The 3-dimensional Lyness map and a self-mirror log Calabi–Yau 3-fold

Abstract

The 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi–Yau surface, given by the complement of an anticanonical pentagon of $(-1)$-curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-${\mathbb {Q}}$-factorial) affine Fano 3-fold of type $V_{12}$, and we show that it is a self-mirror log Calabi–Yau 3-fold.

1 Introduction

1.1 The Lyness map

Definition 1.1

The d-dimensional Lyness map $\sigma _d\in {{\,\textrm{Bir}\,}}({\mathbb {C}}^d)$ is the birational map

$$\begin{aligned} \sigma _d\left( x_1,x_2,\ldots ,x_{d-1},x_d\right) =\left( x_2,x_3,\ldots ,x_d,\frac{1+x_2+\cdots +x_d}{x_1}\right) . \end{aligned}$$

By iterating $\sigma _d^{\pm 1}$, we can consider an associated sequence of rational functions $(x_i: i \in {\mathbb {Z}})$, where $x_i\in {\mathbb {C}}(x_1,\ldots ,x_d)$ is defined inductively for $i>d$ and $i<1$ by using the d-dimensional Lyness recurrence relation

Low dimensional behaviour In dimensions $d\le 3$ the recurrence relation (LR$_d$) is very well behaved, with two very nice and surprising properties: it is periodic, and the sequence of rational functions $(x_i:i\in {\mathbb {Z}})$ possesses a Laurent phenomenon. In other words, this sequence is actually a sequence of Laurent polynomials $x_i\in {\mathbb {C}}[x_1^{\pm 1},\ldots ,x_d^{\pm 1}]$.

Higher dimensional behaviour Unfortunately, for all $d\ge 4$ the recurrence relation (LR$_d$) is much more badly behaved, something that is also reflected in the dynamical behaviour of $\sigma _d$. It is no longer periodic; nor does it have a Laurent phenomenon. Because of this there seems to be no straightforward way of generalising the very attractive log Calabi–Yau varieties (or their scattering diagrams) described in this paper when $d=2$ or 3.

1.2 A tale of two log Calabi–Yau varieties

The two recurrences (LR$_2$) and (LR$_3$) can be used to construct two affine log Calabi–Yau varieties. The first is the famous affine del Pezzo surface of degree 5; the second is an affine Fano 3-fold of type $V_{12}$.¹ As we will recall in Sect. 2, affine log Calabi–Yau varieties (satisfying some suitable technical hypotheses) are expected to enjoy some remarkable properties coming from mirror symmetry. In particular, there is a conjectural involution on the set of such varieties which, for a given log Calabi–Yau variety U, associates a mirror $U^\star$. According to a conjecture of Gross, Hacking & Keel (cf. Conjecture 2.5), one feature of the relationship between U and $U^\star$ is that there is expected to be a special additive basis of the coordinate ring of $U^\star$, called the basis of theta functions, whose elements correspond (roughly speaking) to boundary divisors in compactifications of U, and vice-versa.

1.2.1 Dimension 2 and the del Pezzo surface ${{\,\textrm{dP}\,}}_5$

In dimension 2 the recurrence relation (LR$_2$) generates a 5-periodic sequence $(x_i:i\in {\mathbb {Z}}/5{\mathbb {Z}})$. In terms of $x_1$ and $x_2$, the solution is given by

$$\begin{aligned} x_3 = \frac{x_2+1}{x_1}, \quad x_4 = \frac{x_1 + x_2 + 1}{x_1x_2} \quad \text {and} \quad x_5 = \frac{x_1+1}{x_2}. \end{aligned}$$

These are well-known as the five cluster variables appearing in the simplest nontrivial cluster algebra: the cluster algebra of type $A_2$. As we recall in Sect. 3, the associated cluster variety is an affine surface $U\subset {\mathbb {A}}^5$ whose coordinate ring is generated by $x_1,\ldots ,x_5$ and can be realised as the interior of the log Calabi–Yau pair (X, D), where $X\subset {\mathbb {P}}^5$ is a del Pezzo surface of degree 5 and $D=\sum _{i=1}^5D_i$ is an anticanonical pentagon of $(-1)$-curves. The variety U is known to be self-mirror [17, Example 5.9], i.e. the mirror $U^\star$ is isomorphic to U. In terms of the conjecture mentioned above, this mirror correspondence places the five boundary components $D_i$ in a natural one-to-one correspondence with the five cluster variables $x_i$.

The mirror of a complex projective Fano variety is expected to be a Landau–Ginzburg model (V, w), i.e. a quasiprojective algebraic variety V over ${\mathbb {C}}$ equipped with a holomorphic surjective function $w:V\rightarrow {\mathbb {C}}$, called a Landau–Ginzburg potential. Therefore, for our log Calabi–Yau variety U, adding the boundary divisor D to obtain the del Pezzo surface X corresponds to decorating the mirror $U^\star \cong U$ with a holomorphic function. In this case, summing the five cluster variables that correspond to the components of D defines a potential $w=x_1+\cdots +x_5$ for a Landau–Ginzburg model $w:U\rightarrow {\mathbb {C}}$ which is mirror to X.

Fig. 1

The correspondence between boundary divisors and theta functions on the mirror for a del Pezzo surface X of degree 5, b the Fano 3-fold $X_P$ of type $V_{12}$ and c the Fano 3-fold $X_Q$ of type $V_{16}$

1.2.2 Dimension 3 and the Fano 3-fold $V_{12}$

In dimension 3 the recurrence relation (LR$_3$) generates an 8-perodic sequence $\left( x_i:i\in {\mathbb {Z}}/8{\mathbb {Z}}\right)$. Expanding in terms of $x_1$, $x_2$, $x_3$ we have

$$\begin{aligned} x_4&= \frac{x_2+x_3+1}{x_1} \quad x_5 = \frac{(x_1 + 1)(x_3 + 1) + x_2}{x_1x_2}\\ x_6&= \frac{(x_1 + x_2 + 1)(x_2 + x_3 + 1)}{x_1x_2x_3} \\ x_7&= \frac{(x_1 + 1)(x_3 + 1) + x_2}{x_2x_3} \quad \text {and} \quad x_8 = \frac{x_1+x_2+1}{x_3}. \end{aligned}$$

These rational functions satisfy eight relations of the form $x_{i-1}x_{i+2}=1+x_i+x_{i+1}$ and we observe that $x_1,\ldots ,x_8$ satisfy two further relations: $x_1x_5=x_3x_7$ and $x_2x_6=x_4x_8$. In analogy to the dimension 2 case, we will see in Sect. 4 that the ring generated by the eight variables $x_1,\ldots ,x_8$ defines an algebraic variety $U\subset {\mathbb {A}}^8$ which is also a kind of cluster variety (in the sense of Definition 2.6(3)). This variety U is the interior of a log Calabi–Yau pair $(X_P,D_P)$, where $X_P$ is a special (non-${\mathbb {Q}}$-factorial) Fano 3-fold of type $V_{12}$ and $D_P\in |{-K_X}|$ is a boundary divisor with ten components. Eight components are isomorphic to ${\mathbb {P}}^2$, two components are isomorphic to ${\mathbb {P}}^1\times {\mathbb {P}}^1$ and they meet along their toric boundary strata according to the polytope P shown in Fig. 1b.

Like the affine del Pezzo surface before it, we will see that this 3-fold U is also self-mirror. However, in contrast to the two dimensional case, there is now a problem: we have ten boundary divisors but only eight cluster variables. Moreover the Landau–Ginzburg model $w_P:U\rightarrow {\mathbb {C}}$ defined by the potential $w_P:=x_1+\cdots +x_8$ does not have the right period sequence to be a mirror for a Fano 3-fold of type $V_{12}$. Instead the period sequence of $w_P$ suggests that it is a mirror to a Fano 3-fold of type $V_{16}$ (cf. Sect. 4.4.1).

The explanation for this difference is that the eight ${\mathbb {P}}^2$ components of $D_P$ correspond to the eight cluster variables $x_i$, but the two ${\mathbb {P}}^1\times {\mathbb {P}}^1$ components correspond to two ‘missing’ cluster variables

$$\begin{aligned} q_1&= x_1x_5 - 1 = \frac{(x_1 + 1)(x_3 + 1)}{x_2} \quad \text {and}\\ q_2&= x_2x_6 - 1 =\frac{(x_1 + x_2 + x_3 + 1)(x_2 + 1)}{x_1x_3}, \end{aligned}$$

which are invariants for the squared Lyness map $\sigma _3^2$. The new variables $q_1$ and $q_2$ appear in the derivation of a cluster algebra-like exchange graph for U (cf. Sect. 4.2.2). Adjoining $q_1$ and $q_2$ to the ring ${\mathbb {C}}[X_P]$ corresponds to a birational map (an unprojection in the languange of Miles Reid) mapping $X_P$ onto a 3-fold $X_Q$

$$\begin{aligned} \psi :X_P\subset {\mathbb {P}}^8_{(x_0:x_1:\cdots :x_8)} \dashrightarrow X_Q\subset {\mathbb {P}}^{10}_{(x_0:x_1:\cdots :x_8:q_1:q_2)} \end{aligned}$$

which blows up and then contracts the two (non-Cartier) Weil divisors in the boundary of $X_P$ corresponding to $q_1$ and $q_2$. This extends to a birational map of pairs $\psi :(X_P,D_P)\dashrightarrow (X_Q,D_Q)$, where $X_Q$ is a degenerate Fano 3-fold of type $V_{16}$ and $D_Q$ is an anticanonical boundary divisor with eight ${\mathbb {P}}^1\times {\mathbb {P}}^1$ components meeting according to the polytope Q displayed in Fig. 1c.

Finally, summing the ten cluster variables that generate the coordinate ring of $X_Q$ gives a potential $w_Q:=x_1+\cdots +x_8+q_1+q_2$ for a Landau–Ginzburg model $w_Q:U\rightarrow {\mathbb {C}}$ with the correct period sequence to be mirror to a Fano 3-fold of type $V_{12}$. Thus in this example the Landau–Ginzburg model $(U,w_P)$ is mirror to the Fano variety $X_Q$ and the Landau–Ginzburg model $(U,w_Q)$ is mirror to the Fano variety $X_P$.

1.2.3 Generalising Borisov–Batyrev duality

We note that the two polytopes P and Q appearing in Fig. 1 are combinatorially dual to one another. We will go one step further and interpret them as dual reflexive polytopes in the tropicalisation of U, a self-dual integral affine manifold with singularities, which we denote by $N_U$. This duality can then be generalised for any pair of dual reflexive polytopes in $N_U$ and is the analogue of Borisov–Batyrev duality in the ordinary toric setting.

1.3 Summary

1.3.1 Contents

In Sect. 2 we give a brief overview of mirror symmetry for log Calabi–Yau varieties, which is intended as motivational context for the rest of the paper. In particular we explain how to construct the tropicalisation $N_U$ of a log Calabi–Yau variety U which is an integral affine manifold with singularities. We explain how to construct polytopes inside this space $N_U$ and how they can be dualised to give polytopes in $N_V$, a dual integral affine manifold with singularities corresponding to the mirror log Calabi–Yau variety $V=U^\star$.

In Sect. 3 we discuss the well-known affine del Pezzo surface U of degree 5 related to the 2-dimensional Lyness recurrence (LR$_2$). In particular we describe the cluster structure on U, the tropicalisation $N_U$ and we show that $N_U$ is self-dual as an integral affine manifold with singularities. We then give some examples of mirror symmetry for pairs of reflexive polygons in $N_U$. In Sect. 4 we repeat all of this to do a similar analysis for the affine 3-fold of type $V_{12}$ related to the 3-dimensional Lyness recurrence (LR$_3$).

1.3.2 The significance of being self-mirror

Assuming the Strominger–Yau–Zaslow conjecture, a log Calabi–Yau variety U and its mirror $U^\star$ are fibred by dual Lagrangian tori. Since real 2-tori are self-dual, every log Calabi–Yau surface is diffeomorphic to its mirror, and thus the fact that the affine del Pezzo surface $U=X{{\setminus }} D$ of Sect. 1.2.1 is self-mirror is not that surprising. In fact, all maximal positive log Calabi–Yau surfaces are deformation equivalent to their mirror [22]. However, for a higher dimensional log Calabi–Yau variety to be self-mirror is unusual, and even in dimension 3 not many examples are known. Thus, from the point of view of mirror symmetry, the log Calabi–Yau 3-fold that we study in Sect. 4 is very special.

2 Mirror symmetry and log Calabi–Yau varieties

This section is intended as a motivational context for the rest of the paper and as such we refer the reader to [14, 17] for more detailed accounts.

2.1 Log Calabi–Yau varieties

Definition 2.1

A log Calabi–Yau pair is a pair (X, D) consisting of a smooth complex projective variety X and a reduced effective integral anticanonical divisor $D\in |{-K_X}|$ with simple normal crossings.² We call the interior of a log Calabi–Yau pair $U=X{\setminus } D$ a log Calabi–Yau variety.

A log Calabi–Yau variety U is naturally equipped with a nonvanishing holomorphic volume form $\Omega _U$ in the following way. The boundary divisor $D = {{\,\textrm{div}\,}}s_X$ is cut out by a section $s_X\in H^0(X,{-K_X})$ so, after restricting to U, we get the volume form $\Omega _U = \left( {s_X}|_U\right) ^{-1}\in H^0(U,K_U)$ which has simple poles along the components of D. Since $\Omega _U$ is only determined up to rescaling by a constant, we may assume that $\Omega _U$ is normalised so that $\int _\gamma \Omega _U=1$, where $\gamma$ is the class of the real n-torus which expected to be the fibre in the SYZ fibration.³ The natural framework within which we would now like to work is the category of log Calabi–Yau varieties and volume preserving birational maps, i.e. birational maps $\phi :U \dashrightarrow V$ such that $\phi ^*\Omega _V=\Omega _U$.

Example 2.2

The prototypical example of a log Calabi–Yau variety is the algebraic torus ${\mathbb {T}}_N = {\mathbb {C}}^\times \otimes _{\mathbb {Z}}N$ associated to a lattice $N\cong {\mathbb {Z}}^d$. The volume form on ${\mathbb {T}}_N$ is given by $\Omega _{{\mathbb {T}}_N} = (2\pi i)^{-d}\frac{dz_1}{z_1}\wedge \cdots \wedge \frac{dz_d}{z_d}$. There are two dual lattices associated to ${\mathbb {T}}_N$: the cocharacter lattice N, whose points correspond to divisorial valuations along toric boundary divisors, and the character lattice $M={{\,\textrm{Hom}\,}}(N,{\mathbb {Z}})$, whose points $m\in M$ correspond to monomial functions $z^m=z_1^{m_1}\ldots z_d^{m_d}$ on ${\mathbb {T}}_N={{\,\textrm{Spec}\,}}{\mathbb {C}}[M]$. These monomials form a natural additive basis for the coordinate ring of ${\mathbb {T}}_N$.

Note that the roles of the two lattices N and M are interchanged when we replace the torus ${\mathbb {T}}_N$ by its dual (or ‘mirror’) torus ${\mathbb {T}}_M$. A key idea of Gross, Hacking & Keel is that one can introduce an object that serves as a generalisation of the cocharacter lattice for an arbitrary log Calabi–Yau variety.

Definition 2.3

Given a log Calabi–Yau variety U with volume form $\Omega _U$, the set of integral tropical points of U is given by

$$\begin{aligned} N_U({\mathbb {Z}}) = \left\{ \text {divisorial valuations } \nu :{\mathbb {C}}(U){{\setminus }} \{0\} \rightarrow {\mathbb {Z}}\text { such that } \nu (\Omega _U) <0 \right\} \cup \{0\}. \end{aligned}$$

The set $N_U({\mathbb {Z}})$ is more commonly referred to as $U^{{{\,\textrm{trop}\,}}}({\mathbb {Z}})$ in the literature, but our notation is chosen to emphasise the fact that $N_U({\mathbb {Z}})$ is supposed to be a generalisation of the cocharacter lattice N when $U={\mathbb {T}}_N$. Indeed, as a set we have that $N_{{\mathbb {T}}_N}({\mathbb {Z}})=N$.

2.1.1 The conjecture of Gross, Hacking & Keel

Gross, Hacking & Keel have conjectured that an analogue for the character lattice also exists for U, which generalises the duality enjoyed by tori to pairs of mirror log Calabi–Yau varieties. Before stating their conjecture we need to introduce two technical hypotheses.

Definition 2.4

Let (X, D) be a d-dimensional log Calabi–Yau pair with interior $U=X{\setminus } D$. We call U positive if D supports an ample divisor and we say that U has maximal boundary if D contains a 0-stratum (i.e. a point at which d components of D meet transversely).

From now on we will assume that all log Calabi–Yau varieties are positive and have maximal boundary. The assumption that U is positive is useful because it implies that U is affine. The maximal boundary condition is introduced to ensure that the set of integral tropical points $N_U({\mathbb {Z}})$ is as big as possible. In Sect. 2.3.1 we will realise $N_U({\mathbb {Z}})$ as the set of integral points in a real affine manifold with singularities $N_U$. In general, the dimension of $N_U$ will be equal to the codimension of the smallest stratum of D.

Conjecture 2.5

([12, Conjecture 0.6], cf. [14, 17]) Given a positive log Calabi–Yau variety U with maximal boundary, then there exists a mirror positive log Calabi–Yau variety V with maximal boundary whose coordinate ring ${\mathbb {C}}[V]$ has an additive basis

$$\begin{aligned} {\mathcal {B}}_V = \{ \vartheta _n \in {\mathbb {C}}[V]: n \in N_U({\mathbb {Z}}) \} \end{aligned}$$

parameterised by the integral tropical points of U. The elements of this basis are called the theta functions of V and are canonically determined up to multiplication by scalars. The multiplication rule for theta functions $\vartheta _a\vartheta _b=\sum _{c\in N_U} \alpha _{abc}\vartheta _c$ has structure constants $\alpha _{abc}$ which can be obtained as certain counts of rational curves in U.

Without fixing a complexified Kähler form on U, the mirror variety V is not expected to be unique. Indeed, since changing the choice of complexified Kähler form on U corresponds to changing the complex structure on V, the variety V should appear as a fibre in a deformation family of mirrors to U. On this level, mirror symmetry is then an involution in the sense that the mirror of V is a family of log Calabi–Yau varieties deformation equivalent to U. If U and V are a pair of mirror log Calabi–Yau varieties as in the statement of the conjecture then let us write $M_U({\mathbb {Z}}):=N_V({\mathbb {Z}})$ and $M_V({\mathbb {Z}}):=N_U({\mathbb {Z}})$ (indicating that $N_V({\mathbb {Z}})$ is supposed to be a generalisation of the character lattice for U, and vice-versa).

2.2 Cluster varieties

2.2.1 Toric models

We can create new examples of log Calabi–Yau varieties from a given log Calabi–Yau pair (X, D) by considering volume preserving blowups of X.

Definition 2.6

Consider a log Calabi–Yau pair (X, D) with boundary divisor $D=\sum _{i=1}^kD_i$ and interior $U=X{\setminus } D$.

