&(-C\cdot D_{j})_{j\in J}&\Id&J\\ \end{array}\quad\textrm{ and }\left(\begin{matrix}v_{i}\\ v_{i_{\infty}}\\ v_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}^{T}\left(\begin{matrix}v_{i}\\ v_{i_{0}}\\ v_{j}\end{matrix}\right).$$

Denote by $((\varepsilon_{i})_{i\in L_{\sigma\sigma^{\prime}}},\varepsilon_{i_{0}},(\varepsilon_{j})_{j\in J})$ the basis of $\hat{M}\coloneqq\Hom(\hat{N},\mathbb{Z})$ dual to $\mathcal{B}$, and $((\varepsilon^{\prime}_{i})_{i\in L_{\sigma\sigma^{\prime}}},\varepsilon^{\prime}_{i_{\infty}},(\varepsilon^{\prime}_{j})_{j\in J})$ the basis dual to $\mathcal{B}^{\prime}$. It follows that

(2.4.1)

$$\left(\begin{matrix}\varepsilon^{\prime}_{i}\\ \varepsilon^{\prime}_{i_{\infty}}\\ \varepsilon^{\prime}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}\varepsilon_{i}\\ \varepsilon_{i_{0}}\\ \varepsilon_{j}\end{matrix}\right).$$

The isomorphisms $\beta$ and $\beta^{\prime}$ allow us to view

$$W^{\sigma}\coloneqq((W^{\sigma}_{i})_{i\in L_{\sigma\sigma^{\prime}}},W^{\sigma}_{i_{0}},(W^{\sigma}_{j})_{j\in J})\,\in\,(\mathbb{Z}^{r}\oplus\mathbb{Z}^{J})\otimes\Div_{0}(\mathscr{X})\simeq(\Div_{0}(\mathscr{X}))^{n+1}$$

and $W^{\sigma^{\prime}}$ as elements of $\widehat{N}\otimes\Div_{0}(\mathscr{X})$, that we will still denote by $W^{\sigma}$ and $W^{\sigma^{\prime}}$ .

The inverse $(s^{\sigma}_{i_{0}})^{-1}$ is a section on $\mathfrak{X}$ of $\mathscr{L}^{\sigma^{\prime}}_{i_{\infty}}$, so by Lemma 2.4.2 the sections

$$\begin{cases}s^{\sigma^{\prime}}_{i}\coloneqq s^{\sigma}_{i}\cdot(s^{\sigma}_{i_{0}})^{-(C\cdot D_{i})}&\textrm{ \quad for }i\in L_{\sigma\sigma^{\prime}}\\ s^{\sigma^{\prime}}_{i_{\infty}}\coloneqq(s^{\sigma}_{i_{0}})^{-1}&\\ s^{\sigma^{\prime}}_{j}\coloneqq s^{\sigma}_{j}\cdot(s^{\sigma}_{i_{0}})^{-(C\cdot D_{j})}&\textrm{ \quad for }j\in J\end{cases}$$

are sections on $\mathfrak{X}$ of the line bundles $\mathscr{L}^{\sigma^{\prime}}_{i}$ and $\mathscr{L}^{\sigma^{\prime}}_{j}$. By Lemma 2.3.1 these give equations for $D_{i}$ and $D_{j}$ on the open subscheme $\mathfrak{X}_{\sigma^{\prime}},$ and on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ we have

(2.4.3)

$$\left(\begin{matrix}s^{\sigma^{\prime}}_{i}\\ s^{\sigma^{\prime}}_{i_{\infty}}\\ s^{\sigma^{\prime}}_{j}\end{matrix}\right)=M_{\mathcal{B}^{\prime}\mathcal{B}}\left(\begin{matrix}s^{\sigma}_{i}\\ s^{\sigma^{\prime}}_{i_{0}}\\ s^{\sigma}_{j}\end{matrix}\right)$$

where the additive notation on the matrix corresponds to the multiplicative notation on the sections. Moreover, on $\mathfrak{X}_{\sigma}\cap\mathfrak{X}_{\sigma^{\prime}}$ we have

(2.4.4)

$\displaystyle\begin{split}w_{\sigma^{\prime}}&\coloneqq t\cdot\prod_{j\in J}(s_{j}^{\sigma^{\prime}})^{-1}\cdot\prod_{i\in L_{\sigma^{\prime}}}(s_{i}^{\sigma^{\prime}})^{-1}=t\cdot\prod_{j\in J}(s^{\sigma}_{j})^{-1}(s^{\sigma}_{i_{0}})^{C\cdot D_{j}}\cdot\prod_{i\in L_{\sigma\sigma^{\prime}}}(s^{\sigma}_{i})^{-1}(s^{\sigma}_{i_{0}})^{C\cdot D_{i}}\cdot(s^{\sigma}_{i_{0}})\\ &=t\cdot\prod_{j\in J}(s^{\sigma}_{j})^{-1}\cdot\prod_{i\in L_{\sigma\sigma^{\prime}}}(s^{\sigma}_{i})^{-1}\cdot(s^{\sigma}_{i_{0}})^{\sum_{j\in J}C\cdot D_{j}+\sum_{i\in L_{\sigma\sigma^{\prime}}}C\cdot D_{i}+1}=w_{\sigma},\end{split}$

hence the invertible function $w_{\sigma}$ on $\mathfrak{X}_{\sigma}$ extends to $\mathfrak{X}_{\sigma}\cup\mathfrak{X}_{\sigma^{\prime}}$ by $w_{\sigma^{\prime}}$.