E_{13}: x_1 = 1 + x_2 = x_3 = 0 && E_{57}: 1 +x_1+x_2+x_3+x_1x_3 = 0 && F_{37}: 1+x_3 = 0 \\
E_{24}: x_2 = 1 + x_3 = 0 && E_{68}: 1 + x_1 + x_2 = 0 && F_{48}: 1+x_1+x_2+x_3 = 0
\end{array}$

at (-0.8,0.8) $E_{71}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
[gray] at (0,1/2) $\bullet$;
[very thick] (0,1/2) – (sin(240)/2,cos(240)/2);

at (-0.8,0.8) $E_{82}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
(0,1/2) – (sin(240)/2,cos(240)/2);
[very thick] (sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
[gray] at (0,1/2) $\bullet$;

at (-0.8,0.8) $E_{13}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
(0,1/2) – (sin(240)/2,cos(240)/2);
at (0,1/2) $\bullet$;

at (-0.8,0.8) $E_{24}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(0,1/2) – (sin(240)/2,cos(240)/2);
[very thick] (sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
[gray] at (0,1/2) $\bullet$;

at (-0.8,0.8) $E_{35}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(240)/2,cos(240)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
[gray] at (0,1/2) $\bullet$;
[very thick] (0,1/2) – (sin(120)/2,cos(120)/2);

at (-0.8,0.8) $E_{46}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
[gray] at (0,1/2) $\bullet$;
[very thick, fill = gray, fill opacity = 0.2] (sin(120),cos(120)+1/2) – (0,1/2) – (sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);

at (-0.8,0.8) $E_{57}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
[gray] at (0,1/2) $\bullet$;
[very thick, fill = gray, fill opacity = 0.2] (sin(120)+sin(240)/2,cos(120)+cos(240)/2) – (sin(240)/2,cos(240)/2) – (0,1/2) – (sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));

at (-0.8,0.8) $E_{68}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
(0,1/2) – (sin(240)/2,cos(240)/2);
[gray] at (0,1/2) $\bullet$;
[very thick, fill = gray, fill opacity = 0.2] (sin(240),1/2+cos(240)) – (0,1/2) – (sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));

at (-0.8,0.8) $F_{15}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
(0,1/2) – (sin(240)/2,cos(240)/2);
[gray, opacity = 0.2] (sin(120)/2,1+cos(120)/2) – (sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240)) – (sin(120)/2+sin(240),1+cos(120)/2+cos(240)) – cycle;
[very thick] (sin(120)/2,1+cos(120)/2) – (sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
[gray] at (0,1/2) $\bullet$;

at (-0.8,0.8) $F_{26}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
(0,1/2) – (sin(240)/2,cos(240)/2);
[gray, opacity = 0.2] (sin(240),cos(240)+1/2) – (0,1/2) – (sin(120),cos(120)+1/2) – (0,-1/2) – cycle;
[gray] at (0,1/2) $\bullet$;
[very thick] (sin(240),cos(240)+1/2) – (0,1/2) – (sin(120),cos(120)+1/2);

at (-0.8,0.8) $F_{37}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(0,1/2) – (sin(120)/2,cos(120)/2);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(0,1/2) – (sin(240)/2,cos(240)/2);
[gray, opacity = 0.2] (sin(240)/2,1+cos(240)/2) – (sin(240)/2,cos(240)/2) – (sin(240)/2+sin(120),cos(240)/2+cos(120)) – (sin(240)/2+sin(120),1+cos(240)/2+cos(120)) – cycle;
[very thick] (sin(240)/2,1+cos(240)/2) – (sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
[gray] at (0,1/2) $\bullet$;

at (-0.8,0.8) $F_{48}$;
[gray,dashed] (sin(120),cos(120)) – (0,0) – (sin(240),cos(240)) (0,0) – (0,1);
(sin(120)/2,cos(120)/2) – (sin(120)/2+sin(240),cos(120)/2+cos(240));
(sin(240)/2,cos(240)/2) – (sin(120)+sin(240)/2,cos(120)+cos(240)/2);
[gray] at (0,1/2) $\bullet$;
[very thick, fill = gray, fill opacity = 0.2] (0,1/2) – (sin(240)/2,cos(240)/2) – (sin(120)/2,cos(120)/2) – cycle;

§.§ The tropicalisation of $U_{\lambda,\mu}$

To simplify the notation in this section we let $U=U_{\lambda,\mu}$ so that the dependence on $\lambda,\mu$ is now left implicit. We now want to construct the tropicalisation $N_U$ of $U$ (as in <ref>), the dual space $M_U$ and the dual intersection pairing $\langle {\cdot},{\cdot}\rangle\colon N_U\times M_U\to \RR$.

§.§.§ The intersection pairing

Recall that we have a set of ten cluster variables $\{x_1,\ldots,x_8,q_1,q_2\}$ and a set of ten boundary components $\{D_{123},\ldots,D_{812},D_{1357},D_{2468}\}$ in the boundary divisor $D$ of our projective compactification $(X,D)$ of $U$. For any cluster monomial $\vartheta_m\in \CC[U]$ and any boundary component $D_n$ we let $\langle D_n,\vartheta_m \rangle := \ord_{D_n}(\vartheta_m)$ denote the order of vanishing of $\vartheta_m$ along $D_n$. We start by computing the analogous matrix to Table <ref> that we will use to define the intersection pairing $\langle {\cdot},{\cdot} \rangle\colon N_U\times M_U\to \RR$. Since this matrix is symmetric, this realises a one-to-one correspondence between the ten cluster variables $x_1,\ldots,x_8,q_1,q_2$ in $\CC[U]$ and the ten boundary divisors $D_{123},\ldots, D_{812},D_{1357},D_{2468}$ of $(X,D)$.

Considered as rational functions on $X$, the cluster variables $x_1$ and $q_1$ have divisor

\begin{align*}
\divv x_1 &= E_1 + D_{456} - D_{781} - D_{812} - D_{123} - D_{1357} \\
\divv q_1 &= F_1 + D_{2468} - D_{123} - D_{345} - D_{567} - D_{781} - 2D_{1357}
\end{align*}

and similarly for the other cluster variables, up to the $\Dih_8$ action. In particular, each cluster variable vanishes along a unique boundary component of $(X,D)$, giving a one-to-one correspondence between the ten cluster variables of $U$ and the ten components of $D$. With respect to this ordering, the pairing between these cluster variables and boundary components is represented by the symmetric $10\times 10$ matrix given in Table <ref>.

In terms of the homogeneous equations that define $X$ we are required to compute $\divv(x_0^{-1}x_1)$ and $\divv(x_0^{-2}q_1)$, where $x_0\in|\sO_X(1)|$ is the homogenising variable. Since $\divv x_0 = D = D_{123} + \ldots + D_{812} + D_{1357} + D_{2468}$, the first formula follows from showing that, considered as a section $x_1\in|\sO_X(1)|$, we have

\[ \divv x_1 = E_1 + D_{234} + D_{345} + 2D_{456}  + D_{567}  + D_{678} + D_{2468} \]

and the second formula follows similarly for $q_1\in |\sO_X(2)|$. This now follows from an explicit calculation with the equations of $X$.

The intersection numbers $\langle D_i,x_j \rangle$ for $U$.

$D_{456}$     $D_{567}$     $D_{678}$     $D_{781}$     $D_{812}$     $D_{123}$     $D_{234}$     $D_{345}$     $D_{2468}$     $D_{1357}$
$x_1$     $ 1$     $ 0$     $ 0$     $-1$     $-1$     $-1$     $ 0$     $ 0$     $ 0$     $-1$

$x_2$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$     $-1$     $-1$     $ 0$     $-1$     $ 0$

$x_3$     $ 0$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$     $-1$     $-1$     $ 0$     $-1$

$x_4$     $-1$     $ 0$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$     $-1$     $-1$     $ 0$

$x_5$     $-1$     $-1$     $ 0$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$     $ 0$     $-1$

$x_6$     $-1$     $-1$     $-1$     $ 0$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$     $ 0$

$x_7$     $ 0$     $-1$     $-1$     $-1$     $ 0$     $ 0$     $ 1$     $ 0$     $ 0$     $-1$

$x_8$     $ 0$     $ 0$     $-1$     $-1$     $-1$     $ 0$     $ 0$     $ 1$     $-1$     $ 0$

$q_1$     $ 0$     $-1$     $ 0$     $-1$     $ 0$     $-1$     $ 0$     $-1$     $ 1$     $-2$

$q_2$     $-1$     $ 0$     $-1$     $ 0$     $-1$     $ 0$     $-1$     $ 0$     $-2$     $ 1$

Given the correspondence in Table <ref> it is now convenient to rename the boundary divisors as $D_1,\ldots, D_8,D_{q_1},D_{q_2}$ so that $D_i = D_{i+3,i+4,i+5}$ corresponds to $x_i$ for $i\in \ZZ/8\ZZ$ and $D_{q_1}=D_{2468}$ and $D_{q_2}=D_{1357}$ correspond to $q_1$ and $q_2$. We will us Table <ref> to define a complete fan $\mathcal F$ in $\RR^3$ which will give us both a toric model and a scattering diagram for $U$.

§.§.§ A toric model for $U$

The fan $\mathcal F$.
In order to tropicalise $U$, we must first start by choosing a cluster torus chart for $U$. We consider the cluster torus $j\colon \TT_{1q3}\hookrightarrow U$ with coordinates $x_1,q_1,x_3$, determined by the seed $Z\subset \Aa^3$ described in Remark <ref>.[We could have chosen to consider the `standard' torus chart $\TT_{123}$ given to us by the Lyness map, but we prefer $\TT_{1q3}$ since (unlike the case in Proposition <ref>) the centres $Z\subset \Aa^3$ that we are required to blow up are already disjoint, without having to first blow up any of the 1-dimensional strata of $\Aa^3$.] Now consider the ten primitive integral vectors $v_1,\ldots,v_8,w_1,w_2\in\ZZ^3$ corresponding to the following columns

\[ \begin{array}{cccccccccc}
v_1 & v_2 & v_3 & v_4 & v_5 & v_6 & v_7 & v_8 & w_1 & w_2 \\ \hline
-1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 &-1 & 2 \\
0 & 0 &-1 & 0 & 0 & 1 & 1 & 1 & 0 & 1
\end{array} \]

read off as the negative of the rows corresponding to $x_1,q_1,x_3$ in Table <ref>. We can define a complete fan $\mathcal F$ in $\RR^3$ which has rays generated by these ten vectors and whose cones are dual to the exchange graph in Figure <ref>. Thus $\mathcal F$ has 16 3-dimensional cones corresponding to the 16 cluster torus charts of $U$ and 24 2-dimensional cones corresponding to the 24 possible mutations between cluster torus charts. The fan $\mathcal F$ is displayed in Figure <ref>.