• 1. A toric blowup of (X, D) is a blowup of X along a stratum of D. A nontoric blowup of (X, D) is a blowup of X along a smooth subvariety $Z\subset X$ of codimension 2, where Z is contained in one of the boundary components $Z\subset D_i$ and meets the other components of D transversely.

• 2. We call a log Calabi–Yau pair (X, D) a toric model for U if there exists a map $\pi :(X,D) \rightarrow (T,B)$ to a toric pair (T, B) where $\pi$ is a composition of nontoric blowups.

• 3. We call U a (generalised) cluster variety if it has a toric model.

We use the terminology ‘generalised cluster variety’ since the more commonly accepted definition of a cluster variety (cf. [17, Definition 3.1]) imposes a second condition: in addition to the toric model $\pi :(X,D)\rightarrow (T,B)$, the variety U is required to have a nondegenerate holomorphic 2-form $\sigma \in H^0\left( X,\Omega ^2_X(\log D)\right)$. The existence of this 2-form $\sigma$ imposes a very strong condition on types of centre $Z\subset T$ that can be blown up by $\pi$. In particular it forces $Z=\bigcup _{i=1}^k Z_i$ to be a union of hypertori. If we let N and M be the (co)character lattices of the torus ${\mathbb {T}}=T{\setminus } B$, then a hypertorus is a subvariety of T of the form $Z_i = {\mathbb {V}}(z^{m_i} + \lambda _i)\subset B_{n_i}$, where $\lambda _i\in {\mathbb {C}}^\times$ is some coefficient (usually assumed to be chosen generically), $n_i\in N$ determines the boundary divisor $B_{n_i}\subset T$ containing $Z_i$ and $m_i\in n_i^\perp \subset M$. Thus the centres $Z_i$ are determined by pairs $(n_i,m_i)\in N\times M$. The advantage of this more restrictive situation is that there is a very simple candidate for the mirror to U, called the Fock–Goncharov dual of U: for each $i=1,\ldots ,k$ we simply swap the roles of $n_i\in N$ and $m_i\in M$ to obtain a ‘mirror’ toric model.

Nevertheless, as we will soon see, the cluster algebra-like combinatorics of seeds and mutation can be used to describe any log Calabi–Yau variety with a toric model, and do not rely on the 2-form $\sigma$. This fact has led Corti to suggest that this should be the ‘true’ definition of a cluster variety.

Remark 2.7

The name ‘generalised cluster variety’ is not ideal, since there are many other proposals for a ‘generalised cluster variety’ appearing in the literature. Of these, it is perhaps closest in spirit to the definition of a Laurent phenomenon algebra given by Lam and Pylyavskyy [23]. Translated into our language, a Laurent phenomenon algebra is essentially the coordinate ring of a d-dimensional log Calabi–Yau variety with a toric model given by blowing up d centres $Z_i\subset H_i \subset {\mathbb {A}}^d$, one in each of the coordinate hyperplanes $H_i$ of ${\mathbb {A}}^d$.

2.2.2 Mutations

A toric model $\pi :(X,D)\rightarrow (T,B)$ for U is determined by a set of centres

$$\begin{aligned} S=\{ Z_i \subset T: i = 1,\ldots ,k \} \end{aligned}$$

which comprise the locus $Z=\bigcup _{i=1}^kZ_i$ blown up by $\pi$. Note that the choice of toric model for U is not uniquely determined by the set S, since we can modify our chosen pairs (X, D) and (T, B) by compatible choices of toric blowups or blowdowns. However the set S does specify a unique torus ${\mathbb {T}}_S:=T{\setminus } B$ and a volume preserving embedding $j_S:=(\pi ^{-1})|_{{\mathbb {T}}_S} :{\mathbb {T}}_S \hookrightarrow U$. (This inclusion of the torus ${\mathbb {T}}_S$ into U is the geometric manifestation of the Laurent phenomenon for U.)

Similarly to the (strict) cluster case described above, we can represent each component of Z in the form $Z_i = Z(n_i,f_i):= {\mathbb {V}}(f_i) \subset B_{n_i}$, where $n_i\in N$ is a primitive vector⁴ and $f_i\in {\mathbb {C}}[n_i^\perp \cap M]$ is a Laurent polynomial. The difference is now that $f_i$ is not constrained to be simply binomial.

Definition 2.8

• 1. We call the set S a seed for U and the torus embedding $j_S :{\mathbb {T}}_S \hookrightarrow U$ a cluster torus chart of U.

• 2. Given a pair (n, f), which represents a centre $Z=Z(n,f)$ in the boundary of a toric compactification of a torus ${\mathbb {T}}$, we define the mutation along Z to be the birational map $\mu _{(n,f)}:{\mathbb {T}}\dashrightarrow {\mathbb {T}}$ of the form $\mu _{(n,f)}^*(z^m) = f^{-\langle n,m\rangle }z^m$.

Example 2.9

The geometry of a mutation $\mu =\mu _{(n,f)}$ is described by Gross et al. [14, §3.1]. In particular, they show that $\mu$ can be viewed as a kind of elementary transformation of ${\mathbb {P}}^1$-bundles. Indeed, by blowing up if necessary, we can arrange for T to contain the two boundary divisors $B_+:=B_n$ and $B_-:=B_{-n}$ and then consider the toric subvariety $T_0\cong \left( {\mathbb {P}}^1\times ({\mathbb {C}}^\times )^{d-1}\right) \subset T$ consisting of the big open torus ${\mathbb {T}}$ and the relative interior of $B_+$ and $B_-$. Let $Z_+=Z(n,f)$ and let $E_-:={\mathbb {V}}(f)\subset T_0$. Then they show that the extension of $\mu$ to a birational map $\mu :T_0\dashrightarrow T_0$ is resolved by blowing up the locus $Z_+\subset B_+$ and contracting the strict transform of the divisor $E_-$ to the locus $Z_-=Z({-n},f)\subset B_-$.

Changing coordinates so that $n=(1,0,\ldots ,0)$, then $\mu$ can be assumed to be of the form

$$\begin{aligned} \mu ^*(z_1,z_2,\ldots ,z_d) = \left( f(z_2,\ldots ,z_d)^{-1}z_1, z_2,\ldots ,z_d \right) . \end{aligned}$$

It is convenient to introduce $z_1' = z_1^{-1}f(z_2,\ldots ,z_d)$ and to distinguish the domain and codomain of $\mu :{\mathbb {T}}\dashrightarrow {\mathbb {T}}'$ so that ${\mathbb {T}}=({\mathbb {C}}^\times )^d_{z_1,z_2,\ldots ,z_d}$, ${\mathbb {T}}'=({\mathbb {C}}^\times )^d_{z_1',z_2,\ldots ,z_d}$ and $\mu ^*(z_1',z_2,\ldots ,z_d) = \left( z_1^{-1},z_2,\ldots ,z_d\right)$. However note that this map is volume negating, in the sense that $\mu ^*\Omega _{{\mathbb {T}}'}=-\Omega _{{\mathbb {T}}}$, so the torus ${\mathbb {T}}'$ should be considered to come equipped with the negative of its standard volume form. Now $U = Bl_{Z_+}T_0{\setminus } \left( B_+\cup B_-\right)$ is the affine variety

$$\begin{aligned} U = {\mathbb {V}}\big (z_1z_1' - f(z_2,\ldots ,z_d)\big )\subset {\mathbb {A}}^2_{z_1,z_1'} \times ({\mathbb {C}}^\times )^{d-1}_{z_2,\ldots ,z_d} \end{aligned}$$

and this is covered, up to the complement of a subset $\Sigma ={\mathbb {V}}(z_1,z_2,f)\subset U$ of codimension two in U, by the two torus charts ${\mathbb {T}},{\mathbb {T}}'\hookrightarrow U$ which are glued together by $\mu$. The locus $\Sigma \subset U$ not covered by the two torus charts is the intersection of the two divisors $\Sigma =E_+\cap E_-$, although since U is normal and affine we have that $U={{\,\textrm{Spec}\,}}H^0(U{{\setminus }} \Sigma ,{\mathcal {O}}_{U{{\setminus }} \Sigma })$, so these two charts provide enough information to recover U.

Mutating a seed Example 2.9 gives a complete description of the cluster structure for a log Calabi–Yau variety U which is determined by a seed $S=\{Z(n,f)\}$ with only one centre. In particular there are two cluster torus charts ${\mathbb {T}}_S={\mathbb {T}}$ and ${\mathbb {T}}_{S'}={\mathbb {T}}'$ where $S'$ is the seed $S=\{Z(-n,f)\}$, and the mutation $\mu :{\mathbb {T}}_S\dashrightarrow {\mathbb {T}}_{S'}$ provides the transition map between them.

In general, for a seed with several centres $S = \{ Z_i=Z(n_i,f_i): i \in 1,\ldots ,k \}$ we can consider applying a mutation $\mu _i=\mu _{(n_i,f_i)}$ for each $i\in \{1,\ldots ,k\}$. For a given i, we can define a new seed $\mu _i S$, such that the mutation becomes a map of cluster torus charts $\mu _i:{\mathbb {T}}_S \dashrightarrow {\mathbb {T}}_{\mu _iS}$. To do this, extend $\mu$ to a birational map of projective toric varieties $\mu :(T,B)\dashrightarrow (T',B')$. The mutated seed is the set of centres $\mu _iS=\{\mu _iZ_j: j \in 1,\ldots k\}$, such that

$$\begin{aligned} \mu _i Z_j = \left\{ \begin{array}{ll} Z(-n_i,f_i) &{} \quad j=i \\ C_{Z_j}T' &{} \quad j\ne i \end{array}\right. \end{aligned}$$

where $C_{Z_j}T' = \overline{\mu _i(\eta _{Z_j})}$ denotes the centre of $Z_j$ on $T'$ (the closure of the image of the generic point $\eta _{Z_j}$ under $\mu _i$). As $\mu _i$ is volume preserving, it follows that $C_{Z_j}T'$ is contained in the boundary of $T'$ and is guaranteed to have the form $\mu _iZ_j=Z(n_j',f_j')$ for some $n_j'\in N$ and $f_j'\in {\mathbb {C}}[n_j'\cap M]$.

The exchange graph of U The exchange graph for U is the k-regular graph whose set of vertices is the set of seeds for U and whose set of edges is given by mutations.

An atlas of torus charts for U We can think of the set of seeds for U as giving an atlas of cluster torus charts $j_S:{\mathbb {T}}_S\hookrightarrow U$ which can be glued together by transition maps which are the mutations. By [14, Proposition 2.4], from any initial seed S we can consider the scheme

$$\begin{aligned} U^0 = {\mathbb {T}}_S \cup \bigcup _{i=1}^k {\mathbb {T}}_{\mu _i S} \end{aligned}$$

obtained by gluing all the cluster tori which are one mutation away from S. In general there are some issues with this construction. In particular, $U^0$ may not be separated if two centres have nonempty intersection $Z_i\cap Z_j\ne \emptyset$. However, if all of the centres $Z_i$ are disjoint then the maps $j_{\mu _iS}:{\mathbb {T}}_{\mu _iS}\hookrightarrow U$ glue to give an embedding $j:U^0\hookrightarrow U$, which covers U up to a set $\Sigma =U{\setminus } U^0$ of codimension at least 2 [14, Lemma 3.5]. If U is positive (and hence affine) then we have $U={{\,\textrm{Spec}\,}}H^0(U^0,{\mathcal {O}}_{U^0})$ as in Example 2.9.

2.2.3 Examples

It is natural to wonder whether there is a more explicit combinatorial formula describing the effect of a mutation $\mu _i$ on the set of pairs $(n_j,f_j)$ that define the other centres $Z_j=Z(n_j,f_j)$ in the seed, as there is in the ordinary cluster case [14, Equation 2.3]. Unfortunately, at this level of generality the mutations are somewhat more complicated to keep track of, and the location of the centre $\mu _i Z_j$ depends crucially on the exact form of $f_i$ and $f_j$. It is natural to hope that if $Z_j\subset B_{n_j}$, then $\mu _i {Z_j}\subset \mu _i B_{n_j}$, i.e. that $Z_j$ remains in the same boundary component after the mutation. However this is not necessarily the case. We give a couple of examples to see the kind of difficulties that can occur.

Definition 2.10

Consider a seed $S=\{Z_1,\ldots ,Z_k\}$, a choice of mutation $\mu _i$ and a centre $Z_j$ with $j\ne i$. We say that $Z_j$ makes an unexpected jump if $\mu _i {Z_j}\not \subset \mu _i B_{n_j}$.

Example 2.11

We consider a toric model $\pi :(X,D)\rightarrow ({\mathbb {A}}^3,B)$ obtained by blowing up three centres which are contained in the coordinate hyperplanes of ${\mathbb {A}}^3$:

$$\begin{aligned} Z_1&= Z\big ( (1,0,0), \; 1+z_2 \big ), \quad Z_2 = Z\big ((0,0,1), \; 1+z_2\big ), \\ Z_3&= Z\big ((0,0,1), \; 1+z_1\big ). \end{aligned}$$

Consider the mutation along $Z_1$. From the geometric description given above, $\mu _1$ blows up $Z_1$ and contracts the strict transform of the divisor $E_1={\mathbb {V}}(1+z_2)\subset {\mathbb {A}}^3$ to the centre $\mu _1Z_1 = Z\left( (-1,0,0), \; 1+z_2\right)$ at infinity. Since $\mu _1^*(1+z_1^{-1}+z_2)=(1+z_1^{-1})(1+z_2)$, it is also straightforward to see that the mutation sends $Z_3$ to $\mu _1 Z_3 = Z\left( (0,0,1), \; 1+z_1^{-1}+z_2\right)$. However, since $Z_2\subset E_1$, the contraction of the strict transform of $E_1$ moves the centre of $Z_2$ out of the divisor $B_{(0,0,1)}$. We get $\mu _1 Z_2 = Z\left( (-1,0,1), \; 1+z_2\right)$ and thus $Z_2$ makes an unexpected jump. We illustrate the effect of the mutation on the centres in the following diagram, where the centres $Z_i$ have been drawn ‘tropically’.

The bad behaviour demonstrated in Example 2.11 looks like it can be fixed by introducing a generic coefficient $\lambda \in {\mathbb {C}}^\times$ and deforming the centre $Z_2$ to $Z\left( (0,0,1),\lambda +z_2\right)$. Then we will no longer have the inclusion $Z_2\subset E_1$ and the mutation $\mu _1$ will keep $Z_2$ and $Z_3$ inside the same boundary component. Unfortunately, as we show in the next example, it is not always possible to avoid unexpected jumps, even if we start from a seed where all of the centres have generic coefficients.

Example 2.12

Consider the toric model $\pi :(X,D)\rightarrow ({\mathbb {A}}^3,B)$ obtained by blowing up three centres

$$\begin{aligned} Z_1&= Z\big ((1,0,0),\, a_1 + a_2z_2 + a_3z_3\big ), \\ Z_2&= Z\big ((0,1,0),\, b_1z_1 + b_2 + b_3z_3\big ), \\ Z_3&= Z\big ((0,0,1),\, c_1z_1 + c_2z_2 + c_3\big ) \end{aligned}$$

where $a_i,b_i,c_i\in {\mathbb {C}}^\times$ are generic coefficients. This seed consists of three general lines, one in each of the three coordinate hyperplanes of ${\mathbb {A}}^3$. We can apply the mutation $\mu _1$ along $Z_1$ to obtain a new seed:

$$\begin{aligned} Z_1' := \mu _1Z_1&= Z\big ((-1,0,0),\, a_1 + a_2z_2 + a_3z_3 \big ), \\ Z_2' := \mu _1Z_2&= Z\big ((0,1,0),\, b_1(a_1 + a_3z_3) + z_1^{-1} (b_2 + b_3z_3)\big ), \\ Z_3' := \mu _1Z_3&= Z\big ((0,0,1),\, c_1(a_1 + a_2z_2) + z_1^{-1} (c_2z_2 + c_3)\big ). \end{aligned}$$

We note that $Z_1'$ and $Z_2'$ intersect in a point $p={\mathbb {V}}(z_1^{-1},z_2,a_1+a_3z_3)$ belonging to the intersection of their boundary components. If we now apply the mutation $\mu _2$ along $Z_2'$ then we compute that the total transform of the centre $Z_1'$ is reducible and given by

$$\begin{aligned} Z\big ((-1,0,0),\, z_2^{-1} + a_2b_1 \big ) \cup Z\big ((-1,0,0), \, a_1+a_3z_3 \big ) \end{aligned}$$

where the first component is the centre $\mu _2Z_1'$ and the second component is the exceptional line $p'$ over the point p (represented by the dashed vertical line in the third diagram below). Therefore mutating the seed at $Z_2'$ gives

$$\begin{aligned} Z_1'' := \mu _2Z_1'&= Z\big ((-1,0,0),\, z_2^{-1} + a_2b_1 \big ), \\ Z_2'' := \mu _2Z_2'&= Z\big ((0,-1,0),\, b_1(a_1 + a_3z_3) + z_1^{-1} (b_2 + b_3z_3)\big ), \\ Z_3'' := \mu _2Z_3'&= Z\big ((0,0,1),\, (a_2c_1 + c_2z_1^{-1}) (a_1b_1 + b_2z_1^{-1}) + z_2^{-1}(a_1c_1 + c_3z_1^{-1}) \big ). \end{aligned}$$

We draw the sequence of mutations with the centres represented tropically, as before.

We now see that $Z_1''$ is contained in the hypersurface determined by the equation of $Z_3''$ and so, in exactly the same style as Example 2.11, if we mutate this last seed at $Z_3''$ then $Z_1''$ will make an unexpected jump.

2.3 The tropicalisation of a log Calabi–Yau variety

2.3.1 The tropicalisation of U

The cocharacter lattice of a torus comes with some extra structure that we would like to generalise to $N_U({\mathbb {Z}})$. There is no group structure on $N_U({\mathbb {Z}})$ in general. Instead (as hinted above) the right way to proceed is to try and realise $N_U({\mathbb {Z}})$ as the integral points of a real affine manifold $N_U$, in the same way that the lattice N can be realised as the set of integral points of the real vector space $N_{\mathbb {R}}:= N\otimes _{\mathbb {Z}}{\mathbb {R}}$. This can be done, but only by introducing singularities into the affine structure on $N_U$. The space $N_U$ thus obtained is an integral affine manifold with singularities and is referred to as the tropicalisation of U.

One can build $N_U$ directly, by choosing a compactification⁵ (X, D) of U and defining $N_U$ to be the cone over the dual intersection complex of the boundary divisor D [17]. However, for log Calabi–Yau varieties with a toric model $\pi :(X,D)\rightarrow (T,B)$ then, by generalising the 2-dimensional case covered in [12, §1.2], there is a natural way to build the space $N_U$ by altering the affine structure on the cocharacter space $N_{\mathbb {R}}$ of the torus ${\mathbb {T}}_S=T{\setminus } B$. In particular, if U has a toric model then, as a real manifold, $N_U$ is homeomorphic to $N_{\mathbb {R}}\cong {\mathbb {R}}^d$.

