Let $D$ be a component of $\mathscr{X}_{k}$, with boundary $\Delta_{D}=\sum_{i=1}^{r}C_{i}$. Writing $b_{i}=-(C_{i}^{2})_{D}$, the monodromy $T_{\rho_{\mathscr{X}}}$ of the $\mathbb{Z}$-affine structure induced by $\rho_{\mathscr{X}}$ around $v_{D}$ is given by

$$T_{\rho_{\mathscr{X}}}=\left(\begin{matrix}b_{r}&1\\ -1&0\end{matrix}\right)\cdot\ldots\cdot\left(\begin{matrix}b_{2}&1\\ -1&0\end{matrix}\right)\cdot\left(\begin{matrix}b_{1}&1\\ -1&0\end{matrix}\right)$$

with respect to the basis $(v_{D_{r}},v_{D_{1}})$ and origin $v_{D}$.