2.1.1 The $\mathsf {A}_2$, $\mathsf {C}_2$ and $\mathsf {G}_2$ Looijenga pairs

We will consider $(Y,D)$ to be one of the following three cases.Footnote ³

1. (A₂) Let $k=5$ and $(-D_i^{2} : i=1,\ldots ,5)=(1,1,1,1,1)$.

2. (C₂) Let $k=6$ and $(-D_i^{2} : i=1,\ldots ,6)=(2,1,2,1,2,1)$.

3. (G₂) Let $k=8$ and $(-D_i^{2} : i=1,\ldots ,8)=(3,1,3,1,3,1,3,1)$.

For convenience, in the $\mathsf {C}_2$ case we relabel the boundary divisors $D_1,D_{12},D_2,\ldots ,D_{31}$, so that $D_i^{2}=-2$ and $D_{ij}^{2}=-1$, and similarly in the $\mathsf {G}_2$ case.

2.1.2 Toric models

Any Looijenga pair can be obtained, possibly after a sequence of toric blow-ups, as the blow-up of a toric surface $(\bar {Y},\bar {D})$ at points along the toric boundary divisor $\bar {D}$, such that $D\subset Y$ is the strict transform of $\bar {D}\subset \bar {Y}$ (cf. [Reference Gross, Hacking and KeelGHK15a, Proposition 1.3]). We can realise special casesFootnote ⁴ of the three examples above by considering blow-ups

\[ \pi_{\mathsf{A}_2}\colon Y_{\mathsf{A}_2}\to\mathbb{P}^{2}, \quad \pi_{\mathsf{C}_2}\colon Y_{\mathsf{C}_2}\to\mathbb{P}^{2}, \quad \pi_{\mathsf{G}_2}\colon Y_{\mathsf{G}_2}\to\mathbb{P}^{1}\times\mathbb{P}^{1} \]

at the configurations of points described below, and shown in Figure 3.

Figure 3.

The configurations of points blown up to obtain $Y_{\mathsf {A}_2}$, $Y_{\mathsf {C}_2}$ and $Y_{\mathsf {G}_2}$.

Let $\operatorname {exc}(p)$ be the exceptional divisor above a point $p$, let $\pi ^{-1}(\bar {C})$ be the strict transform of a curve $\bar C$ under a birational map $\pi$, let $L_{p,q}$ be the line in $\mathbb {P}^{2}$ which passes through two points $p,q$, and let $M_{p,q,r}$ be the curve of bidegree $(1,1)$ in $\mathbb {P}^{1}\times \mathbb {P}^{1}$ which passes through three points $p,q,r$. Then the blow-ups we consider are given by the following.

1. (A₂) Let $\bar {D}_2+\bar {D}_4+\bar {D}_5$ be the toric boundary components of $\mathbb {P}^{2}$. We obtain $Y_{\mathsf {A}_2}$ by blowing up the two intersection points $d_1=\bar {D}_5\cap \bar {D}_2$, $d_3=\bar {D}_2\cap \bar {D}_4$ and two general points $e_4\in \bar {D}_4$, $e_5\in \bar {D}_5$.

   The anticanonical cycle $D\subset Y_{\mathsf {A}_2}$ is given by $D_1 = \operatorname {exc}(d_1)$, $D_2 = \pi ^{-1}(\bar {D}_2)$, $D_3 = \operatorname {exc}(d_3)$, $D_4 = \pi ^{-1}(\bar {D}_4)$ and $D_5 = \pi ^{-1}(\bar {D}_5)$. Moreover, we note that $Y_{\mathsf {A}_2}$ contains five interior $(-1)$-curves $E_1 = \pi ^{-1}(L_{d_1,e_4})$, $E_2 = \pi ^{-1}(L_{e_4,e_5})$, $E_3 = \pi ^{-1}(L_{d_3,e_5})$, $E_4 = \operatorname {exc}(e_4)$ and $E_5 = \operatorname {exc}(e_5)$.

2. (C₂) Let $\bar {D}_1+\bar {D}_2+\bar {D}_3$ be the toric boundary components of $\mathbb {P}^{2}$. We obtain $Y_{\mathsf {C}_2}$ by blowing up the three intersection points $d_{ij}=\bar {D}_i\cap \bar {D}_j$ and three points $e_i\in \bar {D}_i\cap \bar {F}$, where $\bar {F}$ is a line in general position with respect to $\bar {D}$.

   Let $(i,j,k)$ vary over all $\operatorname {Dih}_6$-permutations of $(1,2,3)$. Then the anticanonical cycle $D\subset Y_{\mathsf {A}_2}$ is given by $D_i = \pi ^{-1}(\bar {D}_i)$ and $D_{ij} = \operatorname {exc}(d_{ij})$. Moreover, we note that $Y_{\mathsf {C}_2}$ contains six interior $(-1)$-curves $E_i = \operatorname {exc}(e_i)$ and $E_{ij} = \pi ^{-1}(L_{d_{ij},e_k})$, and one interior $(-2)$-curve $F=\pi ^{-1}(\bar {F})$.