For simplicity (and because it applies in the case we study later on) we assume that (X, D) and (T, B) are smooth, and the centres $Z_i\subset T$ blown up by $\pi$ are smooth and disjoint. Let ${\mathcal {F}}$ be the fan in $N_{\mathbb {R}}$ that defines (T, B) and consider a pair of maximal smooth cones $\sigma _1,\sigma _2$ of ${\mathcal {F}}$ that meet along a codimension 2 face $\tau = \sigma _1 \cap \sigma _2$. We can write $\sigma _1 = \langle v_1,\ldots , v_d\rangle$, $\sigma _2 = \langle v_2,\ldots ,v_{d+1}\rangle$ and $\tau = \langle v_2,\ldots , v_d\rangle$ for some choice of primitive vectors $v_1,\ldots ,v_{d+1}\in N$. Now the cone $\tau$ corresponds to a torus invariant curve $\overline{C}_\tau \cong {\mathbb {P}}^1\subset T$ and therefore also a curve $C_\tau =\pi ^{-1}(\overline{C}_\tau )$ in the boundary of (X, D). Similarly the vectors $v_i$ correspond to the set of boundary divisors $D_i=D_{v_i}\subset D$ meeting $C_\tau$. Consider the linear map $\psi :\sigma _1\cup \sigma _2 \rightarrow {\mathbb {R}}^d$ defined by

$$\begin{aligned} \psi (v_i) = e_i\;\text {for}\; i=1,\ldots ,d, \quad \text {and} \quad \psi (v_{d+1}) = -e_1-\sum _{i=2}^d (D_i\cdot C_\tau )e_i, \end{aligned}$$

where $e_1,\ldots , e_d$ are the standard basis vectors of ${\mathbb {R}}^d$. This sends $\sigma _1$ onto the positive orthant $\sigma _1'\subset {\mathbb {R}}^d$ and $\sigma _2$ onto an appropriately chosen cone $\sigma _2'\subset {\mathbb {R}}^d$, which has been cooked up so that the toric variety defined by the fan with the two maximal cones $\sigma _1',\sigma _2'$ contains a projective curve, $C_{\tau '}$ for $\tau '=\sigma _1'\cap \sigma _2'$, which has identical intersection numbers with boundary divisors as the curve $C_\tau \subset X$.

To define the tropicalisation $N_U$ we let $N_U^{\textrm{sing}}$ be the union of cones of ${\mathcal {F}}$ of codimension $\ge 2$ and $N^0_U=N_U{\setminus } N_U^{\textrm{sing}}$. To define the affine structure on $N_U^0$, for any pair of maximal cones $\sigma _1,\sigma _2$ in ${\mathcal {F}}$ as above we consider the integral affine structure on $\textrm{int} (\sigma _1\cup \sigma _2)$ given by pulling back the integral affine structure on ${\mathbb {R}}^d$ by the map $\psi |_{\textrm{int} (\sigma _1\cup \sigma _2)}$. In particular, we have the following condition, in terms of the intersection theory on X, to tell when a piecewise linear function is actually linear along some codimension 2 cone of ${\mathcal {F}}$ [24, §1.3]. Suppose that $\phi :\textrm{int} (\sigma _1\cup \sigma _2)\rightarrow {\mathbb {R}}$ is a piecewise-linear function which is linear on each of the cones $\sigma _1, \sigma _2$. This determines a Weil divisor $\Xi _\phi = \sum _{i=1}^{d+1} \phi (v_i)D_{v_i}$ on X. Then $\phi$ is linear along the interior of $\tau =\sigma _1\cap \sigma _2$ if $\Xi _\phi \cdot C_\tau =0$.

The sets of the form $\textrm{int} (\sigma _1\cup \sigma _2)$ cover $N_U^0$, and these glue together to give an affine structure on $N_U^0$. It may or may not be possible to extend this over some, or all, of the cones of $N_U^{\textrm{sing}}$, so at this point it is customary to redefine $N_U^0$ to be the maximal subset of $N_U$ on which this affine structure extends and then set $N_U^{\textrm{sing}} = N_U{{\setminus }} N_U^0$. We call $N_U^{\textrm{sing}}$ the singular locus of $N_U$. We note that the subset of integral tropical points $N_U({\mathbb {Z}})\subset N_U$ is identified with the cocharacter lattice $N\cong {\mathbb {Z}}^d$ of $N_{\mathbb {R}}$ by this construction.

2.3.2 Scattering diagrams

Given a log Calabi–Yau variety U, the approach pioneered in [12] to proving Conjecture 2.5 in the 2-dimensional setting is to construct the ring ${\mathbb {C}}[V]$ directly from the tropicalisation $N_U$, by equipping $N_U$ with the structure of a consistent scattering diagram.

We begin by working in $N_{\mathbb {R}}$, where $N={\mathbb {Z}}^d$ is a lattice with dual lattice $M={{\,\textrm{Hom}\,}}(N,{\mathbb {Z}})$. In general, one has to be able to work with formal power series that have exponents in N, for example by choosing a strictly convex cone $\sigma \subset N$ and working with the ring ${\mathbb {C}}[\![\sigma \cap N]\!]$ (or else by introducing some formal parameters to control the convergence). Let ${\mathfrak {m}}\subset {\mathbb {C}}[\![\sigma \cap N]\!]$ denote the maximal ideal. A wall in $N_{\mathbb {R}}$ is a rational polyhedral cone ${\mathfrak {d}}_i\subset N_{\mathbb {R}}$ of codimension 1. Roughly speaking, a scattering diagram is then a collection ${\mathfrak {D}}=\{({\mathfrak {d}}_i, f_i): i \in I\}$ of walls ${\mathfrak {d}}_i$ decorated with wall functions $f_i$. If $u_i\in M$ is a primitive vector such that ${\mathfrak {d}}_i\subset u_i^\perp$, then the wall function $f_i\in {\mathbb {C}}[\![u_i^\perp \cap \sigma \cap N]\!]$ is a monic power series, i.e. $f_i \equiv 1 \mod \mathfrak {m}$. The collection of walls is usually not finite and may accumulate in certain regions of $N_{\mathbb {R}}$, or even all of $N_{\mathbb {R}}$. Moreover the wall functions are almost always infinite power series, rather than polynomials, in which case there is a further finiteness condition specifying that only finitely many $f_i\not \equiv 1 \mod \mathfrak {m}^k$ for each $k\ge 1$. However, the scattering diagrams constructed for the examples discussed in this paper are always finite: the set of walls form a finite complete fan ${\mathcal {F}}$ in $N_{\mathbb {R}}$ and the wall functions are all polynomials.

Given any $({\mathfrak {d}}, f)\in {\mathfrak {D}}$, crossing the wall ${\mathfrak {d}}$ in the direction of the normal vector u specifies a wall crossing automorphism

$$\begin{aligned} \theta _{({\mathfrak {d}}, f)} :{\mathbb {C}}[\![\sigma \cap N]\!] \rightarrow {\mathbb {C}}[\![\sigma \cap N]\!] \quad \theta _{({\mathfrak {d}}, f)}(z^n) =z^n f^{-\langle n, u \rangle }. \end{aligned}$$

The scattering diagram is then called consistent if, for any loop $\gamma :[0,1]\rightarrow N_{\mathbb {R}}$ that begins and ends in a chamber of ${\mathfrak {D}}$ and crosses all walls of ${\mathfrak {D}}$ transversely, the composition of all the wall crossing automorphisms is the identity.⁶

Log Calabi–Yau varieties with a toric model Suppose that U is log Calabi–Yau variety with a toric model $\pi :(X,D)\rightarrow (T,B)$ determined by a seed $S = \{Z_i: i = 1,\ldots ,k \}$, where each $Z_i$ is a general smooth hypersurface in a component of B. Let N be the cocharacter lattice of the torus $T{\setminus } B$. Under these assumptions Argüz and Gross [2] give a general inductive method to construct a consistent scattering diagram ${\mathfrak {D}}_S$ in $N_{\mathbb {R}}$. They define an initial scattering diagram ${\mathfrak {D}}_{S,\textrm{in}}$ which is supported on the walls of the fan of T which are affected by the nontoric blowup $\pi$. Then they give an inductive procedure which can be used to compute a consistent scattering diagram ${\mathfrak {D}}_S$ from ${\mathfrak {D}}_{S,\textrm{in}}$ [2, Theorem 1.1].

Broken lines In order to use ${\mathfrak {D}}={\mathfrak {D}}_S$ to construct the coordinate ring ${\mathbb {C}}[V]$ of the mirror $V=U^\star$, it is important to fix the integral affine structure on $N_{\mathbb {R}}$ by reconsidering ${\mathfrak {D}}$ as being a scattering diagram in $N_U$. Now the basis of theta functions ${\mathcal {B}}_V=\{ \vartheta _n \in {\mathbb {C}}[V]: n \in N_U({\mathbb {Z}}) \}$ of Conjecture 2.5 are computed by counting broken lines in $N_U$, which are tropicalisations of certain rational curves in U. Roughly speaking, a broken line for $n\in N_U({\mathbb {Z}})$ which starts at $q\in N_U$ is a parameterised piecewise linear path $\ell :[0,\infty )\rightarrow N_U$ such that

• 1. $\ell (0)=q$,

• 2. $\ell$ is allowed to bend finitely many times as it crosses walls of ${\mathfrak {D}}$ and,

• 3. after it makes its last bend, $\ell$ exits $N_U$ with tangent vector $\ell '(t) = n$.

If we let $t_0=0$, $t_{k+1}=\infty$ and $t_1,\ldots ,t_k\in {\mathbb {R}}_{>0}$ be the times at which $\ell$ bends, and $({\mathfrak {d}}_i,f_i)$ be the corresponding walls of ${\mathfrak {D}}$. Then we can associate a monomial $m_i\in {\mathbb {C}}[N]$ to each domain of linearity $(t_i,t_{i+1})$ of $\ell$. This is done inductively by labelling the last domain of linearity $(t_k,\infty )$ with $m_k:= z^n$, and then labelling $(t_{i-1},t_i)$ with a monomial $m_{i-1}$ appearing in the expansion of $\theta _{({\mathfrak {d}}_i,f_i)}(m_i)$ which is dictated to us by the bend that $\ell$ makes along the ith wall.

Theta functions Fix an initial point $q\in N_U$, which is assumed to be a suitably generic (i.e. irrational) point of $N_U$ to ensure that $\ell$ crosses all walls of ${\mathfrak {D}}$ transversely. The theta function $\vartheta _n$ can be expanded as a Laurent power series⁷$\vartheta _n = \sum _{\ell } m_\ell \in {\mathbb {C}}[\![N]\!]$ obtained by taking the sum over all broken lines for n which start at q, where $m_\ell := m_0$ is the monomial attached to the first domain of linearity of $\ell$ obtained by the process described above. These theta functions then form an additive basis for a ring which is expected to be the coordinate ring of the mirror variety $V=U^\star$.

2.3.3 Generalising toric geometry

Once we have constructed the mirror log Calabi–Yau variety $V=U^\star$, we can treat the tropicalisations $N_U$ and $M_U:= N_V$ as an analogue of the cocharacter space and character space for U respectively. These integral affine manifolds with singularities are equipped with a dual intersection pairing and Mandel [24] has used this to generalise many of the traditional techniques of toric geometry to this setting, particularly in the 2-dimensional case.

The intersection pairing The dual intersection pairing is given by

$$\begin{aligned} \langle {\cdot },{\cdot }\rangle :N_U({\mathbb {Z}})\times M_U({\mathbb {Z}})\rightarrow {\mathbb {Z}}\qquad \langle n,m \rangle = \nu _{D_n}(\vartheta _m) \end{aligned}$$

which is given by evaluating the order of vanishing of a theta function $\vartheta _m\in {\mathbb {C}}[U]$ along a boundary component $D_n$ appearing in a compactification of U. This can be extended to a pairing $\langle {\cdot },{\cdot }\rangle :N_U\times M_U\rightarrow {\mathbb {R}}$ by first extending to the rational points of $N_U$ and $M_U$ by bilinearity, and then extending to the real valued points of $N_U$ and $M_U$ by continuity. At least in the 2-dimensional setting, the intersection pairing obtained this way is the same the intersection pairing given by switching the roles of U and V [24, Theorem 1.5].

Convexity in $N_U$ Given the intersection pairing, Mandel now defines the (strong) convex hull of a subset $S\subset N_U$ to be

$$\begin{aligned} {{\,\textrm{conv}\,}}(S):= \left\{ n\in N_U: \langle n,s \rangle \ge \inf _{s\in S} \langle s, m\rangle , \, \forall m\in M_U \right\} \end{aligned}$$

and similarly for subsets of $M_U$. Then a (strongly) convex subset $S\subset N_U$ is a subset for which $S={{\,\textrm{conv}\,}}(S)$. A polytope $P\subset N_U$ is the convex hull of a finite set S, and we call P integral if $P={{\,\textrm{conv}\,}}(P\cap N_U({\mathbb {Z}}))$. Moreover we can define the Newton polytope ${{\,\textrm{Newt}\,}}(\vartheta )\subset M_U$ for a function $\vartheta \in {\mathbb {C}}[U]$ to be

$$\begin{aligned} {{\,\textrm{Newt}\,}}(\vartheta ) = {{\,\textrm{conv}\,}}\left\{ m\in M_U({\mathbb {Z}}): \vartheta =\sum _{m\in M_U} \alpha _m\vartheta _m, \, \alpha _m\ne 0 \right\} . \end{aligned}$$

For any $c\in {\mathbb {R}}$ and any $m\in M_U$ we let $(m)^{\ge c}$ denote the ‘halfspace’ of $N_U$ given by

$$\begin{aligned} (m)^{\ge c}:= \{ n \in N_U: \langle n, m \rangle \ge c \} \end{aligned}$$

which, in contrast to the classical setting, may be a bounded subset of $N_U$, or even empty. Then the polar polytope of a set $S\subset M_U$ is defined to be $S^\circ := \bigcap _{s\in S} (\vartheta _s)^{\ge -1}$, and these are precisely the convex polytopes of $N_U$ that contain the origin [24, Corollary 5.9]. Note that $(P^\circ )^\circ )=P$. Finally, as in the toric setting, we can define an integral polytope $P\subset N_U$ to reflexive if $P^\circ \subset M_U$ is an integral polytope in $M_U$.

2.4 Mirror symmetry

2.4.1 Mirror symmetry for Fano varieties

Let X be a smooth projective d-dimensional Fano variety over ${\mathbb {C}}$. As mentioned in the introduction Sect. 1.2.1, mirror symmetry predicts the existence of a mirror Landau–Ginzburg model (V, w) to X. Following [20, §2.1], to make the correspondence more precise we must decorate the two sides of the mirror with some extra data

where $\omega _X$ and $\omega _V$ are symplectic forms, $s_X\in H^0(X,{-K_X})$ is an anticanonical section and $\Omega _V\in H^0(V,{K_V})$ is a volume form. In particular, the section $s_X$ specifies a boundary divisor $D={{\,\textrm{div}\,}}s_X\in |{-K_X}|$. In the case that (X, D) is a Fano compactification of a positive log Calabi–Yau variety $U=X{\setminus } D$ with maximal boundary, the mirror Landau Ginzburg model to X will be defined on the mirror to U, i.e. $V=U^\star$. In terms of the mirror correspondence above, deleting the boundary divisor of X corresponds to forgetting the potential on V.

From the point of view of homological mirror symmetry, there are now various flavours of the Fukaya category and derived category that one can associate to either side of this correspondence which are conjectured to be equivalent. However we take a slightly more low-tech point of view championed by the Fanosearch program [7], as we now describe.

2.4.2 Landau–Ginzburg mirrors

There is a test one can apply to check whether a given Landau–Ginzburg model (V, w) is a possible mirror to a given Fano variety X, which is to compute the (classical) period of w,

$$\begin{aligned} \pi _{w}(t) = \int _{V}\frac{\Omega _V}{1-tw} \in {\mathbb {C}}[\![t]\!]. \end{aligned}$$

This calculation is particularly simple in the case that V has a toric model, corresponding to the inclusion of a cluster torus chart $j:{\mathbb {T}}\hookrightarrow V$. After restricting w to ${\mathbb {T}}$, we obtain a Laurent polynomial $w|_{\mathbb {T}}\in {\mathbb {C}}[x_1^{\pm 1},\ldots ,x_d^{\pm 1}]$ and it follows that

$$\begin{aligned} \pi _w(t) = \pi _{w|_{\mathbb {T}}}(t) = \sum _{n=0}^\infty \textrm{const}{\left( \left( w|_{\mathbb {T}}\right) ^n\right) }t^n, \end{aligned}$$

which can be seen by expanding $\left( 1-tw|_{\mathbb {T}}\right) ^{-1}$ as a power series in t and repeatedly applying Cauchy’s residue theorem. Under mirror symmetry, $\pi _{w}(t)$ is expected to equal the regularised quantum period $\widehat{G}_X(t)$ of X, which is a power series whose coefficients encode certain Gromov–Witten invariants for X (see [8, §A] for details). For all of the 105 smooth 3-dimensional Fano varieties X, the regularised quantum period $\widehat{G}_X(t)$ has been computed by Coates et al. [8].

2.4.3 A generalisation of Batyrev–Borisov duality

Let U and $V=U^\star$ be two mirror log Calabi–Yau varieties and let ${\mathcal {B}}_U=\{ \vartheta _m: m\in M_U \}$ and ${\mathcal {B}}_V=\{ \varphi _m: m\in N_U \}$ be the bases of theta functions on U and V. Given a pair of dual reflexive polygons $P\subset N_U$ and $Q=P^\star \subset M_U$ we can now make the following constructions, which generalise Batyrev–Borisov mirror symmetry for toric Fano varieties.⁸

• 1. We define a Landau–Ginzburg model $(V,w_P)$, where the potential $w_{P} = \sum _{p\in P} a_p\varphi _p$ is obtained by summing the theta functions on V that correspond to the integral points of P with a specific choice of coefficients $a_p\in {\mathbb {Z}}_{\ge 0}$ (see Remark 2.14).

• 2. The polytope $Q\subset M_U$ determines a grading on the ring ${\mathbb {C}}[U]$, where $\deg \vartheta _{m}$ is the least integer k such that $m\in kQ$, for each $m\in M_U({\mathbb {Z}})$. We let $R_Q$ be the homogenisation of ${\mathbb {C}}[U]$ with respect to this grading, with homogenising variable $\vartheta _0$. Let $(X_Q,D_Q)$ be the pair $X_Q={{\,\textrm{Proj}\,}}R_Q$ with boundary divisor $D_Q={\mathbb {V}}(\vartheta _0)$.

Note that $X_Q$ is an anticanonically embedded, possibly degenerate (i.e. non-${\mathbb {Q}}$-factorial) Fano variety compactifying $U=X_Q{\setminus } D_Q$, where the sections of $|-K_{X_Q}|$ correspond to the integral points of Q. The expectation is that $(X_Q,D_Q)$ and $(V,w_P)$ are mirror in the following sense.

Conjecture 2.13

If $(X_Q,D_Q)$ admits a ${\mathbb {Q}}$-Gorenstein deformation to a pair (X, D), where X is a ${\mathbb {Q}}$-factorial Fano variety, then there exists a choice of positive integral coefficients on the lattice points of P such that the Landau–Ginzburg model $(V,w_P)$ is mirror to (X, D).

Note that P may admit more than one (or even no) such choice of coefficients if $X_P$ admits more than one (or no) deformation to a Fano variety X. In general P is expected to support a Landau–Ginzburg potential corresponding to each deformation component of $X_Q$ (cf. Example 4.17).

Remark 2.14

We should explain how to choose the coefficients $a_p$ in the construction of the Landau–Ginzburg model $(V,w_P)$. In dimension $d=2$, the Minkowski ansatz of the Fanosearch program [7, §6] suggests that the correct choice of coefficients on a reflexive polygon is to label the origin with coefficient $a_0=0$ and to label all of the other lattice points with binomial edge coefficients, i.e. we label the ith lattice point on each edge of lattice length k with the coefficient $\left( {\begin{array}{c}k\\ i\end{array}}\right)$. In higher dimensions we expect that the correct formulation will be given by a generalisation to this setting of the 0-mutable polynomials, introduced by Corti et al. [10]. In the toric case, the 0-mutable polynomials supported on a face $F\subset P$ are special labellings of F with positive integral coefficients which are conjecturally in bijection with the smoothing components of the corresponding strata $D_F\subset X_Q$. The expectation is that the potential $w_P$ obtained by labelling the faces of P with a compatible system of 0-mutable polynomials will then specify the deformation of $X_Q$ described in Conjecture 2.13.

3 Dimension 2: the del Pezzo surface ${{\,\textrm{dP}\,}}_5$

As an illustration before we tackle the 3-dimensional case, we begin by briefly recalling the famous tale of the del Pezzo surface ${{\,\textrm{dP}\,}}_5$.

3.1 The affine del Pezzo surface U

Let $x_1,\ldots ,x_5$ be the five terms of the 2-dimensional Lyness recurrence (LR$_2$) given in Sect. 1.2.1. Geometrically, $x_1,\ldots ,x_5$ are regular functions on an affine surface

$$\begin{aligned} U = {{\,\textrm{Spec}\,}}{{\mathbb {C}}[x_1,\ldots ,x_5]/\left( x_{i-1}x_{i+1} - x_i - 1: i \in {\mathbb {Z}}/5{\mathbb {Z}}\right) }, \end{aligned}$$

the vanishing locus of the five relations obtained from (LR$_2$). We can homogenise the equations with respect to a new variable $x_0$ to obtain the projective closure $X=\overline{U}\subset {\mathbb {P}}^5_{x_0,x_1,\ldots ,x_5}$ which is a smooth projective surface. The projection map $p_{12}:X\rightarrow {\mathbb {P}}^2_{x_0,x_1,x_2}$ is birational, and by resolving the baselocus of $p_{12}^{-1}$ we find that X is a blowup of ${\mathbb {P}}^2$ in four points

$$\begin{aligned} e_1 = (0:1:-1), \, e_2 = (1:0:-1), \, d_3 = (1:0:0), \, d_5 = (0:1:0) \end{aligned}$$

making X a del Pezzo surface of degree 5. As is classically known, there are ten $(-1)$-curves on X. We call them $D_1,\ldots ,D_5,E_1,\ldots , E_5$, where $E_1,E_2,D_3,D_5$ are the exceptional curves over the points $e_1,e_2,d_3,d_5$ and the remaining six curves are the strict transform of the lines passing through any two of these four points, labelled according to Fig. 2. These ten curves have dual intersection diagram given by the Petersen graph.

Fig. 2

a A realisation of $X\subset {\mathbb {P}}^5$ as a blowup of ${\mathbb {P}}^2$. b The dual intersection graph of the ten $(-1)$-curves in X

A log Calabi–Yau pair (X, D) The complement to U is the divisor $D:= X{\setminus } U$, which is an anticanonical cycle $D = \bigcup _{i=1}^5D_i \in |{-K_X}|$ of five $(-1)$-curves, corresponding to the outside ring of the Petersen graph. In other words, U is the interior of a log Calabi–Yau pair (X, D).

The interior $(-1)$-curves The other five $(-1)$-curves $E = \bigcup _{i=1}^5E_i$ form a complementary anticanonical pentagram, and are called interior $(-1)$-curves. They are given by the locus in U where a cluster variable vanishes, i.e. $E_i|_U = {\mathbb {V}}(x_i)\subset U$ for $i=1,\ldots ,5$.

3.2 The Grassmannian ${{\,\textrm{Gr}\,}}(2,5)$

The five equations defining the affine surface U can be written as the five maximal Pfaffians of the following skewsymmetric $5\times 5$ matrix (where, for simplicity, we have omitted the diagonal of zeroes and the antisymmetry)

$$\begin{aligned} {{\,\textrm{Pf}\,}}_4\begin{pmatrix} 1 &{} x_1 &{} x_2 &{} 1 \\ &{} 1 &{} x_3 &{} x_4 \\ &{} &{} 1 &{} x_5 \\ &{} &{} &{} 1 \end{pmatrix} = 0 \quad \implies \quad x_{i-1}x_{i+1} = x_i + 1 \quad \forall i\in {\mathbb {Z}}/5{\mathbb {Z}}. \end{aligned}$$

We can homogenise (LR$_2$) by introducing five parameters $y_1,\ldots ,y_5$ in a particularly nice and symmetric way:

After setting $x_i=-1$ for all i, we see that $\left( y_{2i}: i \in {\mathbb {Z}}/5{\mathbb {Z}}\right)$ is also a solution to (LR$_2$).

The homogenised equations define a 7-dimensional variety ${\mathcal {U}}\subset {\mathbb {A}}^{10}$, which is the affine cone over ${{\,\textrm{Gr}\,}}(2,5)$ in the Plücker embedding. The projection $p :{\mathcal {U}}\rightarrow {\mathbb {A}}^5_{y_1,\ldots ,y_5}$ realises ${\mathcal {U}}$ as a flat family of affine del Pezzo surfaces. For any point in the big open torus $y\in ({\mathbb {C}}^\times )^5\subset {\mathbb {A}}^5$, the fibre $U_y = p^{-1}(y)$ is isomorphic to $U=X{\setminus } D$ after making the change of variables $z_i=\frac{x_iy_i}{y_{i+2}y_{i-2}}$ for all $i\in {\mathbb {Z}}/5{\mathbb {Z}}$. However, above the coordinate strata of ${\mathbb {A}}^5$ the fibres begin to degenerate. The most degenerate fibre, appearing over $0\in {\mathbb {A}}^5$, is the 5-vertex

$$\begin{aligned} p^{-1}(0) = {\mathbb {A}}^2_{x_1,x_2} \cup {\mathbb {A}}^2_{x_2,x_3} \cup {\mathbb {A}}^2_{x_3,x_4} \cup {\mathbb {A}}^2_{x_4,x_5} \cup {\mathbb {A}}^2_{x_5,x_1}, \end{aligned}$$

a cycle of 5 coordinate planes glued together along their toric boundary strata.

Remark 3.1

This fibration $p:{\mathcal {U}}\rightarrow {\mathbb {A}}^5$ is (very nearly) the mirror family to (X, D) constructed by Gross et al. [12], in which the variables $x_1,\ldots ,x_5$, the compactifying parameters $y_1,\ldots ,y_5$ and the equations ($*$) are interpreted in terms of the Gromov–Witten theory of (X, D). Indeed, the mirror family constructed in [12] is fibred over the toric base scheme ${{\,\textrm{Spec}\,}}{\mathbb {C}}[{{\,\mathrm{\overline{NE}}\,}}(X)]$. In terms of their construction, our parameters correspond to $y_i=z^{D_i}$, and hence our family can be viewed as a fibration over ${\mathbb {A}}^5={{\,\textrm{Spec}\,}}{\mathbb {C}}[z^{D_1},\ldots , z^{D_5}]$. In other words, their family can be pulled back from our one by the inclusion ${\mathbb {C}}[z^{D_1},\ldots , z^{D_5}]\hookrightarrow {\mathbb {C}}[{{\,\mathrm{\overline{NE}}\,}}(X)]$.

In general, the fibres of the family ${\mathcal {U}}$ are mirror to the interior of the pair (X, D), so the fact that U appears in both roles here due to the fact that it is self-mirror.

3.3 The tropicalisation of U

3.3.1 A toric model for U

Consider the toric pair (T, B) obtained by blowing up the two points $d_5,d_3\in {\mathbb {P}}^2$ in Fig. 2a above, so that T is a smooth projective toric surface whose boundary divisor B consists of a cycle of five rational curves, with self-intersection numbers $(0,0,-1,-1,-1)$. This gives a toric model $\pi :(X,D)\rightarrow (T,B)$ for U which is a composition of two nontoric blowups of T (the blowup of the image of the two points $e_1,e_2\in T$). In particular, the induced map $\pi |_D:D\rightarrow B$ on boundary divisors is an isomorphism and so we can label the components of $B=\bigcup _{i=1}^5B_i$ by $i\in {\mathbb {Z}}/5{\mathbb {Z}}$, where $B_i=\pi (D_i)$.

The seed for this toric model is the set of points $S=\{e_1, e_2\}$, so let us denote the corresponding inclusion of a cluster torus by $j_{12}:{\mathbb {T}}_{12}={{\,\textrm{Spec}\,}}{\mathbb {C}}[x_1^{\pm 1},x_2^{\pm 1}]\hookrightarrow U$. Now the Lyness map and its inverse are given by

$$\begin{aligned} \sigma _2(x_1,x_2) = \left( x_2,\frac{1+x_2}{x_1}\right) = \left( x_2,x_3\right) , \quad \sigma _2^{-1}(x_1,x_2) = \left( \frac{1+x_1}{x_2},x_1\right) = \left( x_5,x_1\right) \end{aligned}$$

and these are (up to a permutation of the coordinates) the mutation at $e_1\in B_1$ and the mutation at $e_2\in B_2$ respectively. Consider the mutation at $e_1$. It is given by blowing up $e_1$ and contracting the strict transform of the curve $E_3$ to a point $e_3\in B_3$. This gives a toric model $\pi ':(X,D) \rightarrow (T',B')$ which is isomorphic to the original one, except that now all the labels have been shifted by one, $i\mapsto i+1\in {\mathbb {Z}}/5{\mathbb {Z}}$. In particular the dense open torus is $T'{{\setminus }} B'={\mathbb {T}}_{23}={{\,\textrm{Spec}\,}}{\mathbb {C}}[x_2^{\pm 1},x_3^{\pm 1}]$. Thus the Lyness map can be interpreted as a mutation of toric models.

By the five-periodicity of the Lyness map, the surface U contains five cluster torus charts ${\mathbb {T}}_{i,i+1}:={{\,\textrm{Spec}\,}}{\mathbb {C}}[x_i^{\pm 1},x_{i+1}^{\pm 1}]$ for $i\in {\mathbb {Z}}/5{\mathbb {Z}}$ and U is given by the union of all five of them, identified according to the mutations.

The exchange graph of U Each toric model for U has two nontoric centres to blow up in, so the graph is 2-valent. The 5-periodicity implies that G is a pentagon.

3.3.2 The tropicalisation $N_U$

Consider the complete fan ${\mathcal {F}}$ in ${\mathbb {R}}^2$ determined by the five rays $\rho _i={\mathbb {R}}_{\ge 0} v_i$ for $i\in {\mathbb {Z}}/5{\mathbb {Z}}$, where the $v_i$ are the five points

$$\begin{aligned} v_1 = (-1,0), \quad v_2 = (0,-1), \quad v_3 = (1,0), \quad v_4 = (1,1), \quad v_5 = (0,1). \end{aligned}$$

This is the fan of the toric pair (T, B) which appears in our toric model (Fig. 3). We can now construct $N_U$, the tropicalisation of U, by taking this fan ${\mathcal {F}}$ and altering the integral affine structure in ${\mathbb {R}}^2$ as described in Sect. 2.3.1. This changes the affine structure along the rays $\rho _1$ and $\rho _2$, corresponding to the two components of (T, B) along which a nontoric blowup occurs, and has the effect of bending lines that pass through $\rho _1$ and $\rho _2$ towards the origin. As a result it introduces a singularity at the origin $0\in N_U$.

Fig. 3

a The fan for (X, D) in $N_U$. b The scattering diagram for U in $M_U$

3.3.3 A scattering diagram for U

The same fan ${\mathcal {F}}$ in $N_U$ also supports the structure of a scattering diagram ${\mathfrak {D}}$ which can be used to construct the coordinate ring of (the mirror to) U. The construction of ${\mathfrak {D}}$ and its relationship to ${\mathbb {C}}[U]$ is described in [12, Example 3.7]. To obtain ${\mathfrak {D}}$, we attach the following five scattering functions $f_i\in {\mathbb {C}}[z_1^{\pm 1},z_2^{\pm 1}]$ to the rays $\rho _i$ for $i\in {\mathbb {Z}}/5{\mathbb {Z}}$:

$$\begin{aligned} f_1=1 + z_1, \quad f_2=1 + z_2, \quad f_3=1 + z_1, \quad f_4=1 + z_1z_2, \quad f_5=1 + z_2. \end{aligned}$$

It is now straightforward to check that the collection of walls ${\mathfrak {D}}= \{ \left( \rho _i,f_i\right) : i \in {\mathbb {Z}}/5{\mathbb {Z}}\}$ defines a consistent scattering diagram, i.e. that starting from any point in the interior of a chamber of ${\mathcal {F}}$ and composing the five wall crossing automorphisms corresponding to a loop around $0\in N_U$ yields the identity.

To get from the scattering diagram back to ${\mathbb {C}}[U]$ we can consider the cluster monomials $x_i=\vartheta _{v_i}$ which are the theta functions corresponding to the points $v_i\in M_U$ for $i\in {\mathbb {Z}}/5{\mathbb {Z}}$. As described in Sect. 2.3.2 these can be expanded as Laurent polynomials $x_i=\sum _\ell c_\ell z^{m_\ell }$ where $\ell$ ranges over all of the broken lines for $v_i$ starting at some given point $q\in N_U$. In our case the coefficients of all the scattering functions are all equal to 1 so all $c_\ell =1$. Suppose that q is chosen near to the point $(-1+\varepsilon ,1+\varepsilon )$ for some small irrational $\varepsilon >0$, which lies in the chamber $\langle v_5,v_1 \rangle$. Then the five expansions of the cluster variables are shown as in Fig. 4, and these five terms generate the coordinate ring ${\mathbb {C}}[U]$.

Fig. 4

The monomials $x_i$ obtained by counting broken lines

For any $a,b\in {\mathbb {Z}}_{\ge 0}$ now consider the point $m=av_i + bv_{i+1}$ in the chamber $\langle v_i,v_{i+1} \rangle$. Since any broken line that leaves a chamber of ${\mathfrak {D}}$ can never return to it (cf. [12, Example 3.7]), it is easy to see that there is only one broken line for m that starts at a point q very near to m, namely the straight line that leaves q in the direction of m. Thus if we choose to expand the theta function $\vartheta _m$ by counting broken lines starting at q we obtain $\vartheta _m=\vartheta _i^a\vartheta _{i+1}^b$. This completely determines all of the theta functions associated to the points of $M_U({\mathbb {Z}})$.

3.3.4 The intersection pairing

Since the mirror of U constructed from ${\mathfrak {D}}$ is isomorphic to U, it follows that the tropicalisation $N_U$ is self-dual, i.e. that $N_U\cong M_U$. The duality between the two affine manifolds $N_U$ and $M_U$ is given by extending the intersection pairing

$$\begin{aligned} \langle {\cdot },{\cdot } \rangle :N_U({\mathbb {Z}})\times M_U({\mathbb {Z}}) \rightarrow {\mathbb {Z}}\quad \langle n,m \rangle := \nu _{D_n}(\vartheta _m) \end{aligned}$$

to an intersection pairing $\langle {\cdot },{\cdot } \rangle :N_U\times M_U \rightarrow {\mathbb {R}}$ which can be calculated ${\mathbb {R}}$-linearly in each cone of the fan ${\mathcal {F}}$. Given that, as a rational function on X, the divisor of $x_i$ is

$$\begin{aligned} \textrm{div}\,x_i = E_i + D_i - D_{i-2} - D_{i+2}, \end{aligned}$$

the pairing $N_U\times M_U\rightarrow {\mathbb {R}}$ can be determined by Table 1.

Table 1 Table of intersection numbers $\langle D_i,x_j \rangle$ for $i,j\in {\mathbb {Z}}/5{\mathbb {Z}}$

For example, we can now compute the five halfspaces $(x_i)^{\ge -1}\subset N_U$, which are shown in Fig. 5. Equivalently, exactly the same diagrams show the five halfspaces $(D_i)^{\ge -1}\subset M_U$.

Fig. 5

The halfspaces $(x_i)^{\ge -1}\subset N_U$ for $i\in {\mathbb {Z}}/5{\mathbb {Z}}$

3.4 Applications to mirror symmetry

3.4.1 Reflexive polygons in $N_U$

Now that we have built the spaces $N_U$ and $M_U$ as integral affine manifolds with singularitiesand their their dual intersection pairing $\langle {\cdot },{\cdot }\rangle :N_U\times M_U\rightarrow {\mathbb {R}}$, we can use them to construct examples of del Pezzo pairs and their mirror Landau–Ginzburg models, as in Sect. 2.4.3.

Theorem 3.2

Up to automorphism there are 23 reflexive polygons in $N_U$, which are displayed in Fig. 6. For each reflexive polytope P, the Landau–Ginzburg model $w_Q:U\rightarrow {\mathbb {C}}$ obtained by labelling $Q=P^\star$ with binomial edge coefficients has the right period to be a mirror to the del Pezzo pair $(X_P,D_P)$ (where the del Pezzo surface of degree 8 represented by the unique polygon with eight vertices is ${\mathbb {F}}_1$).

Proof

The classification of reflexive polygons in $N_U$ proceeds in an equivalent manner to the classification of reflexive polygons in the ordinary toric setting. Beginning from any one such polygon, such as the polygon P of Example 3.3, we can add or remove vertices corresponding to blowing up or blowing down $(-1)$-curves in the boundary of the associated Looijenga pair $(X_P,D_P)$. Disregarding polygons with nonzero interior lattice points produces the 23 examples of Fig. 6.

