Let $\mathscr{X}/R$ be a minimal model of $X$; we assume that the special fiber $\mathscr{X}_{k}=\sum_{i\in I}D_{i}$ is reduced. We consider a one-dimensional stratum $C=D_{1}\cap\ldots\cap D_{n}$ of $\mathscr{X}_{k}$, which is therefore a smooth rational curve, and is such that $(\mathscr{X},\mathscr{X}_{k})$ is an snc pair in a formal neighbourhood of $C$ by [NXY19, Corollary 4.6]. Since $(C,\Delta_{C})$ is log Calabi–Yau, we may write its boundary as $\Delta_{C}=p_{0}+p_{\infty}$, where $p_{0}=C\cap D_{0}$ and $p_{\infty}=C\cap D_{\infty}$ for two irreducible components $D_{0},D_{\infty}$ of $\mathscr{X}_{k}$ meeting $C$ transversally.

Following [NXY19], we write $b_{i}=-(C\cdot D_{i})$ for $i=1,\ldots,n$; from $C\cdot\mathscr{X}_{k}=0$ we infer $\sum_{i=1}^{n}b_{i}=2$. The $\Star(\tau_{C})$ consists on the union of two maximal faces corresponding to the zero-dimensional strata $p_{0},p_{\infty}$, meeting along $\tau_{C}$. The goal of this section is to describe the integral affine structure on $\Star(\tau_{C})$ in terms of the intersection numbers $b_{i}$’s, with no assumption on their positivity.

We start with a general definition.

For instance, if $m=2$, $\Gamma_{1}$ is the union of the three line segments joining the barycenter of the triangle to the barycenters of the edges.

We return to the setting of Section 3, that is, let $X/K$ be a smooth $n$-dimensional maximally degenerate Calabi–Yau variety. Assume we are given two minimal models $\mathscr{X}$, $\mathscr{X}^{\prime}$ of $X$ such that $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$, so that $\Sk(\mathscr{X})=\Sk(\mathscr{X}^{\prime})=\Sk(X)$, not only as sets but also with the same triangulation. We fix an ordered labelling $(1,\ldots,s)$ of the vertices of $\Sk(X)$, equivalently of the irreducible components of the special fiber of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$).
Fix a codimension 1 face $\tau$ of $\mathcal{D}(\mathscr{X}_{k})=\mathcal{D}(\mathscr{X}^{\prime}_{k})$, with vertices $v_{i_{1}},\ldots,v_{i_{n}}$. We write $C$ (resp. $C^{\prime}$) for the corresponding strata curves of $\mathscr{X}$ (resp. $\mathscr{X}^{\prime}$), and $D_{i_{l}}$ (resp. $D^{\prime}_{i_{l}}$), $l=1,\ldots,n$ the corresponding components. We then have $C=D_{i_{1}}\cap\ldots\cap D_{i_{n}}$ , and similarly for $C^{\prime}$. We write

the intersection number computed inside $\mathscr{X}$, and similarly:

for $l=1,\ldots,n$. The $(n-1)$-dimensional face $\tau$ is contained in two maximal faces $\tau_{p_{0}}$ and $\tau_{p_{\infty}}$ of $\mathcal{D}(\mathscr{X}_{k})$, since the boundary of $C$ in $(\mathscr{X},\mathscr{X}_{k})$ consists of two strata points $p_{0}=C\cap D_{i_{0}}$ and $p_{\infty}=C\cap D_{i_{\infty}}$; we assume $i_{0}<i_{\infty}$. We set $\mathcal{S}\coloneqq\tau_{p_{0}}\cup\tau_{p_{\infty}}\subset\Sk(X)$, and $\Gamma\coloneqq\Gamma_{n-2}\subset\tau_{C}$ as in Definition 3.2.1.
Given two vertices $v_{i_{m}},v_{i_{m^{\prime}}}$ of $\tau_{C}$, with corresponding components $D_{i_{m}}$ and $D_{i_{m^{\prime}}}$ of $\mathscr{X}_{k}$ containing $C$, we assume $i_{m}<i_{m^{\prime}}$ and construct a loop $\gamma$ as follows:

• -

  $\gamma$ is contained in $\mathcal{S}\setminus\Gamma$;

• -

  $\gamma$ goes around the segment joining the barycenter of $\tau_{C}$ with the barycenter of the edge between $v_{i_{m}}$ and $v_{i_{m^{\prime}}}$;

• -

  $\gamma$ has an orientation induced by the fixed ordered labelling on the vertices of $\Sk(X)$ in the following way: in $\mathcal{S}\setminus\Gamma$, $\gamma$ is homotopy equivalent to the closed path given by the edges which connect in order

  $$v_{D_{i_{0}}},v_{D_{i_{m}}},v_{D_{i_{\infty}}},v_{D_{i_{m^{\prime}}}}.$$

Suppose we are given a retraction $\rho:X^{\an}\longrightarrow\Sk(X)$ such that

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

• 1.

  the special fiber $\mathscr{X}_{k}$ is reduced, consisting of four Weil divisors, i.e. $D_{i}=\{z_{i}=0,t=0\}$;

• 2.

  $\mathscr{X}$ has $24$ singular points, given by $\{z_{i}=0,z_{j}=0,t=0,F_{4}=0\}$, hence 4 on each $D_{i}\cap D_{j}$; the local model around a singular point is given by $\mathscr{U}=\{xy=zt\}\subset\mathbb{A}^{3}_{R}$;

• 3.

  the pair $(\mathscr{X},\mathscr{X}_{k})$ is dlt. Indeed, it is snc away from the singularities, and around a singular point $(\mathscr{U},\mathscr{U}_{k})$ is log canonical by [CLS11, Proposition 11.4.24];

• 4.

  the dual complex $\mathcal{D}(\mathscr{X}_{k})$ is a PL-isomorphic to a tetrahedron, and hence homeomorphic to $\mathbb{S}^{2}$;

• 5.

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

We conclude that $\mathscr{X}$ is a minimal dlt model of the K3 surface $X\coloneqq\mathscr{X}_{K}$, 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.

Our goal is to construct some explicit minimal models of $X$ starting from $\mathscr{X}$, and then to study the integral affine structure on $\Sk(X)$ induced by these models or by combining several of them. To this purpose, we will apply Corollary 3.1.6 and Corollary 3.2.4.

Some good minimal models of $X$ are obtained by performing the following small resolutions of $\mathscr{X}$. For any triple of elements $i,j,k$ in $\{1,\ldots,4\}$ and any fixed order $(i,j,k)$ on them, we blow-up in order the divisors $D_{i}$, $D_{j}$ and $D_{k}$, and denote the resulting model by $\mathscr{X}_{ijk}$ and the morphism by

The exceptional locus of $g_{ijk}$ consists of $24$ smooth rational curves whose images via $g_{ijk}$ are the singular points of $\mathscr{X}$. In particular, the strict transform of $D_{i}$ is isomorphic to the blow-up of $D_{i}$ along the $12$ singular points in $D_{i}$; similarly for $D_{j}$ at $8$ points, and for $D_{k}$ at the $4$ remaining singular points. Instead, for $h\neq i,j,k$, $D_{h}$ is isomorphic to its strict transform. These facts follow from local computations on $\mathscr{U}$. Blowing-up $D_{x}\coloneqq\{x=t=0\}$ induces an exceptional curve inside the strict transform $\tilde{D}_{x}$, which is isomorphic to the blow-up of $D_{x}$ along $q$. The above claims now follow, since the singularities of $\mathscr{X}$ are isolated.
If we denote by $\tilde{D}_{m}$ for $m\in\{1,\ldots,4\}$ the irreducible components of the special fiber of $\mathscr{X}_{ijk}$, and by $C_{mm^{\prime}}=\tilde{D}_{m}\cap\tilde{D}_{m^{\prime}}$ the strata curves, then the intersection numbers in $\mathscr{X}_{ijk}$ are