Let $u_{l}=(u_{l,1},\ldots,u_{l,r})$ for $l\in\{1,\ldots,s\}$. Then $\Pic(Z)$ is generated by the line bundles $\mathcal{O}_{Z}(Z_{l})$, with the $r$ relations:

$$\mathcal{O}_{Z}(\sum_{l=1}^{m}u_{l,1}Z_{l})=\ldots=\mathcal{O}_{Z}(\sum_{l=1}^{m}u_{l,r}Z_{l})=0.$$

In particular, the divisors in the $r$-tuple

$$\Delta=\sum_{l\in L}u_{l}\otimes Z_{l}\,\in\,N\otimes\Div^{\mathbb{T}}(Z)\simeq(\Div^{\mathbb{T}}(Z))^{r}$$

are principal.