Let $\mathcal{X}$ be an SNC model of $X$ and let $\Delta^{\prime}$ be a simplicial projective subdivision of $\Delta_{\mathcal{X}}$. Then there exists a vertical blow-up $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ with the following properties:

• (i)

  $\mathcal{X}^{\prime}$ is normal and vertically $\mathbf{Q}$-factorial.

• (ii)

  The vertices $(e^{\prime}_{i})_{i\in I^{\prime}}$ of $\Delta^{\prime}$ are in bijection with the irreducible components $(E^{\prime}_{i})_{i\in I^{\prime}}$ of $\mathcal{X}^{\prime}_{0}$, in such a way that $c_{\mathcal{X}^{\prime}}(\emb_{\mathcal{X}}(e^{\prime}_{i}))$ is the generic point of $E^{\prime}_{i}$ for each $i\in I^{\prime}$.

• (iii)

  If $J^{\prime}\subset I^{\prime}$, then $E^{\prime}_{J^{\prime}}:=\bigcap_{j\in J^{\prime}}E^{\prime}_{j}$ is normal, irreducible, and nonempty iff the corresponding vertices $e^{\prime}_{j}$, $j\in J^{\prime}$ of $\Delta^{\prime}$ span a face $\sigma^{\prime}_{J^{\prime}}$ of $\Delta^{\prime}$. In this case, $E^{\prime}_{J^{\prime}}$ has codimension $|J^{\prime}|$ and its generic point is the center of $\emb_{\mathcal{X}}(s)$ on $\mathcal{X}^{\prime}$ for all $s$ in the relative interior of $\sigma^{\prime}_{J^{\prime}}$.

• (iv)

  For each $D\in\Div_{0}(\mathcal{X}^{\prime})$ the function $\varphi_{D}\circ\emb_{\mathcal{X}}$ is affine on the faces of $\Delta^{\prime}$.