We consider $\mathscr{X}=\{z_{1}z_{2}z_{3}z_{4}z_{5}+tF_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})=0\}\subset\mathbb{P}^{4}_{R}$, where $F_{5}$ is a generic homogeneous polynomial of degree $5$. The degeneration $\mathscr{X}$ has the following properties:

• 1.

  the special fiber $\mathscr{X}_{k}$ is reduced, consisting of five Weil divisors, i.e. $D_{i}=\{z_{i}=t=0\}$. We denote $D_{i}^{\prime}:=\{z_{i}=F_{5}=0\}$;

• 2.

  the singular locus ${\mathscr{X}}^{\text{sing}}$ of the total space $\mathscr{X}$ is contained in the special fiber, and is the intersection in $\mathbb{P}_{k}^{4}$ of $\{F_{5}=0\}$ and the union of surfaces $S_{ij}=\{z_{i}=z_{j}=0\}$ for $i\neq j$. In particular, each $D_{i}$ intersects ${\mathscr{X}}^{\text{sing}}$ along the union of four quintic curves $C_{ij}$ and by genericity of $F_{5}$, we may assume that $C_{ij}$ does not intersect the torus fixed points of $D_{i}$; $${\mathscr{X}}^{\text{sing}}\cap D_{i}=\bigcup_{\begin{subarray}{c}j=1\\ j\neq i\end{subarray}}^{5}C_{ij}$$ $$C_{ij}\subseteq D_{i}\cap D_{j}$$ $$C_{ij}\cap C_{ij^{\prime}}=\{5\text{ points}\}\text{ for }j\neq j^{\prime}$$

  Figure 2: *

  Irreducible component $D_{i}$

• 3.

  the pair $(\mathscr{X},\mathscr{X}_{k})$ is dlt, in particular snc away from ${\mathscr{X}}^{\text{sing}}$; we refer to Section 4.2 for a local study of the pair at the singular points;

• 4.

  the dual complex $\mathcal{D}(\mathscr{X}_{k})$ of the special fiber is homeomorphic to the $3$-sphere $\mathbb{S}^{3}$, and the triangulation of $\mathcal{D}(\mathscr{X}_{k})$ is the same as the standard one on the boundary of a $4$-simplex;

• 5.

  by adjunction the canonical bundle $K_{\mathscr{X}}$ is trivial.

We conclude that $\mathscr{X}$ is a minimal dlt model of the quintic $3$-fold $X\coloneqq\mathscr{X}_{K}$, but it is not good in the sense of Section 1.1 since the prime components of the special fiber are not $\mathbb{Q}$-Cartier. In particular, even if the dual complex $\mathcal{D}(\mathscr{X}_{k})$ is well-defined, $\mathscr{X}$ does not induce a well-defined retraction of $X^{\text{an}}$ onto $\mathcal{D}(\mathscr{X}_{k})=\Sk(X)$.

Similarly to Section 3.3, the aim is to explicitly construct several explicit minimal models of $X$ starting from $\mathscr{X}$, then apply the results from Section 3.1 and Section 3.2 to study the integral affine structures on $\Sk(X)$, induced by Berkovich retractions or their combinations. We will proceed as follows.

• -

  (Section 4.5) For any order $(i,j,k,l,h)$ on $\{1,\ldots,5\}$, we construct a small resolution of $\mathscr{X}$ by blowing-up in order the four divisors $D_{i}$, $D_{j}$, $D_{k}$ and $D_{l}$. The resulting resolution is denoted $\mathscr{X}_{ijkl}$, is a minimal model of $X$ and comes equipped with the Berkovich retraction

  $$\rho_{\mathscr{X}_{ijkl}}:X^{\an}\rightarrow\Sk(\mathscr{X}_{ijkl})=\Sk(X)\simeq\mathbb{S}^{3}.$$

  The skeleton $\Sk(\mathscr{X}_{ijkl})$ coincides with $\mathcal{D}(\mathscr{X}_{k})$ as simplicial complex; thus, independently on the order, all skeletons $\Sk(\mathscr{X}_{ijkl})$ define the same simplicial structure on $\Sk(X)$.

• -

  (Section 4.6) We construct a model $\mathscr{Z}$ of $X$ which dominates any model $\mathscr{X}_{ijkl}$, so that $\rho_{\mathscr{X}_{ijkl}}$ factors through $\rho_{\mathscr{Z}}$. We then define a combinatorial retraction $\pi^{\prime}$ which contracts the skeleton $\Sk(\mathscr{Z})$ onto $\Sk(X)$. This allows us to consider the composition

  $$\pi:X^{\an}\xrightarrow{\rho_{\mathscr{Z}}}\Sk(\mathscr{Z})\xrightarrow{\pi^{\prime}}\Sk(X)$$

  which is at the core of the statement of Theorem A. The retraction $\pi^{\prime}$ is constructed so that the composition $\pi$ is locally equal to a $\rho_{\mathscr{X}_{ijkl}}$, the order depending on the region of $\Sk(X)$.

• -

  (Section 4.2 to Section 4.4) The constructions and properties of $\mathscr{X}_{ijkl},\mathscr{Z}$ and $\pi^{\prime}$ rely on a local study of the model $\mathscr{X}$: étale locally around each point of ${\mathscr{X}}^{\text{sing}}\cap D_{i}\cap D_{j}\cap D_{j^{\prime}}$, $\mathscr{X}$ is isomorphic to a toric variety and the resolutions are given by refinements of the associated fan.

We will show that