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

We consider a point in ${\mathscr{X}}^{\text{sing}}\cap D_{1}\cap D_{2}\cap D_{3}$; the singular points in the other strata curves can be treated analogously. Étale locally around such a point, $\mathscr{X}$ is isomorphic to the toric variety $\mathscr{U}\coloneqq V(xyz-wt)\subset\mathbb{A}^{5}_{k}$, where

are the components of the special fiber $\{t=0\}$ in $\mathscr{U}$. We still denote these by $D_{1},D_{2}$ and $D_{3}$; they form the toric boundary of $\mathscr{U}$ together with

The pair $(\mathscr{X},\mathscr{X}_{k})_{|\mathscr{U}}=(\mathscr{U},\sum_{i=1,2,3}D_{1}|_{\mathscr{U}})$ is log canonical by [CLS11, Proposition 11.4.24].

We denote the strata surfaces of the special fiber by $D_{ij}\coloneqq D_{j}\cap D_{j}$, for $i,j\in\{1,2,3\}$ and $i\neq j$, and the stratum curve by $D_{123}\coloneqq\{x=y=z=t=0\}$. The singular locus ${\mathscr{U}}^{\text{sing}}$ consists of the torus invariant curves

which intersect each other at the torus invariant point $p\coloneqq\{x=y=z=w=t=0\}$.

The toric blow-up $G_{1}:\mathscr{U}_{1}\coloneqq\Bl_{D_{1}}\mathscr{U}\rightarrow\mathscr{U}$ along $D_{1}$ resolves the singularities along $C_{12}$ and $C_{13}$ except at the point $p$. The exceptional locus of $G_{1}$ consists of two surfaces $S_{12}$ and $S_{13}$ intersecting each other along a curve: these surfaces are mapped by $G_{1}$ to the respective singular curves, are contained in the strict transform of $D_{1}$, and correspond to two new edges in the slice of the fan of $\mathscr{U}_{1}$. The intersection of $S_{12}$ and $S_{13}$ corresponds to a new $2$-dimensional face in the slice of the fan. With a slight abuse of notation we keep the same notation for the strict transforms in $\mathscr{U}_{1}$.

A resolution of $\mathscr{U}$ is given by the composition of $G_{1}$ and the toric blow-up $G_{12}:\mathscr{U}_{12}\coloneqq\Bl_{D_{2}}\mathscr{U}_{1}\rightarrow\mathscr{U}_{1}$ along $D_{2}$; the latter indeed resolves the singularities along $C_{23}$. The exceptional locus of $G_{12}$ is a surface $S_{23}$, which is mapped by $G_{12}$ to $C_{23}$ and is contained in the strict transform of $D_{2}$. The morphism $G_{12}$ induces a new $2$-dimensional face in the slice of the fan of $\mathscr{U}_{12}$, and a new edge corresponding to the surface $S_{23}$. In particular, after the blow-up $G_{12}$, the strict transforms of the surface $S_{12}$ and of the divisor $D_{3}$ have empty intersection.

The small resolution $\mathscr{U}_{12}$ of $\mathscr{U}$ induces an isomorphism on the strict transform of $D_{13}$ and of $D_{23}$, while its restriction to the strict transform of $D_{12}$ is the blow-up of $D_{12}$ along a general point. These facts can be checked computing the charts of the blow-ups $G_{1}$ and $G_{12}$. Alternatively, they can be verified looking at the fans of the strata surfaces $D_{ij}$ in the slice of the fans of $\mathscr{U}$, $\mathscr{U}_{1}$ and $\mathscr{U}_{12}$; indeed, the fan of $D_{ij}$ is induced by the intersection of the slice with a normal plane to the edge corresponding to $D_{ij}$.

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

• -

  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 $<v_{2},v_{13}>$.

• -

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

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

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

• -

  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 $t$:

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

Other resolutions of $\mathscr{U}$ can be obtained by blowing-up the divisors of the special fiber in a different order. Given any order $(i_{1},i_{2},i_{3})$ on $\{1,2,3\}$, we denote

$\begin{array}[]{ccccccccc}\mathscr{V}_{i_{1}i_{2}i_{3}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{3}},S_{i_{2}i_{3}}\end{subarray}}&\mathscr{V}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{2}}\end{subarray}}&\mathscr{U}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ D_{i_{1}},D_{i_{2}}\end{subarray}}&\mathscr{U}.\\ \cup&&\cup&&\cup&&\\ E_{i_{1}i_{3}},E_{i_{2}i_{3}}&&E_{i_{1}i_{2}}&&S_{i_{1}i_{2}},S_{i_{1}i_{3}},S_{i_{2}i_{3}}&&\end{array}$

The refinement of the fan of $\mathscr{U}$ corresponding to $\mathscr{V}_{i_{1}i_{2}i_{3}}$ is such that the skeleton $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})=\Sk(\mathscr{V}_{123})$ as subspaces in the Berkovich space of $\mathscr{U}_{K}$; it is independent on the chosen order so that we simply denote this subspace by $\Sk(\mathscr{V})$. However, the models $\mathscr{V}_{i_{1}i_{2}i_{3}}$ and $\mathscr{V}_{123}$ induce in general different simplicial subdivisions and different retractions onto $<v_{1},v_{2},v_{3}>$. For instance, the only edge in the interior of $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})$ is $<v_{i_{2}},v_{i_{1}i_{3}}>$, which indeed depends on the chosen order. Here below we illustrate the skeletons and the Berkovich retractions in a couple of examples.

