The statement on the multiplicities follows from the definition of $W^{\sigma}_{i}$, as

$$W^{\sigma}_{i}=-\sum_{l\in L}\frac{\det(u_{l},(u_{l^{\prime}})_{l^{\prime}\in L_{\sigma}\setminus\{i\}})}{\det(u_{i},(u_{l^{\prime}})_{l^{\prime}\in L_{\sigma}\setminus\{i\}})}D_{l}.$$

Moreover, $W^{\sigma}_{i}$ is a linear combination of the divisors of the $r$-tuple $\Delta$, hence its restriction to $Z$ is principal by Corollary 1.2.2. ∎