We return to the setting of Section 4.1. The purpose of this section is to compute various intersection numbers on the small resolutions of $\mathscr{X}$ used to define the retraction $\pi$, as this will allow us to compute the monodromy of the associated $\mathbb{Z}$-affine structure on $\Sk(X)$.
Fix an order $(i,j,k,l,h)$ on $\{1,\ldots,5\}$ and consider the small resolution $\mathscr{X}_{ijkl}$, obtained from $\mathscr{X}$ by blowing-up the divisors $D_{i}$, $D_{j}$, $D_{k}$, $D_{l}$ in that order. It now follows from the local study of the singularities of $\mathscr{X}$ that this is indeed a small resolution of $\mathscr{X}$.
In $\mathscr{X}_{ijkl}$ we still denote the strict transforms of the strata of $\mathscr{X}_{k}$ by $D_{m}$, by $D_{mm^{\prime}}=D_{m}\cap D_{m^{\prime}}$ and by $D_{mm^{\prime}m^{\prime\prime}}=D_{m}\cap D_{m^{\prime}}\cap D_{m^{\prime\prime}}$ with $m,m^{\prime},m^{\prime\prime}\in\{1,\ldots,5\}$. By the study of the local model in Section 4.2, the exceptional locus of $g_{ijkl}:\mathscr{X}_{ijkl}\rightarrow\mathscr{X}$

$\begin{array}[]{cccccccccc}g_{ijkl}:&\mathscr{X}_{ijkl}&\xrightarrow[G_{ijkl}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{l}\end{subarray}}&\mathscr{X}_{ijk}&\xrightarrow[G_{ijk}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{k}\end{subarray}}&\mathscr{X}_{ij}&\xrightarrow[G_{ij}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{j}\end{subarray}}&\mathscr{X}_{i}&\xrightarrow[G_{i}]{\begin{subarray}{c}\text{blow-up}\\ \text{of }D_{i}\end{subarray}}&\mathscr{X}\\ &\cup&&\cup&&\cup&&\cup&&\\ &S_{lh}&&S_{kl},S_{kh}&&S_{jk},S_{jl},S_{jh}&&S_{ij},S_{ik},S_{il},S_{ih}&&\end{array}$

consists of ten surfaces $S_{mm^{\prime}}$, with $m,m^{\prime}\in\{1,\ldots,5\}$ and $m<m^{\prime}$ in the order $(i,j,k,l,h)$. The surface $S_{mm^{\prime}}$ is mapped via $g_{ijkl}$ to the singular curve $C_{mm^{\prime}}$, and is contained in the strict transform of $D_{m}$. The component $D_{h}$ (corresponding to the biggest index in the chosen order) is the only one isomorphic to its strict transform.

Given a pair $(m,m^{\prime})$ with $m<m^{\prime}$, the morphism $g_{ijkl}$ induces on $D_{mm^{\prime}}$ the blow-up along $5$ distinct general points on each $D_{mm^{\prime}m^{\prime\prime}}$ with $m^{\prime}<m^{\prime\prime}$. Thus, the intersection numbers between strata curves and strata divisors in $\mathscr{X}_{ijkl}$ are:

	$D_{i}$	$D_{j}$	$D_{k}$	$D_{l}$	$D_{h}$
$C_{ijk}$	1	1	-4	1	1
$C_{ijl}$	1	1	1	-4	1
$C_{ijh}$	1	1	1	1	-4
$C_{ikl}$	1	1	1	-4	1
$C_{ikh}$	1	1	1	1	-4
$C_{ilh}$	1	1	1	1	-4
$C_{jkl}$	1	1	1	-4	1
$C_{jkh}$	1	1	1	1	-4
$C_{jlh}$	1	1	1	1	-4
$C_{klh}$	1	1	1	1	-4