3. (G₂) Let $\bar {D}_1+\bar {D}_2+\bar {D}_3+\bar {D}_4$ be the toric boundary components of $\mathbb {P}^{1}\times \mathbb {P}^{1}$. We obtain $Y_{\mathsf {G}_2}$ by blowing up the four intersection points $d_{ij}=\bar {D}_i\cap \bar {D}_j$ and four points $e_i\in \bar {D}_i\cap (\bar {F}_{13}\cup \bar {F}_{24})$, where $\bar {F}_{13}$ and $\bar {F}_{24}$ are two curves of bidegree $(1,0)$ and $(0,1)$ which are in general position with respect to $\bar {D}$.

   Let $(i,j,k,l)$ vary over all $\operatorname {Dih}_8$-permutations of $(1,2,3,4)$. Then the anticanonical cycle $D\subset Y_{\mathsf {G}_2}$ is given by $D_i = \pi ^{-1}(\bar {D}_i)$ and $D_{ij}=\operatorname {exc}(d_{ij})$. Moreover, we note that $Y_{\mathsf {G}_2}$ contains eight interior $(-1)$-curves $E_i = \operatorname {exc}(e_i)$ and $E_{ij}=\pi ^{-1}(M_{d_{ij},e_k,e_l})$, and two interior $(-2)$-curves $F_{ik} = \pi ^{-1}(\bar {F}_{ik})$.

2.1.3 The Mori cone $\operatorname {NE}(Y)$

A Looijenga pair $(Y,D)$ is said to be positive if the cycle $D$ supports an ample divisor. In particular, this implies that the Mori cone $\operatorname {NE}(Y)\subset N_1(Y)_{\mathbb{Q}}$ is a closed finite polyhedral cone.

In the $\mathsf {A}_2$, $\mathsf {C}_2$ and $\mathsf {G}_2$ cases, $(Y,D)$ is positive and the Mori cone $\operatorname {NE}(Y)$ is spanned by $10$, $13$ and $18$ extremal rays respectively, corresponding to the classes $[D_i],[D_{ij}],[E_i],[E_{ij}],[F],[F_{ik}]$ described above. A $(-1)$-curve contained in $Y\setminus D$ must intersect the boundary divisor $D$ at precisely one point, in the interior of a component $D_i$ or $D_{ij}$. In each of the three cases there is precisely one $(-1)$-curve $E_i$ which intersects $D_i$ and one $(-1)$-curve $E_{ij}$ which intersects $D_{ij}$. Figure 4 depicts the dual intersection diagrams for the curves in $Y$ belonging to the extremal rays of $\operatorname {NE}(Y)$.Footnote ⁵

Figure 4.

Dual intersection diagrams for the extremal curves on $Y_{\mathsf {A}_2}$, $Y_{\mathsf {C}_2}$ and $Y_{\mathsf {G}_2}$.

2.1.4 The intersection pairing and the Looijenga roots

We have the usual intersection pairing

\[ ({\:}\cdot{\:}) \colon N_1(Y)_\mathbb{Q}\times N^{1}(Y)_\mathbb{Q} \to \mathbb{Q} . \]

Let $\boldsymbol {D}\subset N^{1}(Y)_\mathbb {Q}$ be the sublattice $\boldsymbol {D}=\bigoplus _{i=1}^{k}\mathbb {Z} [D_i]$, spanned by the components of $D$. Then elements $\alpha$ of the subspace

\[ \boldsymbol{D}^{\perp}=\{ [C]\in N_1(Y)_\mathbb{Q} : [D]\cdot[C] = 0 \text{ for all }D\in\boldsymbol{D} \}\subset N_1(Y)_\mathbb{Q} \]

satisfying $\alpha ^{2}=-2$ are called Looijenga roots.Footnote ⁶ In the $\mathsf {A}_2$ case $\boldsymbol {D}^{\perp }=\emptyset$, in the $\mathsf {C}_2$ case $\boldsymbol {D}^{\perp }=\mathbb {Z}[F]$ forms a root system of type $\mathsf {A}_1$, and in the $\mathsf {G}_2$ case $\boldsymbol {D}^{\perp }=\mathbb {Z}\langle [F_{13}],[F_{24}]\rangle$ forms a root system of type $\mathsf {A}_2$.

2.2.1 The mirror family $\mathcal {X}$

The mirror family $\mathcal {X}$, for a Looijenga pair $(Y,D)$, is a deformation of the vertex

\[ \mathbb{V}_k = (\mathbb{A}^{2}_{\theta_1,\theta_2} \cup \mathbb{A}^{2}_{\theta_2,\theta_3} \cup \cdots \cup \mathbb{A}^{2}_{\theta_{k-1},\theta_k} \cup \mathbb{A}^{2}_{\theta_k,\theta_1} ) \subseteq \mathbb{A}^{k}_{\theta_1,\ldots,\theta_k}, \]

with $k$ equal to the number of components of $D=\bigcup _{i=1}^{k}D_i$, defined by introducing theta functions and using the machinery of scattering diagrams. If $(Y,D)$ is positive, then the construction yields an algebraic variety $\mathcal {X}$ with the following nice properties.

1. (i) $\mathcal {X}/B$ is a deformation of $\mathbb {V}_k$ over the affine toric variety:

   \[ B := \operatorname{Spec}(\mathbb{C}[\operatorname{NE}(Y)]) = \operatorname{Spec} (\mathbb{C}[ z^{C} : [C] \in \operatorname{NE}(Y) ] ). \]

2. (ii) $\mathcal {X}/B$ is a flat family of affine Gorenstein surfaces with at worst semi-log canonical singularities.

3. (iii) The action of the torus $\mathbb {T}^{D} = \boldsymbol {D}\otimes \mathbb {C}^{*}$ on $B$, given by

   \[ \lambda_i\cdot (z^{C}) = \lambda_i^{D_i\cdot C}z^{C} \quad \text{for } i=1,\ldots,k, \]
   extends uniquely to a $\mathbb {T}^{D}$-action on $\mathcal {X}$.

Our only problem with trying to use $\mathcal {X}/B$ as a key variety directly is that the total space $\mathcal {X}$ is not Gorenstein, but only relatively Gorenstein.

2.2.2 The cluster variety $X$

For that reason we consider a slightly different family, by first taking the restriction $\mathcal {X}|_{\mathbb {T}^{n}}$ to the structure torus $\mathbb {T}^{n}\subset B$ and then considering the variety $X/\mathbb {A}^{n}$, obtained by the compactification $\mathbb {T}^{n}\subset \mathbb {A}^{n}$ with respect to some natural choice of coordinates on $\mathbb {T}^{n}$:

This choice of coordinates is described in § 3.1 for the $\mathsf {C}_2$ cluster variety and in § 4.1 for the $\mathsf {G}_2$ cluster variety.

2.4.1 Equations for the mirror family $\mathcal {X}_{\mathsf {A}_2}$

The equations for $\mathcal {X}_{\mathsf {A}_2}/B_{\mathsf {A}_2}$ are worked out in [Reference Gross, Hacking and KeelGHK15a, Example 3.7]. To simplify the notation we let $A_i=z^{[D_i]}$ and $B_i=z^{[E_i]}$. The base variety $B_{\mathsf {A}_2}$ is a toric variety defined by 10 equations,

\[ A_iB_i = A_{i-2}A_{i+2} = B_{i-1}B_{i+1}, \]

and there are five relative equations,

\[ \theta_{i-1}\theta_{i+1} = A_i\theta_i + A_iB_i, \]

which define $\mathcal {X}_{\mathsf {A}_2}$ as a scheme over $B_{A_2}$. Therefore the total space $\mathcal {X}_{\mathsf {A}_2}\subset \mathbb {A}^{5}_{\theta _i}\times \mathbb {A}^{10}_{A_i,B_i}$ is an affine variety of codimension 8 defined by 15 equations. This variety is Cohen–Macaulay, but not Gorenstein.

2.4.2 The cluster variety $X_{\mathsf {A}_2}$

To recover the Grassmannian $\operatorname {Gr}(2,5)$ we restrict $\mathcal {X}$ to the locus $\mathcal {X}|_{\mathbb {T}^{5}}\subset \mathcal {X}$ over the structure torus $\mathbb {T}^{5}\subset B$. After writing all of the elements of $N_1(Y)$ in terms of the basis $[D_i]$, the equations become

\[ \theta_{i-1}\theta_{i+1} = A_i\theta_i + A_{i-2}A_{i+2} \]

which we see to be ideal, given by the $4\times 4$ maximal Pfaffians of a $5\times 5$ skew-symmetric matrix

\[ \operatorname{Pf}_4\begin{pmatrix} A_5 & \theta_1 & \theta_2 & A_3 \\ & A_2 & \theta_3 & \theta_4 \\ & & A_4 & \theta_5 \\ & & & A_1 \end{pmatrix}. \]

Taking the closure of $\mathcal {X}|_{\mathbb {T}^{5}}$ over $\mathbb {T}^{5}\subset \mathbb {A}^{5}_{A_i}$ gives the cluster variety $X_{\mathsf {A}_2}$. Indeed, we see that $X_{\mathsf {A}_2}$ is nothing other than the affine cone over the Grassmannian $\operatorname {Gr}(2,5)$ in its Plücker embedding.

2.4.3 Symmetries

$X_{\mathsf {A}_2}$ has the action of the group $\operatorname {Dih}_{10}\times \mathbb {T}^{D}$, where $\operatorname {Dih}_{10}$ permutes the indices $\{1,\ldots ,5\}$. The characters for the $\mathbb {T}^{D}$-action are given by $\chi _i(z^{C})=D_i\cdot C$, as shown in Table 1.

Table 1.

The character table for $\mathbb {T}^{D} \curvearrowright X_{\mathsf {A}_2}$.

3.1.1 The toric base $B_{\mathsf {C}_2}$

In this case $B_{\mathsf {C}_2}$ is a singular affine toric variety with 18 equations of the form $z^{X}=z^{Y}$, where $X=Y$ is a linear relation in $N_1(Y_{\mathsf {C}_2})$ for some classes $X,Y\in \operatorname {NE}(Y_{\mathsf {C}_2})$. We only write down six of these 18 equations, which will be relevant to the following calculation:

(3.1)\begin{equation} \begin{aligned} [D_i]+2[E_i]+[F] &= [D_j]+2[D_{jk}]+[D_k], \\ [D_{ij}]+[E_{ij}] &= [D_{jk}]+[D_k]+[D_{ki}]. \end{aligned} \end{equation}

3.1.2 The mirror family $\mathcal {X}_{\mathsf {C}_2}$

In this case, the mirror family $\mathcal {X}_{\mathsf {C}_2}/B_{\mathsf {C}_2}$ is defined by nine relative equations. These nine equations are determined by the six tag equations

\begin{align*} \theta_i\theta_j &= z^{D_{ij}}(\theta_{ij} + z^{E_{ij}}), \\ \theta_{ij}\theta_{jk} &= z^{D_j}(\theta_j + z^{E_j})(\theta_j + z^{E_j+F}), \end{align*}

where the monomials appearing in the right-hand side of these equations come from counting certain classes of rational curves on $Y_{\mathsf {C}_2}$. In general, the expectation that the coefficients appearing in the mirror algebra can be interpreted in terms of the enumerative geometry of $Y$ is described in [Reference Gross and SiebertGS18].

In the case above, the first tag equation is of the form $\theta _i\theta _j=\sum _{m=1}^{2}z^{[\Sigma _m]}\theta _{ij}^{-D_{ij}\cdot \Sigma _m}$, where $[\Sigma _1]=[D_{ij}]$ and $[\Sigma _2]=[D_{ij}]+[E_{ij}]$ are the two classes of an effective rational curve $\Sigma \subset Y_{\mathsf {C}_2}$ such that $\Sigma \cdot D_i=\Sigma \cdot D_j =1$, and $\Sigma \cdot D'=0$ for all other irreducible components in the boundary $D'\subset D$. Similarly, the second tag equation is $\theta _{ij}\theta _{jk}=\sum _{m=1}^{4}z^{[\Sigma _m]}\theta _j^{-D_j\cdot \Sigma _m}$, where $[\Sigma _1]=[D_j]$, $[\Sigma _2]=[D_j]+[E_j]$, $[\Sigma _3]=[D_j]+[E_j]+[F]$ and $[\Sigma _4]=[D_j]+2[E_j]+[F]$ are the four classes of an effective rational curve $\Sigma \subset Y_{\mathsf {C}_2}$ such that $\Sigma \cdot D_{ij}=\Sigma \cdot D_{jk} =1$, and $\Sigma \cdot D'=0$ for all other irreducible components in the boundary $D'\subset D$. The remaining equations, which are of the form $\theta _i\theta _{jk}=\cdots$, can be obtained either by a similar calculation (i.e. finding the relevant classes of rational curves passing between $D_i$ and $D_{jk}$) or by simply calculating the relation which is implied birationally from the tag equations.Footnote ⁷

3.1.3 The cluster variety $X_{\mathsf {C}_2}$

We write $N_1(Y_{\mathsf {C}_2})=\boldsymbol {D} \oplus \mathbb {Z}[\delta ]$, according to the $\mathbb {Q}$-basis $[D_1],\ldots ,[D_{31}],\delta$, where

\[ \delta=\tfrac 12([D_1]+[D_2]+[D_3]-[F]) = \pi_{\mathsf{C}_2}^{*}H - D_{12} - D_{23} - D_{31}, \]

in which $\pi _{\mathsf {C}_2}$ is as in § 2.1.2 and $H$ is the hyperplane class on $\mathbb {P}^{2}$. The reason for this choice of basis is that, by the equations for $B_{\mathsf {C}_2}$ (3.1), we have:

\begin{align*} [E_i] &= [D_{jk}] - [D_i] + \delta, \\ [E_{ij}] &= [D_{jk}] + [D_k] + [D_{ki}] - [D_{ij}], \end{align*}

which allows us to eliminate the coefficients $z^{E_i}$, $z^{E_{ij}}$ in a $\operatorname {Dih}_6$-invariant way. After doing this, and setting $A_i:=z^{D_i}$, $A_{ij}:=z^{D_{ij}}$ and $\lambda :=z^{\delta }(1+z^{F})$ to simplify the notation, we arrive at our desired equations (1.1), albeit defined over the torus $\mathbb {T}^{7}_{A_1,\ldots ,A_{31},\lambda }$. Since all the exponents that appear in the equations are positive and integral, the equations defining ${\mathcal {X}_{\mathsf {C}_2}}|_{\mathbb {T}^{7}}$ immediately extend to obtain the cluster variety $X_{\mathsf {C}_2}/\mathbb {A}^{7}$.

