We set

$$\Delta:=\sum_{l\in L}u_{l}\otimes D_{l}\,\in\,N\otimes\Div_{0}(\mathscr{X})\simeq(\Div_{0}(\mathscr{X}))^{r};$$

this is an $r$-tuple of divisors on $\mathscr{X}$. Moreover, the restriction of any of these to $Z$ is a principal divisor by Corollary 1.2.2. Given a maximal cone $\sigma$ of $\Sigma$, for any $i\in L_{\sigma}$, we define

$$W^{\sigma}_{i}\coloneqq-\frac{\det(\Delta,(u_{l})_{l\in L_{\sigma}\setminus\{i\}})}{\det(u_{i},(u_{l})_{l\in L_{\sigma}\setminus\{i\}})}\in\Div_{0}(\mathscr{X})$$

where the column vectors $u_{l}$ are in the same order in the numerator and in the denominator, and the denominator has value $\pm 1$ by Lemma 2.1.3.

For $j\in J$, we define the divisor on $\mathscr{X}$

(2.2.2)

$\displaystyle\begin{split}W^{\sigma}_{j}&\coloneqq-D_{j}-\sum_{l\in L}\lambda_{j,l}D_{l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}W^{\sigma}_{i}\\ &=-D_{j}-\sum_{l\in L\setminus L_{\sigma}}\lambda_{j,l}D_{l}-\cancel{\sum_{l\in L_{\sigma}}\lambda_{j,l}D_{l}}-\sum_{i\in L_{\sigma}}\lambda_{j,i}\big(\cancel{-D_{i}}+\sum_{l\in L\setminus L_{\sigma}}c_{i,l}D_{l}\big)\\ &=-D_{j}+\sum_{l\in L\setminus L_{\sigma}}d_{j,l}D_{l}\quad\textrm{ with }\quad d_{j,l}=-\lambda_{j,l}-\sum_{i\in L_{\sigma}}\lambda_{j,i}c_{i,l}.\end{split}$

The restriction of $W^{\sigma}_{j}$ to $Z$ is a principal divisor, as the ${W^{\sigma}_{i}}_{|Z}$ are principal and ${-D_{j}}_{|Z}$ is linearly equivalent to $\sum_{l\in L}\lambda_{j,l}Z_{l}$ by Eq. 2.1.4.