Let $R$ be a stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ corresponding to the open face $\tau$ of $\mathfrak{D}$.

1. (a)

   $\dim(\tau)=\codim(R,\tilde{\mathfrak{X}}^{\prime\prime})$.

2. (b)

   $S:=\tilde{\iota}(R)$ is a stratum of $\tilde{\mathfrak{X}}$.

3. (c)

   $R\overset{\tilde{\iota}}{\rightarrow}S$ is a fibre bundle with fibre $T$ where $T$ is the $\dim(R)-\dim(S)$ dimensional torus orbit from the proof of Proposition 2.8.

4. (d)

   Every stratum of $\tilde{\mathfrak{X}}^{\prime\prime}$ is smooth.

5. (e)

   The closure $\bar{R}$ is the union of all strata of $\tilde{\mathfrak{X}}^{\prime\prime}$ corresponding to open faces $\sigma$ of $\mathfrak{D}$ with $\tau\subseteq\bar{\sigma}$.

6. (f)

   For an irreducible component $Y$ of $\tilde{\mathfrak{X}}^{\prime\prime}$, let $\zeta_{Y}$ be the unique point of $\mathfrak{X}^{\textup{an}}$ with reduction equal to the generic point of $Y$. Then $Y\mapsto\zeta_{Y}$ is a bijection between the irreducible components of $\tilde{\mathfrak{X}}^{\prime\prime}$ and the vertices of $\mathfrak{D}$.