3.1.4 Symmetries

The cluster variety $X_{\mathsf {C}_2}$ has the action of $\operatorname {Dih}_6\times \mathbb {T}^{6}$, where $\operatorname {Dih}_6$ permutes the indices $\{1,2,3\}$. The torus action $\mathbb {T}^{6}=\mathbb {T}^{D} \curvearrowright X_{\mathsf {C}_2}$ is determined by the six characters $\chi _i$, $\chi _{ij}$, as defined in § 2.2.1. Since $\delta \cdot [D_i]=-1$ for all $i$ and $\delta \cdot [D_{ij}]=1$ for all $i,j$, the character table for $\mathbb {T}^{D} \curvearrowright X_{\mathsf {C}_2}$ is given by Table 2.

Table 2.

The character table for $\mathbb {T}^{D}\curvearrowright X_{\mathsf {C}_2}$.

3.2.1 Crazy Pfaffian format

The equations can be written in a $6\times 6$ crazy Pfaffian format:

\[\operatorname{Pf}_4\begin{pmatrix} A_3A_{31} & \theta_1 & \theta_{12} & A_2\theta_2 + \lambda A_{31} & A_2A_{23}A_3 + \lambda \theta_1 \\ & A_{12} & \theta_2 & \theta_{23} & A_3\theta_3 + \lambda A_{12} \\ & & A_{23} & \theta_3 & \theta_{31} \\ & & & A_{31}A_1 & A_1\theta_1 \\ & & & & A_1A_{12}A_2 \end{pmatrix} \]

where the variables $A_1,A_2,A_3$ are floating factors. In other words, after expanding these Pfaffians some of the relations are found to be divisible by $A_1$, $A_2$ or $A_3$. In crazy Pfaffian format we allow ourselves to divide by these floating factors wherever possible. In particular, if we set $A_1=A_2=A_3=1$ and $\lambda =0$ then we recover the codimension 4 extrasymmetric format which first appeared in Dicks’ thesis [Reference DicksDic88], and now in many other places.

3.2.2 Triple unprojection structure

Eliminating $\theta _{12},\theta _{23},\theta _{31}$ from $\mathcal {A}_{\mathsf {C}_2}$ gives a Gorenstein projection $X_{\mathsf {C}_2}\dashrightarrow Z$ where $Z$ is the hypersurface:

\[ \theta_1\theta_2\theta_3 = A_{31}A_1A_{12}\theta_1 + A_{12}A_2A_{23}\theta_2 + A_{23}A_3A_{31}\theta_3 + \lambda A_{12}A_{23}A_{31}. \]

This variety $Z$ is a family of affine cubic surfaces over $\mathbb {A}^{7}_{A_i,A_{ij},\lambda }$ whose general member has three lines at infinity meeting at three $\tfrac 12(1,1)$ singularities, obtained by contracting the three $(-2)$-curves in the boundary divisor of $Y_{\mathsf {C}_2}$. The variable $\theta _{ij}$ can be recovered from $Z$ as a serial Gorenstein type I unprojection of the divisor $\Pi _{ij}=V(A_{ij},\theta _k)$. This gives rise to the following description as an interlaced $4\times 4$-Pfaffian format for the three matrices:

(3.2)\begin{equation} \operatorname{Pf}_4 \begin{pmatrix} A_kA_{ki} & \theta_i & \theta_{ij} & A_j\theta_j + \lambda A_{ki} \\ & A_{ij} & \theta_j & \theta_{jk} \\ & & A_{jk} & \theta_k \\ & & & A_{ki}A_i \end{pmatrix} \end{equation}

where two Pfaffian equations in each matrix are repeated in one of the other two matrices. From any one of these three matrices, $X_{\mathsf {C}_2}$ is given by unprojecting the Tom$_{3}$ ideal $(A_{ki},\theta _{ij},\theta _{j},\theta _{jk})$ with unprojection variable $\theta _{ki}$.

3.2.3 Papadakis and Neves’ $\binom {n}{2}$ Pfaffian format

Papadakis and Neves [Reference Papadakis and NevesPN09] define the $\binom n2$ Pfaffian format as a series of parallel type I unprojections from a certain codimension 1 ring. When $n=3$ (and in different notation from [Reference Papadakis and NevesPN09]) it is given by the parallel unprojection of the three ideals $(u_1,v_1)$, $(u_2,v_2)$, $(u_3,v_3)$ contained in the hypersurface:

\[ Cu_1u_2u_3 - D_1v_1u_2u_3 \!-\! D_2u_1v_2u_3 - D_3u_1u_2v_3 \!+\! E_1u_1v_2v_3 + E_2v_1u_2v_3 \!+\! E_3v_1v_2u_3 - Fv_1v_2v_3 = 0. \]

The result is a Gorenstein ring in codimension 4 with $9\times 16$ equations and syzygies. For $(i,j,k)$ any $\operatorname {Dih}_6$-permutation of $(1,2,3)$, the nine equations are

(3)\begin{align*} w_iu_i &= D_iu_ju_k - E_ju_jv_k - E_ku_kv_j + Fv_jv_k, \\ w_iv_i &= Cu_ju_k - D_ju_kv_j - D_ku_jv_k + E_iv_jv_k, \\ w_jw_k &= (D_jD_k - CE_i)u_i^{2} + (CF + D_iE_i - D_jE_j - D_kE_k)u_iv_i + (E_jE_k - D_iF)v_i^{2}, \end{align*}

and (as can be seen from the hypersurface model $Z$ of § 3.2.2) if we set

\[ (u_i,\; v_i,\; w_i;\; C,\; D_i,\; E_i,\; F)\mapsto(\theta_i,\; A_{jk},\; \theta_{jk};\; 1,\; 0,\; -A_i,\; \lambda) \]

then we recover $X_{\mathsf {C}_2}$. The $\binom 32$ Pfaffians ring has symmetry groupFootnote ⁸ $\operatorname {BDih}_6\times \mathbb {T}^{7}$ which is slightly larger than $\operatorname {Dih}_6\times \mathbb {T}^{6}$, the symmetry group of $X_{\mathsf {C}_2}$.

4.1.1 The toric base

The base variety $B_{\mathsf {G}_2}$ is a singular affine toric variety with 40 equations of the form $z^{X}=z^{Y}$, where $X=Y$ is a linear relation for some classes $X,Y\in \operatorname {NE}(Y_{\mathsf {G}_2})$. We only write down eight of the 40 equations, which will be relevant to our calculations:

(4.1)\begin{equation} \begin{aligned} [D_i]+3[E_i]+2[F_{ik}]+[F_{jl}] &= [D_j]+3[D_{jk}]+2[D_k]+3[D_{kl}]+[D_l], \\ [D_{ij}]+[E_{ij}] &= [D_{jk}]+[D_k]+2[D_{kl}]+[D_l]+[D_{li}]. \end{aligned} \end{equation}

4.1.2 The mirror family

In this case $\mathcal {X}_{\mathsf {G}_2}/B_{\mathsf {G}_2}$ is given by 20 relative equations, determined by eight tag equations:

\begin{align*} \theta_i\theta_j &= z^{D_{ij}}(\theta_{ij} + z^{E_{ij}}), \\ \theta_{ij}\theta_{jk} &= z^{D_j}(\theta_j + z^{E_j}) (\theta_j + z^{E_j+F_{jl}})(\theta_j + z^{E_j+F_{jl}+F_{ik}}), \end{align*}

where the monomials appearing in the equations are counting certain classes of rational curves on $Y_{\mathsf {G}_2}$. More precisely, the first tag equation is derived from $\theta _i\theta _j=\sum _{m=1}^{2}z^{[\Sigma _m]}\theta _{ij}^{-D_{ij}\cdot \Sigma _m}$, where $[\Sigma _1]=[D_{ij}]$ and $[\Sigma _2]=[D_{ij}]+[E_{ij}]$ are the two classes of an effective rational curve $\Sigma \subset Y_{\mathsf {G}_2}$ such that $\Sigma \cdot D_i=\Sigma \cdot D_j =1$, and $\Sigma \cdot D'=0$ for all other irreducible components in the boundary $D'\subset D$. The second tag equation comes from $\theta _{ij}\theta _{jk}=\sum _{m}z^{[\Sigma _m]}\theta _{j}^{-D_j\cdot \Sigma _m}$, where $[\Sigma _m]$ runs over the classes of an effective rational curve $\Sigma \subset Y_{\mathsf {G}_2}$ such that $\Sigma \cdot D_{ij}=\Sigma \cdot D_{jk} =1$, and $\Sigma \cdot D'=0$ for all other irreducible components in the boundary $D'\subset D$. As before, the other 14 equations defining $\mathcal {X}_{\mathsf {G}_2}$ can be found from the tag equations by working birationally.

As explained in Remark 3.1, to give a rigorous proof that this description holds we could use the machinery of scattering diagrams. However, we skip this since we are only interested in the existence of $X_{\mathsf {G}_2}$ in order for us to use it as a key variety.

4.1.3 The cluster variety

We write $N_1(Y)=\boldsymbol {D}\oplus \mathbb {Z}\langle \delta _{13},\delta _{24}\rangle$, according to the $\mathbb {Q}$-basis $[D_1],\ldots ,[D_{41}]$, $\delta _{13}$, $\delta _{24}$, where

\[ \delta_{ik} = \tfrac 13(2[D_i]+[D_j]+2[D_k]+[D_l]-2[F_{ik}]-[F_{jl}]). \]

The $\delta _{ik}$ were chosen so that

\begin{align*} [E_i] &= [D_{jk}]+[D_{kl}]-[D_i]+\delta_{ik}, \\ [E_{ij}] &= [D_{jk}]+[D_k]+2[D_{kl}]+[D_l]+[D_{li}]-[D_{ij}], \end{align*}

and therefore we can eliminate all of the coefficients $z^{E_i}$, $z^{E_{ij}}$ in a $\operatorname {Dih}_8$-equivariant way. Writing $A_i:=z^{D_i}$, $A_{ij}:=z^{D_{ij}}$, $\lambda _{ik}:=z^{\delta _{ik}}(1 + z^{F_{ik}} + z^{F_{ik}+F_{jl}})$, and noting that

\[ 2\delta_{ik} = \delta_{jl}+[D_i]+[D_k]-[F_{ik}], \]

we recover our desired equations (1.2). Since all powers are positive and integral we can easily extend all these equations to get an irreducible affine Gorenstein variety $X_{\mathsf {G}_2}/\mathbb {A}^{10}$.