[inner sep = -2pt] (o) at (0,0) ;
(x) at (cos(330)+0.25,sin(330)) ;
(y) at (cos(90),sin(90)) ;
(z) at (cos(210)+0.25,sin(210)+0.1) ;
[dashed] ($(o)-(y)-({0.55*(cos(330)+0.25)},{0.55*sin(330)})$) – ($(o)-(y)$);
[dashed] ($(o)-(y)$) – ($(o)$) – ($(o)-(z)$) – ($({0.32*(cos(330)+0.25)},{0.32*(sin(330)+1)})-(z)$);
[dashed] ($(o)-(x)$) – ($(o)-(x)-(z)$) – ($({0.45*(cos(330)+0.25)},{0.45*(sin(330))})-(z)$);
[dashed] (0,0.87) – (o);
[dashed] (-0.55*(cos(330)+0.25),-0.55*sin(330)) – (0.12*(cos(330)+0.25),0.12*sin(330));
[dashed] ($(0,0.32)-(x)$) – ($(o)-(0,0.68)-(x)$);
[dashed] ($(o)$) – (0.49*(cos(330)/2+cos(210)/2+0.25),0.49*(sin(330)/2+sin(210)/2+1.05));
[dashed] ($(y)$) – ($(x)+(y)$) ($(o)$) – ($({0.45*(cos(330)+0.25)},{0.45*(sin(330)+1)})$) ($(o)-(y)$) – ($(o)-(y)-(z)$) – (-cos(210)-0.25,-sin(210+0.1)+0.22);
[gray,opacity=0.2] ($(o)-(x)-(z)$) – ($(o)-(x)+(z)$) – ($(o)+(x)+(z)$) – ($(o)+(x)-(z)$) – cycle;
[gray,opacity=0.2] ($(o)+(x)$) – ($(x)+(y)+(z)$) – ($(o)-(x)+(y)+(z)$) – ($(o)-(x)$) – cycle;
[gray,opacity=0.2] ($(o)+(z)$) – ($(x)+(y)+(z)$) – ($(o)-(z)+(y)+(x)$) – ($(o)-(z)$) – cycle;
[gray,opacity=0.2] ($(o)-(y)+(z)$) – ($(o)+(y)+(z)$) – ($(o)+(y)-(z)$) – ($(o)-(y)-(z)$) – cycle;
[gray,opacity=0.2] ($(o)-(y)+(x)$) – ($(o)+(y)+(x)$) – ($(o)+(y)-(x)$) – ($(o)-(y)-(x)$) – cycle;
[gray,opacity=0.2] ($(o)$) – ($(x)+(y)$) – ($(x)+(y)+(y)+(z)$) – ($(y)+(z)$) – cycle;
[gray,opacity=0.2] ($(o)$) – ($(x)+(y)+(z)$) – ($(x)+(y)+(y)+(z)$) – ($(y)$) – cycle;
($(o)+(x)+(y)-(z)$) – ($(o)+(x)+(y)+(z)$) – ($(o)-(x)+(y)+(z)$) – ($(o)-(x)$) – ($(o)-(x)+(z)$) – ($(o)+(x)+(z)$) – ($(o)+(x)-(z)$);
($(o)-(x)+(y)$) – ($(y)$) ($(o)+(x)+(y)$) – ($(o)+(x)-(y)$) ($(o)-(z)+(y)$) – ($(o)+(z)+(y)$) – ($(o)+(z)-(y)$) – ($(o)-(y)$) ($(o)+(x)+(z)+(y)$) – ($(o)+(z)$) – ($(o)$) – ($(o)+(x)+(y)+(z)$) – ($(x)$);
($(o)+(x)+(y)$) – ($(o)+(x)+(y)+(y)+(z)$) – ($(o)+(z)+(y)$) – ($(o)$);
($(o)+(x)-(y)$) – ($(o)-(y)$) – (0,-0.67);
($(o)+(x)$) – (0.12*(cos(330)+0.25),0.12*sin(330));
($(o)+(y)$) – (0,0.87);
($(o)+(y)-(x)$) – ($(0,0.32)-(x)$);
($(o)+(y)-(z)$) – (-cos(210)-0.25,-sin(210+0.1)+0.22);
($(o)+(x)+(y)-(z)$) – ($({0.32*(cos(330)+0.25)},{0.32*(sin(330)+1)})-(z)$);
($(o)+(x)-(z)$) – ($({0.45*(cos(330)+0.25)},{0.45*(sin(330))})-(z)$);
($(o)+(x)+(y)$) – ($({0.45*(cos(330)+0.25)},{0.45*(sin(330)+1)})$);
($(o)-(x)$) – (-0.55*(cos(330)+0.25),-0.55*sin(330));
($(o)-(0,0.68)-(x)$) – ($(o)-(y)-(x)$) – ($(o)-(y)-({0.55*(cos(330)+0.25)},{0.55*sin(330)})$);
($(y)$) – ($(x)+(y)+(y)+(z)$) – ($(x)+(y)+(z)$);
(0.49*(cos(330)/2+cos(210)/2+0.25),0.49*(sin(330)/2+sin(210)/2+1.05)) – ($(x)+(y)+(y)+(z)$);

[inner sep = -2pt] (x1) at ($(o)-(x)$) [label=[label distance=0pt]180:$v_1$] $\bullet$;
[inner sep = -2pt] (x2) at ($(o)-(y)$) [label=[label distance=0pt]270:$w_1$] $\bullet$;
[inner sep = -2pt] (x3) at ($(o)-(z)$) [label=[label distance=0pt]60:$v_3$] $\bullet$;
[inner sep = -2pt] (x4) at ($(x)$) [label=[label distance=0pt]-30:$v_5$] $\bullet$;
[inner sep = -2pt] (x5) at ($(x)+(y)$) [label=[label distance=0pt]-30:$v_4$] $\bullet$;
[inner sep = -2pt] (x6) at ($(x)+(y)+(z)$) [label=[label distance=0pt]-90:$v_6$] $\bullet$;
[inner sep = -2pt] (x7) at ($(y)+(z)$) [label=[label distance=0pt]90:$v_8$] $\bullet$;
[inner sep = -2pt] (x8) at ($(z)$) [label=[label distance=0pt]235:$v_7$] $\bullet$;
[inner sep = -2pt] (q1) at ($(y)$) [label=[label distance=0pt]90:$v_2$] $\bullet$;
[inner sep = -2pt] (q2) at ($(x)+(y)+(y)+(z)$) [label=[label distance=0pt]90:$w_2$] $\bullet$;

at ($(o)$) $\times$;

The fan $\mathcal F$ giving both a toric model of $U$, and the scattering diagram for $\CC[U]$.

The toric model for $U$.
We consider the toric variety $(T,B)$ defined by the fan $\mathcal F$, and the locus $Z\subset T$ determined by the seed

\[ S = \left\{ \left( (-1,0,0), \; \lambda q_1 + 1 \right), \; \left( (0,-1,0), \; \mu x_1x_3 + x_1 + x_3 + 1 \right), \; \left( (0,0,-1), \; q_1 + 1\right) \right\} \]

described in Remark <ref>.

The pair $\pi\colon (\widetilde X,\widetilde D) \to (T,B)$ obtained by blowing up $Z\subset T$ is a toric model for $U$. This compactification $(\widetilde X,\widetilde D)$ is a small resolution of the projective compactification $(X,D)$ of $U$, and identifies the boundary components $D_1,\ldots, D_8,D_{q_1},D_{q_2}$ of $D$ with the ten components of $\widetilde D$ corresponding to the vectors $v_1,\ldots,v_8,w_1,w_2$ respectively.

The proof that $\pi \colon (X,D)\to (T,B)$ is a toric model for $U$ follows from extending a similar calculation to the proof of Proposition <ref> from the toric variety $\Aa^3$ to the toric variety $T$. To see that the boundary components are identified as claimed, we can check that the identifications hold for the affine chart $\Aa^3_\sigma = \Spec\CC[\sigma^\vee\cap M]\subset T\dashrightarrow X$ for each maximal cone $\sigma\subset \mathcal F$.

After blowing up the locus $Z\subset T$ we see that the boundary divisor $\widetilde D$ is isomorphic to $B$, except for the fact that each of $\widetilde D_1,\widetilde D_3,\widetilde D_5$ and $\widetilde D_7$ are blown-up in two boundary points, turning these four boundary components into del Pezzo surfaces of degree 5 with an anticanonical pentagon of $(-1)$-curves (as in Example <ref>). The model $(X,D)$ is now obtained from $(\widetilde X,\widetilde D)$ by contracting sixteen $\mathcal{O}(-1,-1)$-curves in the boundary (giving the sixteen nodes of Lemma <ref>). These are the eight boundary strata corresponding to the cones $\langle x_i,q_i \rangle$ for $i\in\ZZ/8\ZZ$ and eight more curves, four in each of the boundary components $\widetilde D_1$, $\widetilde D_3$, $\widetilde D_5$, $\widetilde D_7$, giving the eight nodes along the lines $\PP^1_{x_i,x_{i+1}}$. These sixteen curves are indicated by the dashed lines in Figure <ref>.

