We construct a third singular integral affine structure on $\Sk(X)$ pushing forward the techniques developed so far. This can be viewed as a dispersion of singularieties with respect to the integral affine structure studied in Section 3.3.2: on each edge we pass from one singular point around which the monodromy is $\begin{pmatrix}1&0\\ 4&1\end{pmatrix}$, to $4$ singular points around each of which the monodromy is $\begin{pmatrix}1&0\\ 1&1\end{pmatrix}$. in the literature Such singularities are called focus-focus and are the most standard examples of singularities for $\mathbb{Z}$-affine structures in dimension 2. Those arise for instance when considering the hyperkähler rotation of a generic elliptic K3 surface $f:S\longrightarrow\mathbb{C}\mathbb{P}^{1}$, with $f$ an elliptic fibration: the hyperkähler rotation $S^{\HK}$ is a complex surface with same underlying topological space as $S$, and hence comes with a map $f^{\HK}:S^{\HK}\longrightarrow\mathbb{S}^{2}$, induced by $f$ at the level of topological spaces. The map $f^{\HK}$ is no longer holomorphic, but is a symplectic torus fibration inducing a $\mathbb{Z}$-affine structure with 24 focus-focus singularities on $\mathbb{S}^{2}$ and acting as an SYZ fibration for $S^{\HK}$. We refer the reader to [GW00] for more details.

Let $e$ be an edge of $\Sk(\mathscr{X})$, let $C_{e}=D_{e_{1}}\cap D_{e_{2}}$ be the corresponding stratum curve in $\mathscr{X}_{k}$. We recall that as the degree four polynomial $F_{4}$ is generic, $C_{e}$ contains four singular points $p_{1},\ldots,p_{4}$ of $\mathscr{X}$, which are ordinary double points. Around each $p_{i}$, $\mathscr{X}$ is étale locally of the form $\{xy=wt\}\subset\mathbb{A}^{3}_{R}$, with $x$ and $y$ being local equations for $D_{e_{1}}$ and $D_{e_{2}}$ away from $p_{i}$. Blowing-up the singular point $p_{i}$ yields an exceptional divisor $E\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$. Contracting one or the other ruling of $E$, we obtain two distinct small resolutions of $\widehat{\mathscr{X}_{/C_{e}}}$ around $p_{i}$, respectively with an exceptional curve inside $D_{e_{1}}$ or $D_{e_{2}}$.

For $j\in\{0,\ldots,4\}$, we denote by $\mathscr{X}_{e,j}$ the following small resolution of $\widehat{\mathscr{X}_{/C_{e}}}$: around $p_{i}$ for $i\leqslant j$ we consider the small resolution such that the exceptional curve over $p_{i}$ lies in $D_{e_{1}}$, while for $i>j$ the small resolution such that the exceptional curves lie in $D_{e_{2}}$. The gluing of these local small resolutions is done in the étale topology, so that in general the obtained models are no longer schemes but only algebraic spaces. Nevertheless, $\mathscr{X}_{e,j}$ is dominated by $\mathscr{Z}$ (defined in Section 3.3.2) and still induces a Berkovich retraction $\rho_{\mathscr{X}_{e,j}}$, as described in Section 5. In particular, by Proposition 5.0.4, $\rho_{\mathscr{X}_{e,j}}$ is an affinoid torus fibration over $\Star(\tau_{C_{e}})$.

We construct the following continuous retraction

where $\rho_{\mathscr{Z}}$ is the Berkovich retraction onto the skeleton $\Sk(\mathscr{Z})$, and $\overline{\rho}^{\prime}$ is a retraction of the 24 wings of $\Sk(\mathscr{Z})$ to the sphere given as follows. We fix four distinct, ordered, interior points $a_{e,1},\ldots,a_{e,4}$ of each edge $e$. Then the map $\overline{\rho}^{\prime}$ on the wing $W_{i}$ attached to $e$ is defined as the map $\rho^{\prime}$ of Section 3.3.2, setting $a_{e}=a_{e,i}$, for each $i\in\{1,\ldots,4\}.$

Note that for a generic family of quartic surfaces $X/K$, the metric aspects of the Kontsevich-Soibelman conjecture suggest that there should exist a distinguished singular affine structure on $\Sk(X)$ (coming from the Gromov-Hausdorff limit of the family), and hence a canonical choice of interior points $a_{e,i}$ for each edge. To the authors’ knowledge there does not exist a way to produce such a canonical set of $a_{e,i}$’s using non-archimedean techniques.