# Explicit equations for mirror families to log Calabi-Yau surfaces

Lawrence Jack Barrott

###### Abstract

Mirror symmetry for del Pezzo surfaces was studied in [3] where they suggested
that the mirror should take the form of a Landau-Ginzburg model with a
particular type of elliptic fibration. This argument came from symplectic
considerations of the derived categories involved. This problem was then
considered again but from an algebro-geometric perspective by Gross, Hacking
and Keel in [16]. Their construction allows one to construct a formal mirror
family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and
$D$ a certain type of Weil divisor supporting an ample or anti-ample class. In
the case where the self intersection matrix for $D$ is not negative semi-
definite it was shown in [16] that this family may be lifted to an algebraic
family over an affine base.

In this paper we perform this construction for all smooth del Pezzo surfaces
of degree at least two and obtain explicit equations for the mirror families
and explain some of the motivation for their construction. In the end we will
provide enumerative data supporting the claim that these are in fact the
desired mirror families.

##  1 Introduction

Our starting materials will be a pair $(S,D)$ where $S$ is a smooth del Pezzo
surface of degree at least two and $D\in\mid-K_{S}\mid$ is a cycle of rational
curves. Such pairs were first studied in [23] as relating to certain cusp
singularities and therefore are called _Looijenga pairs_. When viewed another
way they are _log Calabi-Yau surfaces_. Our goal will be to explicitly
construct the mirror to $(S,D)$ in a variety of cases using the Gross-Siebert
program.

Mirror symmetry predicts, according to Givental’s 1994 ICM lecture, that
associated to a toric Fano variety $V$ there should exist a mirror Landau-
Ginzburg model (LG model). This is a pair $(\check{V},\check{W})$ where
$\check{V}$ is a smooth variety and $\check{W}:\check{V}\rightarrow\mathbb{C}$
the _super-potential_ , a regular function. This was extended in work of Cho
and Oh, who in [9] realised that a choice of effective anti-canonical divisor
played a role in determining the Landau-Ginzburg model. We will introduce the
toric construction and then explain how to correct this to the non-toric case
to produce an explicit description of the mirror family of LG models in terms
of algebraic equations over a known base.

The paper [16] constructs a formal smoothing of the $n$-vertex, a cycle of
planes meeting along their axes, over $\operatorname{Spec}\;k[\\![NE(S)]\\!]$.
The key result for us of [16] is the following.

######  Theorem 1.1.

Let $(S,D)$ be a pair as above and suppose that the self intersection matrix
of $D$ is not negative semi-definite. Then the above formal smoothing lifts to
an algebraic family over the entirety of $\operatorname{Spec}\;k[NE(S)]$.

###### Proof.

This is Theorem 0.2 of [16].

∎

The entire Gross-Siebert program is driven by tropicalising the SYZ
conjecture. We briefly describe this philosophy and will explain throughout
the body of the paper how it motivates each step of the construction. First
let us recall the SYZ conjecture:

######  Conjecture 1.2 (The SYZ conjecture).

Let $X,\check{X}$ be a mirror pair of _Calabi-Yau varieties_. Then there is an
affine manifold with singularities $B$ and maps $\phi:X\rightarrow B$,
$\check{\phi}:\check{X}\rightarrow B$ which are dual special Lagrangian
fibrations. This is shown schematically below

$X$$\check{X}$$T$$T^{\vee}$$\phi$$\phi^{\vee}$

There is a natural collection of varieties with such a special Lagrangian
fibration, toric varieties with their momentum maps. The momentum map fibres a
toric variety over a polytope for that variety and the fibres degenerate in a
natural way over the boundary. These toric varieties form an important
laboratory in which we can study mirror symmetry. Indeed Batyrev and Borisov
used this structure to construct mirror Calabi-Yau pairs for “good”
hypersurfaces in toric varieties. Here the duality is induced by replacing the
polytope by the dual polytope.

The philosophy of Gross and Siebert is to assume the existence of such a
fibration $X\rightarrow B$ to a known base supporting a choice of piecewise
linear function $\phi$ taking values in $NE(S)$ and understand how
singularities of the fibration affect the mirror $\check{X}$. For example the
image of a curve under the fibration is a topological space, called a
_tropical amoeba_ , retracting onto a balanced piecewise linear graph, as
described in [24]. Such a skeleton is called a _tropical curve_ , and the
process of passing to this limit is called _tropicalisation_. Gross and
Siebert use these tropical curves to replace curves living on $X$, calling
their analogue _broken lines_.

An important symplectic invariant of a symplectic manifold $(S,\omega)$ is the
symplectic cohomology. This is a ring whose objects are classes of Lagrangian
submanifolds and whose multiplication counts certain Maslov index two disks
filling the gap between specified Lagrangians. Now points of $B$ correspond to
special Lagrangians and the image of one of these triangles is a trivalent
graph on $B$, bending where Maslov index zero disks have been glued on. The
expectation is that the underlying scheme of the mirror should be the spectrum
of this ring. This fails to be true once one moves away from the dimension two
case. There are examples due to Pomerleano where every fibre of the family is
singular and one must resolve these singularities to obtain the correct mirror
family. In general we expect that the family will be projective over an affine
base.

In the toric case the choice of base $B$ embeds into $\mathbb{R}^{2}$ and
there are no index zero disks. Furthermore the base $B$ has one-strata
$v_{1},\ldots,v_{n}$ corresponding to toric invariant divisors,
$D_{1},\ldots,D_{n}$. The construction becomes the following, for every point
$P\in B_{\mathbb{Z}}$ introduce a variable $\vartheta_{P}$. They define a
multiplication rule on the $\mathbb{C}$ linear span of these by declaring that
$\vartheta_{P}\vartheta_{Q}=z^{\phi(P+Q)-\phi(P)-\phi(Q)}\vartheta_{P+Q}$.
This precisely replicates the counts of disks mentioned above. With this
product rule one produces an associative ring whose spectrum fibres over
$\operatorname{Spec}\;k[NS(S)]$ and the mirror family is this family together
with the superpotential $\sum\vartheta_{v_{i}}$.

Now in the general case the presence of singularities of the SYZ fibration
produces Maslov index zero disks in a controlled way. Gross, Hacking and Keel
mimic this by introducing the _canonical scattering diagram_ on $B$, a
combinatorial structure which records the gluing possibilities for Maslov
index zero disks. The techniques themselves were introduced by Kontsevich and
Soibelman in their paper [22] to study $K3$ surfaces. These were applied by
Gross in [14] to study the prototype example of the construction, mirror
symmetry for $\mathbb{P}^{2}$, and by Gross and Siebert in [19] to construct
mirror partners to Calabi-Yau threefolds. It was expanded upon in work of both
[18] and [7] to construct mirrors to Fano varieties.

We begin by constructing the base of the SYZ fibration, as described in [16]
and then perform combinatorics to limit the information we must calculate. The
salient point for the non negative semi-definite case is that there is a
positivity requirement that restricts the possible pairs of broken lines which
may contribute in the product. One may think of this as analogous to the
convergence of quantum cohomology for Fano manifolds. For high degree the
calculations may be explicitly checked by hand, whilst for low degree we must
rely on computer algebra packages. The techniques mentioned here can be
applied to $dP_{1}$ though the explicit equations produced will potentially be
too complicated to reproduce in a meaningful way.

Having found such equations this we will provide numerological evidence that
the constructed families are indeed the correct mirror family. One prediction
from mirror symmetry would be an isomorphism between (an appropriate
enhancement of) the Fukaya category of $V$ and the category of matrix
factorisations of the potential $\check{W}$. To make this more precise one can
pass to the Hochschild cohomology of these categories.

Conjecturally, and with promising results by Bourgeois, Ekholm and Eliashberg
in [6], the Hochschild cohomology of the Fukaya category of $S$ is canonically
isomorphic to the quantum cohomology of $V$. Recall that the quantum
cohomology is the deformation over $\mathbb{C}[H_{2}(S)]$ of the usual
intersection form on the cohomology ring $H^{*}(V)$. The quantum product
$H_{1}*H_{2}$ of classes $H_{1}$ and $H_{2}$ is defined by

$\langle H_{1}*H_{2},H_{3}\rangle=\sum_{\alpha\in
H_{2}(V)}I_{0,3}(V,\alpha,H_{1},H_{2},H_{3})z^{\alpha}$

Where $I_{0,3}(X,\alpha,H_{1},H_{2},H_{3})$ counts rational curves in class
$\alpha$ with three marked points lying on classes Poincaré dual to $H_{1}$,
$H_{2}$ and $H_{3}$. This product converges under some mild positivity
constraints and remains associative with the central fibre reproducing the
classical cup product. The details of this construction are described in [10]
Theorem 8.1.4.