(0,4) – (2,3) (0,4) – (-2,3) (-4,-2) – (0,0) (4,-2) – (0,0);
[dashed] (4,1) – (4,-2) (-4,1) – (-4,-2) (0,0) – (0,4);
[very thick] (0,2) – (-4,0) (-1.05,-0.525) – (-0.5,-0.75) – (0.5,-0.75) – (1.05,-0.525) (0,1) – (4,-1) (-0.5,-0.75) – (-3.5,-2.25) (0.5,-0.75) – (3.5,-2.25);
(-2,3) – (-4,1) (2,3) – (4,1)  (-4,-2) – (0,-4) – (4,-2);
at (-2.5,0) $(-1,0,0)$;
at (0,-5/2) $(0,-1,0)$;
at (2.5,0.5) $(0,0,-1)$;
[dashed] (-1.1,3.45) – (-1.1,-0.55) (1.1,3.45) – (1.1,-0.55);
[dashed] (-1,-0.5) – (-1,0.5) (0,1) – (-1,0.5) – (-2.75,2.25);
[dashed] (1,-0.5) – (1,1.5) (0,2) – (1,1.5) – (2.25,2.75);

(4,1) – (2,3) – (0,4) – (-2,3) – (-4,1) (-4,-2) – (0,-4) – (4,-2);
[dashed] (4,-2) – (4,1) (-4,-2) – (-4,1) (-2,3) – (-1,5/2) (1,5/2) – (2,3) (-2,0) – (-1,1) (1,1) – (2,0) (0,-2) – (0,-4);
(-1,5/2) – (-1,1) – (1,1) – (1,5/2) – cycle;
(-4,1) – (-2,0) (2,0) – (4,1) (-2,0) – (0,-2) – (2,0);
at (0,3.25) $(0,1,0)$;
at (0,0) $(1,1,1)$;
at (0,1.75) $(1,2,1)$;
at (-2,1.5) $(0,1,1)$;
at (2,1.5) $(1,1,0)$;
at (-1.5,-2) $(0,0,1)$;
at (1.5,-2) $(1,0,0)$;
[dashed] (3.05,-2.475) – (3.05,0.525);
[dashed] (-3.05,-2.475) – (-3.05,0.525);
[dashed] (-4,-1) – (-2.95,-1.525) (-2.95,-2.525) – (-2.95,-1.525) – (-1.5,-0.5);
[dashed] (4,-1) – (2.95,-1.525) (2.95,-2.525) – (2.95,-1.525) – (1.5,-0.5);

A toric model $\pi\colon (\widetilde X,\widetilde D)\to (T,B)$ for $U$ is constructed by blowing up the locus $Z\subset B$ which is a union of three rational curves (drawn tropically with thick lines). The model $(X,D)$ is obtained from $(\widetilde X,\widetilde D)$ as a small contraction of the sixteen $\mathcal O(-1,-1)$-curves (drawn tropically with dashed lines) to ordinary nodes.

The tropicalisation $N_U$.
We can now construct $N_U$, the tropicalisation of $U$ with respect to this seed, by altering the integral affine structure along the codimension 1 cones of $\mathcal F$ by following the process described in <ref>. We arrive at the following description.

The integral affine structure on $N_U$ is obtained by bending any line that passes through one of the following seven walls of $\mathcal F$:

\[ \langle v_7,v_1\rangle, \: \langle v_1,v_3\rangle, \: \langle v_3,v_5\rangle, \: \langle v_1,w_1\rangle, \: \langle v_3,w_1\rangle, \: \langle v_5,w_1\rangle, \: \langle v_7,w_1\rangle. \]

If $M_{i,i+2}$ represents the bend from passing from the cone $\langle v_i,w_i,v_{i+2}\rangle$ to the cone $\langle v_i,v_{i+1},v_{i+2}\rangle$ through the wall $\langle v_i,v_{i+2}\rangle$ for $i=7,1,3$, then

\[
M_{71} = \begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}, \quad M_{13} = \begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}, \quad M_{35} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 1 & 1
\end{pmatrix}, \]

and if $M_{iq}$ represents the bend from passing from the cone $\langle v_{i-2},v_i,w_i\rangle$ to the cone $\langle v_i,v_{i+2},w_i\rangle$ through the wall $\langle v_i,w_i\rangle$ for $i=1,3,5,7$, then

\[ M_{1q} = \begin{pmatrix}
1 & 0 & 0 \\
-1 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}, \quad M_{3q} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 1 \\
0 & 0 & 1
\end{pmatrix}, \quad M_{5q} = M_{1q}^{-1}, \quad M_{7q} = M_{3q}^{-1}. \]

The singular locus of $N_U$ is given by the four rays $\RR_{\geq0}v_i$ for $i=1,3,5,7$, corresponding to the four non-toric boundary divisors of $(\widetilde X,\widetilde D)$.

§.§.§ A scattering diagram for $U$

The finite cluster structure on $U$ is equivalent to the existence of a consistent scattering diagram with a finite number of chambers. As described in <ref>, by [2] this scattering diagram can be computed inductively from some initial scattering diagram for any log Calabi–Yau variety with a suitably nice toric model. Consider the fan $\mathcal F$ defined above. Let $\mathfrak d_{ij}$ be the wall spanned by $x_i$ and $x_j$, and let $\mathfrak d_{iq}$ be the wall spanned by $x_i$ and $q_i$. In what follows, when we refer to a wall $\mathfrak d_{ij}$ it is convenient to implicitly allow $j=q$.

Decorating the walls of the fan $\mathcal F$ with the wall functions given in Table <ref> defines a consistent scattering diagram $\mathfrak D_{\lambda,\mu}$ in $N_U$.

Wall functions for the scattering diagram $\mathfrak D_{\lambda,\mu}$.

!Wall     Function     Wall     Function     Wall     Function
$\mathfrak d_{12}$     $1 + z_2 + z_1z_2$
$\mathfrak d_{13}$     $1 + z_1 + z_3 + \mu z_1z_3$
$\mathfrak d_{1q}$     $1 + z_2$

$\mathfrak d_{23}$     $1 + \lambda z_2 + \lambda z_2z_3$
$\mathfrak d_{24}$     $1 + z_2 + z_1z_2 + \lambda z_1z_2^2$
$\mathfrak d_{2q}$     $1 + \lambda z_1z_2^2z_3$

$\mathfrak d_{34}$     $1 + \lambda z_2z_3 + \lambda\mu z_1z_2z_3$
$\mathfrak d_{35}$     $1 + z_1 + z_3 + \mu z_1z_3$
$\mathfrak d_{3q}$     $1 + \lambda z_2$

$\mathfrak d_{45}$     $1 + z_2 + z_1z_2$
$\mathfrak d_{46}$     $1 + \lambda z_1z_2 + \lambda\mu z_1z_2z_3 + \lambda\mu z_1^2z_2^2z_3$
$\mathfrak d_{4q}$     $1 + \lambda\mu z_1z_2^2z_3$

$\mathfrak d_{56}$     $1 + z_2z_3 + \mu z_1z_2z_3$
$\mathfrak d_{57}$     $1 + z_1 + z_3 + \mu z_1z_3$
$\mathfrak d_{5q}$     $1 + z_2$

$\mathfrak d_{67}$     $1 + \lambda z_2z_3 + \lambda\mu z_1z_2z_3$
$\mathfrak d_{68}$     $1 + z_2z_3 + \mu z_1z_2z_3 + \lambda\mu z_1z_2^2z_3^2$
$\mathfrak d_{6q}$     $1 + \lambda z_1z_2^2z_3$

$\mathfrak d_{78}$     $1 + \lambda z_2 + \lambda z_2z_3$
$\mathfrak d_{71}$     $1 + z_1 + z_3 + \mu z_1z_3$
$\mathfrak d_{7q}$     $1 + \lambda z_2$

$\mathfrak d_{81}$     $1 + z_2z_3 + \mu z_1z_2z_3$
$\mathfrak d_{82}$     $1 + \lambda z_2 + \lambda z_2z_3 + \lambda z_2^2z_3$
$\mathfrak d_{8q}$     $1 + \lambda\mu z_1z_2^2z_3$

It is straightforward to check (via computer algebra) that composing the wall-crossing automorphisms corresponding to an oriented loop around any joint (i.e. a codimension 2 cone of $\mathcal F$) in $\mathfrak D_{\lambda,\mu}$ gives the identity.

The wall functions can be easily read off from the mutation relations. First note that, when expanded as a Laurent polynomial in $x_1,q_1,x_3$, the cluster variables all have a monic leading monomial[This is not a happy accident. Indeed the rescaling that happens in <ref> whilst deriving the equations of $U$ was specially chosen in order to achieve this very fact.], i.e. the constant term of their numerator is equal to $1$. Now the scattering function on the wall $\mathfrak d_{ij}$ is obtained by substituting the leading monomial of each cluster variable (written in terms of $z_1,z_2,z_3$) into the righthand side of the corresponding mutation relation and clearing the denominators. For example crossing the wall $\mathfrak{d}_{68}$ corresponds to the mutation $x_7q_2 = \lambda\mu + x_6 + \mu x_8 + x_6x_8$. If we substitute $x_6\mapsto (z_1z_2z_3)^{-1}$ and $x_8\mapsto (z_2z_3)^{-1}$ into the righthand side and clear the denominator we get the scattering function $f_{68}$ in Table <ref>. Of course in general the scattering diagram is usually introduced in order to derive the mutation relations, not the other way round.

Since the mutation relations generating the cluster exchange graph for $U$ are now easily read off as the wall-crossing automorphisms in $\mathfrak D_{\lambda,\mu}$, this shows that the cluster variables $x_1,\ldots,x_8,q_1,q_2$ can now be recovered from $\mathfrak D_{\lambda,\mu}$ as the theta functions attached to the integral points $v_1,\ldots,v_8,v_{q_1},v_{q_2}\in N_U(\ZZ)$ respectively. Moreover, these could be computed by counting broken lines in $\mathfrak D_{\lambda,\mu}$.

Since the tropicalisation $N_U$ also has the nice property that once a straight line has left a cone of $\mathcal F$ it can never return, which can been seen by an explicit computation with affine structure of $N_U$ as described in Lemma <ref>. Therefore, by a similar argument to the dimension 2 case <ref>, for all $a,b,c\geq0$ the integral point $av_1+bv_2+cv_3\in N_U(\ZZ)$ belonging to the cone $\langle v_1,v_2,v_3 \rangle$ of $\mathcal F$ must parameterise the theta function $\vartheta_{av_1+bv_2+cv_3} = \vartheta_{v_1}^a\vartheta_{v_2}^b\vartheta_{v_3}^c$, and similarly for all of the other cones of $\mathcal F$. This gives a complete description of the theta functions on $U$.

