# Cluster Varieties and Toric Specializations
of Fano Varieties

Alessio Corti  Department of Mathematics, Imperial College London, London SW7
2AZ UK<EMAIL_ADDRESS>
###### Abstract.

I state a conjecture describing the set of toric specializations of a Fano
variety with klt singularities. The conjecture asserts that for all generic
Fano varieties $X$ with klt singularities there exists a polarized cluster
variety $U$ and a surjection from the set of torus charts on $U$ to the set of
toric specializations of $X$.

I outline the first steps of a theory of the cluster varieties that I use. In
dimension $2$, I sketch a proof of the conjecture after Kasprzyk–Nill–Prince,
Lutz and Hacking by way of work of Lai–Zhou. This reveals a surprising
structure to the classification of log del Pezzo surfaces that was first
conjectured in [1]. In higher dimensions, I survey the evidence from the
Fanosearch program, cluster structures for Grassmannians and flag varieties,
and moduli spaces of conformal blocks.

###### 2020 Mathematics Subject Classification:

14J33 (Primary); 14J45; 13F60 (Secondary)

###### Contents

  1. 1 Introduction
    1. 1.1 Cluster Varieties and toric specializations of Fano varieties
    2. 1.2 Contents of the rest of the paper
  2. 2 Cluster varieties: a short introduction
    1. 2.1 A near-atlas of torus charts
    2. 2.2 The mutation graph
  3. 3 The case of surfaces
  4. 4 Examples and evidence
    1. 4.1 Evidence from the Fanosearch program
    2. 4.2 Examples from cluster algebras
    3. 4.3 Conformal blocks

##  1\. Introduction

I work in characteristic zero only; I work over $\mathbb{C}$ unless stated
otherwise.

A first aim of this paper is to state Conjecture 18 below, and sketch a proof
in dimension $2$. The conjecture asserts that for all generic Fano varieties
$X$ with klt singularities there exists a cluster variety $U$ and a surjection
from the set of torus charts of $U$ to the set of toric specializations***Here
specialization — see Definition 10 below — is synonymous with degeneration. I
only consider projective normal specializations: since these are mild, I
prefer the term specialization to degeneration. of $X$.

All these words will be defined shortly, but I warn you in advance that the
cluster varieties in use here are more general than those that can be found
elsewhere in the literature, see below for additional comments on this point.
(The most important difference is that our exchange relations involve Laurent
polynomials, not just binomials.)

The paper leaves untouched the question of whether or not, or under what
conditions, such a specialization exists.

If algebra is your thing, as opposed to geometry, please read on! The ring of
global functions $H^{0}(U,\mathcal{O}_{U})$ is a _Laurent phenomenon algebra_
[25], and a torus chart on $U$ corresponds to a _seed_ in the language of
cluster algebras, see Sec. 2. Our conjecture is a vast generalization of the
correspondence between toric specializations of a (Fano) variety and seeds of
a cluster algebra, of which there are many examples in the literature on
cluster algebras. I, however, arrived at the statement coming from a different
place: the Fanosearch program, whose aim is to classify Fano varieties by
first classifying their toric specializations.

A second aim was to give a definition of cluster varieties more general than
that previously given in [13, Sec. 2], and to outline a theory of these
varieties: this is done in Sec. 2. This material is not very difficult, and
not really original — for example, it was known independently by (at least)
Sean Keel — but it is not, I think, in print.

Sec. 3 sketches a proof of Conjecture 18 in dimension $2$. I also show that,
in dimension $2$, the conjecture is equivalent to [1, Conjecture A]. That
conjecture was one of two cornerstones of a new point of view on the
classification of log del Pezzo surfaces inspired by mirror symmetry — the
other being [1, Conjecture B]. The results of this paper complete a
substantial part of that program, revealing a surprising structure to the
classification of log del Pezzo surfaces.

In the final Sec. 4 I collect some evidence.

###  1.1. Cluster Varieties and toric specializations of Fano varieties

The goal of this section is to state our main Conjecture 18. The statement
relates the worlds of Fano varieties and cluster varieties. I begin by
introducing cluster varieties.

Fix a torsion free $\mathbb{Z}$-module $N\cong\mathbb{Z}^{n}$ and dual
$\mathbb{Z}$-module $M=\operatorname{Hom}(N,\mathbb{Z})$. At all time in the
paper this choice remains in force. Below I denote by $\mathbb{T}^{\star}$ the
$n$-dimensional torus with character group $N$:

$\mathbb{T}^{\star}=\operatorname{Spec}\mathbb{C}[N]$

(I sometimes refer to this torus as the _dual torus_. I make an effort of not
confusing this torus with the torus
$\mathbb{T}=\operatorname{Spec}\mathbb{C}[M]$, referred to simply as _the
torus_.)

######  Definition 1.

Let $A$ be an affine variety and $U$ a variety. A rational map $j\colon
A\dasharrow U$ is a _near-inclusion_ if there exists a Zariski closed subset
$Z\subset A$ of codimension $\geq 2$ such that $j\colon A\setminus
Z\hookrightarrow U$ is an open inclusion. The subset $Z$ is called the
_irregular set_.

######  Definition 2.

A _cluster variety_ is a variety $U=Y\setminus D$ for a triple $(Y,D,j)$
where:

  1. (1)

$Y$ is a projective $\mathbb{Q}$-factorial normal variety and $D\subset Y$ is
a reduced integral Weil divisor such that the pair $(Y,D)$ has dlt (divisorial
log terminal) singularities;

  2. (2)

The divisor $K_{Y}+D$ is Cartier and linearly equivalent to $0$ — in other
words, $(Y,D)$ is a _Calabi–Yau pair_ ;

  3. (3)

$j\colon\mathbb{T}^{\star}\dasharrow U=Y\setminus D$ is a near-inclusion.

######  Remark 3.

Here are a few comments on the definition.

  1. (1)

Recall that, by definition, $Y$ is $\mathbb{Q}$-factorial if all Weil divisors
on $Y$ are $\mathbb{Q}$-Cartier. This notion is local in the Zariski topology
but not in the étale, or the analytic, topology. Hence, it would be misleading
to say things like “$Y$ has $\mathbb{Q}$-factorial singularities.”

  2. (2)

If you are not at home with the various classes of singularities of pairs that
are in use in higher dimensional complex algebraic geometry, then it is
permitted, _en première lecture_ , to replace the condition that the pair
$(Y,D)$ have dlt singularities with the assumption that $Y$ is nonsingular and
$D\subset Y$ is a simple normal crossing (snc) divisor.

I refer to [23, Sec. 2.3] for the definition and first properties of _dlt
(divisorial log terminal) and klt (Kawamata log terminal) singularities_ of
pairs, and a discussion of the role that these singularities play in the
minimal model program for algebraic varieties. It follows from the definition,
and the fact that $K_{Y}+D$ is Cartier, that $U=Y\setminus D$ has Gorenstein
canonical singularities. In particular, in dimension $2$ this means that $U$
has _rational double point_ singularities. In many cases of interest, $U$ is
in fact nonsingular.

The point about the class of dlt pairs $(Y,D)$ where $Y$ is
$\mathbb{Q}$-factorial is that it is well-behaved under volume preserving
birational maps — see [8].

  3. (3)

I work with near-inclusions, as opposed to _tout court_ inclusions, because of
the subtleties of birational geometry in dimension $\geq 3$. As shown in
Proposition 21, this distinction in unnecessary in dimension $2$. Please don’t
let this detail distract you.

######  Definition 4.

Let $U=Y\setminus D$ be a cluster variety with torus near-inclusion
$j\colon\mathbb{T}^{\star}\dasharrow U$. A _torus chart_ on $U$ is a near-
inclusion $j^{\prime}\colon\mathbb{T}^{\star}\dasharrow U$.