On the mirror side the Hochschild cohomology of the category of matrix
factorisations is isomorphic to the Jacobian ring $Jac(\tilde{W})$ of the
critical locus of $\check{W}$, at least in the case where the critical locus
is isolated. Recall that this is the ring

$k[x_{1},\ldots x_{n}]/\langle\partial\check{W}/\partial x_{i}\rangle$

Since mirror symmetry is supposed to provide us with an isomorphism between
the two $dg$-categories it should also provide an isomorphism between the
corresponding Hochschild cohomologies. In the final section we will look for
numerological evidence that this is indeed the case. A general proof of the
existence of this isomorphism should follow from a deeper understanding of the
gluing requirements for Gromov-Witten invariants.

####  1.0.1 Acknowledgements

This work was an initial project in my PhD thesis, as suggested by Mark Gross.
It was from him that I learnt much of the background and motivation for the
Gross-Siebert program. Various discussions about the material of [16] with
Tyler Kelly, Zhi Jin, Ben Morely and Andrea Petracci proved immensely helpful
to my understanding of the material. This project was included in my thesis
and I must thank Tom Coates and Pelham Wilson for examining me, and ironing
out various parts of the exposition and of course Mark Gross who helped to
review the initial draft of this paper and suggesting the problem.

This project was funded by a variety of sources: by an Internal Graduate
Studentship provided by Trinity College, by a research studentship provided by
the Cambridge Philosophical Society and by Final Term Funding provided by the
Department for Pure Mathematics and Mathematical Statistic (DPMMS) in
Cambridge. I wrote this paper as a research assistant at the NCTS in Taipei.

##  2 Constructions of mirror families to del Pezzo surfaces

Throughout this paper let $S$ be a smooth complex del Pezzo surface and
$D=D_{1}\cup D_{2}\cup\ldots D_{n}$ an anti-canonical cycle of rational curves
with $n$ at least three (we will explain later how to deal with the case of
$dP_{2}$). Recall that by the classification of del Pezzo surfaces this
ensures that $S$ is isomorphic to one of the following:

  * •

$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

  * •

The blow-up of $\mathbb{P}^{2}$ in zero through eight points in general
position.

The respective Chow groups $A_{1}(S)$ are generated by

  * •

The pullback of a point under the two projection maps.

  * •

A hyperplane class $H$ and the exceptional curves $E_{1},\ldots,E_{9-d}$.

whilst the groups $A_{0}(S)$ and $A_{2}(S)$ are isomorphic to $\mathbb{Z}$
generated by a point and a fundamental class respectively.

This pair can be thought of in many ways, but perhaps the most appealing to
those studying mirror symmetry is as a _log Calabi-Yau varieties_ , varieties
such that there is a non-vanishing differential form on $S\setminus D$ with at
worst logarithmic singularities along $D$. Of these the surfaces
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, $\mathbb{P}^{2},dP_{8},dP_{7}$ and
$dP_{6}$ are all toric and mirror families can be constructed using techniques
we mentioned in the introduction. The next three surfaces were studied in [4].
To build up our intuition with the Gross-Siebert program we will show that
these agree with the predictions there. We will then handle one of the
remaining two cases, the blow up in seven points. Precisely the same
techniques can be applied in the case of $dP_{1}$ but given the frankly
atrocious nature of the equations for $dP_{2}$ we do not pursue this.

In the toric case the base is a fan for the variety in the sense of [11].
Recall that a fan is a simplicial complex of cones in $\mathbb{R}^{n}$, where
a cone is the $\mathbb{R}^{+}$-linear span of a collection of vectors
$v_{1},\ldots,v_{k}$. Then according to [11] there is an equivalence of
categories between the category of toric varieties and toric morphisms and the
category of fans and their morphisms. The base of the SYZ fibration for a
toric del Pezzo surface is just the fan. This can be recovered, up to a choice
of embedding into $\mathbb{R}^{2}$, as the dual intersection complex of $D$.

######  Construction 2.1.

Let $D=D_{1}\cup D_{2}\cup\ldots\cup D_{n}$ be a cycle of rational curves on a
smooth surface $S$ such that $D_{i}\cap D_{j}$ is a single point just when $i$
and $j$ differ by 1 mod $n$ and otherwise is empty (in the case $n=2$ we relax
this to saying that there are two points in the unique intersection $D_{1}\cap
D_{2}$). Then $\Delta_{(S,D)}$ contains precisely one zero-dimensional cell
$\\{0\\}$ corresponding to the interior $S\setminus D$. For each component
$D_{i}$, $\Delta_{(S,D)}$ contains a one-dimensional cone with $v_{i}$ its
primitive generator. Attach the zero-dimensional cone as 0 inside each of
these rays. Now introduce a two-dimensional cone in $\Delta_{(S,D)}$ for each
intersection point of $D_{i}\cap D_{j}$, spanned by $v_{i}$ and $v_{j}$. This
produces the _dual intersection complex_ $\Delta_{(S,D)}$ as a cone complex
but it carries more structure. We write $B$ to be the underlying topological
space of the cone complex $\Delta_{(S,D)}$. It is homeomorphic to
$\mathbb{R}^{2}$. We give $B\setminus\\{0\\}$ an affine manifold structure by
defining an affine coordinate chart by embedding the union of the cones
$\mathbb{R}^{\geq 0}v_{i+1}\oplus\mathbb{R}^{\geq 0}v_{i}$ and
$\mathbb{R}^{\geq 0}v_{i}\oplus\mathbb{R}^{\geq 0}v_{i-1}$ into
$\mathbb{R}^{2}$ via the relations $v_{i-1}\mapsto(1,0)$, $v_{i}\mapsto(0,1)$
and $v_{i+1}\mapsto(-1,-D_{i}^{2})$. This expresses $\Delta_{(S,D)}$ not just
as a complex of sets but an affine manifold with singularities (indeed a
single singularity at the origin).

This construction makes perfect sense outside of the toric case, and morally
should give the correct base for the mirror, except that it may not embed via
an affine map into $\mathbb{R}^{2}$. This fits into the general picture that
the base of the SYZ fibration should be an affine manifold with singularities,
a topological manifold with charts away from codimension two such that the
transition functions lie in the affine transformation group
$\mathbb{R}^{n}\ltimes Sl(\mathbb{Z}^{n})$. That the dual intersection complex
does embed into $\mathbb{R}^{2}$ in the toric case is a consequence of the
following lemma:

######  Lemma 2.2 (Toric reconstruction).

Let $S$ be a compact toric surface and $D$ the toric boundary. Then $(S,D)$ is
a Looijenga pair, $\Delta_{(S,D)}$ embeds into $\mathbb{R}^{2}$ and any choice
of embedding gives the fan for a toric variety isomorphic to $S$.

###### Proof.

See [11] chapter 3.

∎

An integral affine manifold $M$ carries a sheaf of integral vector fields,
which we write $\Lambda_{M}$. On a chart this is isomorphic to the constant
sheaf with coefficients $\mathbb{Z}^{n}$. The transition functions naturally
give identifications of these constant sheaves. In addition, we define
$M(\mathbb{Z})$ to be the set of points of $M$ with integer coordinates in
some, hence all, integral affine coordinate chart.

Given a choice of ample divisor $H$ on $S$ toric, the fan $\Delta_{(S,D)}$
carries a natural choice of piecewise linear function up to a choice of global
linear function and we denote this $\phi$. Such a function is defined by how
it changes where it is non-linear. If $\phi_{v_{i},v_{i+1}}$ denotes the
linear extension of $\phi$ restricted to $\mathbb{R}^{\geq
0}v_{i}+\mathbb{R}^{\geq 0}v_{i+1}$, then we determine $\phi$ by insisting
that $\phi_{v_{i},v_{i+1}}-\phi_{v_{i-1},v_{i}}=(H.D_{i})n_{i}$, where $n_{i}$
is a primitive element of the dual space vanishing on $v_{i}$ and positive on
$v_{i}\mathbb{R}^{\geq 0}+v_{i+1}\mathbb{R}^{\geq 0}$.

Let us calculate the mirror in one of the toric cases, $\mathbb{P}^{2}$. As we
said in the introduction that in the toric case there are no corrections from
singularities of the fibration. We will see that the conjectured isomorphism
exists.

######  Example 2.3.

Our first goal is to find the fan. Fortunately we learnt a fan for
$\mathbb{P}^{2}$ back in infancy: it has one cells generated by $(1,0),(0,1)$
and $(-1,-1)$ as shown in Figure 2.1.