§.§ Applications to mirror symmetry

In this section, and until the end of the paper, we restrict to looking at the special fibre $U:=U_{1,1}$ of our family $\mathcal U$ of cluster varieties, corresponding to the values $\lambda=\mu=1$. This is the most degenerate fibre, since it gains the two extra interior nodal singularities, and it is precisely the affine variety defined by the original recurrence relation (LR$_3$). The reason for this specialisation is that the expansions of the cluster variables as Laurent polynomials on a given cluster torus $\TT_{123}$ all now satisfy the binomial edge coefficient condition of Remark <ref>, which we expect for our mirror Landau–Ginzburg potentials. We continue to let $(X,D)$ be the pair obtained by taking the projective closure $X = \overline{U}\subset \PP^8$ with boundary divisor $D$.

§.§.§ Landau–Ginzburg models mirror to $V_{12}$ and $V_{16}$

We now describe how to construct explicit Landau–Ginzburg models which are mirror to the Fano 3-fold pairs considered in <ref>.

The polytopes $P = \conv \left( x_1,\ldots,x_8 \right)$ and $Q = \conv \left( x_1,\ldots,x_8, q_1, q_2 \right)$ are a pair of dual polytopes in $M_U$. Moreover they are, combinatorially, a realisation of the two polytopes of Figure <ref> when considered in the affine structure of $M_U$.

When drawn with respect to the fan $\mathcal F$ in $\RR^3$, the polytopes $P$ and $Q$ are pictured as follows

at (-1.5,1) $P$;
[inner sep = -2pt] (o) at (0,0) ;
(x) at (cos(330)+0.25,sin(330)) ;
(y) at (cos(90),sin(90)) ;
(z) at (cos(210)+0.25,sin(210)+0.1) ;

[gray,opacity = 0.2] ($(y)$) – ($(x)+(y)$) – ($(x)$) – (0,-1/2) – ($(z)$) – ($(o)-(x)$) – cycle;
($(y)$) – ($(x)+(y)$) – ($(x)+(y)+(z)$) – ($(z)+(y)$) – cycle;
($(o)-(x)$) – ($(o)-(z)$) – ($(x)$) – ($(z)$) – cycle;
($(o)-(x)$) – ($(y)$) – ($(o)-(z)$) – ($(x)+(y)$) – ($(x)$) – ($(x)+(y)+(z)$) – ($(z)$) – ($(z)+(y)$) – cycle;
[dashed] ($(o)-(x)$) – (0,-1/2) – ($(x)$) ($(o)-(z)$) – (0,-1/2) – ($(z)$);
at (o) $\times$;
[inner sep = -2pt] (x1) at ($(o)-(x)$) [label=[label distance=0pt]180:$x_1$] $\bullet$;
[inner sep = -2pt] (x2) at ($(y)$) [label=[label distance=0pt]90:$x_2$] $\bullet$;
[inner sep = -2pt] (x3) at ($(o)-(z)$) [label=[label distance=2pt]90:$x_3$] $\bullet$;
[inner sep = -2pt] (x4) at ($(x)+(y)$) [label=[label distance=0pt]0:$x_4$] $\bullet$;
[inner sep = -2pt] (x5) at ($(x)$) [label=[label distance=0pt]0:$x_5$] $\bullet$;
[inner sep = -2pt] (x6) at ($(x)+(y)+(z)$) [label=[label distance=0pt]270:$x_6$] $\bullet$;
[inner sep = -2pt] (x7) at ($(z)$) [label=[label distance=0pt]180:$x_7$] $\bullet$;
[inner sep = -2pt] (x8) at ($(y)+(z)$) [label=[label distance=2pt]0:$x_8$] $\bullet$;

at (-1.5,1) $Q$;
[inner sep = -2pt] (o) at (0,0) ;
(x) at (cos(330)+0.25,sin(330)) ;
(y) at (cos(90),sin(90)) ;
(z) at (cos(210)+0.25,sin(210)+0.1) ;

[gray,opacity = 0.2] ($(y)$) – ($(x)+(y)+(y)+(z)$) – ($(x)+(y)$) – ($(x)$) – ($(o)-(y)$) – ($(z)$) – ($(o)-(x)$) – cycle;

($(x)+(y)$) – ($(x)+(y)+(y)+(z)$) – ($(x)+(y)+(z)$);
($(z)+(y)$) – ($(x)+(y)+(y)+(z)$) – ($(y)$);
($(o)-(x)$) – ($(o)-(y)$) – ($(x)$);
($(o)-(z)$) – ($(o)-(y)$) – ($(z)$);
($(o)-(x)$) – ($(y)$) – ($(o)-(z)$) – ($(x)+(y)$) – ($(x)$) – ($(x)+(y)+(z)$) – ($(z)$) – ($(z)+(y)$) – cycle;
[dashed] ($(z)$) – ($(o)-(x)$) – ($(o)-(z)$) – ($(x)$);

at (o) $\times$;
[inner sep = -2pt] (x1) at ($(o)-(x)$) [label=[label distance=0pt]180:$x_1$] $\bullet$;
[inner sep = -2pt] (x2) at ($(y)$) [label=[label distance=0pt]90:$x_2$] $\bullet$;
[inner sep = -2pt] (x3) at ($(o)-(z)$) [label=[label distance=2pt]90:$x_3$] $\bullet$;
[inner sep = -2pt] (x4) at ($(x)+(y)$) [label=[label distance=0pt]0:$x_4$] $\bullet$;
[inner sep = -2pt] (x5) at ($(x)$) [label=[label distance=0pt]0:$x_5$] $\bullet$;
[inner sep = -2pt] (x6) at ($(x)+(y)+(z)$) [label=[label distance=0pt]270:$x_6$] $\bullet$;
[inner sep = -2pt] (x7) at ($(z)$) [label=[label distance=0pt]180:$x_7$] $\bullet$;
[inner sep = -2pt] (x8) at ($(y)+(z)$) [label=[label distance=2pt]0:$x_8$] $\bullet$;
[inner sep = -2pt] (q1) at ($(o)-(y)$) [label=[label distance=0pt]-90:$q_1$] $\bullet$;
[inner sep = -2pt] (q2) at ($(x)+(y)+(y)+(z)$) [label=[label distance=0pt]90:$q_2$] $\bullet$;

and so we are required to show that the dashed edges in the diagram are really flat when considered with respect to the affine structure on $M_U$.

The polytope $P$ gives an integral piecewise-linear function $\phi_P\colon N_U\to \RR$ given by $\phi_P(n) = \min\{ c\in \RR : n\in cP \}$, which is linear in each cone of our fan $\mathcal F$. Since $\phi_P(v_i) = 1$ for all $i\in\ZZ/8\ZZ$ and $\phi_P(w_1)=\phi_P(w_2)=2$, this determines a Weil divisor $\Xi_P=D_1 + \cdots + D_8 + 2D_{q_1} + 2D_{q_2}$ on $\widetilde X$. Now any codimension 2 cone $\tau$ of $\mathcal F$, corresponds to a curve $C_\tau$ in the boundary of $\widetilde X$. Using the criterion described in <ref>, the map $\phi_P$ is linear with respect to the affine structure on $N_U$ if $\Xi_P\cdot C_\tau=0$. Now we easily check that $\Xi_P$ is only nonzero on the curve classes corresponding to the cones $\langle x_i,x_{i+1} \rangle$ and $\langle x_i,q_i \rangle$.

The computation is similar for the polytope $Q$, which corresponds to the Weil divisor $\Xi_Q = D_1 +\cdots + D_8 + D_{q_1} + D_{q_2}$ on $\widetilde X$.

We now consider the two compactifications $(X_P,D_P)$ and $(X_Q,D_Q)$ of $U$ which were discussed in <ref>. Note that $(X_P,D_P)=(X,D)$ is the standard projective compactification of $U$ that we have been considering throughout this section.

An unprojection.

Before describing the mirror Landau-Ginzburg models to $(X_P,D_P)$ and $(X_Q,D_Q)$, we explain how to go from one model to the other by making a birational modification $\psi\colon X_P\dashrightarrow X_Q$, called an unprojection. Geometrically, this contracts the two $\PP^1\times\PP^1$ divisors $D_{1357},D_{2468}\subset D$. This is done by adjoining $q_1 = x_0^{-1}(x_1x_5 - x_0^2)$ and $q_2 = x_0^{-1}(x_2x_6 - x_0^2)$, which are rational sections of $\sO_X(1)$, as generators to the homogeneous coordinate ring $\CC[X_P]$.

Let $X_Q = \overline{\left\{ (x_0:x_1:\ldots:x_8:q_1:q_2) \in \PP^{10} : (x_0:x_1:\ldots:x_8)\in X_P \right\}}$ be the graph of the two rational sections $q_1,q_2$ on $X_P$. Then $X_Q\subset\PP^{10}$ is a projectively Gorenstein 3-fold of codimension 7 determined by the 21 equations

\begin{align}
x_ix_{i+3} &= x_0(x_0 + x_{i+1} + x_{i+2}) \tag{$\times8$} \\
x_ix_{i+4} &= x_0(q_i + x_0) \tag{$\times4$} \\
x_iq_{i+1} &= (x_0+x_{i+1})(x_0+x_{i-1})  \tag{$\times8$} \\
q_1q_2 &= x_0(4x_0+x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8) \tag{$\times1$}
\end{align}

and has graded resolution with Betti numbers $(1,21,64,70,70,64,21,1)$.

This is an application of the theory of unprojection developed by Papadakis–Reid [17]. The divisor $D_{q_1} = \VV(x_0,x_2,x_4,x_6,x_8,x_1x_5-x_3x_7)\subset \PP^8$ is a complete intersection of codimension 6 (hence projectively Gorenstein) and contained in $X_P\subset \PP^8$ which is projectively Gorenstein of codimension 5. By adjunction we have $K_{X_P}=\sO_{X_P}(-1)$ and $K_{D_{q_1}}=\sO_{D_{q_1}}(-2)$, and since $(-1)-(-2)=1>0$ we can apply <cit.> to obtain an unprojection variable $q_1$ as a rational section of $\sO_{X_P}(1)$. We compute that $q_1=x_0^{-1}(x_1x_5-x_0^2)$ works as an unprojection variable, and satisfies the equations

\[ x_0q_1 = x_1x_5 - x_0^2 = x_3x_7 - x_0^2, \qquad x_{i}q_1 = (x_0+x_{i-1})(x_0+x_{i+1}) \quad \text{for } i=2,4,6,8 \]

and thus we obtain an unprojection $(D_{q_1}\subset X_P)\dashrightarrow(P_{q_1}\in X_P')$ which contracts $D_{q_1}$ to an ordinary node at the coordinate point $P_{q_1}\in X_P'$. Now we repeat the procedure to the strict transform of $D_{q_1}$ in $X_P'$ to get $q_2=x_0^{-1}(x_2x_6-x_0^2)$ and an unprojection $(D_{q_2}\subset X_P')\dashrightarrow (P_{q_2}\in X_Q)$.

Geometry of the unprojection.
The unprojection factors as a blowup of the (non-Cartier) Weil divisor $D_{q_1}$, which resolves four nodes of $X_P$, followed by the contraction of the strict transform to an ordinary node. The result of this operation is depicted below. To construct $X_Q$ we do this unprojection to both $D_{q_1}$ and $D_{q_2}\subset X_P$ (i.e. to both the front and back squares of the diagram).

[gray,dashed] (cos(45),sin(45)) – (cos(135),sin(135)) – (cos(225),sin(225)) – (cos(315),sin(315)) – cycle;
[thick] (cos(0),sin(0)) – (cos(90),sin(90)) – (cos(180),sin(180)) – (cos(270),sin(270)) – cycle;

ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
at (cos(45*ı),sin(45*ı)) $\bullet$;
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;

[gray,dashed] (cos(45),sin(45)) – (cos(135),sin(135)) – (cos(225),sin(225)) – (cos(315),sin(315)) – cycle;
[thick] (0.5*cos(0),0.5*sin(0)) – (0.5*cos(90),0.5*sin(90)) – (0.5*cos(180),0.5*sin(180)) – (0.5*cos(270),0.5*sin(270)) – cycle;

ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
at (cos(90*ı+ 45),sin(90*ı+ 45)) $\bullet$;
(cos(90*ı),sin(90*ı)) – (0.5*cos(90*ı),0.5*sin(90*ı));
[->] (-1.25,-0.25) to node[yshift=0.65cm] blowup (-1.75,-0.5);
[->] (1.25,-0.25) to node[yshift=0.65cm] contract (1.75,-0.5);

[gray,dashed] (cos(45),sin(45)) – (cos(135),sin(135)) – (cos(225),sin(225)) – (cos(315),sin(315)) – cycle;
at (0,0) $\bullet$;

ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
at (cos(90*ı+ 45),sin(90*ı+ 45)) $\bullet$;
(cos(90*ı),sin(90*ı)) – (0,0);

Note that the unprojection $\psi\colon X_P\dashrightarrow X_Q$ restricts to an isomorphism on the open set $U=X_P\setminus D_P$, since $q_1$ and $q_2$ are regular functions on $U$. Therefore if $D_Q$ is the strict transform of the boundary divisor $D_P$ then $(X_Q,D_Q)$ is also a log Calabi–Yau compactification of $U$.

The potential $w_Q$.

According to Remark <ref>, since the ten nonzero integral points of $Q$ are all vertices, the Landau–Ginzburg potential supported on $Q$ that we consider is uniquely determined by summing the corresponding ten theta functions with coefficient 1, i.e.

\[ w_Q = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + q_1 + q_2. \]

We note that this is $\sigma_3$-invariant function and, indeed, restricting to the cluster torus chart $\TT_{123}\subset U$ gives

\[ (w_Q + 5)|_{\TT_{123}} = \frac{(1 + x_1 + x_2)(1 + x_2 + x_3)(1 + x_1 + x_2 + x_3 + x_1x_3)}{x_1x_2x_3} \]

which is well-known to be an invariant function for Lyness map $\sigma_3$.

The period $\pi_{w_Q}(t)$.
Consider the period

\[ \pi_{w_Q}(t) = 1  + 48t^2 + 600t^3 + 13176t^4 + 276480t^5 + 6259800t^6 + 146064240t^7 + \cdots \]

which agrees with the regularised quantum period $\widehat{G}_X(t)$ for the Fano 3-fold $X$ of type $V_{12}$, as listed in <cit.>. Alternatively, the shifted potential $w_Q+5$ has period

\[ \pi_{w_Q+5}(t) = 1 + 5t + 73t^2 + 1445t^3 + 33001t^4 + 819005t^5 + 21460825t^6+ \cdots \]

which can be recognised as the Apéry series $\pi_{w+5}(t) = \sum_{n=0}^\infty\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2 t^n$ and is also well-known as a mirror period sequence for the Fano 3-fold $V_{12}$, cf. [12].

A K3 fibration.
In order to appreciate some beautiful geometry hiding here, it is convenient to consider extending the potential $w_Q$ to the compactification $X_Q$. In analogy to the critical value $-3$ for the Landau–Ginzburg potential on the del Pezzo surface, the factorisation of $(w_Q + 5)|_{\TT_{123}}$ corresponds to the fact that the interior divisor $E_Q := w_Q^{-1}(-5) \subset U$ is reducible.

at (-1,1) $D_P\cap E_P$;
[thick] (cos(0),sin(0)) – (cos(90),sin(90)) – (cos(180),sin(180)) – (cos(270),sin(270)) – cycle;
ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
ıin 0,...,7
at (cos(45*ı),sin(45*ı)) $\bullet$;
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
[very thick] (0.92*cos(90*ı+22.5),0.92*sin(90*ı+22.5)) – (0.8*cos(90*ı+45),0.8*sin(90*ı+45)) – (0.92*cos(90*ı+45+22.5),0.92*sin(90*ı+45+22.5));
[very thick] (0.8*cos(45),0.8*sin(45)) – (0.8*cos(180+45),0.8*sin(180+45));
[very thick] (0.8*cos(135),0.8*sin(135)) – (0.8*cos(270+45),0.8*sin(270+45));

[rotate = 45, opacity=0.5, dashed]
[thick] (cos(0),sin(0)) – (cos(90),sin(90)) – (cos(180),sin(180)) – (cos(270),sin(270)) – cycle;
ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
ıin 0,...,7
at (cos(45*ı),sin(45*ı)) $\bullet$;
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
[very thick] (0.92*cos(90*ı+22.5),0.92*sin(90*ı+22.5)) – (0.8*cos(90*ı+45),0.8*sin(90*ı+45)) – (0.92*cos(90*ı+45+22.5),0.92*sin(90*ı+45+22.5));
[very thick] (0.8*cos(45),0.8*sin(45)) – (0.8*cos(180+45),0.8*sin(180+45));
[very thick] (0.8*cos(135),0.8*sin(135)) – (0.8*cos(270+45),0.8*sin(270+45));

[xshift = -3cm]
at (-1,1) $D_Q\cap E_Q$;
[thick] (cos(0),sin(0)) – (cos(180),sin(180));
[thick] (cos(90),sin(90)) – (cos(270),sin(270));
ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
[very thick] (0.92*cos(90*ı+22.5),0.92*sin(90*ı+22.5)) – (0.5*cos(90*ı+90),0.5*sin(90*ı+90));
[very thick] (0.92*cos(90*ı-22.5),0.92*sin(90*ı-22.5)) – (0.5*cos(90*ı-90),0.5*sin(90*ı-90));

[xshift = -3cm, rotate = 45, opacity=0.5, dashed]
[thick] (cos(0),sin(0)) – (cos(180),sin(180));
[thick] (cos(90),sin(90)) – (cos(270),sin(270));
ıin 0,...,7
[thick] (cos(45*ı),sin(45*ı)) – (cos(45*ı+45),sin(45*ı+45));
at (0.92*cos(45*ı+22.5),0.92*sin(45*ı+22.5)) $\bullet$;
ıin 0,...,3
[very thick] (0.92*cos(90*ı+22.5),0.92*sin(90*ı+22.5)) – (0.5*cos(90*ı+90),0.5*sin(90*ı+90));
[very thick] (0.92*cos(90*ı-22.5),0.92*sin(90*ı-22.5)) – (0.5*cos(90*ı-90),0.5*sin(90*ı-90));

The sixteen lines of intersection for the boundary divisor $D_Q$ and the degenerate interior divisor $E_Q$ represented inside $D_Q$. (b) Similarly for the twelve lines of intersection of $D_P$ and $E_P$.

Then, in the notation of Proposition <ref>, the reducible fibre is $E_Q = \cup_{i=1}^8 E_{i,i+2}$ which is the union of eight copies of $\PP^1\times \PP^1$ and happens to be isomorphic to $D_Q$ (i.e. these eight components also meet like the faces of the polytope $Q$). The intersection of the divisors $D_Q\cap E_Q$ is a set of sixteen lines, given by the pair of lines on each face of $D_Q$ which pass through the eight nodes $(1:-1)\in \PP^1_{x_i,x_{i+1}}$ in the boundary of $X_Q$. These lines are shown in Figure <ref>.

Now the fibres of $w_Q\colon X_Q\dashrightarrow \PP^1$ are given by the anticanonical pencil $|D_Q,E_Q|$ with baselocus the sixteen lines described above. The general such member of this pencil is a K3 surface $S_t := w_Q^{-1}(t)$ which is smooth, apart from eight ordinary double points which lie at the eight nodes of $X$ that are contained in $D_Q\cap E_Q$. Therefore, after taking its minimal resolution, the K3 surface $S_t$ contains a configuration $\Sigma=\bigcup_{i=1}^{24}\Sigma_i$ of $24$ $(-2)$-curves with the dual intersection diagram displayed in Figure <ref>(a), where the black nodes correspond to the sixteen lines of $D_Q\cap E_Q$ and the white nodes correspond to the eight ordinary double points. Since the 24 $(-2)$-curves $\Sigma$ span a lattice of rank $19$ and $S_t$ deforms in a nontrivial family, it follows that $S_t$ has Picard rank 19.

ıin 1,...,8

(cos(45*ı+22.5),sin(45*ı+22.5)) – (2*cos(45*ı+22.5),2*sin(45*ı+22.5));
(2*cos(45*ı+22.5),2*sin(45*ı+22.5)) – (2*cos(45*(ı+1)+22.5),2*sin(45*(ı+1)+22.5));
(cos(45*ı+22.5),sin(45*ı+22.5)) – (cos(45*(ı+3)+22.5),sin(45*(ı+3)+22.5));

[fill=black] (cos(0+22.5),sin(0+22.5)) circle (2pt);
[fill=black] (cos(45+22.5),sin(45+22.5)) circle (2pt);
[fill=black] (cos(90+22.5),sin(90+22.5)) circle (2pt);
[fill=black] (cos(135+22.5),sin(135+22.5)) circle (2pt);
[fill=black] (cos(180+22.5),sin(180+22.5)) circle (2pt);
[fill=black] (cos(225+22.5),sin(225+22.5)) circle (2pt);
[fill=black] (cos(270+22.5),sin(270+22.5)) circle (2pt);
[fill=black] (cos(315+22.5),sin(315+22.5)) circle (2pt);
[fill=white] (3/2*cos(0+22.5),3/2*sin(0+22.5)) circle (2pt);
[fill=white] (3/2*cos(45+22.5),3/2*sin(45+22.5)) circle (2pt);
[fill=white] (3/2*cos(90+22.5),3/2*sin(90+22.5)) circle (2pt);
[fill=white] (3/2*cos(135+22.5),3/2*sin(135+22.5)) circle (2pt);
[fill=white] (3/2*cos(180+22.5),3/2*sin(180+22.5)) circle (2pt);
[fill=white] (3/2*cos(225+22.5),3/2*sin(225+22.5)) circle (2pt);
[fill=white] (3/2*cos(270+22.5),3/2*sin(270+22.5)) circle (2pt);
[fill=white] (3/2*cos(315+22.5),3/2*sin(315+22.5)) circle (2pt);
[fill=black] (2*cos(0+22.5),2*sin(0+22.5)) circle (2pt);
[fill=black] (2*cos(45+22.5),2*sin(45+22.5)) circle (2pt);
[fill=black] (2*cos(90+22.5),2*sin(90+22.5)) circle (2pt);
[fill=black] (2*cos(135+22.5),2*sin(135+22.5)) circle (2pt);
[fill=black] (2*cos(180+22.5),2*sin(180+22.5)) circle (2pt);
[fill=black] (2*cos(225+22.5),2*sin(225+22.5)) circle (2pt);
[fill=black] (2*cos(270+22.5),2*sin(270+22.5)) circle (2pt);
[fill=black] (2*cos(315+22.5),2*sin(315+22.5)) circle (2pt);

[xshift = 6cm]
ıin 1,...,8

(2*cos(45*ı+22.5),2*sin(45*ı+22.5)) – (2*cos(45*(ı+1)+22.5),2*sin(45*(ı+1)+22.5));

(2*cos(45*0+22.5),2*sin(45*0+22.5)) – (2*cos(45*4+22.5),2*sin(45*4+22.5));
(2*cos(45*1+22.5),2*sin(45*1+22.5)) – (2*cos(45*5+22.5),2*sin(45*5+22.5));
(2*cos(45*2+22.5),2*sin(45*2+22.5)) – (2*cos(45*6+22.5),2*sin(45*6+22.5));
[white, fill=white] (0,0) circle (5pt);
(2*cos(45*3+22.5),2*sin(45*3+22.5)) – (2*cos(45*7+22.5),2*sin(45*7+22.5));

[white,fill=white] (0.71*cos(315+22.5),0.71*sin(315+22.5)) circle (3pt);
[white,fill=white] (0.71*cos(180+22.5),0.71*sin(180+22.5)) circle (3pt);

(cos(0+22.5),sin(0+22.5)) – (cos(270+22.5),sin(270+22.5));
(cos(135+22.5),sin(135+22.5)) – (cos(225+22.5),sin(225+22.5));

[fill=black] (cos(0+22.5),sin(0+22.5)) circle (2pt);
[fill=black] (cos(135+22.5),sin(135+22.5)) circle (2pt);
[fill=black] (cos(225+22.5),sin(225+22.5)) circle (2pt);
[fill=black] (cos(270+22.5),sin(270+22.5)) circle (2pt);
[fill=white] (1.84*cos(0),1.84*sin(0)) circle (2pt);
[fill=white] (1.84*cos(45),1.84*sin(45)) circle (2pt);
[fill=white] (1.84*cos(90),1.84*sin(90)) circle (2pt);
[fill=white] (1.84*cos(135),1.84*sin(135)) circle (2pt);
[fill=white] (1.84*cos(180),1.84*sin(180)) circle (2pt);
[fill=white] (1.84*cos(225),1.84*sin(225)) circle (2pt);
[fill=white] (1.84*cos(270),1.84*sin(270)) circle (2pt);
[fill=white] (1.84*cos(315),1.84*sin(315)) circle (2pt);
[fill=black] (2*cos(0+22.5),2*sin(0+22.5)) circle (2pt);
[fill=black] (2*cos(45+22.5),2*sin(45+22.5)) circle (2pt);
[fill=black] (2*cos(90+22.5),2*sin(90+22.5)) circle (2pt);
[fill=black] (2*cos(135+22.5),2*sin(135+22.5)) circle (2pt);
[fill=black] (2*cos(180+22.5),2*sin(180+22.5)) circle (2pt);
[fill=black] (2*cos(225+22.5),2*sin(225+22.5)) circle (2pt);
[fill=black] (2*cos(270+22.5),2*sin(270+22.5)) circle (2pt);
[fill=black] (2*cos(315+22.5),2*sin(315+22.5)) circle (2pt);

(a) The configuration $\Sigma$ of 24 $(-2)$-curves in the boundary of $w_Q^{-1}(t)$. (b) The configuration $\Sigma$ of 20 $(-2)$-curves in the boundary of $w_P^{-1}(t)$.

Returning to our original Landau–Ginzburg model $w_Q\colon U\to C$, we see that the fibres are a family of affine K3 surfaces $w_Q^{-1}(t) = S_t\setminus \Sigma$, given by the complement of this configuration of lines $\Sigma$ in $S_t$. These can be represented by as affine quartics

\[ (t + 5)x_1x_2x_3 = (1 + x_1 + x_2)(1 + x_2 + x_3)\left(1 + x_1 + x_2 + x_3 + x_1x_3\right) \]

and there are precisely four degenerate fibres. Two reducible fibres corresponding to the values $E_Q=w_Q^{-1}(-5)$ and $D_Q=w_Q^{-1}(\infty)$, and two values $\lambda_\pm = 24(1 \pm \sqrt{2})$ for which the Picard rank jumps to 20. We get a picture as in the 2-dimensional case.

[gray,dashed] (cos(45),2+sin(45)) – (cos(225),2+sin(225))  (cos(135),2+sin(135)) – (cos(-45),2+sin(-45));
[thick] (cos(0),2+sin(0)) – (cos(45),2+sin(45)) – (cos(90),2+sin(90)) – (cos(135),2+sin(135)) – (cos(180),2+sin(180)) – (cos(225),2+sin(225)) – (cos(270),2+sin(270)) – (cos(315),2+sin(315)) – cycle;
[thick] (cos(0),2+sin(0)) – (cos(180),2+sin(180)) (cos(90),2+sin(90)) – (cos(270),2+sin(270));

[thick] (4,2) circle (1cm);
[dashed] (3,2) to [in = 150, out = 30] (5,2);
(3,2) to [in = -150, out = -30] (5,2);

[thick] (4,2) circle (1cm);
[dashed] (3,2) to [in = 150, out = 30] (5,2);
(3,2) to [in = -150, out = -30] (5,2);

[gray,dashed] (cos(45),2+sin(45)) – (cos(225),2+sin(225))  (cos(135),2+sin(135)) – (cos(-45),2+sin(-45));
[thick] (cos(0),2+sin(0)) – (cos(45),2+sin(45)) – (cos(90),2+sin(90)) – (cos(135),2+sin(135)) – (cos(180),2+sin(180)) – (cos(225),2+sin(225)) – (cos(270),2+sin(270)) – (cos(315),2+sin(315)) – cycle;
[thick] (cos(0),2+sin(0)) – (cos(180),2+sin(180)) (cos(90),2+sin(90)) – (cos(270),2+sin(270));

(-1,0.5) – (10,0.5);
at (0,0.5) [label=below:$-5$] $\times$;
at (3,0.5) [label=below:$\lambda_-$] $\times$;
at (6,0.5) [label=below:$\lambda_+$] $\times$;
at (9,0.5) [label=below:$\infty$] $\times$;
at (3,1.5) $\rho=20$;
at (6,1.5) $\rho=20$;

The potential $w_P$.
We can rerun this entire analysis with the potential $w_P=x_1+\ldots+x_8$ in place of $w_Q$. In summary, the period of $w_P$ is

\[ \pi_{w_P}(t) = 1  + 24t^2 + 192t^3 + 2904t^4 + 40320t^5 + 611520t^6 + 9515520t^7 + \cdots \]

which agrees with the regularised quantum period $\widehat{G}_X(t)$ for the Fano 3-fold $X$ of type $V_{16}$, as listed in <cit.>. There is a factorisation

\[ (w_P + 4)|_{\TT_{123}} = \frac{(1 + x_1)(1 + x_2)(1 + x_3)(1 + x_1 + x_2 + x_3)}{x_1x_2x_3} \]

which is another $\sigma_3$-invariant Laurent polynomial, and corresponds to a reducible fibre given by $E_P := w_P^{-1}(-4) = \bigcup_{i=1}^4F_{i,i+4}$ in the notation of Proposition <ref>. When extended to the compactification $X_P$, the boundary divisor $D_P$ meets $E_P$ in a collection of 12 lines, given by the $\Dih_8$-orbits of $\VV(x_1+x_2+x_3)\subset D_{123}$ and $\VV(x_1+x_3)\subset D_{1357}$. These are displayed (tropically) in Figure <ref>(b) and pass through the same eight nodes in the boundary of $X_P$ as the previous case. The general member in the anticanonical pencil $|D_P,E_P|$, corresponding to the fibres of $w_Q$, is a K3 surface $S_t = w_Q^{-1}(t)$ which passes through the twelve lines and is smooth apart from eight ordinary double points at the eight nodes of $X_P$. The minimal resolution of $S_t$ is a K3 surface containing a configuration $\Sigma$ of 20 $(-2)$-curves generating a lattice of Picard rank 19, and which correspond to the dual intersection diagram of Figure <ref>(b). The fibres of the original Landau–Ginzburg model $w_P\colon U\to \CC$ are affine K3 surfaces of the form $S_t\setminus \Sigma$. There are exactly four degenerate fibres: the two reducible fibres $D_P=w_Q^{-1}(\infty)$ and $E_P=w_Q^{-1}(-4)$, and a further two values where the Picard rank jumps to 20.