To verify that the potential $w_Q$ has the right period to be mirror to $(X_P,D_P)$, we can restrict to the cluster torus ${\mathbb {T}}_{12}\subset U$ to get a Laurent polynomial $w_0:= w_Q|_{{\mathbb {T}}_{12}}\in {\mathbb {C}}[x_1^{\pm 1},x_2^{\pm 1}]$. The only difference between the Landau–Ginzburg model $(U,w_Q)$ and the toric Landau–Ginzburg model $({\mathbb {T}}_{12},w_0)$ is that we have extended the domain of definition of $w_0$ along the two exceptional curves of $U{\setminus } {\mathbb {T}}_{12}$. It is now just a case of verifying that $w_0$ is mutation equivalent to a known Laurent polynomial mirror to $X_Q$ (see [1, Figure 1]). $\square$

Fig. 6

Representatives for the 23 classes of reflexive polygons in $N_U$. The gray lines indicate the two rays along which we bend the affine structure of $N_U$ and the arrows between polygons denote the addition or removal of a vertex. Since $N_U$ is isomorphic to its dual space $M_U$, given a polygon $P\subset N_U$ we can identify the dual polygon $P^\star \subset M_U$ with a polygon in $N_U$. Duality between these reflexive polygons is then given by top-to-bottom reflection in the diagram. In particular there are three self-dual pentagons in the central row

If d(P) denotes the number of boundary points of a reflexive polygon $P\subset N_U$, then for a dual pair of reflexive polygons we have $d(P)+d(P^\star )=10$. This is in contrast to the usual formula $d(P)+d(P^\star )=12$ that holds in the ordinary toric setting, and corresponds to the fact that we have made two nontoric blowups in the boundary.

3.4.2 Examples

We illustrate Theorem 3.2 with a few examples.

Example 3.3

Suppose we take the standard projective compactification (X, D) of U, which corresponds to the pair $(X_P,D_P)$ where $P=P^\star$ is the self-dual polygon

which, according to Remark 2.14, has been labelled with binomial edge coefficients. Note that P is cut out by the five halfspaces of Fig. 5. Let $w = w_P$ be the potential on the mirror Landau–Ginzburg model $w:U\rightarrow {\mathbb {C}}$, which is given by

$$\begin{aligned} w = x_1 + x_2 + x_3 + x_4 + x_5. \end{aligned}$$

This is a $\sigma _2$-invariant function, and we note that this has a nice alternative representation as $w+3 = x_1x_2x_3x_4x_5$, corresponding to the fact that the fibre $w^{-1}(-3)\subset U$ breaks up as the union of the five interior $(-1)$-curves $E=\bigcup _{i=1}^5 E_i$.

The period $\pi _{w}(t)$ By restricting w to the cluster torus chart ${\mathbb {T}}_{12}$ (i.e. expanding it as a Laurent polynomial in terms of $x_1,x_2$), we can compute the period $\pi _{w}(t)$ which we see to be equal to the regularised quantum period for ${{\,\textrm{dP}\,}}_5$. The first few terms are given by

$$\begin{aligned} \pi _{w}(t) = 1 + 10t^2 + 30t^3 + 270t^4 + 1560t^5 + 11350t^6 + 77700t^7 + \cdots \end{aligned}$$

Alternatively, we can compute the period of the shifted potential $w+3$, which is given by the change of variables $\pi _{w+3}(t) =\frac{1}{1-3t}\pi _{w}\left( \frac{t}{1-3t}\right)$. The series is

$$\begin{aligned} \pi _{w+3}(t) = 1 + 3t + 19t^2 + 147t^3 + 1251t^4 + 11253t^5 + 104959t^6 + \cdots \end{aligned}$$

which can be recognised as one of the famous Apéry series $\pi _{w+3}(t) = \sum _{n=0}^\infty \sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) ^2 \left( {\begin{array}{c}n+k\\ k\end{array}}\right) t^n$ and is well-known as a period for the del Pezzo surface of degree 5.

The elliptic fibration After extending to a birational map $w:X\dashrightarrow {\mathbb {P}}^1$, the fibres of w belong to the anticanonical pencil $|D,E|\subset |{-K_X}|$ with baselocus given by the five points $D_i\cap E_i\in X$. Blowing up these five points $\phi :\widetilde{X}\rightarrow X$ resolves w into an elliptic fibration $\widetilde{w}=w\circ \phi :\widetilde{X}\rightarrow {\mathbb {P}}^1$, which appears in Beauville’s classification of rational elliptic surfaces with four singular fibres [4]. There are two $I_5$ fibres over $\infty$ and $-3$, corresponding to D and E respectively, and two further $I_1$ fibres over the values $-\frac{5}{\varphi }$ and $5\varphi$, where $\varphi =\frac{1+\sqrt{5}}{2}$ is the golden ratio.

The Landau–Ginzburg model $w:U\rightarrow {\mathbb {C}}$ is obtained from this elliptic fibration $\widetilde{w}$ by deleting the fibre at $\infty$ and five horizontal sections given by the five $\phi$-exceptional curves. In terms of mirror symmetry, the significance of the five deleted sections is due to the fact that the anticanonical divisor $D\subset X$ has five corners [20, Remark 2.1]. Changing the choice of boundary divisor by smoothing a corner of D corresponds to extending w along one of the missing horizontal sections, so the Landau–Ginzburg model obtained by extending w to $\widetilde{X}{\setminus } \widetilde{w}^{-1}(\infty )$ is mirror to a del Pezzo surface of degree 5 with a smooth anticanonical divisor, as expected [3].

The fibration $\widetilde{w}$ has some very interesting arithmetic properties. The five $\phi$-exceptional curves of $\widetilde{w}$ are permuted by the ${\mathbb {Z}}/5{\mathbb {Z}}$-symmetry. For each smooth fibre ${\widetilde{X}}_t$ these sections give rise to the orbit of a rational 5-torsion point under translation. Moreover, the monodromy action on $H^1({\widetilde{X}}_t,{\mathbb {Z}})\cong {\mathbb {Z}}^2$ around the four singular fibres generates the congruence subgroup $\Gamma _1(5)\subset {{\,\textrm{SL}\,}}(2,{\mathbb {Z}})$, as shown in [5].

Example 3.4

To give a slightly more exotic example, consider the following polygon $P\subset N_U$ and its dual $Q=P^\star \subset M_U$, which have also been labelled with binomial edge coefficients.

Note that both P and Q are both bigons; they each have precisely two vertices when considered in the affine structure of $N_U$ or $M_U$.

Graded ring calculation for $(X_Q,D_Q)$ Consider the theta functions $\vartheta _1,\ldots ,\vartheta _4$ corresponding to the lattice points $(-1,-1),(1,1)\in Q$ and $(-1,0),(0,-1)\in 2Q$ respectively. In terms of the basis of theta functions of ${\mathbb {C}}[U]$, these are

$$\begin{aligned} \vartheta _1 = x_1x_2, \quad \vartheta _2 = x_4, \quad \vartheta _3 =x_1, \quad \vartheta _4 = x_2 \end{aligned}$$

and, after homogenising, they can be used to give a presentation of the graded ring $R_Q = {\mathbb {C}}[\vartheta _0,\vartheta _1,\vartheta _2,\vartheta _3,\vartheta _4]/I$ with generators in degrees 1, 1, 1, 2, 2 respectively. The ideal of equations I defining $R_Q$ is generated by two relations

$$\begin{aligned} \vartheta _0^3\vartheta _1=\vartheta _3\vartheta _4 \quad \text {and} \quad \vartheta _1\vartheta _2 = \vartheta _0^2 + \vartheta _3 + \vartheta _4. \end{aligned}$$

Using the second equation to eliminate $\vartheta _4$, we get a quartic hypersurface

$$\begin{aligned} X_Q\cong {\mathbb {V}}\left( \vartheta _1\vartheta _2\vartheta _3 - \vartheta _0^3\vartheta _1 - \vartheta _0^2\vartheta _3 - \vartheta _3^2 \right) \subset {\mathbb {P}}(1,1,1,2)_{\vartheta _0,\vartheta _1,\vartheta _2,\vartheta _3}. \end{aligned}$$

As may be expected from the spanning fan of $P\subset N_U$, this defines a singular del Pezzo surface of degree 2, with an $A_2$ singularity at the coordinate point $P_1 = (0:1:0:0)$ and an $A_4$ singularity at the coordinate point $P_2 = (0:0:1:0)$. The boundary divisor $D_Q={\mathbb {V}}(\vartheta _0)$ has two components

$$\begin{aligned} D_1 = {\mathbb {V}}\left( \vartheta _0, \vartheta _3\right) , \qquad D_2= {\mathbb {V}}\left( \vartheta _0, \vartheta _1\vartheta _2-\vartheta _3\right) \end{aligned}$$

which meet at the two singular points $P_1,P_2$.

The Landau–Ginzburg model $w_P:U\rightarrow {\mathbb {C}}$ We form the Landau–Ginzburg potential

$$\begin{aligned} w_P = x_2x_3^2 + 3x_3 + 3x_5 + x_5^2x_1 + 5x_5x_1 + 10x_1 + 10x_2 + 5x_2x_3 \end{aligned}$$

obtained by considering binomial edge coefficients on P. By restricting to the torus chart ${\mathbb {T}}_{12}\subset U$ we obtain a Laurent polynomial, which satisfies the following identity

$$\begin{aligned} \left( w_P+12\right) |_{{\mathbb {T}}_{12}} = \frac{(1+x_1+x_2)^2(x_1+x_2)^3}{x_1^2x_2^2}. \end{aligned}$$

By Sect. 2.4.2, the nth coefficient $\alpha _n$ of the period $\pi _{w+12}(t)=\sum _{n\ge 0}\alpha _nt^n$ is equal to the coefficient of $x_1^{2n}x_2^{2n}$ in

$$\begin{aligned} (1+x_1+x_2)^{2n}(x_1+x_2)^{3n} = \sum _{k=0}^{2n} \left( {\begin{array}{c}2n\\ k\end{array}}\right) (x_1+x_2)^{3n+k}, \end{aligned}$$

where the righthand side was obtained by expanding the first bracket by using the binomial formula, treating $1+x_1+x_2$ as the sum of 1 and $x_1+x_2$. The term $x_1^{2n}x_2^{2n}$ can only appear in the righthand side if $k=n$, and then we easily see that

$$\begin{aligned} \alpha _n = \left( {\begin{array}{c}2n\\ n\end{array}}\right) \left( {\begin{array}{c}4n\\ 2n\end{array}}\right) \implies \pi _{w_P+12}(t) = \sum _{k\ge 0} \frac{(4n)!}{n!n!(2n)!}t^n. \end{aligned}$$

This function $\pi _{w_P+12}(t)$ is equal to the regularised I-function $\widehat{I}_{X_4}(t)$ for a hypersurface $X_4\subset {\mathbb {P}}(1,1,1,2)$, and it is known to be a shift of the regularised quantum period $\widehat{G}_{X_4}(t)$ of $X_4$ [8, Proposition D.9]. In order to recover $\widehat{G}_{X_4}(t)$ from $\widehat{I}_{X_4}(t)$, the appropriate shift is by the unique constant term required to kill the coefficient $\alpha _1$ of $\widehat{I}_{X_4}(t)$. Since $\alpha _1=\left( {\begin{array}{c}2\\ 1\end{array}}\right) \left( {\begin{array}{c}4\\ 2\end{array}}\right) =12$ we see that $\pi _{w_P}(t) = \widehat{G}_{X_4}(t)$.

This shows that $w_P$ has the right period to be mirror to $X_Q$, but, more precisely, we expect that the particular Landau–Ginzburg model that we have constructed is actually a mirror to the (singular) del Pezzo surface $X_Q$ with its boundary divisor $D_Q$. Note that $w_P$ has a reducible fibre $w_P^{-1}(-12)$ with two disjoint components; one component of multiplicity 2 and one of multiplicity 3. After extending $w_P$ to the dual compactification $w_P:X_P\dashrightarrow {\mathbb {P}}^1$, blowing up two basepoints and then resolving singularities, we obtain an extremal rational elliptic fibration with three singular fibres of type $I_1$, $I_2$ and $\widetilde{E}_7$ over the points 52, $\infty$ and $-12$ respectively.

The two components of multiplicity 2 and 3 are given by the two solid lines in the $\widetilde{E}_7$ fibre, shown in the diagram above. The surface U is obtained by deleting the two horizontal $(-1)$-curves, as well as the six dotted $(-2)$-curves in the $\widetilde{E}_7$ fibre. We saw above that the presence of an ordinary node in the boundary divisor $D_Q\subset X_Q$ corresponds to the deletion of a horizontal $(-1)$-curve in the mirror to $(X_Q,D_Q)$. This example suggests that the presence of an $A_k$ singularity at a node corresponds to the deletion of a chain of $k+1$ rational curves of self-intersection $(-1,-2,\ldots ,-2)$.

Example 3.5

One can do the computations of Example 3.4 with the roles of P and Q reversed. The graded ring $R_P$ is the homogeneous coordinate ring of a smooth surface $X_P\subset {\mathbb {P}}^8$, which is an anticanonically polarised ${\mathbb {F}}_1$. The boundary divisor $D_P=D_1\cup D_2$ is the union of two smooth rational curves of self-intersection $D_1^2=1$ and $D_2^2=3$. By restricting to ${\mathbb {T}}_{12}\subset U$ we see that the Landau–Ginzburg potential satisfies $w_Q|_{{\mathbb {T}}_{12}} = x_1x_2 + \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_1x_2}$ which is a mirror Laurent polynomial for ${\mathbb {F}}^1$.

4 Dimension 3: the Fano 3-fold $V_{12}$

We now describe a parallel story in the 3-dimensional setting which generalises

• 1. the cluster structure on U the affine del Pezzo surface of degree 5,

• 2. the fibration on the affine cone over ${{\,\textrm{Gr}\,}}(2,5)$ by such surfaces,

• 3. the self-dual integral affine manifold with singularities $N_U$ obtained by tropicalising U,

• 4. the Borisov–Batyrev style mirror symmetry constructions of Sect. 3.4.

Indeed, the corresponding actors will be the orthogonal Grassmannian ${{\,\textrm{OGr}\,}}(5,10)$ and the Fano 3-fold $V_{12}$. We will begin by generalising the second statement by giving a description of ${{\,\textrm{OGr}\,}}(5,10)$ which leads to a homogenisation of the 3-dimensional Lyness recurrence (LR$_3$).

4.1 The orthogonal Grassmannian ${{\,\textrm{OGr}\,}}(5,10)$

4.1.1 ${{\,\textrm{OGr}\,}}(5,10)$ as a homogeneous variety

The orthogonal Grassmannian ${{\,\textrm{OGr}\,}}(5,10)$ is one of the two isomorphic irreducible components in the space of 5-planes ${\mathbb {C}}^5 \subset {\mathbb {C}}^{10}$ which are isotropic with respect to a given quadratic form. It is the homogeneous variety for the group ${{\,\textrm{SO}\,}}(10)$ of type ${\textsf{D}}_5$ with a ‘half-spinor embedding’ ${{\,\textrm{OGr}\,}}(5,10)\subset {\mathbb {P}}(S^+)$, where $S^+$ is one half of the spin representation $S^+\oplus S^-$ of ${{\,\textrm{SO}\,}}(10)$. Given the symmetry broken by choosing $S^+$ over $S^-$, it is perhaps better to think of the isomorphic Grassmannian ${{\,\textrm{OGr}\,}}(4,9)\cong {{\,\textrm{OGr}\,}}(5,10)$ instead. This is a homogeneous variety for the group ${{\,\textrm{SO}\,}}(9)$ of type ${\textsf{B}}_4$, with spinor embedding ${{\,\textrm{OGr}\,}}(4,9)\subset {\mathbb {P}}(S)$ corresponding to the (irreducible) spin representation $S=\bigoplus _{i=0}^4\bigwedge ^i{\mathbb {C}}^4$ of ${{\,\textrm{SO}\,}}(9)$.

The representation S is 16-dimensional with weights $\tfrac{1}{2}(\pm 1,\pm 1,\pm 1,\pm 1)$, which are the vertices of a 4-dimensional cube C in the weight lattice for ${\textsf{B}}_4$. The Weyl group $W({\textsf{B}}_4)$ acts as the full symmetry group of C and a Coxeter element in $W({\textsf{B}}_4)$ acts on C as a rotation of order 8. Taking the orthogonal projection onto the Coxeter plane (shown in Fig. 7a) we see this 8-fold rotational symmetry of C which splits the vertices into two groups of size 8.

Fig. 7

a The sixteen spinor variables labelling the 4-cube C. b The face of C which corresponds to the equation $x_1x_4=x_2y_5+x_3y_8+y_2y_3$

4.1.2 The equations of ${{\,\textrm{OGr}\,}}(5,10)$

Let $R = {\mathbb {C}}[S]$ and name the 16 spinor variables $x_1,\ldots ,x_8,y_1,\ldots ,y_8\in R$, indexed by ${\mathbb {Z}}/8{\mathbb {Z}}$, according to the labels in Fig. 7a. The corresponding vertices of C are represented by the columns of the following matrix (where the minus signs in front of $y_3$ and $y_7$ are only chosen to make the equations displayed below more beautiful).

$$\begin{aligned} \begin{array}{cccccccc|cccccccc} x_1&{}x_2&{}x_3&{}x_4&{}x_5&{}x_6&{}x_7&{}x_8&{}y_1&{}y_2&{}-y_3&{}y_4&{}y_5&{}y_6&{}-y_7&{}y_8 \\ \hline +&{}-&{}-&{}-&{}-&{}+&{}+&{}+&{} -&{}+&{}-&{}-&{}+&{}-&{}+&{}+\\ +&{}+&{}-&{}-&{}-&{}-&{}+&{}+&{} +&{}-&{}+&{}-&{}-&{}+&{}-&{}+\\ +&{}+&{}+&{}-&{}-&{}-&{}-&{}+&{} +&{}+&{}-&{}+&{}-&{}-&{}+&{}-\\ +&{}+&{}+&{}+&{}-&{}-&{}-&{}-&{} -&{}+&{}+&{}-&{}+&{}-&{}-&{}+ \end{array} \end{aligned}$$

The ten quadratic equations defining ${{\,\textrm{OGr}\,}}(5,10)$ are now obtained by swapping minus signs in columns that differ in three or more places. For example,

$$\begin{aligned} \left( \begin{array}{c} + \\ + \\ + \\ + \end{array}\right) \left( \begin{array}{c} + \\ - \\ - \\ - \end{array}\right) = \left( \begin{array}{c} + \\ + \\ + \\ - \end{array}\right) \left( \begin{array}{c} + \\ - \\ - \\ + \end{array}\right) - \left( \begin{array}{c} + \\ - \\ + \\ - \end{array}\right) \left( \begin{array}{c} + \\ + \\ - \\ + \end{array}\right) + \left( \begin{array}{c} + \\ + \\ - \\ - \end{array}\right) \left( \begin{array}{c} + \\ - \\ + \\ + \end{array}\right) \end{aligned}$$

corresponds to the equation $x_1x_6 = x_8y_5 + y_7y_8 + x_7y_2$. The full ideal I defining ${{\,\textrm{OGr}\,}}(5,10)$ is given by

$$\begin{aligned} x_1x_4&= x_2y_5 + x_3y_8 + y_2y_3 \\ x_5x_8&= x_6y_1 + x_7y_4 + y_6y_7\\ x_1x_5 - x_3x_7&= y_1y_5 - y_3y_7 \\ x_2x_5&= x_3y_6 + x_4y_1 + y_3y_4\\ x_6x_1&= x_7y_2 + x_8y_5 + y_7y_8\\ x_2x_6 - x_4x_8&= y_2y_6 - y_4y_8 \\ x_3x_6&= x_4y_7 + x_5y_2 + y_4y_5\\ x_7x_2&= x_8y_3 + x_1y_6 + y_8y_1 \\ x_4x_7&= x_5y_8 + x_6y_3 + y_5y_6\\ x_8x_3&= x_1y_4 + x_2y_7 + y_1y_2 \end{aligned}$$

where the first two columns give the eight periodic relations

$$\begin{aligned} x_ix_{i+3} = x_{i+1}y_{i+4} + x_{i+2}y_{i-1} + y_{i+1}y_{i+2} \quad i\in {\mathbb {Z}}/8{\mathbb {Z}}\end{aligned}$$

which are a homogenisation of (LR$_3$) by the variables $y_1,\ldots ,y_8$. The binomials appearing in these eight equations correspond to the antipodal vertices in the eight 3-cube faces of C, as shown in Fig. 7b. The last two equations are implied from the first eight and correspond to the two orthoplexes (4-dimensional octahedra) inscribed on the two bipartite decompositions on the set of vertices of C.