In particular the given near-inclusion $j\colon\mathbb{T}^{\star}\dasharrow U$
is tautologically a torus chart, which I call the _reference chart_.

######  Remark 5.

My definition of cluster variety is more general than the one given in [13,
Sec. 2]. It is still true, however, that a cluster variety in my sense is
nearly covered — see Definition 22 and the short discussion just after the
following Definition 6 — by finitely many torus charts: this is proved in
Theorem 23 of Sec. 2 below.

######  Definition 6.

An _(algebraic) mutation data_ is a pair $(u,h)$ consisting of a primitive
vector $u\in M$ and a Laurent polynomial $h\in\mathbb{C}[u^{\perp}\cap N]$
that is either irreducible or the power of an irreducible.

An algebraic mutation data induces a birational map

$\mu=\mu_{u,h}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$

called an _(algebraic) mutation_ by setting

$\mu^{\sharp}(x^{n})=x^{n}h^{\langle u,n\rangle}$

and extending to a ring homomorphism
$\mu^{\sharp}\colon\mathbb{C}[N]\to\mathbb{C}(N)$ — where $\mathbb{C}(N)$
denotes the fraction field of $\mathbb{C}[N]$ — in the obvious way.

If $U=Y\setminus D$ is a cluster variety,
$j_{2},j_{1}\colon\mathbb{T}^{\star}\dasharrow U$ are torus charts and
$j_{2}=j_{1}\circ\mu_{u,h}$, I say that $j_{2}$ is the _mutation_ of $j_{1}$
and write $j_{2}=\mu_{u,h}(j_{1})$.

In Theorem 23 of Sec. 2, given a cluster variety $U=Y\setminus D$ with
reference chart $j=j_{0}\colon\mathbb{T}^{\star}\dasharrow U$, we will
construct additional torus charts
$j_{1},\dots,j_{r}\colon\mathbb{T}^{\star}\dasharrow U$ such that the union

$\bigcup_{m=0}^{r}j_{m}(\mathbb{T}^{\star}\setminus Z_{m})\subset U$

— where $Z_{m}\subset\mathbb{T}^{\star}$ is the irregular set — _nearly
covers_ $U$, in the sense that the complement has codimension $\geq 2$. In
this construction for all $m\geq 1$ the transition map $j_{0}^{-1}\circ
j_{m}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$ will be an
algebraic mutation in the sense just defined.

Thus our algebraic mutations correspond to the exchange relations in cluster
algebras, and our Laurent polynomials $h$ to the exchange binomials in cluster
algebras.

I next introduce Fano varieties and their specializations.

######  Definition 7.

A _Fano variety_ is a normal projective variety $X$ such that the
anticanonical divisor $-K_{X}$ is $\mathbb{Q}$-Cartier and ample.

A Fano variety of dimension $2$ is called a _del Pezzo surface_.

I aim to discuss the irreducible components of the moduli space of Fano
varieties. This is a coarse enough discussion that I want to have it without
even speaking the words “moduli space”. Stability, or $K$-polystability, is
not relevant: I am not into the finer question of what exactly are the points
of the moduli space and what Fano varieties they correspond to. My Definition
9 below captures the Fano varieties that correspond to the generic points of
irreducible components of the moduli space.

######  Definition 8.

A _family of Fano varieties_ is a projective flat morphism
$f\colon\mathfrak{X}\to T$ where $T$ is reduced and irreducible and:

  1. (1)

every fibre of $f$ is a normal Fano variety;

  2. (2)

$f$ is qG (i.e., _$\mathbb{Q}$ -Gorenstein_), that is, there exist a line
bundle $\mathcal{L}$ on $\mathfrak{X}$ and a strictly positive integer $r$
such that for all $t\in T$,
$\mathcal{L}_{|\mathfrak{X}_{t}}=\omega_{\mathfrak{X}_{t}}^{[r]}$.†††Here
$\omega_{\mathfrak{X}_{t}}^{[r]}=\bigl{(}\omega_{\mathfrak{X}_{t}}^{\otimes\,r}\bigr{)}^{\star\star}$
denotes the saturation of $\omega_{\mathfrak{X}_{t}}^{\otimes\,r}$. (In
particular $\mathcal{L}$ is $f$-ample.)

For more information on the $\mathbb{Q}$-Gorenstein condition, see [22, Chap.
2]; the key point of the condition is to ensure that natural numerical
invariants are locally constant in $T$, see [22, Sec. 2.5]. The condition is
satisfied in many concrete situations of interest.

The following definition makes precise what I mean by a _generic Fano variety_
, that is, informally, one that corresponds to the generic point of an
irreducible component of the moduli space of Fano varieties.

In the definition, a fg field is a field of characteristic $0$ that is
finitely generated (as a field) over $\mathbb{Q}$.

######  Definition 9.

  1. (1)

Let $X$ and $Y$ be Fano varieties defined over fg fields $K$, $L$. I say that
$X$ is a _specialization_ of $Y$ — and that $Y$ is a _generization_ of $X$ —
if there is a family of Fano varieties $f\colon\mathfrak{X}\to T$ such that:
There are points $t_{1},t_{2}\in T$ where $t_{2}$ is a specialization of
$t_{1}$, inclusions $k(t_{1})\subset L$, $k(t_{2})\subset K$, and isomorphisms
$Y=f^{\star}(t_{1})\otimes L$, $X=f^{\star}(t_{2})\otimes K$.

  2. (2)

A Fano variety $X$ defined over a fg field $K$ is _generic_ if: For all Fano
varieties $Y$ defined over a fg field $L$, if $X$ is a specialization of $Y$,
then there exist a fg field $F$, field extensions $K,L\subset F$, and an
isomorphism $Y\otimes F=X\otimes F$.

  3. (3)

A Fano variety $X$ over $\mathbb{C}$ is generic if it can be defined over a fg
field $K\subset\mathbb{C}$, and it is generic as a Fano variety over $K$.

######  Definition 10.

Let $Y$ be a Fano variety. A _toric specialization_ of $Y$ is a toric Fano
variety $X$ that is a specialization of $Y$.

I want to understand all toric specializations of a given generic Fano variety
$X$ with klt singularities, up to isomorphism. I require that $X$ has klt
singularities [23, Sec. 2.3] because a Fano variety with klt singularities in
many ways still behaves like a Fano variety is expected to do.

In the following discussion all toric varieties are toric varieties for the
torus with character group $M$:

$\mathbb{T}=\operatorname{Spec}\mathbb{C}[M]$

######  Definition 11.

A convex lattice polytope‡‡‡I abuse notation and write $P\subset N$ for a
lattice polytope to mean that $P\subset N\otimes\mathbb{R}$ and it has its
vertices in $N$. $P\subset N$ is _Fano_ if it has the origin in its strict
interior (in particular $P$ is full dimensional) and all of its vertices are
primitive lattice elements.

A _Fano polygon_ is a Fano polytope of dimension $2$.

If $P$ is a Fano polytope, I denote by $X_{P}$ the toric variety with fan the
_face fan_ (aka _spanning fan_) of $P$: $X_{P}$ is a toric Fano variety and
$P$ is its _anticanonical polytope_.§§§The polar polytope $Q=P^{\star}\subset
M$ is the _moment polytope_ of $X_{P}$ anticanonically polarized. It is
obvious that, conversely, if $X$ is a toric Fano variety with anticanonical
polytope $P$, then $X=X_{P}$.

######  Definition 12.

A _(combinatorial) mutation data_ is a pair $(u,H)$ consisting of a primitive
vector $u\in M$ and lattice polytope $H\subset u^{\perp}\cap N$.

A combinatorial mutation data induces an operation [2, Definition 5] on the
set of Fano polytopes in $N$ called a _combinatorial mutation_. When
defined,¶¶¶A combinatorial condition must be satisfied for the mutation to be
defined. Informally, for all $k>0$, $kH$ needs to be a Minkowski factor of the
level set $P\cap u^{-1}(-k)$, see [2] for a complete discussion. the mutation
of $P$ is denoted by $\mu_{u,H}(P)$.

######  Remark 13.

The definition of combinatorial mutation is subtle and not super-easy to
digest. Two properties help to understand it:

  1. (1)

Let $(u,h)$ be an algebraic mutation data, and $f\in\mathbb{C}[N]$ such that
$\mu_{(u,h)}^{\sharp}(f)\in\mathbb{C}[N]$ — when this happens I say that $f$
is _mutable_. Consider the combinatorial mutation data $(u,H)$ where
$H=\operatorname{Newt}h$ (the _Newton polytope_ of $h$), and let
$P=\operatorname{Newt}f$. Then $\mu_{(u,H)}(P)$ is defined, and it equals
$\operatorname{Newt}\bigr{(}\mu_{(u,h)}^{\sharp}(f)\bigl{)}$.

  2. (2)

The mutation of the polar polytope $P^{\star}\subset M$ is induced by an
integral piece-wise linear transformation of $M$.

######  Theorem 14.

[20, 29] Let $P_{2}=\mu_{u,H}(P_{1})$ be a combinatorial mutation of Fano
lattice polytopes; denote by $X_{P_{1}}$, $X_{P_{2}}$ the corresponding toric
Fano varieties. There exists a family of Fano varieties
$f\colon\mathfrak{X}\to\mathbb{P}^{1}$ such that

$X_{P_{1}}=f^{\star}(0)\quad\text{and}\quad X_{P_{2}}=f^{\star}(\infty)$

######  Definition 15.

The family of Fano varieties of the theorem above is called an _Ilten pencil_.

I next define the concept of a polarization on a cluster variety. The notion
is closely related to that of a polytope in a _tropical mutation scheme_ in
forthcoming work by Laura Escobar, Megumi Harada and Chris Manon.

######  Definition 16.

A _polarization_ of a cluster variety $U=Y\setminus D$ is a set-theoretic
function:

$p\colon\bigl{\\{}j\colon\mathbb{T}^{\star}\dasharrow U\mid\text{$j$ is a
torus chart}\bigr{\\}}\to\bigl{\\{}P\subset N\mid\text{$P$ is a Fano
polytope}\bigr{\\}}$

that is mutation-preserving in the sense that, for all torus charts
$j_{1},j_{2}\colon\mathbb{T}^{\star}\dasharrow U$:

If $j_{2}=\mu_{u,h}(j_{1})$, then
$p(j_{2})=\mu_{u,\operatorname{Newt}h}\bigl{(}p(j_{1})\bigr{)}$

where I denote by $\operatorname{Newt}h$ the Newton polytope of $h$. A
_polarized cluster variety_ is a pair of a cluster variety and a polarization.

######  Remark 17.

Polarizations arise from the following construction. Let $U=Y\setminus D$ be a
cluster variety. Let $H$ be an effective divisor on $Y$, such that

  1. (1)

$H$ meets $D$ properly, and

  2. (2)

$H\sim B$, where $B\geq 0$ is supported on $D$.

Let $f\in H^{0}(U,\mathcal{O}_{U})$ be a global function such that
$\operatorname{div}_{Y}f=H-B$. These data induce a polarization by setting

$p(j)=\operatorname{Newt}(f\circ j)$

Thus, a polarization is the same as a mobile boundary divisor on $X$.

I am now ready to state our main conjecture. In Sec. 3 I will prove the
assertion in dimension $n=2$ and show that it is equivalent to [1, Conjecture
A].

######  Conjecture 18.

Let $X$ be a generic Fano variety with klt singularities. There is a polarized
cluster variety $U=Y\setminus D$ with polarization $p$, such that the set of
toric Fano varieties

$\bigl{\\{}X_{p(j)}\mid\text{$j\colon\mathbb{T}^{\star}\dasharrow U$ is a
torus chart}\bigr{\\}}$

up to isomorphism is precisely the set of toric specializations of $X$ up to
isomorphism.

######  Example 19.

The first nontrivial example of Conjecture 18 concerns toric specializations
of $\mathbb{P}^{2}$. It is well-known [27, 19] that these are the weighted
projective planes $\mathbb{P}(a^{2},b^{2},c^{2})$ where $(a,b,c)$ is a triple
of integers that solve the _Markov equation_

$a^{2}+b^{2}+c^{2}-3abc=0$

but even in this case the explicit correspondence with the set of torus charts
on a polarized cluster surface follows from results of Kasprzyk–Nill–Prince
[21], see also Lutz [26] and Sec. 3 below.

######  Remark 20.

  1. (1)

In dimension $>2$, I don’t know an example of a cluster variety where an
explicit description of the set of _all_ torus charts is known. The difficulty
has to do with the far greater subtlety of birational geometry in dimension
$\geq 3$.

  2. (2)

If $U=Y\setminus D$ is a cluster variety, then it is often possible to
construct finitely many toric degenerations of $Y$ corresponding to points
$x\in D$ of the _zero stratum of $D$_. These toric degenerations are
constructed from the Cox rings made from the irreducible components
$D_{i}\subset D$ such that $x\in D_{i}$.

This type of construction is often used in the representation theory and
combinatorial geometry community to produce toric degenerations, but it is
_not_ the mechanism at work behind Conjecture 18. Rather, the mechanism there
is the concept of polarization of a cluster variety.

###  1.2. Contents of the rest of the paper

The remaining sections of this note depend logically on the Introduction but
they do not depend on each other and thus they can be read in any order.

In Sec. 2 I develop some consequences of the definition and provide the first
steps of a theory of cluster varieties. This material is not very difficult,
and not especially original — for example, it was known independently by (at
least) Sean Keel — but it is not, I think, in print.

Sec. 3 sketches a proof of Conjecture 18 in dimension $2$. I also show that,
in dimension $2$, the conjecture is equivalent to [1, Conjecture A]. That
conjecture was one of two cornerstones of our new point of view on the
classification of log del Pezzo surfaces inspired by mirror symmetry — the
other being [1, Conjecture B]. Thus the results of this paper complete a
substantial part of that program, revealing a surprising structure to the
classification of log del Pezzo surfaces.

In Sec. 4 I collect some of the evidence. My motivation for stating the
conjecture comes from the Fanosearch program, and the most striking evidence —
consisting of large numerical experiments with $\mathbb{Q}$-Fano 3-folds [7,
6] — is presented in Sec. 4.1. In Sec. 4.2 I briefly touch on toric
specializations of Grassmannians and their relation with seeds in certain
cluster algebras that have been discovered in the representation theory
community. In Sec. 4.3, I sketch work by Manon [28] and
Belmans–Galkin–Mukhopadhyay [3] on toric specializations of the Fano varieties
$M_{C}(2,\text{odd})$, the moduli space of rank two vector bundles on a curve
$C$ with determinant a line bundle of odd degree on $C$. This example is
psychologically important because the moduli space of these Fanos in genus $g$
is birational to the moduli space of curves of genus $g$ — in particular, for
large $g$, it is far from being unirational.

### Acknowledgements

It was a privilege and a great honor to be invited to submit a paper to this
volume on the occasion of the 70th anniversary of V. V. Shokurov.

It will be clear to all those familiar with the issues that this note owes a
very significant intellectual debt to the Gross–Siebert program in general,
and in particular work by Gross–Pandharipande–Siebert [17], Gross–Hacking–Keel
and Lai–Zhou [14, 24], and Gross–Hacking–Keel–Kontsevich [16].

My ideas here have been developed in the Fanosearch group and in particular in
discussions with Tom Coates, Liana Heuberger, Al Kasprzyk and Andrea Petracci.
My treatment of cluster varieties in Sec. 2 benefits greatly from
conversations with Sean Keel. I thank Paul Hacking for allowing me to include
in Sec. 3 a sketch of his proof of Conjecture 18 in the case of surfaces,
personally communicated at the _Retrospective Workshop on Calabi–Yau
Varieties: Arithmetic, Geometry and Physics_ , Herstmonceux Castle, 20–25 June
2016. (The proof depends crucially on the results of the recent paper [24],
which explains why it took so long to write it up.)

This paper was finalized during my stay at the _Banff International Research
Station_ to attend the Workshop on _Toric Degenerations_ , 4–9 December 2022.
There I met Laura Escobar, Megumi Harada and Chris Manon and learned about
their forthcoming work. I thank Alfredo Nájera Chávez and Lara Bossinger for
discussions about cluster structures on, and toric degenerations of,
Grassmannians and more general flag varieties, and Chris Manon for explaining
his work on moduli spaces of conformal blocks, during and after the Workshop.
I am very grateful to the participants, and the staff at BIRS, for creating an
atmosphere where ideas were exchanged between diverse mathematical
communities.

I thank Lara Bossinger, Tom Bridgeland, Tom Coates, Laura Escobar, Megumi
Harada, Liana Heuberger, Sean Keel, Jonathan Lai, Wendelin Lutz, Chris Manon,
Helge Ruddat, Jenia Tevelev and Yan Zhou for helpful feedback on earlier
versions.

Finally I thank the anonymous referee for some very helpful remarks and
corrections on the submitted version.

##  2\. Cluster varieties: a short introduction

In this section, I develop some consequences of the definition and provide the
first steps of a theory of cluster varieties. I start off by explaining that a
cluster variety is always nearly covered by a union of torus charts. Then I
briefly speak of the mutation graph of a cluster variety: this is the graph
with vertices the torus charts and where two vertices are connected by an edge
if they differ by a mutation.

###  2.1. A near-atlas of torus charts

######  Proposition 21.

If $U=Y\setminus D$ is a $2$-dimensional cluster variety, and
$j\colon\mathbb{T}^{\star}\dasharrow U$ is a torus chart, then in fact $j$ is
defined everywhere and it is an open embedding of $\mathbb{T}^{\star}$ in $U$.

###### Proof.

Let $G\to\mathbb{T}^{\star}\times U$ be the normalization of the graph and
denote by $p\colon G\to\mathbb{T}^{\star}$, $q\colon G\to U$ the obvious
morphisms. I need to show that $p\colon G\to\mathbb{T}^{\star}$ is finite-to-
one. Assume for a contradiction that $E\subset G$ is a curve such that
$p(E)=t\in\mathbb{T}^{\star}$ is a point. Then $q(E)\subset U$ is a divisor,
necessarily with discrepancy $a(E,\mathbb{T}^{\star})>0$ — see Definition 24 —
and this contradicts the Claim in the proof of Lemma 28 below, stating that
all these discrepancies must be $0$. ∎

The difference in dimension $\geq 3$ is that there can be divisors $E\subset
G$ that are both $p$\- and $q$-exceptional. Simple examples show that in
dimension $\geq 3$ there are torus charts $j\colon\mathbb{T}^{\star}\dasharrow
U$ where $j$ is not defined everywhere.

######  Definition 22.

Let $X$ be a variety. A Zariski closed subset $Z\subset X$ is _small_ if every
component of $Z$ has codimension $\geq 2$ in $X$.

A Zariski open subset $U\subset X$ is _large_ if the complement $Z=X\setminus
U$ is small.

I say that $X$ is _nearly covered_ by $\\{U_{m}\mid m=0,\dots,r\\}$ if
$\cup_{m=0}^{r}\,U_{m}\subset X$ is large.

######  Theorem 23.

Let $U=Y\setminus D$ be a cluster variety with reference torus chart
$j\colon\mathbb{T}^{\star}\dasharrow U$. There exists a finite set of torus
charts:

$j_{0}=j,j_{1},\dots,j_{r}\colon\mathbb{T}^{\star}\dasharrow U$

such that, denoting by $Z_{m}\subset\mathbb{T}^{\star}$ the irregular set of
$j_{m}$:

  1. (1)

the union $\cup_{m=0}^{r}\,j_{m}(\mathbb{T}^{\star}\setminus Z_{m})\subset U$
is large — that is, $U$ is nearly covered by

$\\{j_{m}(\mathbb{T}^{\star}\setminus Z_{m})\mid m=0,\dots,r\\}$

  2. (2)

for all $m=1,\dots,r$, $j_{0}^{-1}\circ j_{m}$ is a mutation.

The proof — given below — is based on the notion of discrepancy of a
divisorial valuation of the fraction field $\mathbb{C}(N)$, and the
classification of divisorial valuations of discrepancy $0$ given by Lemma 26
below.

######  Definition 24.

A _divisorial valuation_ of $\mathbb{C}(N)$ is a rank $1$ discrete valuation
corresponding to a prime divisor $E\subset Y$, where $Y$ is a normal variety
birational to $\mathbb{T}^{\star}$. I often abuse language and identify $E$
with the associated valuation.

Choose a proper toric variety $\mathbb{T}^{\star}\subset X$ compactifying
$\mathbb{T}^{\star}$, and let $B=X\setminus\mathbb{T}^{\star}$ be the toric
boundary. We have two possibilities:

  1. (1)

$E$ is a divisor on $X$, in which case I take $Y=X$, or

  2. (2)

$E$ is not a divisor on $X$. By restricting $Y$, we may assume that the
birational map $Y\dasharrow X$ is a birational morphism $f\colon Y\to X$ whose
only exceptional divisor is $E\subset Y$.

In either case we can write:

$K_{Y}=f^{\star}(K_{X}+B)+aE$

where the integer $a=a(E,\mathbb{T}^{\star})$ is independent of choices and is
called the _discrepancy_ of $E$ over $\mathbb{T}^{\star}$.

######  Remark 25.

  1. (1)

In the language of [23, Sec. 2.3], $a(E,\mathbb{T}^{\star})=a(E,X,B)$ where
$\mathbb{T}^{\star}\subset X$ is any proper toric variety with boundary
$B=X\setminus\mathbb{T}^{\star}$.

  2. (2)

The notion is not to be confused with the discrepancy $a(E,W)$ defined in [23,
Sec. 2.3], where $W$ is a variety and $E$ a divisorial valuation of the
fraction field $k(W)$ with centre on $W$. There is no conflict, however,
because if both discrepancies are defined — that is, if $E$ has centre on
$\mathbb{T}^{\star}$ — then they are equal.

######  Lemma 26.

(Classification of divisorial valuations $E$ with
$a(E,\mathbb{T}^{\star})=0$.)

  1. (1)

Let $(u,g^{k})$ be algebraic mutation data where $g\in\mathbb{C}[u^{\perp}\cap
N]$ is irreducible. Denote by $\Sigma\subset M$ the fan consisting of the
cones $\\{0\\},\langle u\rangle_{+}\subset M_{\mathbb{R}}$, and by
$(\overline{X},\overline{B})$ the associated toric manifold with irreducible
boundary divisor $\overline{B}$ — note that
$\overline{B}=\operatorname{Spec}\mathbb{C}[u^{\perp}\cap N]$ — and let
$R=\\{g=0\\}\subset\overline{B}$. Let $Y\to\overline{X}$ be the blow up of
$kR$, and $E\subset Y$ the exceptional divisor. Then
$a(E,\mathbb{T}^{\star})=0$.

  2. (2)

Conversely, let $E$ be a divisorial valuation of $\mathbb{C}(N)$ of
discrepancy $a(E,\mathbb{T}^{\star})=0$. There exists an algebraic mutation
data $(u,g^{k})$ such that $E$ is given by the above construction.

######  Definition 27.

Let $E$ be as in the lemma above. The pair $(u,g^{k})$ is the mutation data of
$E$. Conversely, $E$ is the divisor of the mutation data $(u,g^{k})$.

###### Proof.

The first part is straightforward, so let us prove Part (2). Let
$\mathbb{T}^{\star}\subset X$ be any projective toric manifold with toric
boundary $B$.

If $z=z_{X}E\in X$ is the centre of $E$ on $X$ ($z$ is a scheme-theoretic
point), then in fact $z\in B$ — otherwise $z\in\mathbb{T}^{\star}$ and,
because $\mathbb{T}^{\star}$ is smooth, we would have
$a(E,\mathbb{T}^{\star})>0$. Note that $z$ can never be the generic point of a
component $B_{0}$ of $B$ because all these have discrepancy
$a(B_{0},\mathbb{T}^{\star})=-1$.

I claim that, perhaps after a toric blow up, I can arrange that $z\in B_{0}$
for a unique irreducible component $B_{0}\subset B$. Indeed, failing this, $z$
lies in a toric stratum of codimension $\geq 2$, and let $f\colon X_{1}\to X$
be the blow-up of that stratum. Replacing $X$ with the toric manifold $X_{1}$,
the claim follows by induction since $a(E,X_{1})<a(E,X)$. The toric divisor
$B_{0}$ does not depend on choices, and I identify the corresponding valuation
of the fraction field $\mathbb{C}(N)$ with a primitive lattice vector $u\in
M$.

Now in fact $z\in B_{0}$ has codimension $1$ — otherwise by blowing up the
Zariski closure $\overline{z}\subset X$ we see that
$a(E,\mathbb{T}^{\star})>0$; but note that $B_{0}$ is a toric variety for the
torus $\operatorname{Spec}\mathbb{C}[u^{\perp}\cap N]$ and by what I said $z$
lies in the torus hence the Zariski closure $R=\overline{z}\subset B_{0}$ is
the vanishing locus of an irreducible Laurent polynomial
$g\in\mathbb{C}[u^{\perp}\cap N]$. Moreover, we see that $E$ is obtained by
blowing up $kR$ for some $k>0$. ∎

######  Lemma 28.

Let $U=Y\setminus D$ be a cluster variety with reference chart
$j\colon\mathbb{T}^{\star}\dasharrow U$. If $E\subset U\setminus
j(\mathbb{T}^{\star}\setminus Z)$ is a divisor, then:

  1. (1)

$a(E,\mathbb{T}^{\star})=0$.

  2. (2)

Denote by $(u,g^{k})$ the mutation data of $E$ and consider the mutation
$\mu=\mu_{u,g^{k}}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$. The
rational map $j^{\prime}=j\circ\mu\colon\mathbb{T}^{\star}\dasharrow U$ is a
torus chart.

A crucial ingredient in the proof — given below — of the lemma is a well-known
geometric description of algebraic mutations. Because this description is of
independent interest I state it here (without proof):

######  Lemma 29.

Fix algebraic mutation data $(u,h)$ consisting of a primitive vector $u\in M$
and Laurent polynomial $h=\in\mathbb{C}[u^{\perp}]\subset\mathbb{C}[N]$. As in
Definition 6, the data induce a birational map

$\mu=\mu_{u,h}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$

of the dual torus $\mathbb{T}^{\star}=\operatorname{Spec}\mathbb{C}[N]$ by
setting:

$\mu^{\sharp}\colon x^{n}\mapsto x^{n}h^{\langle u,n\rangle}$

The map $\mu$ has the following geometric description. Consider the fan

$\Xi=\\{(0),\mathbb{R}_{+}u,\mathbb{R}_{-}u\\}$

in $M$ and let $\mathcal{A}$ be the corresponding toric variety (for the dual
torus $\mathbb{T}^{\star}$). There is a commutative diagram:

$\textstyle{\widetilde{\mathcal{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\mu}$$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathbb{T}^{\star}/u}$$\textstyle{\mathbb{T}^{\star}/u}$

where

  * •

I interpret $u\in
M=\operatorname{Hom}(\mathbb{C}^{\times},\mathbb{T}^{\star})$ as a
$1$-parameter subgroup of
$\mathbb{T}^{\star}=\operatorname{Spec}\mathbb{C}[N]$ and I denote by
$\mathbb{T}^{\star}/u$ the quotient. The natural morphism
$\pi\colon\mathcal{A}\to\mathbb{T}^{\star}/u$ makes $\mathcal{A}$ into a
$\mathbb{P}^{1}$-bundle over $\mathbb{T}^{\star}/u$. Denoting by $D^{+}$,
$D^{-}\subset\mathcal{A}$ the two boundary divisors corresponding to the
generators $u$, $-u$, note that $\pi$ maps both $D^{+}$ and $D^{-}$
isomorphically to $\mathbb{T}^{\star}/u$.

  * •

Note that $h\in\mathbb{C}[u^{\perp}]=\mathbb{C}[\mathbb{T}^{\star}/u]$ is a
function on $\mathbb{T}^{\star}/u$. Write $h=g^{k}$ where $g$ is irreducible,
and let $R=(g=0)\subset\mathbb{T}^{\star}/u$ and write
$R^{+}=\pi^{\star}(R)\cap D^{+}$ and $R^{-}=\pi^{\star}(R)\cap D^{-}$. Then
$p$ is the blow up of $kR^{+}\subset\mathcal{A}$, and $q$ is the blow up of
$kR^{-}\subset\mathcal{A}$. Denoting by $E^{+}$ the $p$-exceptional divisor
and by $E^{-}$ the $q$-exceptional divisor we have:

$p(E^{+})=R^{+}\;\text{and}\;\;p(E^{-})=\pi^{-1}(R)\quad\text{and}\quad
q(E^{-})=R^{-}\;\text{and}\;\;q(E^{+})=\pi^{-1}(R)$

∎

###### Proof of Lemma 28.

As before choose a compact toric manifold $\mathbb{T}^{\star}\subset X$ with
toric boundary $B\subset X$.

Claim $a(E,\mathbb{T}^{\star})=a(E,X,B)=a(E,Y,D)=0$.

I first conclude the proof assuming the claim. The proof of the Claim is
below.

Consider the mutation
$\mu=\mu_{u,g^{k}}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$ and
the rational map $j^{\prime}=j\circ\mu\colon\mathbb{T}^{\star}\dasharrow U$. I
claim that in fact $j^{\prime}$ is a torus chart. I use the notation and
statement of Lemma 29 with $h=g^{k}$. Denote by
$\varphi\colon\mathcal{A}\dasharrow U$ the induced birational map. By
construction, $\varphi(E^{+})=E$; and $\varphi(E^{-})\subset U$ is also a
divisor: let us denote it by $E_{0}$. It follows from this that $\varphi$ is a
near-inclusion, and hence so is $j^{\prime}\colon\mathbb{T}^{\star}\dasharrow
U$.

Next I prove the Claim. The proof is in several steps.

Step 1: set up Consider a common log resolution:

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{X}$$\textstyle{Y}$

and define the divisors $B_{W}$, $D_{W}$ by the formulas:

(1) $\displaystyle K_{W}+B_{W}$ $\displaystyle=f^{\star}(K_{X}+B)$ (2)
$\displaystyle K_{W}+D_{W}$ $\displaystyle=g^{\star}(K_{Y}+D)$

The key property to note here is that both $B_{W}$ and $D_{W}$ are
_subboundaries_ in the sense of Shokurov: in other words, they are formal
linear combinations of prime divisors with coefficients $\leq 1$. This is of
course just another way to say that the pairs $(X,B)$ and $(Y,D)$ have dlt
singularities. The Claim is equivalent to the statement that $B_{W}=D_{W}$.

Step 2 I show that $\operatorname{Supp}f_{\star}(D_{W})=B$.

Indeed suppose that $G\subset W$ is a prime divisor and that $f(G)\subset X$
is a divisor. If $f(G)\cap\mathbb{T}^{\star}\neq\emptyset$, then also
$f(G)\cap(\mathbb{T}^{\star}\setminus Z)\neq\emptyset$, and then, because
$j\colon\mathbb{T}^{\star}\setminus Z\hookrightarrow U$ is an open inclusion,
it follows that $g(G)$ is a divisor on $Y$ and $g(G)\cap U\neq\emptyset$,
which means that $G$ can not appear in $D_{W}$.

Step 3 In fact $f_{\star}(D_{W})=B$.

This is because $f_{\star}(D_{W})$ is a subboundary and $f_{\star}(D_{W})\sim
B$.

Step 4 $D_{W}=B_{W}$.

By Step 3, $D_{W}-B_{W}\sim 0$ and $D_{W}-B_{W}$ is supported on the
$f$-exceptional set. The statement then follows from the negativity of
contractions.∎

###### Proof of Theorem 23..

Let $E_{1},\dots E_{r}\subset U$ be the prime divisors in $U\setminus
j(\mathbb{T}^{\star}\setminus Z)$. For all $m$ the previous lemma gives
$j_{m}\colon\mathbb{T}^{\star}\dasharrow U$ and it is clear from the
construction that $\cup_{m=0}^{r}\,j_{m}(\mathbb{T}^{\star}\setminus Z_{m})$
is large in $U$. ∎

I briefly sketch the next few steps of a theory of cluster varieties.

######  Definition 30.

A _seed_ is a finite set $S$ of discrete valuations $E$ of $\mathbb{C}(N)$ of
discrepancy $a(E,\mathbb{T}^{\star})=0$. Equivalently — by identifying a
valuation with the mutation data associated to it by Lemma 26 — a seed is a
finite set of mutation data.

Consider a cluster variety $U=Y\setminus D$. The reference chart
$j\colon\mathbb{T}^{\star}\dasharrow U$ determines a seed $S$ whose elements
are precisely the prime divisors $E\subset U\setminus
j(\mathbb{T}^{\star}\setminus Z)$.∥∥∥It follows that
$\operatorname{rk}H^{2}(\mathbb{T}^{\star})+|S|=\binom{n}{2}+|S|=\operatorname{rk}H^{2}(U;\mathbb{Z})$.
In our context $S$ can be arbitrarily large.

Conversely, out of a seed $S=\\{(u_{m},g_{m}^{k_{m}})\mid m=1,\dots,r\\}$, I
construct a cluster variety $U=Y\setminus D$, by blowing up appropriately in
the boundary of a toric manifold $X$ where all $u_{m}$ are boundary divisors.
By blowing up centres one at a time and in any order, it is not hard to verify
inductively by explicit computation that the resulting $Y$ is
$\mathbb{Q}$-factorial and that the pair $(Y,D)$ has dlt singularities. Note
that $Y$, $D$ and $U=Y\setminus D$ are not uniquely determined by the seed but
depend on the blow up order.

An element $s=(u_{m},g_{m}^{k_{m}})\in S$ gives a torus chart
$j_{m}\colon\mathbb{T}^{\star}\dasharrow U$ and, using this chart as reference
chart and going through Theorem 23 starting from this reference chart, I
construct $U$ out of a new seed $S^{\prime}$, which I call the _mutation_ of
$S$ at $s\in S$.

###  2.2. The mutation graph

######  Definition 31.

The _mutation graph_ of a cluster variety $U=Y\setminus D$ is the graph where

  1. (1)

The set of vertices is the set of torus charts
$j\colon\mathbb{T}^{\star}\dasharrow U$;

  2. (2)

Two vertices $j_{1}\colon\mathbb{T}^{\star}\dasharrow U$,
$j_{2}\colon\mathbb{T}^{\star}\dasharrow U$ are connected by an edge if there
is a mutation
$\mu=\mu_{u,h}\colon\mathbb{T}^{\star}\dasharrow\mathbb{T}^{\star}$ such that
$j_{2}=j_{1}\circ\mu$.

######  Remark 32.

I am grateful to Sean Keel and Yan Zhou for telling me about the paper [30],
which contains a remarkable example of a $6$-dimensional cluster variety where
the mutation graph has two connected components.

In all examples I know of cluster varieties that are mirrors of Fano varieties
as in Conjecture 18, I only know of one connected component of the mutation
graph.

It turns out that the mutation graph of a cluster surface is always connected:
the following statement is made in [18, Proposition 3.27], and proved there in
the Appendix by Wendelin Lutz (correcting an error in the arXiv version). The
proof uses the Sarkisov program for volume-preserving birational maps [8]. The
assertion is closely related to the statement — first made by Usnich
(unpublished) and proved by Blanc [4] — that the group
$\operatorname{SCr}_{2}(\mathbb{C})$ of volume-preserving birational maps of
the $2$-dimensional torus is generated by mutations, $\mathbb{C}^{\times\,2}$
and $\operatorname{SL}_{2}(\mathbb{Z})$.

######  Proposition 33.

The mutation graph of a cluster variety $U=Y\setminus D$ of dimension two is
connected.

######  Corollary 34.

Let $P$, $P^{\prime}$ be Fano polygons. If there is a family
$f\colon\mathfrak{X}\to T$ of del Pezzo surfaces with klt general fibre of
which $X_{P}$ and $X_{P^{\prime}}$ are fibres, then there exist Ilten pencils

$f_{0}\colon\mathfrak{X}_{0}\to\mathbb{P}^{1},\dots,f_{k}\colon\mathfrak{X}_{k}\to\mathbb{P}^{1}$

where $f_{0}^{\star}(0)=X_{P}$, $f_{k}^{\star}(\infty)=X_{P^{\prime}}$, and
for all $m=1,\dots,k$ $f_{m}^{\star}(0)=f_{m-1}^{\star}(\infty)$.

###### Proof.

Let $X$ be a generic del Pezzo surface with klt singularities of which both
$X_{P}$ and $X_{P^{\prime}}$ are specialisations. By Conjecture 18, there
exists a polarised cluster surface $U=Y\setminus D$ with polarisation $p$ and
torus charts $j\colon\mathbb{T}^{\star}\dasharrow U$,
$j^{\prime}\colon\mathbb{T}^{\star}\dasharrow U$ such that $p(j)=P$,
$p(j^{\prime})=P^{\prime}$. By Proposition 33, there exist torus charts

$j_{0},\dots,j_{k}\colon\mathbb{T}^{\star}\dasharrow U$

where $j_{0}=j$, $j_{k}=j^{\prime}$, and for all $m=1,\dots,k$
$j_{m}^{-1}j_{m-1}$ is an (algebraic) mutation, that is, there exist algebraic
mutation data $(u_{m},h_{m})$ such that

$j_{m}^{-1}j_{m-1}=\mu_{u_{m},h_{m}}$

According to Conjecture 18, the polarisation is mutation-preserving, that is,
writing $H_{m}=\operatorname{Newt}h_{m}$ and $P_{m}=p(j_{m})$, we have:

$P_{m}=\mu_{u_{m},H_{m}}(P_{m-1})$

Theorem 14 then provides the required family
$f_{m}\colon\mathfrak{X}_{m}\to\mathbb{P}^{1}$ such that
$f_{m}^{\star}(0)=X_{P_{m-1}}$ and $f_{m}^{\star}(\infty)=X_{P_{m}}$. ∎

The corollary seems to assert a sort of Shokurov connectedness principle for
the non-klt locus in the boundary of the moduli stack of del Pezzo surfaces.