§.§.§ Other Fano 3-folds

We do not try to attempt to classify all reflexive polytopes in $N_U$, although it would be interesting to know how many there are in comparison to the 4319 reflexive polytopes in $\RR^3$. Nevertheless, we can consider all of the polytopes obtained from removing vertices from the polytope $Q$ from <ref>.

Consider the Laurent polynomial $w_0=w|_{\TT_{123}}\in\CC[x_1^{\pm1},x_2^{\pm1},x_3^{\pm1}]$ obtained by restricting the Landau–Ginzburg potential $w = \varepsilon_1x_1 + \ldots + \varepsilon_8x_8+\varepsilon_9q_1+\varepsilon_{10}q_2$ with coefficients $\varepsilon_i\in\{0,1\}$ for $i=1,\ldots,10$.

* Of the 1024 possibilities for $w_0$, 705 have 3-dimensional Fano Newton polytopes, i.e. $\Newt(w_0)$ has primitive vertices and $0\in \Int(\Newt(w_0))$.

* These 705 Laurent polynomials give rise to 46 distinct periods.

* Of these 46 periods, 20 are equal to the regularised quantum period of a smooth Fano 3-fold, as described in Table <ref>.

Mirror Landau–Ginzburg potentials on $U$ for 20 smooth Fano 3-folds.

Fano 3-fold     Mirror Landau–Ginzburg potential $w$
$V_{12}$     $x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+q_1+q_2$

$V_{14}$     $x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+q_1$

$V_{16}$     $x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8$

$V_{18}$     $x_1+x_2+x_3+x_4+x_5+x_6+x_7$

$V_{22}$     $x_1+x_2+x_3+x_4+x_5+x_6$
$\MM_{2,9}$     $x_1 + x_2 + x_3 + x_6 + q_1 + q_2$

$\MM_{2,12}$     $x_1 + x_2 + x_3 + x_5 + x_6 + x_7$

$\MM_{2,13}$     $x_1 + x_2 + x_3 + x_4 + x_6 + x_7$

$\MM_{2,14}$     $x_1 + x_2 + x_3 + x_4 + x_5 + x_7 + q_1$

$\MM_{2,16}$     $x_1 + x_2 + x_3 + x_6 + q_1$

$\MM_{2,17}$     $x_1 + x_2 + x_3 + x_5 + x_6$

$\MM_{2,20}$     $x_1 + x_2 + x_3 + x_4 + x_7$

$\MM_{2,21}$     $x_1 + x_2 + x_3 + x_5 + x_7$

$\MM_{2,22}$     $x_1 + x_2 + x_3 + x_6$
$\MM_{3,7}$     $x_1 + x_2 + x_4 + x_6 + x_7$

$\MM_{3,10}$     $x_1 + x_2 + x_3 + x_5 + x_7 + q_1$

$\MM_{3,12}$     $x_1 + x_3 + x_6 + x_7 + q_1$

$\MM_{3,13}$     $x_1 + x_5 + q_2$

$\MM_{3,15}$     $x_1 + x_4 + x_5 + x_7$

$\MM_{3,20}$     $x_1 + x_4 + x_6$

Table <ref> only contains one representative Landau–Ginzburg potential in each case and, even after taking the $\Dih_8$ symmetry into account, the same period sequence can be obtained from several different potentials. For example, in addition to the entry in the Table <ref>, the following five potentials also have the right period to be mirror to $V_{22}$:

\[ x_1+x_2+x_5+x_6+q_2, \quad x_1+x_2+x_3+x_4+x_5+q_2, \quad x_1+x_2+q_1+q_2, \]

\[ x_1+x_2+x_3+x_4+x_5+x_7 \quad \text{and} \quad x_1+x_2+x_3+x_4+x_6+q_2. \]

According to Conjecture <ref>, the different choices of potential correspond to different degenerations of a Fano 3-fold of type $V_{22}$ to a pair $(X_Q,D_Q)$ where $Q$ is the dual polytope to $P=\Newt(w)\subset M_U$. It would be interesting to know whether the remaining 26 period sequences are all period sequences of Fano 3-folds with terminal singularities.

§.§.§ A final example

We end with a final example of mirror duality for a pair of exotic polytopes in $N_U$, which cannot be realised combinatorially as polytopes in Euclidean space. The intersection of the two halfspaces $(q_1)^{\geq-1}\cap (q_2)^{\geq-1}$ is a closed polytope $Q\subset N_U$ with eight vertices $x_ix_{i+1}$ for $i\in \ZZ/8\ZZ$. Therefore the dual polytope $P$ is given by $P = \bigcap_{i\in\ZZ/8\ZZ} (x_ix_{i+1})^{\geq-1}$. This gives a pair of dual polytopes in $N_U$

\[ P=\conv(q_1, q_2) \qquad \text{and} \qquad Q=P^\star=\conv(x_1x_2, x_2x_3, \ldots, x_8x_1). \]

We draw these two polytopes below, where the solid lines denote edges of the polytope and dashed lines are due to the bend in the affine structure of $N_U$. We note that $P$ has two vertices and eight faces (so topologically it looks like a beachball with eight stripes), whereas $Q$ has eight vertices and two faces.

[xshift=0cm, yshift=0.5cm]
[inner sep = -2pt] (o) at (0,0) ;
(x) at (cos(330)+0.25,sin(330)) ;
(x2) at ((cos(330)+0.25)/2,sin(330)/2) ;
(x3) at ((cos(330)+0.25)/3,sin(330)/3) ;
(y) at (cos(90),sin(90)) ;
(z) at (cos(210)+0.25,sin(210)+0.1) ;
(z2) at ((cos(210)+0.25)/2,(sin(210)+0.1)/2) ;
(z3) at ((cos(210)+0.25)/3,(sin(210)+0.1)/3) ;

[gray, opacity=0.2] ($(x)+(y)+(y)+(z)$) – ($(x3)-(z3)$) – ($(x2)$) – ($(o)-(y)$) – ($(z3)-(x3)$) – ($(o)-(x2)$) – cycle;

($(x)+(y)+(y)+(z)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(x2)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(x3)-(z3)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(o)-(z2)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(o)-(x3)-(z3)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(o)-(x2)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(z3)-(x3)$) – ($(o)-(y)$)
($(x)+(y)+(y)+(z)$) – ($(z2)$) – ($(o)-(y)$);
[dashed] ($(x2)$) – ($(x3)-(z3)$) – ($(o)-(z2)$) – ($(o)-(x3)-(z3)$) – ($(o)-(x2)$) – ($(z3)-(x3)$) – ($(z2)$);

at ($(o)$) $\times$;
at ($(o)-(y)$) [label=below:$q_1$] $\bullet$;
at ($(x)+(y)+(y)+(z)$) [label=above:$q_2$] $\bullet$;

[inner sep = -2pt] (o) at (0,0) ;
(x) at (cos(330)+0.25,sin(330)) ;
(y) at (cos(90),sin(90)) ;
(y2) at (cos(90)/2,sin(90)/2) ;
(z) at (cos(210)+0.25,sin(210)+0.1) ;

[gray, opacity=0.2] ($(z)$) – ($(o)-(y2)$) – ($(x)$) – ($(x)+(x)+(y)+(z)$) – ($(x)+(x)+(y)$) – ($(x)+(y)-(z)$) – ($(y)-(z)$) – ($(y)-(x)$) – ($(y)+(z)-(x)$) – ($(y)+(z)+(z)$) – cycle;

($(y)-(x)$) – ($(y)+(z)-(x)$) – ($(y)+(z)+(z)$) – ($(x)+(y)+(z)+(z)$) – ($(x)+(x)+(y)+(z)$) – ($(x)+(x)+(y)$) – ($(x)+(y)-(z)$) – ($(y)-(z)$) – cycle;
[dashed] ($(y)-(x)$) – ($(o)-(x)$) – ($(z)$) – ($(x)+(y)+(z)+(z)$)
($(y)-(z)$) – ($(o)-(z)$) – ($(x)$) – ($(x)+(x)+(y)+(z)$)
($(y)+(z)-(x)$) – ($(o)-(x)$) – ($(o)-(z)$) – ($(x)+(y)-(z)$)
($(y)+(z)+(z)$) – ($(z)$) – ($(o)-(y2)$) – ($(o)-(z)$)
($(y)+(x)+(x)$) – ($(x)$) – ($(o)-(y2)$) – ($(o)-(x)$);

at ($(o)-(x)$) $\bullet$;
at ($(o)-(z)$) $\bullet$;
at ($(x)$) $\bullet$;
at ($(z)$) $\bullet$;
at ($(y)+(z)$) $\bullet$;
at ($(x)+(y)$) $\bullet$;
at ($(x)+(y)+(z)$) $\bullet$;
at ($(y)$) $\bullet$;

at ($(o)$) $\times$;

at ($(y)-(x)$) [label=above:$x_1x_2$] $\bullet$;
at ($(y)+(z)-(x)$) [label=left:$x_8x_1$] $\bullet$;
at ($(y)+(z)+(z)$) [label=left:$x_7x_8$] $\bullet$;
at ($(x)+(y)+(z)+(z)$) [label=below:$x_6x_7$] $\bullet$;
at ($(x)+(x)+(y)+(z)$) [label=below:$x_5x_6$] $\bullet$;
at ($(x)+(x)+(y)$) [label=right:$x_4x_5$] $\bullet$;
at ($(x)+(y)-(z)$) [label=right:$x_3x_4$] $\bullet$;
at ($(y)-(z)$) [label=above:$x_2x_3$]$\bullet$;

We now describe how to set up the mirror correspondence between the (degenerate) Fano 3-fold $(X_P,D_P)$ (resp. $(X_Q,D_Q)$) and the Landau–Ginzburg model $(U,w_Q)$ (resp. $(U,w_P)$).

We start by describing the mirror correspondence for $(X_Q,D_Q)$ and $(U,w_P)$.