The affine cone ${\mathcal {U}}$ We let ${\mathcal {U}}={{\,\textrm{Spec}\,}}(R/I)\subset {\mathbb {A}}^{16}$ be the affine variety defined by these equations, i.e. the affine cone over ${{\,\textrm{OGr}\,}}(5,10)$. Note that there is a map $i:{\mathcal {U}}\rightarrow {\mathcal {U}}$ with $i(x_i)=-y_{3i}$ and $i(y_i)=x_{3i+4}$, which switches the role of the x variables and the y variables. Therefore, after setting $x_i=-1$ for all i we see that $(y_{3i}:i\in {\mathbb {Z}}/8{\mathbb {Z}})$ is also a solution to the Lyness recurrence (LR$_3$).

Anticanonical class The ideal I is a Gorenstein ideal of codimension 5 and has minimal resolution

$$\begin{aligned} R \leftarrow R(-2)^{\oplus 10} \leftarrow R(-3)^{\oplus 16} \leftarrow R(-5)^{\oplus 16} \leftarrow R(-6)^{\oplus 10} \leftarrow R(-8) \leftarrow 0. \end{aligned}$$

From the last module we can read off the adjunction number for ${\mathcal {U}}\subset {\mathbb {A}}^{16}$, giving $-K_{\mathcal {U}}={\mathcal {O}}_{{\mathbb {A}}^{16}}(16-8)|_{\mathcal {U}}={\mathcal {O}}_{\mathcal {U}}(8)$. In particular, if we let ${\mathcal {D}}_i={\mathbb {V}}(y_i)$ and ${\mathcal {E}}_i={\mathbb {V}}(x_i)$ for $i=1,\ldots ,8$, then both ${\mathcal {D}}=\bigcup _{i=1}^8{\mathcal {D}}_i$ and ${\mathcal {E}}=\bigcup _{i=1}^8{\mathcal {E}}_i$ define anticanonical divisors in ${\mathcal {U}}$.

4.1.3 A fibration of ${\mathcal {U}}$ by affine Fano 3-folds

Consider the projection $p:{\mathcal {U}}\rightarrow {\mathbb {A}}^8_{y_1,\ldots ,y_8}$. The fibres of p are a flat family of affine Gorenstein 3-folds which spreads out the components of ${\mathcal {D}}$ above the coordinate strata of ${\mathbb {A}}^8$.

The central fibre $U_0$ The central fibre $U_0:= p^{-1}(0)$ is given by the common intersection of all components of ${\mathcal {D}}\subset {\mathcal {U}}$. It breaks up a reducible affine toric 3-fold with ten components:

$$\begin{aligned} \bigcap _{i=1}^8{\mathcal {D}}_i = {\mathbb {V}}(y_1,\ldots ,y_8) = \bigcup _{i=1}^8 {\mathbb {A}}^3_{x_{i-1},x_i,x_{i+1}} \cup Q_{1357} \cup Q_{2468} \end{aligned}$$

where $Q_{1357} = {\mathbb {V}}(x_1x_5-x_3x_7)\subset {\mathbb {A}}^4_{x_1,x_3,x_5,x_7}$ and $Q_{2468} = {\mathbb {V}}(x_2x_6-x_4x_8)\subset {\mathbb {A}}^4_{x_2,x_4,x_6,x_8}$ are both isomorphic to the cone over the Segre embedding of ${\mathbb {P}}^1\times {\mathbb {P}}^1$. Therefore $U_0$ looks like the cone over a reducible toric surface

$$\begin{aligned} D_P:= D_{123}\cup D_{234} \cup \ldots \cup D_{812} \cup D_{1357} \cup D_{2468}, \end{aligned}$$

with ten components $D_{1357}={\mathbb {P}}(Q_{1357})$, $D_{2468} = {\mathbb {P}}(Q_{2468})$ and $D_{123} = {\mathbb {P}}^2_{x_1,x_2,x_3}$, etc. These components intersect along their toric boundary strata like the polytope P of Fig. 8.

Fig. 8

The polytope P and the dual polytope $Q=P^*$

The general fibre $U_{\lambda ,\mu }$ We now look at a general fibre of this fibration $U_y = p^{-1}(y)$ where $y=(y_1,\ldots ,y_8)\in ({\mathbb {C}}^\times )^8$. We can rescale the coordinates on $U_y$ by

$$\begin{aligned}{} & {} (x_1,\ldots ,x_8)\\{} & {} \quad \mapsto \left( \frac{y_1y_2}{y_4}x_1, \; \frac{y_1y_2}{y_7}x_2, \; \frac{y_2y_3}{y_8}x_3, \; \frac{y_4y_5}{y_7}x_4, \; \frac{y_4y_5}{y_2}x_5, \; \frac{y_4y_5y_8}{y_2y_3}x_6, \; \frac{y_7y_8}{y_2}x_7, \; \frac{y_8y_1}{y_3}x_8 \right) \end{aligned}$$

to find that $U_y \cong U_{\lambda ,\mu }$, for an affine 3-fold $U_{\lambda ,\mu }\subset {\mathbb {A}}^8_{x_1,\ldots ,x_8}$ depending on two parameters $\lambda = \frac{y_3y_7}{y_1y_5}$ and $\mu = \frac{y_2y_6}{y_4y_8}$ with $\lambda ,\mu \in {\mathbb {C}}^\times$. The equations defining $U_{\lambda ,\mu }$ become:

$$\begin{aligned} x_1x_4&= x_2 + \lambda x_3 + \lambda \quad x_5x_8 = x_6 + \lambda x_7 + \lambda \mu \quad x_1x_5 - \lambda x_3x_7 = 1 - \lambda \\ x_2x_5&= \lambda \mu x_3 + x_4 + \lambda \quad x_6x_1 = \lambda x_7 + x_8 + \lambda \quad x_2x_6 - x_4x_8 = \lambda \mu - \lambda \\ x_3x_6&= x_4 + x_5 + 1 \quad x_7x_2 = x_8 + \mu x_1 + 1 \\ x_4x_7&= x_5 + x_6 + \mu \quad x_8x_3 = x_1 + x_2 + 1 \end{aligned}$$

Thus the fibres of p are isomorphic over the intersection of two quadrics in the base given by ${\mathbb {V}}(\lambda y_1y_5- y_3y_7,\; y_2y_6- \mu y_4y_8)\cap ({\mathbb {C}}^\times )^8$.

Lemma 4.1

The projective closure $X_{\lambda ,\mu }:= \overline{U}_{\lambda ,\mu }\subset {\mathbb {P}}^8$ has boundary divisor $D_{\lambda ,\mu }$ isomorphic to $D_P$, and $X_{\lambda ,\mu }$ has 16 ordinary (non-${\mathbb {Q}}$-factorial) nodal singularities along $D_{\lambda ,\mu }$, given by the 8 coordinate points $P_{x_i}$ and the 8 points:

$$\begin{aligned}{} & {} (1:-1)\in {\mathbb {P}}^1_{x_1,x_2}, \quad (\lambda :-1)\in {\mathbb {P}}^1_{x_2,x_3}, \quad (1:-\lambda \mu )\in {\mathbb {P}}^1_{x_3,x_4}, \quad (1:-1)\in {\mathbb {P}}^1_{x_4,x_5},\\{} & {} (1:-1)\in {\mathbb {P}}^1_{x_5,x_6}, \quad (\lambda :-1)\in {\mathbb {P}}^1_{x_6,x_7}, \quad (1:-\lambda )\in {\mathbb {P}}^1_{x_7,x_8}, \quad (\mu :-1)\in {\mathbb {P}}^1_{x_8,x_1}. \end{aligned}$$

Moreover, the interior $U_{\lambda ,\mu }$

• 1. is smooth if both $\lambda ,\mu \ne 1$,

• 2. acquires one node if either $\lambda =1$ or $\mu =1$,

• 3. acquires two nodes if both $\lambda =\mu =1$.

Proof

If $\mu =1$ then the point $(-1,0,-1,0,-1,0,-1,0)$ belongs to $U_{\lambda ,1}$ and we can eliminate the variables $x_1,x_3,x_5,x_7$ in a neighbourhood of this point to find an ordinary nodal singularity with local equation $(x_2x_6-x_4x_8=0) \subset {\mathbb {A}}^4_{x_2,x_4,x_6,x_8}$. Similarly if $\lambda =1$ the point $(0,-1,0,-1,0,-1,0,-1)$ becomes a node of $U_{1,\mu }$. If both $\lambda =\mu =1$ then this gives nodes at two distinct points of $U_{1,1}$. Smoothness elsewhere and the singular locus along $D_{\lambda ,\mu }$ can be checked by computer algebra, e.g. Macaulay2. $\square$

4.1.4 Mirror interpretation of the family ${\mathcal {U}}$

We saw in Remark 3.1 that the fibration we defined in Sect. 3.2 was not quite the same as the mirror family for (X, D), the del Pezzo surface of degree 5, constructed by Gross et al. [12], since their mirror construction produces a family over ${{\,\textrm{Spec}\,}}{\mathbb {C}}[{{\,\mathrm{\overline{NE}}\,}}(X)]$, rather than ${\mathbb {A}}^5$. In a similar vein, the referee of this paper suggested the following more conceptual mirror description of our 3-dimensional family ${\mathcal {U}}$, using the geometry of $U_{1,1}$ that we work out in the rest of this section.

Following Proposition 4.15, there is a compactification $(X_Q,D_Q)$ of the special fibre $U_{1,1}$ whose boundary divisor $D_Q$ consists of eight copies of ${\mathbb {P}}^1\times {\mathbb {P}}^1$ intersecting like the polytope Q. By blowing up the non-Cartier interior divisors of $X_Q$ in some order, we can make a small partial resolution $\psi :\widetilde{X}_Q\rightarrow X_Q$ that resolves 10 of the 12 nodes of $X_Q$ but does not change the intersection complex of $\widetilde{D}_Q$ (and in particular, leaves the two nodes at the two 4-valent vertices of Q intact). The intrinsic mirror symmetry construction of Gross and Siebert [15] then produces a mirror family $p':{\mathcal {U}}'\rightarrow B'$ over the base $B':={{\,\textrm{Spec}\,}}{\mathbb {C}}[\textrm{Nef}(\widetilde{X}_Q)]$ with the same central fibre $U_0$ as our family ${\mathcal {U}}$.

This mirror family is equivariant for the action of the torus ${\mathbb {T}}$ on $B'$, where ${\mathbb {T}}:= {\mathbb {T}}_{\widetilde{D}_Q}\cap {{\,\textrm{Pic}\,}}\widetilde{X}_Q$ and ${\mathbb {T}}_{\widetilde{D}_Q}$ is the torus of rank 8 generated by the irreducible components of $\widetilde{D}_Q$. In this case ${\mathbb {T}}$ has rank 6, since the coefficients of a Cartier divisor supported on $\widetilde{D}_Q$ must satisfy two linear conditions imposed by the two nodes. From the description of $\widetilde{X}_Q$ as the blowup of a toric variety (cf. Proposition 4.2) one can compute that ${{\,\textrm{Pic}\,}}\widetilde{X}_Q$ has rank 8, and thus ${{\,\textrm{Pic}\,}}U_{1,1}\cong {{\,\textrm{Pic}\,}}\widetilde{X}_Q/{\mathbb {T}}$ is a lattice of rank 2, confirming that the fibres of the mirror family ${\mathcal {U}}'$ depend on two deformation parameters $\lambda$ and $\mu$. These two deformation parameters correspond to the two $\psi$-exceptional ${\mathcal {O}}(-1,-1)$-curves over the two interior nodes of $U_{1,1}$ on one side of the mirror, and the two vanishing Lagrangian spheres $S^3$ in the smoothing of $U_{1,1}$ on the other side.

As in Remark 3.1, our family ${\mathcal {U}}$ should now rather be viewed as an extension of this mirror family $p':{\mathcal {U}}'\rightarrow B'$ to a family $p:{\mathcal {U}}\rightarrow {\mathbb {A}}^8$ defined over the base ${\mathbb {A}}^8={{\,\textrm{Spec}\,}}{\mathbb {C}}[\sigma ^\star ]$, where $\sigma ^\star \supseteq \textrm{Nef}(\widetilde{X}_Q)$ is dual to the cone $\sigma \subset A_1(\widetilde{X}_Q)$ generated by the classes of 1-strata in the boundary of $(\widetilde{X}_Q,\widetilde{D}_Q)$. This cone is independent of the choice of small resolution $\psi$ of $X_Q$, and one would expect it to be equal to the cone containing $\textrm{Nef}(\widetilde{X}_Q)$ in the secondary fan of $(\widetilde{X}_Q,\widetilde{D}_Q)$, as defined by Hacking et al. [18].

4.2 The affine Fano-3-fold $U_{\lambda ,\mu }$

We now study the affine Fano 3-fold $U_{\lambda ,\mu } = X_{\lambda ,\mu }{{\setminus }} D_{\lambda ,\mu }$, which has a (generalised) cluster structure and is the 3-dimensional analogue of the affine del Pezzo surface considered in Sect. 3.

4.2.1 Blowup description of $U_{\lambda ,\mu }$

One way that the affine del Pezzo surface can be obtained is by blowing up one point on each of the coordinate axis in ${\mathbb {A}}^2$ and deleting the strict transform of the boundary. We now give a similar description for the 3-folds $U_{\lambda ,\mu }$.

Proposition 4.2

Let $H={\mathbb {V}}(x_1x_2x_3)\subset {\mathbb {A}}^3$ denote the union of the coordinate hyperplanes. There is a locus $Z\subset H\subset {\mathbb {A}}^3$ such that $U_{\lambda ,\mu } \cong Bl_Z({\mathbb {A}}^3) {{\setminus }} \widetilde{H}$—i.e. the blowup of Z minus the strict transform of H. According to the cases of Lemma 4.1:

• 1. if $\lambda ,\mu \ne 1$ then Z consists of a conic and two lines

  $$\begin{aligned} Z = {\mathbb {V}}(x_1, \lambda + x_2 + \lambda x_3)&\cup {\mathbb {V}}(x_2, 1 + x_1 + x_3 + \mu x_1x_3 )\\&\cup {\mathbb {V}}(x_3, 1 + x_1 + x_2), \end{aligned}$$

• 2. if $\lambda \ne 1,\mu =1$ then Z consists of four lines (two lines and a reducible conic)

  $$\begin{aligned} Z = {\mathbb {V}}(x_1, \lambda + x_2 + \lambda x_3)&\cup {\mathbb {V}}(x_2, 1 + x_1 ) \cup {\mathbb {V}}(x_2, 1 + x_3 ) \\&\cup {\mathbb {V}}(x_3, 1 + x_1 + x_2), \end{aligned}$$

• 3. if $\lambda =1,\mu \ne 1$ then Z consists of a conic and two lines meeting at an embedded point

  $$\begin{aligned} Z&= {\mathbb {V}}(x_1, 1 + x_2 + x_3) \cup {\mathbb {V}}(x_2, 1 + x_1 + x_3 + \mu x_1x_3)\\&\cup {\mathbb {V}}(x_3, 1 + x_1 + x_2) \cup {\mathbb {V}}(x_1, 1 + x_2, x_3), \end{aligned}$$

• 4. if $\lambda =\mu =1$ then Z consists of a reducible conic and two lines meeting at an embedded point

  $$\begin{aligned} Z&= {\mathbb {V}}(x_1, 1 + x_2 + x_3) \cup {\mathbb {V}}(x_2, 1 + x_1) \cup {\mathbb {V}}(x_2, 1 + x_3 ) \\&\cup {\mathbb {V}}(x_3, 1 + x_1 + x_2) \cup {\mathbb {V}}(x_1, 1 + x_2, x_3). \end{aligned}$$

Remark 4.3

Before proving the proposition, we briefly describe the effect of the blowups we make and explain how $U_{\lambda ,\mu }$ ends up with the number of nodes expected from Lemma 4.1.

• 1. The blowup $Bl_Z({\mathbb {A}}^3)$ obtains an ordinary node in the fibre over any point $P\in Z$ contained in the intersection of two curves. If P is contained in a 1-stratum of the boundary then after the blowup the node is contained in $\widetilde{H}$ and hence does not appear in $U_{\lambda ,\mu }$. However, if P is contained in the interior of a 2-stratum (as happens in cases 2. and 4.) then the node is not contained in $\widetilde{H}$ and therefore must appear in $U_{\lambda ,\mu }$.

• 2. Blowing up an embedded point in the intersection of two lines also produces a node in the interior $U_{\lambda ,\mu }$ (as happens in cases 3. and 4.). To see how, we can first blow up the embedded point with exceptional divisor E. Let $L\subset E$ be the line that passes through the strict transform of the two lines on either side. Blowing up the lines turns L into a contractible ${\mathcal {O}}(-1,-1)$ curve, which is contracted to an ordinary node.

  

• 3. It is interesting that cases 2. and 3. give two different constructions of varieties that should be isomorphic by exchanging the role of $\lambda$ and $\mu$. Indeed, we can send the configuration 3. to the configuration 2. by the mutating one of the two line components, e.g. we blow up the line ${\mathbb {V}}(x_3,1+x_1+x_2)$ and then contract the strict transform of the divisor ${\mathbb {V}}(1+x_1+x_2)$ to the plane at infinity.

