If $Y$ is a stratum of ${\mathcal{X}}_{0}$ corresponding to a face $\sigma$ of $\Delta({\mathcal{L}})$, then $Y\not\subset\operatorname{supp}D$. It follows that the ${\mathbb{Q}}$-Cartier divisor

$$B^{\mathcal{L}}_{Y}:=B_{Y}-D|_{Y}$$

is well-defined, and we have a canonical identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$ as ${\mathbb{Q}}$-line bundles. Further, if $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$, then the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt.