We now introduce the local model for the dominating model $\mathscr{Z}\longrightarrow\mathscr{X}$ we will construct.
We consider the blow-up of the exceptional surfaces $S_{12},S_{13}$ and $S_{23}$ one after the other

$\begin{array}[]{ccccccccc}\mathscr{V}_{123}&\xrightarrow[H_{23}]{\text{blow-up of }S_{23}}&\mathscr{V}_{13}&\xrightarrow[H_{13}]{\text{blow-up of }S_{13}}&\mathscr{V}_{12}&\xrightarrow[H_{12}]{\text{blow-up of }S_{12}}&\mathscr{U}_{12}&\xrightarrow[G_{12}\circ G_{1}]{}&\mathscr{U}\\ \cup&&\cup&&\cup&&&&\downarrow\\ E_{23}&&E_{13}&&E_{12}&&&&\mathbb{A}^{1}_{t}.\end{array}$

As these surfaces are toric strata of $\mathscr{U}_{12}$, the blow-ups are toric as well and the corresponding fans are refinements of the fan of $\mathscr{U}_{12}$. We note that

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  the dual complexes of the special fibers of $\mathscr{U}_{12},\mathscr{V}_{12},\mathscr{V}_{13}$ and $\mathscr{V}_{123}$ are obtained from the slices of the corresponding fans by removing the vertices corresponding to $D_{1}^{\prime}$, $D_{2}^{\prime}$ and $D_{3}^{\prime}$, as well as each face containing one of these.

  Then $\Sk(\mathscr{V}_{123})$ consists of four $3$-cells: $<v_{13},v_{1},v_{2},v_{3}>$, $<v_{13},v_{1},v_{2},v_{12}>$, $<v_{23},v_{13},v_{2},v_{3}>$ and $<v_{23},v_{13},v_{2},v_{12}>$; it has only one edge in the interior, which is $<v_{2},v_{13}>$.

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  The remaining $3$-dimensional simplices of the slice of the fan of $\mathscr{V}_{123}$ are

  		$\displaystyle<v_{12},v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}>$
  		$\displaystyle<v_{13},v_{1}^{\prime},v_{3}^{\prime},v_{12}>$
  		$\displaystyle<v_{23},v_{12},v_{2}^{\prime},v_{3}^{\prime}>$
  		$\displaystyle<v_{23},v_{13},v_{3}^{\prime},v_{12}>$

  		$\displaystyle<v_{1},v_{1}^{\prime},v_{12},v_{13}>$
  		$\displaystyle<v_{2},v_{2}^{\prime},v_{12},v_{23}>$
  		$\displaystyle<v_{3},v_{3}^{\prime},v_{13},v_{23}>$

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  The Berkovich retractions associated with the models $\mathscr{U}_{12}$, $\mathscr{V}_{12}$ and $\mathscr{V}_{13}$ map $v_{12}$ and $v_{13}$ to $v_{1}$, and $v_{23}$ to $v_{2}$.

We study more in details the retraction $\rho_{\mathscr{V}_{12}}$ near the vertex $v_{3}$, as this will be relevant later in the construction of the local combinatorial retraction (see Eq. 4.4.1). We observe that $\rho_{\mathscr{V}_{12}}$ collapses the convex hull $P(v_{1},v_{2},v_{3},v_{13},v_{23})$ of $v_{1},v_{2},v_{3},v_{13}$ and $v_{23}$ onto the face $<v_{1},v_{2},v_{3}>$. If we identify the skeleton $\Sk(\mathscr{V}_{123})$ with the polyhedron in $\mathbb{R}^{3}_{(x,y,z)}$ below, $\rho_{\mathscr{V}_{12}}$ on $P(v_{1},v_{2},v_{3},v_{13},v_{23})$ is written explicitly as follows:

The function $t$ on $<v_{1},v_{2},v_{3}>$ is the slope of the line segment joining the vertex $v_{3}$ to $tv_{1}+(1-t)v_{2}$ for $t\in[0,1]$. We give a picture of the retraction $\rho_{\mathscr{V}_{12}}$ for various values of $t$:

$\begin{array}[]{ccccccccc}\mathscr{G}&\xrightarrow[H_{123}]{\begin{subarray}{c}\text{blow-up of }\\ D_{2}\cap E_{13},E_{12}\cap D^{\prime}_{3}\end{subarray}}&\mathscr{V}_{123}&\xrightarrow[H_{23}\circ H_{13}\circ H_{12}]{\begin{subarray}{c}\text{blow-up of }\\ S_{12},S_{13},S_{23}\end{subarray}}&\mathscr{U}_{12}&\xrightarrow[G_{12}\circ G_{1}]{\begin{subarray}{c}\text{blow-up of }\\ D_{1},D_{2}\end{subarray}}&\mathscr{U}.\\ \cup&&\cup&&\cup&&\\ E_{123},E^{\prime}_{123}&&E_{12},E_{13},E_{23}&&S_{12},S_{13},S_{23}&&\end{array}$