  

Proof

Consider $g :U_{\lambda ,\mu }\subset {\mathbb {A}}^8 \rightarrow {\mathbb {A}}^3_{x_1,x_2,x_3}$, which is a birational projection of $U_{\lambda ,\mu }$ onto its image. This image is the constructible set

$$\begin{aligned} g(U_{\lambda ,\mu }) = ({\mathbb {C}}^\times )^3_{x_1,x_2,x_3} \cup Z^0, \quad \text {where}\quad Z^0=\left\{ \begin{array}{ll} Z{\setminus }{\mathbb {V}}(x_1,x_3) &{} \quad \text {if}\;\lambda \ne 1 \\ Z &{} \quad \text {if}\;\lambda = 1 \end{array}\right. \end{aligned}$$

and Z is as in the statement of the proposition. (In other words, if $\lambda \ne 1$ the image of $U_{\lambda ,\mu }$ misses the two points of Z that lie on the $x_2$-axis.) Now g restricts to an isomorphism on the open torus $({\mathbb {C}}^\times )^3_{x_1,x_2,x_3}\subset U_{\lambda ,\mu }$ and we can check that the fibres over $z\in Z$ are all affine lines $g^{-1}(z)\cong {\mathbb {A}}^1$, unless $\lambda =1$ and $z=(0,-1,0)$, in which case $g^{-1}(z) \cong {\mathbb {A}}^2$. We can resolve the inverse map $g^{-1}:g(U_{\lambda ,\mu }) \dashrightarrow U_{\lambda ,\mu }$ by blowing up $Z\subset {\mathbb {A}}^3$, as described in Remark 4.3. $\square$

4.2.2 The exchange graph of $U_{\lambda ,\mu }$

Recall our fibration $p:{\mathcal {U}}\rightarrow {\mathbb {A}}^8_{y_1,\ldots ,y_8}$, which we can think of as a family of affine 3-folds over the coefficient ring $R_y={\mathbb {C}}[y_1,\ldots ,y_8]$. We let ${\mathbb {A}}_y:={\mathbb {A}}^8_{y_1,\ldots ,y_8}$ denote the base of this fibration. In the traditional language of cluster algebras, we think of the x variables as cluster variables on ${\mathcal {U}}$ and the y variables as frozen variables. Moreover, we recall the two quadratic terms $q_1,q_2$ which we introduced in Sect. 1.2.2. These are homogenised as follows:

$$\begin{aligned} q_1:= x_1x_5 - y_1y_5 = x_3x_7 - y_3y_7, \quad q_2:= x_2x_6 - y_2y_6 = x_4x_8 - y_4y_8. \end{aligned}$$

For convenience of notation, we let $q_1=q_3=q_5=q_7$ and $q_2=q_4=q_6=q_8$. For reasons that will shortly become clear (see Remark 4.5), we want to add $q_1$ and $q_2$ to our list of cluster variables.

Lemma 4.4

For any $i=1,\ldots ,8$, we have

$$\begin{aligned} x_1,\ldots ,x_8,q_1,q_2&\in R_y[x_{i-1}^{\pm 1},x_i^{\pm 1},x_{i+1}^{\pm 1}] \quad \text {and}\\ x_1,\ldots ,x_8,q_1,q_2&\in R_y[x_{i}^{\pm 1},q_{i}^{\pm 1},x_{i+2}^{\pm 1}]. \end{aligned}$$

Proof

This is straightforward to check by simply expanding all of the cluster variables in the corresponding Laurent ring. $\square$

Fig. 9

The exchange graph G for the affine charts covering ${\mathcal {U}}$

We let ${\mathbb {T}}_{i-1,i,i+1}:= {{\,\textrm{Spec}\,}}R_y[x_{i-1}^{\pm 1},x_i^{\pm 1},x_{i+1}^{\pm 1}]$ and ${\mathbb {T}}_{i,q,i+2} = {{\,\textrm{Spec}\,}}R_y[x_{i}^{\pm 1},q_{i}^{\pm 1},x_{i+2}^{\pm 1}]$ for $i\in {\mathbb {Z}}/8{\mathbb {Z}}$. Then Lemma 4.4 shows that these give a system of 16 open affine charts in ${\mathcal {U}}$ of the form ${\mathbb {T}}=({\mathbb {C}}^\times )^3\times {\mathbb {A}}_y$, which we can represent at the vertices of the exchange graph G shown in Fig. 9. The graph is the 1-skeleton of a 3-dimensional polytope and we can label the faces of this polytope so that the three faces around each vertex give the coordinates on the torus factor of each chart. The edges of G correspond to the three different types of exchange relation:

$$\begin{aligned} x_{i-1}x_{i+2}&= x_iy_{i+3} + x_{i+1}y_{i-2} + y_iy_{i+1}, \\ x_{i-2}x_{i+2}&= q_i + y_{i-2}y_{i+2}, \\ x_iq_{i-1}&= x_{i-1}x_{i+1}y_{i+4} + x_{i-1}y_{i+1}y_{i+2} + x_{i+1}y_{i-1}y_{i-2} + y_{i-1}y_iy_{i+1}. \end{aligned}$$

Remark 4.5

We can also now see the geometrical reason as to why we are led to include $q_1$ and $q_2$ in our list of cluster variables. In the coordinates on the fibre $U_{\lambda ,\mu }$ of ${\mathcal {U}}$ described above, and in the case $\lambda ,\mu \ne 1$, from the blowup up description $U_{\lambda ,\mu }=\textrm{Bl}_Z({\mathbb {A}}^3_{x_1,x_2,x_3}) {{\setminus }} \widetilde{H}$ of Proposition 4.2 we see that the exchange map

$$\begin{aligned} \left( x_1,x_2,x_3\right) \mapsto \left( x_2,x_3,\frac{\lambda x_2+x_3 + \lambda }{x_1}\right) , \end{aligned}$$

is a mutation along one of the two line components of Z. Similarly, the exchange map

$$\begin{aligned} \left( x_1, x_2, x_3 \right) \mapsto \left( x_1, \frac{\mu x_1x_3 + x_1 + x_3 + 1}{x_2}, x_3 \right) , \end{aligned}$$

is a mutation along the conic component of Z. This transforms the centres along which we obtain our blowup description as follows: now $U_{\lambda ,\mu }=\textrm{Bl}_{Z'}({\mathbb {A}}^3_{x_1,q_1,x_3}){{\setminus }} {\widetilde{H}}'$ where $H'={\mathbb {V}}(x_1q_1x_3)$ and $Z'\subset H'$ is the union of the following three components.

From this we see that the final type of exchange map

$$\begin{aligned} \left( x_1, q_1, x_3 \right) \mapsto \left( x_3, q_1, \frac{\lambda q_1 + 1}{x_1} \right) , \end{aligned}$$

is a mutation along one of the two line components of $Z'$. The nice surprise is that this system of torus charts and mutations forms a closed system (i.e. composing mutations around closed cycles in G gives the identity map on the corresponding torus chart).

Remark 4.6

In the cases where one or both of $\lambda ,\mu =1$ the number of centres to blow up in the locus Z increases from three to either four or five respectively. Therefore the valency of the mutation graph also increases. After a fairly involved computation explicitly tracking the centres of Z (which includes having to deal with unexpected jumps similar to those described in Sect. 2.2.3), it can be shown that the exchange graph of $U_{1,\mu }$ (or equivalently $U_{\lambda ,1}$) is 4-valent with 28 vertices. Similarly the exchange graph of $U_{1,1}$ is 5-valent with 48 vertices. In particular, these exchange graphs are also finite. The exchange graph for the rank 5 case is a bit too complicated to draw elegantly, however the rank 4 case gives the following graph

which is the 1-skeleton of a 4-dimensional polytope. Visually, we see that it can be ‘collapsed’ onto the exchange graph G of Fig. 9.

4.2.3 The interior divisors inside $U_{\lambda ,\mu }$

We consider the following ten interior divisors of $U_{\lambda ,\mu }$

$$\begin{aligned} E_i={\mathbb {V}}(x_i) \text { for } i=1,\ldots ,8 \quad \text {and} \quad F_i={\mathbb {V}}(q_i) \text { for } i=1,2. \end{aligned}$$

These are all the divisors that appear in the complement of one of the 16 cluster tori and are the analogue of the five interior $(-1)$-curves of the del Pezzo surface in Sect. 3. Because of the jump in the rank of the class group ${{\,\textrm{Cl}\,}}(U_{\lambda ,\mu })$, something special happens to these divisors when either or both of $\lambda ,\mu =1$.

Proposition 4.7

• 1. If $\lambda ,\mu \ne 1$, then the divisors $E_i$ and $F_i$ are all irreducible.

• 2. If $\mu =1$, the four ‘odd’ divisors $E_1,E_3,E_5,E_7$ break up into four overlapping irreducible components $E_{13},E_{35},E_{57},E_{71}$, where $E_{13} = {\mathbb {V}}(x_1,x_3)$ etc. Moreover $F_2$ breaks into two components as $F_2=F_{26}\cup F_{48}$, where $F_{26} = {\mathbb {V}}(1+x_2,1+x_6)$ and $F_{48} = {\mathbb {V}}(1+x_4,1+x_8)$.

• 3. If $\lambda =1$, the four ‘even’ divisors $E_2,E_4,E_6,E_8$ and $F_1$ all break into two components, with the analogous description to (2).

In particular, if $\lambda =\mu =1$ then all ten divisors break up into a total of twelve components.

Proof

We consider the image of these divisors under the projection $g:U_{\lambda ,\mu }\rightarrow {\mathbb {A}}^3_{x_1,x_2,x_3}$ with the discriminant locus $Z\subset {\mathbb {A}}^3$ as in Proposition 4.2. For $\lambda ,\mu \ne 1$ we find that $E_1,E_2,E_3$ are the exceptional divisors over the three components of Z and the remaining divisors are given by the strict transform under $g^{-1}$ of the following (irreducible) divisors in ${\mathbb {A}}^3$.

$$\begin{aligned} \begin{array}{ll} E_1: x_1 = \lambda + x_2 + \lambda x_3 = 0 &{} \quad E_6: (1+x_1+x_2)(\lambda +x_2+\lambda x_3) = \lambda (1-\mu )x_1x_3 \\ E_2: x_2 = 1 + x_1 + x_3 + \mu x_1x_3 = 0 &{} \quad E_7: 1 + x_1 + x_2 + x_3 + \mu x_1x_3 = 0 \\ E_3: x_3 = 1 + x_1 + x_2 = 0 &{} \quad E_8: 1 + x_1 + x_2 = 0 \\ E_4: \lambda + x_2 + \lambda x_3 = 0 &{} \quad F_1: 1 + x_1 + x_3 + \mu x_1x_3 = 0 \\ E_5: \lambda + \lambda x_1 + x_2 + \lambda x_3 + \lambda \mu x_3 = 0 &{} \quad F_2: (\lambda + x_2 + \lambda x_3)(1 + x_1 + x_2) = \lambda x_1x_3 \end{array} \end{aligned}$$

If $\mu =1$ the conic component of Z breaks into two pieces. This has the effect of breaking all the divisors $E_2,E_4,E_6,E_8$ and $F_1$ in two. If $\lambda =1$ then the two line components of Z meet at an embedded point. This has the effect of breaking all the divisors $E_1,E_3,E_5,E_7$ and $F_2$ in two. In the most extreme case when $\lambda =\mu =1$, the ten divisors break up into twelve components as follows.

$$\begin{aligned} \begin{array}{lll} E_{71}: x_1 = 1 + x_2 + x_3 = 0 &{} \quad E_{35}: x_3 = 1 + x_1 + x_2 = 0 &{} \quad F_{15}: 1 + x_1 = 0 \\ E_{82}: x_2 = 1 + x_1 = 0 &{} \quad E_{46}: 1 + x_2 + x_3 = 0 &{} \quad F_{26}: 1 + x_2 = 0 \\ E_{13}: x_1 = 1 + x_2 = x_3 = 0 &{} \quad E_{57}: 1 +x_1+x_2+x_3+x_1x_3 = 0 &{} \quad F_{37}: 1+x_3 = 0 \\ E_{24}: x_2 = 1 + x_3 = 0 &{} \quad E_{68}: 1 + x_1 + x_2 = 0 &{} \quad F_{48}: 1+x_1+x_2+x_3 = 0 \end{array} \end{aligned}$$

$\square$

4.3 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 Sect. 2.3.1), the dual space $M_U$ and the dual intersection pairing $\langle {\cdot },{\cdot }\rangle :N_U\times M_U\rightarrow {\mathbb {R}}$.

4.3.1 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 $\vartheta _m\in {\mathbb {C}}[U]$ and any boundary component $D_n$ we let $\langle D_n,\vartheta _m \rangle := {{\,\textrm{ord}\,}}_{D_n}(\vartheta _m)$ denote the order of vanishing of $\vartheta _m$ along $D_n$. We start by computing the analogous matrix to Table 1 that we will use to define the intersection pairing $\langle {\cdot },{\cdot } \rangle :N_U\times M_U\rightarrow {\mathbb {R}}$. 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 ${\mathbb {C}}[U]$ and the ten boundary divisors $D_{123},\ldots , D_{812},D_{1357},D_{2468}$ of (X, D).

Proposition 4.8

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

$$\begin{aligned} {{\,\textrm{div}\,}}x_1&= E_1 + D_{456} - D_{781} - D_{812} - D_{123} - D_{1357} \\ {{\,\textrm{div}\,}}q_1&= F_1 + D_{2468} - D_{123} - D_{345} - D_{567} - D_{781} - 2D_{1357} \end{aligned}$$

and similarly for the other cluster variables, up to the ${{\,\textrm{Dih}\,}}_8$ 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 2.

Proof

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

$$\begin{aligned} {{\,\textrm{div}\,}}x_1 = E_1 + D_{234} + D_{345} + 2D_{456} + D_{567} + D_{678} + D_{2468} \end{aligned}$$

and the second formula follows similarly for $q_1\in |{\mathcal {O}}_X(2)|$. This now follows from an explicit calculation with the equations of X. $\square$

Table 2 The intersection numbers $\langle D_i,x_j \rangle$ for U

Given the correspondence in Table 2 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 $i\in {\mathbb {Z}}/8{\mathbb {Z}}$ and $D_{q_1}=D_{2468}$ and $D_{q_2}=D_{1357}$ correspond to $q_1$ and $q_2$. We will us Table 2 to define a complete fan ${\mathcal {F}}$ in ${\mathbb {R}}^3$ which will give us both a toric model and a scattering diagram for U.

4.3.2 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 $j:{\mathbb {T}}_{1q3}\hookrightarrow U$ with coordinates $x_1,q_1,x_3$, determined by the seed $Z'\subset {\mathbb {A}}^3$ described in Remark 4.5. This is one mutation away from the original torus chart ${\mathbb {T}}_{123}$ with coordinates $x_1,x_2,x_3$, corresponding to the seed $Z\subset {\mathbb {A}}^3$ which we used to describe a toric model for U in Proposition 4.2. The reason for this change in the choice of seed is that the irreducible components of $Z'\subset {\mathbb {A}}^3$ are all smooth and disjoint (as opposed to those of Z). Thus we can obtain a description of the tropicalisation of U by blowing up $Z'$ directly, without having to blow up any of the 1-dimensional strata of ${\mathbb {A}}^3$.

Consider the ten primitive integral vectors $v_1,\ldots ,v_8,w_1,w_2\in {\mathbb {Z}}^3$ corresponding to the following columns

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

read off as the negative of the rows corresponding to $x_1,q_1,x_3$ in Table 2. We can define a complete fan ${\mathcal {F}}$ in ${\mathbb {R}}^3$ which has rays generated by these ten vectors and whose cones are dual to the exchange graph in Fig. 9. 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 Fig. 10.

Fig. 10

The fan ${\mathcal {F}}$ giving both a toric model of U, and the scattering diagram for ${\mathbb {C}}[U]$

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

$$\begin{aligned} S = \left\{ \left( (-1,0,0), \lambda q_1 + 1 \right) , \left( (0,-1,0), \mu x_1x_3 + x_1 + x_3 + 1 \right) , \left( (0,0,-1), q_1 + 1\right) \right\} \end{aligned}$$

described in Remark 4.5, corresponding to the torus chart with coordinates $x_1,q_1,x_3$.

Proposition 4.9

The pair $\pi :(\widetilde{X},\widetilde{D}) \rightarrow (T,B)$ 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.

Proof

The proof that $\pi :(X,D)\rightarrow (T,B)$ is a toric model for U follows from extending a similar calculation to the proof of Proposition 4.2 from the toric variety ${\mathbb {A}}^3$ 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 ${\mathbb {A}}^3_\sigma = {{\,\textrm{Spec}\,}}{\mathbb {C}}[\sigma ^\vee \cap M]\subset T\dashrightarrow X$ 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 3.3). 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 4.1). These are the eight boundary strata corresponding to the cones $\langle x_i,q_i \rangle$ for $i\in {\mathbb {Z}}/8{\mathbb {Z}}$ 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 ${\mathbb {P}}^1_{x_i,x_{i+1}}$. These sixteen curves are indicated by the dashed lines in Fig. 11. $\square$

Remark 4.10

In Proposition 4.9 we obtained our projective compactification $(\widetilde{X},\widetilde{B})$ of U by blowing up $Z'\subset T$. This construction uses a different seed to the seed $Z\subset {\mathbb {A}}^3$ appearing in Proposition 4.2, which can be used to obtain an affine partial compactification $(\textrm{Bl}_Z({\mathbb {A}}^3),\widetilde{H})$ of U. Nevertheless, $(\widetilde{X},\widetilde{B})$ is a compactification of a small resolution of $\textrm{Bl}_Z({\mathbb {A}}^3)$ in the following way. Note that $\textrm{Bl}_Z({\mathbb {A}}^3)$ is singular, since the components of $Z\subset {\mathbb {A}}^3$ meet at points in the $x_1$-axis and $x_3$-axis. Consider the small resolution $\psi :(W,H_W)\rightarrow (\textrm{Bl}_Z({\mathbb {A}}^3),\widetilde{H})$, where W is obtained by first blowing up the conic component of $Z\subset {\mathbb {A}}^3$, and then the two lines. This $(W,H_W)$ can be embedded inside $(\widetilde{X},\widetilde{B})$ where the boundary components correspond to $B_{(-1,0,0)}$, $B_{(0,1,0)}$ and $B_{(0,0,-1)}$, and the two $\psi$-exceptional lines are the two dashed vertical lines in the relative interiors of $B_{(-1,0,0)}$ and $B_{(0,0,-1)}$, appearing in Fig. 11.

Fig. 11

A toric model $\pi :(\widetilde{X},\widetilde{D})\rightarrow (T,B)$ 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 Sect. 2.3.1. We arrive at the following description.

Lemma 4.11

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

$$\begin{aligned} \langle v_7,v_1\rangle , \, \langle v_1,v_3\rangle , \, \langle v_3,v_5\rangle , \, \langle v_1,w_1\rangle , \, \langle v_3,w_1\rangle , \, \langle v_5,w_1\rangle , \, \langle v_7,w_1\rangle . \end{aligned}$$

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

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

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

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

The singular locus of $N_U$ is given by the four rays ${\mathbb {R}}_{\ge 0}v_i$ for $i=1,3,5,7$, corresponding to the four non-toric boundary divisors of $(\widetilde{X},\widetilde{D})$.

4.3.3 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. 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 $j=q$.

Proposition 4.12

Decorating the walls of the fan ${\mathcal {F}}$ with the wall functions given in Table 3 defines a consistent scattering diagram ${\mathfrak {D}}_{\lambda ,\mu }$ in $N_U$.

