We have $p^{-1}(\operatorname{Sk}({\mathcal{X}}))=\operatorname{Sk}({\mathcal{X}}^{\prime})$. Further, for each face $\sigma$ of $\Delta({\mathcal{X}})$, there exist positive integers $e_{\sigma}$, $f_{\sigma}$ and $g_{\sigma}$ satisfying

$$e_{\sigma}=\frac{m}{\gcd(m,b_{\sigma})}{\quad\text{and}\quad}f_{\sigma}g_{\sigma}=\gcd(m,b_{\sigma})$$

and such that the following properties hold: $p^{-1}(\sigma)$ is a union of $g_{\sigma}$ faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}}^{\prime})$, and these are permuted by $G$. For each $\alpha$:

• (a)

  $p$ induces a ${\mathbb{Q}}$-affine isomorphism $\sigma^{\prime}_{\alpha}\overset{\sim}{\to}\sigma$;

• (b)

  $p$ induces a generically finite map $Y_{\sigma^{\prime}_{\alpha}}\to Y_{\sigma}$, of degree $f_{\sigma}$;

• (c)

  $mp^{*}M_{\sigma}\subset M_{\sigma^{\prime}_{\alpha}}$, and $[M_{\sigma^{\prime}_{\alpha}}:mp^{*}M_{\sigma}]=e_{\sigma}$.

Furthermore, we have:

• (i)

  $M_{\sigma^{\prime}_{\alpha}}=p^{*}\left(mM_{\sigma}+{\mathbb{Z}}1_{\sigma}\right)$;

• (ii)

  $\operatorname{Vol}(\sigma^{\prime}_{\alpha})=m^{\dim\sigma}\operatorname{Vol}(\sigma)$;

• (iii)

  $b_{\sigma^{\prime}_{\alpha}}=b_{\sigma}/\gcd(m,b_{\sigma})$.