4.1.4 Symmetries

The cluster variety $X_{\mathsf {G}_2}$ has the action of $\operatorname {Dih}_8\times \mathbb {T}^{8}$, where $\operatorname {Dih}_8$ permutes the indices $\{1,2,3,4\}$. By calculating $D_i\cdot \delta _{ik}=1$ etc., we get the character table for the torus action $\mathbb {T}^{8}\curvearrowright X_{\mathsf {G}_2}$, as shown in Table 3.

Table 3.

The character table for $\mathbb {T}^{D}\curvearrowright X_{\mathsf {G}_2}$.

4.2.1 Quadruple unprojection structure

Eliminating $\theta _{12}$, $\theta _{23}$, $\theta _{34}$, $\theta _{41}$ from $\mathcal {A}_{\mathsf {G}_2}$ gives a Gorenstein projection $X_{\mathsf {G}_2}\dashrightarrow Z$ where $Z$ is the following complete intersection of codimension2:

\begin{gather*} \theta_1\theta_3 = A_{12}A_2A_{23}\theta_2 + \lambda_{24}A_{12}A_{23}A_{34}A_{41} + A_{34}A_4A_{41}\theta_4, \\ \theta_2\theta_4 = A_{41}A_1A_{12}\theta_1 + \lambda_{13}A_{12}A_{23}A_{34}A_{41} + A_{23}A_3A_{34}\theta_3. \end{gather*}

This variety $Z$ is a family of affine surfaces over $\mathbb {A}^{10}_{A_i,A_{ij},\lambda _{ik}}$ whose general member has a compactification to a singular Del Pezzo surface with four lines at infinity meeting at four $\tfrac 13(1,1)$ singularities; see [Reference Corti and HeubergerCH17, § 2.2.1]. These four $\tfrac 13(1,1)$ singularities are obtained by contracting the four $(-3)$-curves in $Y_{\mathsf {G}_2}$. Each variable $\theta _{ij}$ can be recovered from $Z$ as a serial Gorenstein type I unprojection of the divisor $D_{ij}=V(A_{ij},\theta _k,\theta _l)$, giving the following codimension 3 Pfaffian format $\operatorname {Pf}_4(M_{ij})=0$, where $M_{ij}$ is the matrix

(4.2)\begin{equation} M_{ij} = \begin{pmatrix} A_{jk}A_kA_{kl} & \theta_l & \theta_i & A_{ij} \\ & A_{li}(A_i\theta_i+\lambda_{ik}A_{jk}A_{kl}) & \theta_{ij} & \theta_j \\ & & A_{jk}(A_j\theta_j+\lambda_{jl}A_{kl}A_{li}) & \theta_k \\ & & & A_{kl}A_lA_{li} \end{pmatrix}. \end{equation}

We note that $M_{ij}$ contains three unprojection divisors:

1. (i) the Tom$_5$ ideal $(\theta _l,\theta _i,\theta _{ij},A_{jk})$ for the unprojection variable $\theta _{jk}$;

2. (ii) the Jer$_{24}$ ideal $(\theta _i,\theta _{ij},\theta _j,A_{kl})$ for the unprojection variable $\theta _{kl}$;

3. (iii) the Tom$_1$ ideal $(\theta _{ij},\theta _j,\theta _k,A_{li})$ for the unprojection variable $\theta _{li}$.

Taken all together, these variables give an unprojection cascade, partly shown in Figure 2.

4.2.2 The $\mathsf {G}_2^{(5)}$ and $\mathsf {G}_2^{(4)}$ subformats

It is clear from equations (1.2) that if $\phi ^{*}(A_{ij})=1$ for some regular pullback $\phi$ from $X_{\mathsf {G}_2}$, then the variable $\theta _{ij}$ becomes redundant.

4.2.3 $\mathsf {G}_2^{(5)}$ format

This is a triple Jerry format. In terms of the matrices $M_{ij}$ defined above, the 14 equations are:

\begin{gather*} \operatorname{Pf}_4(M_{23}|_{A_{12}=1})=0,\quad \operatorname{Pf}_4(M_{34}|_{A_{12}=1})=0,\quad \operatorname{Pf}_4(M_{41}|_{A_{12}=1})=0, \\ \theta_{23}\theta_{34} = (\text{long equation}),\quad \theta_{34}\theta_{41} = (\text{long equation}), \quad \theta_{41}\theta_{12} = (\text{long equation}). \end{gather*}

If we wish to keep the variable $\theta _{12}$ with the equation $\theta _{12}= \theta _1\theta _2 - A_{23}A_3A_{34}^{2}A_4A_{41}$, then we call this $\mathsf {G}_2^{(5)}+\operatorname {CI}^{(1)}$ format.