Table 3 Wall functions for the scattering diagram ${\mathfrak {D}}_{\lambda ,\mu }$

Proof

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. $\square$

Remark 4.13

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,⁹ 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 3. 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 $v_1,\ldots ,v_8,v_{q_1},v_{q_2}\in N_U({\mathbb {Z}})$ 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 4.11. Therefore, by a similar argument to the dimension 2 case Sect. 3.3.3, for all $a,b,c\ge 0$ the integral point $av_1+bv_2+cv_3\in N_U({\mathbb {Z}})$ 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.

Relationship to the Argüz–Gross scattering diagram As described in Sect. 2.3.2, for any log Calabi–Yau variety with a suitably nice toric model Argüz and Gross [2] define a consistent scattering diagram by an inductive procedure, starting from an initial scattering diagram. Following this procedure in our case essentially recovers our special scattering diagram ${\mathfrak {D}}_{1,1}$.

Indeed, given the three centres $Z_1={\mathbb {V}}(x_1,1+q_1)$, $Z_2={\mathbb {V}}(q_1,1+x_1+x_3+\mu x_1x_3)$ and $Z_3={\mathbb {V}}(x_3,1+\lambda q_1)$ in our toric variety X, their initial scattering diagram is built by attaching the nontrivial wall functions $1+t_1z_1$ to ${\mathfrak {d}}_{71}$, $1+t_2z_2$ to ${\mathfrak {d}}_{1q}$, ${\mathfrak {d}}_{3q}$, ${\mathfrak {d}}_{5q}$ and ${\mathfrak {d}}_{7q}$, $1+t_3z_3$ to ${\mathfrak {d}}_{35}$ and $(1+t_1z_1)(1+t_3z_3)$ to ${\mathfrak {d}}_{13}$. Here $t_1$, $t_2$ and $t_3$ are formal parameters (one for each centre $Z_i$) introduced to control the convergence of the wall functions in the case that the scattering diagram turns out not to be of finite type. The consistent scattering diagram obtained by their procedure should be the same as the one obtained from our diagram ${\mathfrak {D}}_{1,1}$ after substituting $z_i\mapsto t_iz_i$ for $i=1,2,3$.

Enumerative interpretation of the walls Moreover, given our consistent scattering diagram ${\mathfrak {D}}_{1,1}$ it should now be possible to go one step further and understand the enumerative geometry of the log Calabi–Yau pair $(\widetilde{X},\widetilde{D})$ in similar terms to the example studied in [2, §7]. Namely, for the wall functions $f_{{\mathfrak {d}}}$ appearing in the scattering diagram, one should be able to interpret the coefficients of the power series $\log f_{{\mathfrak {d}}}$ as punctured Gromov–Witten invariants for $(\widetilde{X},\widetilde{D})$. In particular, each of the eight interior divisors $E_{i,i+2}\subset U_{1,1}$ is isomorphic to ${\mathbb {A}}^2_{u,v}$, and hence each one contains a 2-dimensional family of ${\mathbb {A}}^1$-curves $\{u^a=v^b: a,b\in {\mathbb {Z}}_{\ge 0}\}$ (i.e. smooth rational curves which, after further resolving $(\widetilde{X},\widetilde{D})$ by toric blowups, intersect the boundary $\widetilde{D}$ transversely in one point). These should correspond to the eight wall functions of the form $1+x+y+xy$ decorating ${\mathfrak {d}}_{i,i+2}$. The four interior divisors $F_{i,i+4}\subset U_{1,1}$ are all isomorphic to ${\mathbb {A}}^1\times {\mathbb {C}}^\times$, and hence each one contains a unique deformation family of ${\mathbb {A}}^1$-curves. These should correspond to the four wall functions of the form $1+x$ decorating ${\mathfrak {d}}_{i,q}\cup {\mathfrak {d}}_{i+4,q}$. Lastly, the eight wall functions of the form $1+x+y$ should correspond to the eight ${\mathcal {O}}(-1,-1)$-curves in $\widetilde{D}\subset \widetilde{X}$ that are contracted to the eight (non-coordinate point) nodes $p\in D\subset X$ described in Lemma 4.1. Indeed, taking an appropriate affine neighbourhood of any one of these nodes gives the local SYZ singularity $X_{2,1}$ in the notation of [13], and the mirror $X_{1,2}$ has scattering diagram given by a single wall with a wall function of exactly this form.

4.4 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 ${\mathbb {T}}_{123}$ all now satisfy the binomial edge coefficient condition of Remark 2.14, which we expect for our mirror Landau–Ginzburg potentials. We continue to let (X, D) be the pair obtained by taking the projective closure $X=\overline{U}\subset {\mathbb {P}}^8$ with boundary divisor D.

4.4.1 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 Sect. 1.2.2.

Proposition 4.14

The polytopes $P = {{\,\textrm{conv}\,}}\left( x_1,\ldots ,x_8 \right)$ and $Q={{\,\textrm{conv}\,}}\big ( x_1,\ldots , x_8, q_1, q_2 \big )$ are a pair of dual polytopes in $M_U$. Moreover they are, combinatorially, a realisation of the two polytopes of Fig. 8 when considered in the affine structure of $M_U$.

Proof

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

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 $\phi _P:N_U\rightarrow {\mathbb {R}}$ given by $\phi _P(n) = \min \{ c\in {\mathbb {R}}: n\in cP \}$, which is linear in each cone of our fan ${\mathcal {F}}$. Since $\phi _P(v_i) = 1$ for all $i\in {\mathbb {Z}}/8{\mathbb {Z}}$ 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 Sect. 2.3.1, 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 $\widetilde{X}$. $\square$

We now consider the two compactifications $(X_P,D_P)$ and $(X_Q,D_Q)$ of U which were discussed in Sect. 1.2.2. 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 $\psi :X_P\dashrightarrow X_Q$, called an unprojection. Geometrically, this contracts the two ${\mathbb {P}}^1\times {\mathbb {P}}^1$ 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 ${\mathcal {O}}_X(1)$, as generators to the homogeneous coordinate ring ${\mathbb {C}}[X_P]$.

Proposition 4.15

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

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

Proof

This is an application of the theory of unprojection developed by Papadakis–Reid [25]. The divisor $D_{q_1} = {\mathbb {V}}(x_0,x_2,x_4,x_6,x_8,x_1x_5-x_3x_7)\subset {\mathbb {P}}^8$ is a complete intersection of codimension 6 (hence projectively Gorenstein) and contained in $X_P\subset {\mathbb {P}}^8$ which is projectively Gorenstein of codimension 5. By adjunction we have $K_{X_P}={\mathcal {O}}_{X_P}(-1)$ and $K_{D_{q_1}}={\mathcal {O}}_{D_{q_1}}(-2)$, and since $(-1)-(-2)=1>0$ we can apply [25, Theorem 1.5] to obtain an unprojection variable $q_1$ as a rational section of ${\mathcal {O}}_{X_P}(1)$. We compute that $q_1=x_0^{-1}(x_1x_5-x_0^2)$ works as an unprojection variable, and satisfies the equations

$$\begin{aligned} x_0q_1&= x_1x_5 - x_0^2 = x_3x_7 - x_0^2, \\ x_{i}q_1&=(x_0+x_{i-1})(x_0+x_{i+1}) \quad \text {for}\; i=2,4,6,8 \end{aligned}$$

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 $(D_{q_2}\subset X_P')\dashrightarrow (P_{q_2}\in X_Q)$. $\square$

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).

Note that the unprojection $\psi :X_P\dashrightarrow X_Q$ 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 2.14, 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.

$$\begin{aligned} w_Q = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + q_1 + q_2. \end{aligned}$$

We note that this is $\sigma _3$-invariant function and, indeed, restricting to the cluster torus chart ${\mathbb {T}}_{123}\subset U$ gives

$$\begin{aligned} (w_Q + 5)|_{{\mathbb {T}}_{123}} = \frac{(1 + x_1 + x_2)(1 + x_2 + x_3) (1 + x_1 + x_2 + x_3 + x_1x_3)}{x_1x_2x_3} \end{aligned}$$

which is well-known to be an invariant function for Lyness map $\sigma _3$.

The period $\pi _{w_Q}(t)$ Consider the period

$$\begin{aligned} \pi _{w_Q}(t)&= 1 + 48t^2 + 600t^3 + 13176t^4 + 276480t^5 + 6259800t^6\\&\quad + 146064240t^7 + \cdots \end{aligned}$$

which agrees with the regularised quantum period $\widehat{G}_X(t)$ for the Fano 3-fold X of type $V_{12}$, as listed in [8, §13]. Alternatively, the shifted potential $w_Q+5$ has period

$$\begin{aligned} \pi _{w_Q+5}(t) = 1 + 5t + 73t^2 + 1445t^3 + 33001t^4 + 819005t^5 + 21460825t^6+ \cdots \end{aligned}$$

which can be recognised as the Apéry series $\pi _{w+5}(t) = \sum _{n=0}^\infty \sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) ^2 \left( {\begin{array}{c}n+k\\ k\end{array}}\right) ^2 t^n$ and is also well-known as a mirror period sequence for the Fano 3-fold $V_{12}$, cf. [16].

A K3 fibration As we saw in the 2-dimensional examples of Sect. 3.4.2, smoothing a singular log Calabi–Yau pair (X, D) to a pair $(\widetilde{X},\widetilde{D})$ consisting of a smooth Fano 3-fold $\widetilde{X}$ with smooth anticanonical boundary divisor $\widetilde{D}$, corresponds under mirror symmetry to a compactification of the fibres of the mirror Landau–Ginzburg fibration. The Landau–Ginzburg model mirror to the smooth Fano 3-fold $V_{2k}$ of Picard rank 1, is a fibration of K3 surfaces of Picard rank 19 (the fibration $\hat{{\mathcal {X}}}_k$ in the notation of [11]), and our self-mirror log Calabi–Yau 3-fold appears as an open subset in the fibration $\hat{{\mathcal {X}}}_6$, as we now describe.

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 $(w_Q + 5)|_{{\mathbb {T}}_{123}}$ corresponds to the fact that the interior divisor $E_Q:= w_Q^{-1}(-5) \subset U$ is reducible. Then, in the notation of Proposition 4.7, the reducible fibre is $E_Q = \cup _{i=1}^8 E_{i,i+2}$ which is the union of eight copies of ${\mathbb {P}}^1\times {\mathbb {P}}^1$ 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 $(1:-1)\in {\mathbb {P}}^1_{x_i,x_{i+1}}$ in the boundary of $X_Q$. These lines are shown in Fig. 12.

Fig. 12

a 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 $E_P$

Now the fibres of $w_Q:X_Q\dashrightarrow {\mathbb {P}}^1$ 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 Fig. 13a, 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.

Fig. 13

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 $w_Q:U\rightarrow C$, 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

$$\begin{aligned} (t + 5)x_1x_2x_3 = (1 + x_1 + x_2)(1 + x_2 + x_3) \left( 1 + x_1 + x_2 + x_3 + x_1x_3\right) \end{aligned}$$

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 $\lambda _\pm = 24(1 \pm \sqrt{2})$ which acquire an $A_1$ singularity. We get a picture as in the 2-dimensional case.

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

$$\begin{aligned} \pi _{w_P}(t) = 1 + 24t^2 + 192t^3 + 2904t^4 + 40320t^5 + 611520t^6 + 9515520t^7 + \cdots \end{aligned}$$

which agrees with the regularised quantum period $\widehat{G}_X(t)$ for the Fano 3-fold X of type $V_{16}$, as listed in [8, §15].

In this case the pencil of affine K3 surfaces determined by $w_P$ specifies an open subset of the fibration $\hat{{\mathcal {X}}}_7$ of [11], known to be mirror to the smooth Fano 3-fold $V_{14}$. There is a factorisation

$$\begin{aligned} (w_P + 4)|_{{\mathbb {T}}_{123}} = \frac{(1 + x_1)(1 + x_2)(1 + x_3) (1 + x_1 + x_2 + x_3)}{x_1x_2x_3} \end{aligned}$$

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 4.7. This reducible fibre $E_P$ is the union of four rational surfaces intersecting like a tetrahedron in U. When extended to the compactification $X_P$, the boundary divisor $D_P$ meets $E_P$ in a collection of 12 lines, given by the ${{\,\textrm{Dih}\,}}_8$-orbits of ${\mathbb {V}}(x_1+x_2+x_3)\subset D_{123}$ and ${\mathbb {V}}(x_1+x_3)\subset D_{1357}$. These are displayed (tropically) in Fig. 12b 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 Fig. 13b. The fibres of the original Landau–Ginzburg model $w_P:U\rightarrow {\mathbb {C}}$ 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 fibre obtains an $A_1$ singularity.

4.4.2 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 ${\mathbb {R}}^3$. Nevertheless, we can consider all of the polytopes obtained from removing vertices from the polytope Q from Sect. 4.4.1.

Theorem 4.16

Consider the Laurent polynomial $w_0=w|_{{\mathbb {T}}_{123}}\in {\mathbb {C}}[x_1^{\pm 1},x_2^{\pm 1},x_3^{\pm 1}]$ 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 $i=1,\ldots ,10$.

• 1. Of the 1024 possibilities for $w_0$, 705 have 3-dimensional Fano Newton polytopes, i.e. ${{\,\textrm{Newt}\,}}(w_0)$ has primitive vertices and $0\in {{\,\textrm{int}\,}}({{\,\textrm{Newt}\,}}(w_0))$.

• 2. These 705 Laurent polynomials give rise to 46 distinct periods.

• 3. Of these 46 periods, 20 are equal to the regularised quantum period of a smooth Fano 3-fold, as described in Table 4.

Table 4 Mirror Landau–Ginzburg potentials on U for 20 smooth Fano 3-folds

Table 4 only contains one representative Landau–Ginzburg potential in each case and, even after taking the ${{\,\textrm{Dih}\,}}_8$ symmetry into account, the same period sequence can be obtained from several different potentials. For example, in addition to the entry in the Table 4, the following five potentials also have the right period to be mirror to $V_{22}$:

$$\begin{aligned}{} & {} x_1+x_2+x_5+x_6+q_2, \quad x_1+x_2+x_3+x_4+x_5+q_2, \quad x_1+x_2+q_1+q_2, \\{} & {} x_1+x_2+x_3+x_4+x_5+x_7 \quad \text {and} \quad x_1+x_2+x_3+x_4+x_6+q_2. \end{aligned}$$

According to Conjecture 2.13, 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 $P={{\,\textrm{Newt}\,}}(w)\subset M_U$. It would be interesting to know whether the remaining 26 period sequences are all period sequences of Fano 3-folds with terminal singularities.

4.4.3 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 $(q_1)^{\ge -1}\cap (q_2)^{\ge -1}$ is a closed polytope $Q\subset N_U$ with eight vertices $x_ix_{i+1}$ for $i\in {\mathbb {Z}}/8{\mathbb {Z}}$. Therefore the dual polytope P is given by $P = \bigcap _{i\in {\mathbb {Z}}/8{\mathbb {Z}}} (x_ix_{i+1})^{\ge -1}$. This gives a pair of dual polytopes in $N_U$

$$\begin{aligned} P={{\,\textrm{conv}\,}}(q_1, q_2) \quad \text {and} \quad Q=P^\star ={{\,\textrm{conv}\,}}(x_1x_2, x_2x_3, \ldots , x_8x_1). \end{aligned}$$

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.

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. $(U,w_P)$).

Example 4.17

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 2.14 they both receive coefficient 1 and this uniquely determines the potential $w_P=q_1+q_2$. By restricting to the torus chart ${\mathbb {T}}_{123}$ and computing the period sequence of the resulting Laurent polynomial, we find that

$$\begin{aligned} \pi _{w_P}(t) = 1 + 8t^2 + 24t^3 + 240t^4 + 1440t^5 +11960t^6 + 89040t^7 + \cdots \end{aligned}$$

which is the regularised quantum period of the smooth Fano 3-fold ${{\,\textrm{MM}\,}}_{2,21}$ [8, §38].

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.

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

$$\begin{aligned} x_0, \, x_1, \ldots , x_8,\; x_1x_2, \ldots , x_8x_1 \end{aligned}$$

where $x_0$ is the homogenising variable. According to computer algebra $X_Q={{\,\textrm{Proj}\,}}R_Q\subset {\mathbb {P}}^{16}$ is a Gorenstein Fano variety of Fano index 1 and degree $\deg X_Q={{\,\textrm{vol}\,}}Q=28$. This at least agrees with $X_Q$ being a (degeneration of a) Fano 3-fold of type ${{\,\textrm{MM}\,}}_{2,21}$, as predicted by the period sequence of $w_P$. By Conjecture 2.13 we expect that $X_Q$ admits a ${\mathbb {Q}}$-Gorenstein smoothing to a smooth Fano 3-fold of type ${{\,\textrm{MM}\,}}_{2,21}$.

Moreover, the boundary divisor $D_Q={\mathbb {V}}(x_0)$ 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 $D_Q$.

Example 4.18

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. It suggests the possibility of constructing at least three different deformation families of Fano 3-folds with the Hilbert series [6, #14885] in the Graded Ring database. Currently two such families are known: one constructed as a complete intersection in a toric variety [19] and the other via unprojection [26] (although it is not known whether these two families are distinct).

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 $X_P={{\,\textrm{Proj}\,}}R_P\subset {\mathbb {P}}(1^3,2^8)$ in weighted projective space. One of the equations defining $X_P$ is

$$\begin{aligned} q_1q_2 = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + 4x_0^2 \end{aligned}$$

which we could use to eliminate $x_1$ say, and reduce to a model $X_P\subset {\mathbb {P}}(1^3,2^7)$. The Hilbert series of $X_P$ now agrees with [6, #14885], showing that $X_P$ has degree $\deg (X_P)={{\,\textrm{vol}\,}}(P)=4$, Fano index 1 and basket of quotient singularities $\{8\times \tfrac{1}{2}(1,1,1)\}$. The anticanonical boundary divisor $D_P={\mathbb {V}}(x_0)$ has eight components:

$$\begin{aligned} D_{12} = {\mathbb {V}}(x_0,x_3,x_4,x_5,x_6,x_7,x_8,q_1q_2 - x_1 - x_2) \quad \text {etc.} \end{aligned}$$

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

We note that if Y is a Fano 3-fold with Hilbert series [6, #14885], then $h^0(Y,{-K_Y})=3$ 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 2.14. 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 [10], 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.

After taking into account the ${{\,\textrm{Dih}\,}}_8$ symmetry, there are just three ways to decorate Q with coefficients, yielding three Landau–Ginzburg potentials with distinct periods. As described in Remark 2.14, according to Conjecture 2.13 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.

Remark 4.19

Smoothings of Gorenstein toric Fano 3-folds $(X_P,D_P)$ associated to an (honest) reflexive polytope $P\subset {\mathbb {R}}^3$ have been studied in [9], and their construction requires the input of a Minkowski decomposition of each facet of P. Different choices of Minkowski decompositions lead to topologically distinct smoothings.