The blow-up of $\mathscr{V}_{i_{1}i_{2}i_{3}}$ along the toric strata $D_{i_{2}}\cap E_{i_{1}i_{3}}$ and $E_{i_{1}i_{2}}\cap D_{i_{3}}^{\prime}$ yields a refinement of the fan which coincides with the fan of $\mathscr{G}$, constructed at the end of Section 4.3. It follows that the model $\mathscr{G}$ dominates all resolutions $\mathscr{V}_{i_{1}i_{2}i_{3}}$ independently on the order, hence all Berkovich retractions $\rho_{\mathscr{V}_{i_{1}i_{2}i_{3}}}$, $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ and $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ factors through $\rho_{\mathscr{G}}$.

Our goal is to construct a map $\pi$, composing the Berkovich retraction $\rho_{\mathscr{G}}$ with a collapse $\kappa$ of the additional 3-cell $\tau$ and a combinatorial retraction $\rho$

$\begin{array}[]{ccccccccc}&\pi:&\mathscr{U}_{K}^{\an}&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow[\text{collapse}]{\kappa}&\Sk(\mathscr{V})&\xrightarrow[\text{retraction}]{\rho}&\Sk(\mathscr{U})\\ &\rotatebox[origin={c}]{270.0}{$=$}&\cup&&\cup&&\cup&&\cup\\ \text{over }\Star(v_{j})^{\prime}\text{\hskip 10.0pt}&\rho_{\mathscr{U}_{i_{1}i_{2}}}:&\pi^{-1}(\Star(v_{j})^{\prime})&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow{\rho_{\mathscr{V}_{i_{1}i_{2}j}}}&\Sk(\mathscr{V}_{i_{1}i_{2}j})&\xrightarrow{\rho_{\mathscr{U}_{i_{1}i_{2}}}}&\Star(v_{j})^{\prime}\end{array}$

such that, given any vertex $v_{j}$ in $\Sk(\mathscr{U})$, the restriction of $\pi$ over $\Star(v_{j})^{\prime}$ (the $\Star$ is taken with respect to the first barycentric subdivision, as in Definition 3.2.1) is $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ for any order $(i_{1},i_{2},j)$ on $\{1,2,3\}$, i.e. any order where the index $j$ is the biggest. This guarantees that around each $v_{j}$, the map $\pi$ is the Berkovich retraction induced by a small resolution $\mathscr{U}_{i_{1}i_{2}}$ where the strict transform of $D_{j}$ is isomorphic to $D_{j}$, so that we are in the set-up of Corollary C.

• -

  The retraction $\rho$. We identify again the skeleton $\Sk(\mathscr{V})$ with the polyhedron in $\mathbb{R}^{3}$ described in Section 4.3. On the convex hull $P$ of $v_{23},(-1/4,1/4,1),(0,1/3,1),(0,0,0),v_{3}$ and $(0,1/3,0)$, the retraction $\rho$ is given as follows

  (4.4.1)

  $\displaystyle\begin{split}&(x,y,z)\in P\mapsto\begin{cases}\Big(x+\left(1-\frac{t}{2}\right)z,y+\frac{t}{2}z,0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\leqslant 0\\ \Big(0,\frac{y}{x+1},0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\geqslant 0\end{cases}\\ &\text{where }t=\frac{2y}{x+y+1}.\end{split}$

  Here is a pictorial description for certain values of $t$:

  We extend the definition of $\rho$ to $\Sk(\mathscr{V})$ by symmetry along the medians of the triangles $<v_{1},v_{2},v_{3}>$ and $<v_{12},v_{13},v_{23}>$. In particular, we note that the image of $<v_{12},v_{13},v_{23}>$ is the graph in $<v_{1},v_{2},v_{3}>$ of Definition 3.2.1.

• -

  The combinatorial retraction $\pi^{\prime}$. We define the collapse $\kappa$ as the projection of the additional $3$-cell $\tau$ of $\Sk(\mathscr{G})$ onto $<v_{12},v_{13},v_{23}>$ along the $z$-direction. We call $\pi^{\prime}\coloneqq\rho\circ\kappa$ the combinatorial retraction of the skeleton $\Sk(\mathscr{G})$ onto $\Sk(\mathscr{U})=<v_{1},v_{2},v_{3}>$.

• -

  Finally, we check that $\pi^{\prime}=\rho_{\mathscr{U}_{i_{1}i_{2}}}$ over $\Star(v_{j})^{\prime}$. As the preimage of $\Star(v_{j})^{\prime}$ is disjoint from $<v_{12},v_{13},v_{23},v^{\prime}_{123}>$, we have to prove that $\rho=\rho_{\mathscr{U}_{i_{1}i_{2}}}$. By symmetry of $\rho$, it is enough to check this for $v_{3}$. Over $\Star(v_{3})^{\prime}$ we have $\rho_{\mathscr{U}_{i_{1}i_{2}}}=\rho_{\mathscr{V}_{i_{1}i_{2}}}$; there, the expression of $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ determined in Eq. 4.3.1 coincides with the definition of $\rho$ in Eq. 4.4.1, hence we conclude.