The Landau–Ginzburg model $(U,w_P)$.
Using $P$ as the polytope to define a Landau–Ginzburg model $(U,w_P)$ is straightforward, since $P$ contains only two nonzero lattice points which are both vertices. By Remark <ref> they both receive coefficient 1 and this uniquely determines the potential $w_P=q_1+q_2$. By restricting to the torus chart $\TT_{123}$ and computing the period sequence of the resulting Laurent polynomial, we find that

\[ \pi_{w_P}(t) = 1 + 8t^2 + 24t^3 + 240t^4 + 1440t^5 + 11960t^6 + 89040t^7 + \cdots \]

which is the regularised quantum period of the smooth Fano 3-fold $\MM_{2,21}$ <cit.>.

The Fano 3-fold $(X_Q,D_Q)$.
When considered in the integral affine structure of $M_U$, the two faces of $Q$ are two flat lattice octagons labelled with the following theta functions.

[thick,fill=gray,fill opacity=0.2] (1,0) – (2,0) – (3,1) – (3,2) – (2,3) – (1,3) – (0,2) – (0,1) – cycle;
[thick,dashed] (0,2) – (3,2) (1,0) – (1,3) (2,0) – (2,3) (0,1)  – (1,1) – (2,2) (1,2) – (2,1) – (3,1);
at (1,0) [label=[label distance=-5pt]-90:$x_6x_7$]$\bullet$;
at (2,0) [label=[label distance=-5pt]-90:$x_5x_6$]$\bullet$;
at (3,1) [label=[label distance=-5pt]0:  $x_4x_5$]$\bullet$;
at (3,2) [label=[label distance=-5pt]0:  $x_3x_4$]$\bullet$;
at (2,3) [label=[label distance=-5pt]90: $x_2x_3$]$\bullet$;
at (1,3) [label=[label distance=-5pt]90: $x_1x_2$]$\bullet$;
at (0,2) [label=[label distance=-5pt]180:$x_8x_1$]$\bullet$;
at (0,1) [label=[label distance=-5pt]180:$x_7x_8$]$\bullet$;

at (1,1) [label=[label distance=-7pt]-135:$x_7$]$\bullet$;
at (2,1) [label=[label distance=-7pt]-45: $x_5$]$\bullet$;
at (2,2) [label=[label distance=-7pt]45:  $x_3$]$\bullet$;
at (1,2) [label=[label distance=-7pt]135: $x_1$]$\bullet$;

[xshift = 7cm, yshift=-0.66cm, rotate=45]
[thick,fill=gray,fill opacity=0.2] (1,0) – (2,0) – (3,1) – (3,2) – (2,3) – (1,3) – (0,2) – (0,1) – cycle;

at (1,0) [label=[label distance=-5pt]-90:$x_5x_6$]$\bullet$;
at (2,0) [label=[label distance=-5pt]0:$x_4x_5$]$\bullet$;
at (3,1) [label=[label distance=-5pt]0:  $x_3x_4$]$\bullet$;
at (3,2) [label=[label distance=-5pt]90:  $x_2x_3$]$\bullet$;
at (2,3) [label=[label distance=-5pt]90: $x_1x_2$]$\bullet$;
at (1,3) [label=[label distance=-5pt]180: $x_8x_1$]$\bullet$;
at (0,2) [label=[label distance=-5pt]180:$x_7x_8$]$\bullet$;
at (0,1) [label=[label distance=-5pt]-90:$x_6x_7$]$\bullet$;

at (1,1) [label=[label distance=-5pt]-90:$x_6$]$\bullet$;
at (2,1) [label=[label distance=-5pt]0: $x_4$]$\bullet$;
at (2,2) [label=[label distance=-5pt]90:  $x_2$]$\bullet$;
at (1,2) [label=[label distance=-5pt]180: $x_8$]$\bullet$;

Thus to construct $(X_Q,D_Q)$ we consider the graded ring $R_Q$ which is generated in degree 1 by seventeen generators

\[ x_0, \: x_1, \; \ldots,\; x_8,\; x_1x_2,\; \ldots,\; x_8x_1 \]

where $x_0$ is the homogenising variable. According to computer algebra $X_Q=\Proj R_Q\subset \PP^{16}$ is a Gorenstein Fano variety of Fano index 1 and degree $\deg X_Q=\vol Q=28$. This at least agrees with $X_Q$ being a (degeneration of a) Fano 3-fold of type $\MM_{2,21}$, as predicted by the period sequence of $w_P$. By Conjecture <ref> we expect that $X_Q$ admits a $\QQ$-Gorenstein smoothing to a smooth Fano 3-fold of type $\MM_{2,21}$.

Moreover, the boundary divisor $D_Q=\VV(x_0)$ consists of two components $D=D_1\cup D_2$ which are toric surfaces (defined by the two octagons above) which glued together other along an octagon of rational curves. This realises $Q$ as the intersection complex of $D_Q$.

The more interesting (and currently less clear) direction is to understand how to set up the mirror correspondence with the roles of $P$ and $Q$ reversed.

The Fano 3-fold $(X_P,D_P)$.
The polytope $P$ gives degrees $2,2,\ldots,2,1,1$ to the variables $x_1,x_2,\ldots,x_8,q_1,q_2$ respectively. Taking the projective closure of $U$ with respect to this grading we get a Fano 3-fold $X_P=\Proj R_P\subset \PP(1^3,2^8)$ in weighted projective space. One of the equations defining $X_P$ is

\[ q_1q_2 = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + 4x_0^2 \]

which we could use to eliminate $x_1$ say, and reduce to a model $X_P\subset \PP(1^3,2^7)$. The Hilbert series of $X_P$ now agrees with <cit.>, showing that $X_P$ has degree $\deg(X_P)=\vol(P)=4$, Fano index 1 and basket of quotient singularities $\{8\times\tfrac12(1,1,1)\}$. The anticanonical boundary divisor $D_P=\VV(x_0)$ has eight components:

\[ D_{12} = \VV(x_0,x_3,x_4,x_5,x_6,x_7,x_8,q_1q_2 - x_1 - x_2) \qquad \text{etc.} \]

which are all copies of $\PP(1,1,2)$ that are joined to their neighbours along conics meeting at the two coordinate points where $q_1\neq0$ and $q_2\neq0$. Thus $P$ is realised as the intersection complex for the boundary divisor $D_P$. One of the eight $\tfrac12(1,1,1)$ points of $X_P$ lies in the relative interior of $D_{12}$ at the point where $x_1+x_2\neq 0$ and all of the other coordinates vanish. Similarly for the other seven $\tfrac12(1,1,1)$ points.

We note that if $Y$ is a Fano 3-fold with Hilbert series <cit.>, the $h^0(Y,{-K_Y})=3$ and hence $-K_Y$ is ample but not very ample, so there is no Gorenstein toric degeneration of $Y$. Therefore there is no reflexive polytope we can write down which will give us a mirror Laurent polynomial for $Y$.

The Landau–Ginzburg model $(U,w_Q)$.
To cook up a Landau–Ginzburg potential $w_Q$ which is mirror to $(X_P,D_P)$, we need to label the lattice points of $Q$ with appropriate integer coefficients, as described in Remark <ref>. Since we label all vertices of $Q$ with the coefficient 1, the only decisions we need to make are over the lattice points belonging to the relative interior of each face. Following [9], we expect that the right thing to do is label the lattice points of each face $F\subset Q$ with the coefficients of a 0-mutable polynomial supported on $F$. In our case this is one of the following three choices of coefficients, according to the three possible maximal Minkowski decompositions of the octagon.

at (0,0.25) $\begin{matrix}&1&1&\\1&2&2&1\\1&2&2&1\\&1&1&\end{matrix}$;
at (6,0.25) $\begin{matrix}&1&1&\\1&3&2&1\\1&2&3&1\\&1&1&\end{matrix}$;
at (12,0.25) $\begin{matrix}&1&1&\\1&2&3&1\\1&3&2&1\\&1&1&\end{matrix}$;

(-2,2) – (-2,2.5);
(-1,2.25) – (-0.5,2.25);
(0.5,2) – (1,2.5);
(2.5,2) – (2,2.5);

at (-2,2) $\bullet$;
at (-2,2.5) $\bullet$;
at (-1,2.25) $\bullet$;
at (-0.5,2.25) $\bullet$;
at (0.5,2) $\bullet$;
at (1,2.5) $\bullet$;
at (2.5,2) $\bullet$;
at (2,2.5) $\bullet$;

at (-1.5,2.25) $+$;
at (0,2.25) $+$;
at (1.5,2.25) $+$;

[xshift = 6cm]
(-1.75,2) – (-1.75,2.5) – (-1.25,2.5) – cycle;
(-0.25,2) – (0.25,2) – (0.25,2.5) – cycle;
(1.75,2) – (1.25,2.5) – cycle;

at (-1.75,2) $\bullet$;
at (-1.75,2.5) $\bullet$;
at (-1.25,2.5) $\bullet$;
at (-0.25,2) $\bullet$;
at ( 0.25,2) $\bullet$;
at ( 0.25,2.5) $\bullet$;
at ( 1.75,2) $\bullet$;
at ( 1.25,2.5) $\bullet$;

at (-0.75,2.25) $+$;
at (0.75,2.25) $+$;

[xshift = 12cm]
(-1.75,2) – (-1.75,2.5) – (-1.25,2) – cycle;
(-0.25,2.5) – (0.25,2) – (0.25,2.5) – cycle;
(1.25,2) – (1.75,2.5) – cycle;

at (-1.75,2) $\bullet$;
at (-1.75,2.5) $\bullet$;
at (-1.25,2) $\bullet$;
at (-0.25,2.5) $\bullet$;
at ( 0.25,2) $\bullet$;
at ( 0.25,2.5) $\bullet$;
at ( 1.25,2) $\bullet$;
at ( 1.75,2.5) $\bullet$;

at (-0.75,2.25) $+$;
at (0.75,2.25) $+$;

After taking into account the $\Dih_8$ symmetry, there are just three ways to decorate $Q$ with coefficients, yielding three Landau–Ginzburg potentials with distinct periods. As described in Remark <ref>, according to Conjecture <ref> these are conjecturally in one-to-one correspondence with three possible deformations of $X_P$. None of these three periods appear in the classification of quantum periods for smooth Fano 3-folds [8], which is unsurprising since $X_P$ has isolated terminal quotient singularities (which cannot be deformed away by any smoothing). However it suggests that $X_P$ may lie at the intersection of three different components of Fano 3-folds in its Hilbert scheme.

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