##  3\. The case of surfaces

In this section, I sketch a proof of Conjecture 18 for surfaces. The proof
rests heavily on the work of Gross–Pandharipande–Siebert [17],
Gross–Hacking–Keel [14] and Lai–Zhou[24]. Lemma 37 below states everything
that I need from these papers.

#### The mirror surface

Let $0\in P\subset N$ be a Fano lattice polygon and $X_{P}$ the corresponding
toric surface — a toric surface for the torus
$\mathbb{T}=\operatorname{Spec}\mathbb{C}[M]$. It is known, see for example
[1, Lemma 6], that qG-deformations of $X_{P}$ are unobstructed: thus, there is
a unique generic del Pezzo surface of which $X_{P}$ is a toric specialization,
and we prove Conjecture 18 for this generic del Pezzo surface.

The proof is based on the following construction, starting from $P$, of the
mirror cluster surface. Denote by $(\overline{Y}_{P},\overline{D}_{P})$ the
polarized toric surface with moment polygon $P$ — this is a toric surface for
the dual torus $\mathbb{T}^{\star}=\operatorname{Spec}\mathbb{C}[N]$. Let
$E_{1},\dots,E_{r}$ be the edges of $P$ in some order. Abusing notation, I
also denote by $E_{i}$ the corresponding boundary component
$E_{i}\subset\overline{D}_{P}$. Note that this boundary component is
canonically isomorphic to $\mathbb{P}^{1}$: let us denote by $x_{i}\in E_{i}$
the point $(-1:1)\in\mathbb{P}^{1}$. As an edge of $P$, $E_{i}$ has an
integral length $l_{i}$ and an integral distance from the origin $r_{i}$:
write $l_{i}=k_{i}r_{i}+\overline{r}_{i}$ with
$0\leq\overline{r}_{i}<r_{i}$.******Denote by $C_{i}=\langle
E_{i}\rangle_{+}\subset N$ the cone spanned by $E_{i}$. In [1] $C_{i}$ is said
to be of class $T$ if $\overline{r}_{i}=0$. In general, $k_{i}$ is the number
of primitive $T$-cones of $P$. The polygon $P$ is of class $T$ if all
$\overline{r}_{i}=0$: this is so if and only if the surface $X_{P}$ is qG-
smoothable. The cluster surface $Y_{P}$ is the surface obtained by blowing up
the subscheme

$Z_{P}=\sum_{i=1}^{r}k_{i}x_{i}\subset\overline{D}_{P}$

in $\overline{Y}_{P}$; the boundary $D_{P}\subset Y_{P}$ is the strict
transform. The inverse image of the dual torus $\mathbb{T}^{\star}$ is the
reference torus chart, and I denote it by
$j\colon\mathbb{T}^{\star}\hookrightarrow U_{P}=Y_{P}\setminus D_{P}$. Next I
will use maximally mutable Laurent polynomials [1, Def. 4] and [_ibid._ pp.
517–518] to construct a polarization on $Y_{P}$ by the method of Remark 17.
Let $f$ be a generic maximally mutable Laurent polynomial; then $f$ defines a
regular function

$\widetilde{f}\colon U_{P}\to\mathbb{A}^{1}$

and for all torus charts $j^{\prime}\colon\mathbb{T}^{\star}\hookrightarrow
U_{P}$ I define the polarization by setting

$p(j^{\prime})=\operatorname{Newt}(\widetilde{f}\circ j^{\prime})$

The function $p$ is mutation-preserving by Remark 13(1), so indeed it is a
polarization.

######  Definition 35.

Let $\mathbf{0}\in P\subset N$ be a Fano lattice polygon. The _mirror surface_
is the polarized cluster surface $(Y_{P},D_{P})$ just constructed.

It is easy to see that the mirror surface depends only on the combinatorial
mutation class of the polygon $P$.

######  Lemma 36.

In dimension $n=2$, Conjecture 18 is equivalent to [1, Conjecture A].

###### Proof.

Recall the statement of [1, Conjecture A]: there is a canonical injective set-
theoretic function $\iota\colon\mathfrak{P}\to\mathfrak{F}$ where
$\mathfrak{P}$ is the set of mutation equivalence classes of Fano lattice
polygons and $\mathfrak{F}$ is the set of deformation classes of locally qG-
rigid orbifold del Pezzo surfaces, where — as explained in [1] — the function
$\iota$ maps a lattice polygon $P$ to a generic qG-deformation of the toric
surface $X_{P}$ — or indeed any locally qG-rigid qG-deformation of $X_{P}$. To
say that the function is injective is to say that, given two Fano lattice
polygons $P_{1}$, $P_{2}$, if the surfaces $X_{P_{1}}$, $X_{P_{2}}$ have a
common locally qG-rigid qG-deformation, then there exists a chain of
combinatorial mutations starting from $P_{1}$ and ending at $P_{2}$.

Let us prove Conjecture 18 from this. Let $P\subset N$ be a Fano polygon and
denote by $(Y_{P},D_{P})$ the mirror surface. Let $P^{\prime}$ be another Fano
lattice polytope and assume that $X_{P^{\prime}}$ and $X_{P}$ are
specializations of the same generic (necessarily locally qG-rigid) del Pezzo
surface. By [1, Conjecture A], the polygons $P$ and $P^{\prime}$ are mutation
equivalent and hence they give torus charts on the same mirror surface.

Conversely, assume Conjecture 18. I want to show that the function
$\iota:\mathfrak{P}\to\mathfrak{F}$ is injective. Let $P_{1}$ and $P_{2}$ be
Fano lattice polygons. If $X_{P_{1}}$ and $X_{P_{2}}$ are qG-deformation
equivalent, then they are both specializations of the same generic del Pezzo
surface $X$. Let $U=Y\setminus D$ be the polarized cluster variety whose
existence is guaranteed by Conjecture 18 applied to $X$. There are torus
charts $j_{1}\colon\mathbb{T}^{\star}\hookrightarrow U$ and
$j_{2}\colon\mathbb{T}^{\star}\hookrightarrow U$ such that $p(j_{1})=P_{1}$,
$p(j_{2})=P_{2}$. By Proposition 33, there exists a sequence of algebraic
mutations starting at $j_{1}$ and ending at $j_{2}$. The corresponding
sequence of combinatorial mutations starts at $P_{1}$ and ends at $P_{2}$. ∎

######  Lemma 37.

Let $P\subset N$ be a Fano polygon not of class $T$, and let $(Y_{P},D_{P})$
be the mirror surface. Then $X_{P}$ and $Y_{P}$ are qG-deformation equivalent.

###### Sketch of proof.

The proof can be extracted from the papers [17, 14, 24] with some additional
work. If $P$ is a polygon not of class $T$, then the cluster surface
$(Y_{P},D_{P})$ is positive in the sense of [14, 24] and hence the theory in
[24] applies.