$(1,1)$ Figure 2.1:

Calculating the product defined above, using the ample divisor $H$ being the
class of a line in $\mathbb{P}^{2}$, one finds
$\vartheta_{(1,0)}\vartheta_{(0,1)}=z\vartheta_{(-1,-1)}$. As in the basis
constructions of [8] these functions morally ought to produce an embedding of
the mirror into affine space and their sum should be the super-potential.
Doing this we obtain the mirror as being $\mathbb{G}_{m}^{2}$ with
superpotential $x+y+z/xy$. As expected this is the mirror predicted by other
constructions. Calculating the Jacobian ring of the singularity we find that
it is $\mathbb{Z}[X,z]/\langle X^{3}-z\rangle$, isomorphic to the quantum
cohomology of $\mathbb{P}^{2}$.

Gross and Siebert were influenced initially by the Mumford degeneration and we
turn to this for guidance on how to construct a choice of $\phi$ in the non-
toric case.This construction will also show how different choices of ample $H$
can give rise to different mirrors.

We fix the data of all the Looijenga pairs $(S,D)$ we will study, so let us
record here a choice of boundary divisor and affine manifold for all the non-
toric del Pezzo surfaces. Here we record the boundary divisor in
$\mathbb{P}^{2}$ before blowing up. The circles represent points to be blown
up, with the exceptional divisor included in the boundary if the circle is
red. Since we can’t embed the dual complex as an affine manifold in
$\mathbb{R}^{2}$, we instead provide a list of cones in $\mathbb{R}^{2}$ and
an integral affine isomorphism between two of these cones. The dual
intersection complex is then obtained by identifying these two cones using
this isomorphism. In the pictures below the cone shaded grey is identified
integral linearly with the standard first quadrant, whilst the dotted region
is removed entirely.

Surface |  Boundary |  Class of boundary |  Dual complex
---|---|---|---
$dP_{5}$ |  |  $\cup D_{i}=E_{1}\cup(H-E_{1}-E_{2})$ $\cup E_{2}\cup(H-E_{2}-E_{3})\cup(H-E_{1}-E_{4})$ |
$dP_{4}$ |  |  $\cup D_{i}=E_{1}\cup(H-E_{1}-E_{2})\cup$ $(H-E_{3}-E_{4})\cup(H-E_{1}-E_{5})$ |
$dP_{3}$ |  |  $\cup D_{i}=(H-E_{1}-E_{2})\cup$ $(H-E_{3}-E_{4})\cup(H-E_{5}-E_{6})$ |
$dP_{2}$ |  |  $\cup D_{i}=(H-E_{1}-E_{2})\cup$ $(2H-E_{3}-E_{4}-E_{5}-E_{6}-E_{7})$ |

###  2.1 The Mumford degeneration

The construction of the mirror family was inspired by the Mumford degeneration
of an Abelian variety to the union of toric varieties. Let us recall the
construction of [25].

######  Example 2.4.

Let $N\cong\mathbb{N}^{k}$ be a lattice, $B\subset
N_{\otimes\mathbb{N}}\mathbb{R}$ be a lattice polyhedron, $\mathcal{P}$ a
lattice polyhedral decomposition of $B$ and $\phi:B\rightarrow\mathbb{R}$ a
strictly convex piecewise linear integral function. One takes the graph over
$\phi$ to produce a new polyhedron

$\Gamma(B,\phi):=\\{(n,r)\in N_{\mathbb{R}}\oplus\mathbb{R}\mid n\in
B,r\geq\phi(n)\\}$

This produces a lattice polyhedron unbounded in the positive direction on
$\mathbb{R}$. To construct a family from this we perform a cone construction,
let $C(\Gamma)$ be the closure of the cone over $\Gamma(B,\phi)$:

$C(\gamma)=\overline{\\{(n,r_{1},r_{2})\in
N_{\mathbb{R}}\oplus\mathbb{R}\oplus\mathbb{R}\,|\,(n,r_{1})\in
r_{2}\Gamma(B,\phi)\\}}.$

This carries an action of $\mathbb{R}^{+}$ given by translating the second
component of $\Gamma(B,\phi)$. The integral points of $C(\Gamma)$ form a
graded monoid and we can take $Proj$ of this to produce a variety projective
over $\mathbb{A}^{1}$. We write this $\mathbb{P}_{\Gamma(B,\phi)}$. The
general fibre is a toric variety, whilst the fibre over the origin is a union
of toric varieties.

