The first point is a simple consequence of the triviality of the normal bundle ${\mathcal{O}}_{{\mathcal{X}}_{0}}({\mathcal{X}}_{0})$ together with the adjunction formula

$$K_{Y}=(K_{\mathcal{X}}+\sum_{i\in J}E_{i})|_{Y},$$

canonically realized by Poincaré residues once an order on $J$ has been chosen. When $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$, each $E_{i}$ meeting $Y$ properly satisfies $\kappa_{i}>\kappa_{\min}$, which implies that $B^{\mathcal{L}}_{Y}$ has coefficients $<1$. ∎