The divisor $W^{\sigma}_{i}$ has multiplicity $-1$ along $D_{i}$, multiplicity $0$ along $D_{l}$ for $l\in L_{\sigma}\setminus\{i\}$, and along $D_{j}$ for $j\in J$. In other words, we may write:

$$W^{\sigma}_{i}=-D_{i}+\sum_{l\in L\setminus L_{\sigma}}c_{i,l}D_{l}$$

for some coefficients $c_{i,l}\in\mathbb{Z}$. Moreover, the restriction of $W^{\sigma}_{i}$ to $Z$ is principal.