Starting from $(Y_{P},D_{P})$ paper [24, Theorem 1.8] constructs a “mirror”
algebraic family of log Calabi–Yau surfaces:

$\pi\colon(\mathfrak{X},\mathfrak{B})\to\operatorname{Spec}\mathbb{C}[\operatorname{NE}(Y_{P})]$

The positivity of $(Y_{P},D_{P})$ is used to prove that we have a bona fide
algebraic family over
$\operatorname{Spec}\mathbb{C}[\operatorname{NE}(Y_{P})]$ and not just a
formal family over a formal completion. That statement assumes that $Y$ is
smooth but the assumption is not needed.

This is intended to be the mirror family of $(Y_{P},D_{P})$ and hence it is
intended to be a deformation of $(X_{P},B_{P})$ (where $B_{P}$ is the toric
boundary) but, actually, $(X_{P},B_{P})$ is not a fibre — though a Mumford
degeneration of it is. We need to enlarge the family to see the toric
degeneration $(X_{P},B_{P})$ appear as a fibre. The relevant construction is
carried out in [14, Sec. 3.4] and in the corresponding place [24, Sec. 6.1].
In short, the family is enlarged by attaching to it the family previously
constructed in [17]. It can be checked that the resulting enlarged family is
qG.

Finally, as stated in [24, Theorem 1.8], the pair $(Y_{P},D_{P})$ is a fibre
of $\pi$. Quoting verbatim from [14, pg. 74], “The fact that $(Y,D)$ appears
as a fibre is perhaps a bit surprising as, after all, we set out to construct
the mirror and have obtained our original surface back. Note however that dual
Lagrangian torus fibrations in dimension 2 are topologically equivalent by
Poincaré duality, so this is consistent with the SYZ formulation of mirror
symmetry.” ∎

######  Remark 38.

A direct construction of a qG-family of which $X_{P}$ and $Y_{P}$ are fibres
would let us avoid the complicated and convoluted constructions of [14, 24].

######  Theorem 39.

(Hacking††††††Personal communication at the _Retrospective Workshop on
Calabi–Yau Varieties: Arithmetic, Geometry and Physics_ , Herstmonceux Castle,
20–25 June 2016., Kasprzyk–Nill–Prince [21], Lutz [26]) In dimension $n=2$
Conjecture 18 holds.

###### Proof.

I thank Paul Hacking for permitting me to give a sketch of his unpublished
proof here.

I invoke Lemma 36 and show [1, Conjecture A].

The case of smooth del Pezzo surfaces follows directly from the combinatorial
classification of polygons of class $T$ up to mutation, originally due to
Kasprzyk, Nill and Prince [21]; see Lutz [26] for a shorter and more
conceptual geometric proof.

From now on I will work with polygons not of class $T$ — these are precisely
those for which the generic qG-deformation of the associated toric surface is
singular.

Let $P_{1}$ and $P_{2}$ be polygons not of class $T$ and assume that the
surfaces $X_{P_{1}}$, $X_{P_{2}}$ are qG-deformation equivalent: I need to
show that there exists a sequence of combinatorial mutations starting at
$P_{1}$ and ending at $P_{2}$. Consider the mirror surfaces
$(Y_{P_{1}},D_{P_{1}})$ and $(Y_{P_{2}},D_{P_{2}})$. By Lemma 37 $X_{P_{1}}$
is qG-deformation equivalent to $Y_{P_{1}}$, and $X_{P_{2}}$ is qG-deformation
equivalent to $Y_{P_{2}}$; hence it follows that $Y_{P_{1}}$ and $Y_{P_{2}}$
are qG-deformation equivalent.

Writing as usual $U_{P_{1}}=Y_{P_{1}}\setminus D_{P_{1}}$, and similarly
$U_{P_{2}}=Y_{P_{2}}\setminus D_{P_{2}}$, by construction the mixed Hodge
structures $H^{2}(U_{P_{1}};\mathbb{Z})$ and $H^{2}(U_{P_{2}};\mathbb{Z})$ are
split Tate and hence by the Torelli theorem for Looijenga pairs [15] the
surfaces $Y_{P_{1}}$ and $Y_{P_{2}}$ are isomorphic. By choosing an
isomorphism we have that $P_{1}$ and $P_{2}$ provide torus charts on the same
cluster surface. By Proposition 33, we see that there exists a sequence of
combinatorial mutations starting at $P_{1}$ and ending at $P_{2}$. ∎

##  4\. Examples and evidence

###  4.1. Evidence from the Fanosearch program

Many examples and some of the most striking evidence for the conjectures come
from the Fanosearch program, see for example [6, 7].

In Fanosearch, we attempt to classify Fano varieties up to qG-deformation by
looking for their toric degenerations in the form of Fano polytopes.

The paper [7] does not mention cluster varieties. Instead, it defines a class
of Laurent polynomials, called _rigid maximally mutable Laurent polynomials_
(rigid MMLP), and conjectures [7, Conjecture 2] an injective set-theoretic
function from the set of mutation equivalence classes of rigid MMLP in three
variables to the set of qG-deformation families of $\mathbb{Q}$-Fano 3-folds
(that is, $\mathbb{Q}$-factorial Fano 3-folds with terminal singularities).

Out of a Laurent polynomial $f$, we can always form — as sketched in Sec. 2 —
a cluster variety $U=Y\setminus D$ from the seed

$S=\bigl{\\{}\text{mutation data $s=(u,h)$}\mid\text{$f$ is
$s$-mutable}\bigr{\\}}$

If $f$ is a rigid MMLP, the body of numerical data of the Fanosearch program
supports Conjectures 18 for this cluster variety (and the connectedness of its
mutation graph).

###  4.2. Examples from cluster algebras

Cluster algebras — see Sec 6.3 of [9, 10, 11, 12] for an introduction — are
rings of global functions on special cases of our cluster varieties.

Toric degenerations of Grassmannians, flag varieties, more general homogeneous
varieties, and Schubert-type varieties give rise to canonical bases of group
representations.

There are several statements in the literature where a correspondence is
asserted between sets of toric degenerations of a particular variety and sets
of seeds (i.e. torus charts) of a particular cluster algebra. The simplest
such statement asserts that the homogeneous coordinate ring of
$\operatorname{Gr}(2,n)$ is a cluster algebra whose seeds correspond to toric
degenerations, see for example [5, Sec. 4.2]. All of these statements give
examples and evidence for our Conjecture 18.

###  4.3. Conformal blocks

If $C$ is a curve of genus $g$, denote by $M_{C}(2,\text{odd})$ the moduli
space of stable rank two vector bundles on $C$ with determinant a line bundle
of odd degree. It is known that $M_{C}(2,\text{odd})$ is a Fano variety.

Recall that a stable curve is maximally degenerate if every component is
$\mathbb{P}^{1}$ with three distinguished points; in particular, the dual
graph is trivalent. The paper [28] constructs, for all maximally degenerate
genus-$g$ stable curves $\Gamma$, a Fano polytope $P=P_{\Gamma}$ and toric
degeneration $X_{\Gamma}=X_{\Gamma}(2,\text{odd})$. If $\Gamma^{\prime}$ is
obtained from $\Gamma$ by a mutation — the dual graphs undergoing a transition
through a $4$-valent graph — then $P^{\prime}$ is a combinatorial mutation of
$P$. The paper [3] shows — among other things — that there exists a cluster
variety $U=Y\setminus D$ and torus charts on $U$ corresponding to these toric
degenerations $X_{\Gamma}$.

This example is psychologically important because the moduli space of these
Fanos is birational to the moduli space of curves — in particular, for large
genus, it is far from being unirational.

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