4.2.4 $\mathsf {G}_2^{(4)}$ format

This is a double Jerry format (cf. [Reference Brown, Kerber and ReidBKR12, § 9]). The nine equations are:

\[ \operatorname{Pf}_4(M_{23}|_{A_{12}=A_{34}=1})=0, \quad \operatorname{Pf}_4(M_{41}|_{A_{12}=A_{34}=1})=0, \quad \theta_{23}\theta_{41}=(\text{long equation}). \]

If we wish to keep the variables $\theta _{12},\theta _{34}$ and their tag equations, then we call this $\mathsf {G}_2^{(4)}+\operatorname {CI}^{(2)}$ format.

4.6.1 Some further subformats

Let $\widehat Y \subset \mathbb {A}^{10}$ be the four-dimensional affine cone over a quasismooth weighted projective 3-fold $Y$. We refine cases (ii) and (iii) of Proposition 4.3.

Proof. We have that $d(A_{12})=d(A_{34})=0$, so we assume that $\phi ^{*}(A_{12})=\phi ^{*}(A_{34})=1$. By equation ($\ddagger$), if $d(A_1)<0$ or $d(A_3)<0$ then we must have $d(\lambda _{13})=0$. Similarly, if $d(A_2)<0$ or $d(A_4)<0$ then we must have $d(\lambda _{24})=0$.

Suppose we are in case (ii). Then equations ($\dagger$), combined with the above, imply that at least one of $d(A_2)<0$ or $d(A_4)<0$ is negative, and $d(A_1),d(A_3)\geq 0$. Say $d(A_2)<0$ and let $\phi \colon \mathbb {A}^{14}\to \mathbb {A}^{18}$ be defined by $\phi ^{*}(A_2)=0$, $\phi ^{*}(\lambda _{24})=\phi ^{*}(A_{12})=\phi ^{*}(A_{34})=1$, and $\phi ^{*}(z)=z$ otherwise. Since $d(\lambda _{24})=0$, the coordinate changes

\[ A_{23}' = A_4\theta_4 + A_{23}, \quad \theta_2' = \theta_2 + \lambda_{13}A_4A_{41} + A_3A_4\theta_3 \]

are homogeneous, and the ideal defining $\phi ^{*}X_{\mathsf {G}_2}$ is in rolling factors format:

\[ \bigwedge^{2}\begin{pmatrix} \theta_{23} & \theta_3 & \theta_4 & A_{41} \\ \theta_2' & A_{23}' & \theta_{41} & \theta_1 \end{pmatrix} =0, \quad \quad \begin{array}{r@{\,}ll} \theta_{23}\theta_4 &= A_3\theta_3^{2} + \lambda_{13} A_{41}\theta_3 + A_1A_{41}^{2}, \\ \theta_{23}\theta_{41} &= A_3A_{23}'\theta_3 + \lambda_{13} A_{41}A_{23}' + A_1A_{41}\theta_1, \\ \theta_2'\theta_{41} &= A_3A_{23}'^{2} + \lambda_{13} \theta_1A_{23}' + A_1\theta_1^{2}. \\\end{array}\]

If we are in case (iii), then equations ($\dagger$) reduce to $d(\lambda _{13})=d(A_1)+d(A_3)$ and $d(\lambda _{24})=d(A_2)+d(A_4)$. Thus $d(A_1)=-d(A_3)$ and $d(A_2)=-d(A_4)$. So either two consecutive $A_i$ have negative degree, or $d(A_i)=0$ for all $i$.

For the former case, suppose $\phi ^{*}(A_2)=\phi ^{*}(A_3)=0$, $\phi ^{*}(\lambda _{13})=\phi ^{*}(\lambda _{24})=\phi ^{*}(A_{12})=\phi ^{*}(A_{34})=1$ and $\phi ^{*}(z)=z$ otherwise. Then $\phi ^{*}(X_{\mathsf {G}_2})\subset \mathbb {A}^{12}$ is in $\mathbb {P}^{1}\times \mathbb {P}^{1}\times \mathbb {P}^{1}$ format, defined by the $2\times 2$ minors of the following cube after the displayed coordinate changes:

In the latter case, $\phi ^{*}(A_i)=\phi ^{*}(\lambda _{13})=\phi ^{*}(\lambda _{24})=\phi ^{*}(A_{12})=\phi ^{*}(A_{34})=1$ for all $A_i$, and $\phi ^{*}(z)=z$ otherwise. Then $\phi ^{*}(X_{\mathsf {G}_2})\subset \mathbb {A}^{12}$ is in $\mathbb {P}^{1}\times \mathbb {P}^{1}\times \mathbb {P}^{1}$ format, defined by the $2\times 2$ minors of the cube after the displayed coordinate changes:

where $\epsilon$ is a primitive eighth root of unity.