The construction of [16] mimics this in reverse: it claims that the central
fibre should be the $n$-vertex, which for ${n>2}$ is a cycle of $n$ copies of
$\mathbb{A}^{2}$ glued adjacently along their axes, and then attempts to
smooth. We cannot hope to have global coordinates as in the toric Mumford
degeneration, only local coordinates. To incorporate this data the authors of
[16] define a twist of the tangent sheaf which has enough global sections, in
particular the function $\phi$ lifts to a section of this bundle.

######  Definition 2.5.

Let $(S,D)$ be a Looijenga pair with associated dual intersection complex
$\Delta_{(S,D)}$ and let $\eta:NE(S)\rightarrow M$ be a homomorphism of
monoids. We want to construct a multi-valued piecewise linear function on
$\Delta_{(S,D)}$ which will bend only along the one-cells. This will be a
collection of piecewise linear functions defined on open subsets of $B$ which
differ by linear functions on overlaps. Such functions are determined by their
bends at one-cells, which are encoded as follows. For a one-cell
$\tau=\mathbb{R}^{\geq 0}v_{i}$, choose an orientation $\sigma_{+}$ and
$\sigma_{-}$ of the two two-cells separated by $\tau$ and let $n_{\tau}$ be
the unique primitive linear function positive on $\sigma_{+}$ and annihilating
$\tau$. We want to construct a representative $\phi_{i}$ for $\phi$ on
$\sigma_{+}\cup\sigma_{-}$. Writing $\phi_{+}$ and $\phi_{-}$ for the linear
function defined by $\phi_{i}$ on $\sigma_{+}$ and $\sigma_{-}$, the function
$\phi_{i}$ is then defined up to a linear function by the requirement that

$\phi_{+}-\phi_{-}=n_{\tau}\otimes\eta([D_{i}])$

Such a function is convex in the sense of [16] Definition 1.11, and one says
that it is _strictly convex_ if $\eta([D_{i}])$ is not invertible for any $i$.

This function determines a $M^{gp}\otimes\mathbb{R}$-torsor $\mathbb{P}$ as
defined in [21], Construction 1.14, on $B\setminus\\{0\\}$ which is trivial on
each $(\sigma_{+}\cup\sigma_{-})\setminus\\{0\\}$, i.e., is given by
$((\sigma_{+}\cup\sigma_{-})\setminus\\{0\\})\times(M^{gp}\otimes\mathbb{R})$.
These trivial torsors are glued on the overlap of two adjacent such sets,
namely on $\mathbb{R}^{\geq 0}v_{i}\oplus\mathbb{R}^{\geq 0}v_{i+1}$, via the
map

$(x,p)\mapsto(x,p+\phi_{i+1}(x)-\phi_{i}(x))$

which induce isomorphisms on the monoid of points lying above $\phi$. This
construction is designed to allow us to run the Mumford construction locally,
even if we cannot run it globally. Write $\pi:\mathbb{P}\rightarrow
B\setminus\\{0\\}$, and we write $\mathcal{M}$ for the sheaf
$\pi_{*}(\Lambda_{\mathbb{P}})$, bearing in mind that $\mathbb{P}$ also has
the structure of an affine manifold. Then there is a canonical exact sequence

$0\rightarrow\underline{M^{gp}}\rightarrow\mathcal{M}\xrightarrow{r}\Lambda_{B}\rightarrow
0$

We write $r$ for the second map in this sequence. This will not be mentioned
again until we define the canonical scattering diagram so keep this in mind
until then. Furthermore, [15], Definition 1.16 gives a subsheaf of monoids
$\mathcal{M}^{+}\subset\mathcal{M}$.

If one performs this construction in the case of a toric variety with the
function $\phi$ and pairs it with an ample class one obtains the height
function we used in the Mumford degeneration. From this Gross, Hacking and
Keel use such a $\phi$ to produce a canonical deformation of the $n$-vertex
corresponding to the case where there are no corrections. We introduce the
corrected version directly.

##  3 Scattering diagrams

In Mumford’s degeneration of an Abelian variety above all the fibres of the
SYZ torus fibration on a generic member of the family are smooth. This cannot
be the case for a special Lagrangian fibration of a general variety, there
must be some singular fibres in the interior and around these singular fibres
there will be some monodromy action. This monodromy is an obstruction to a
naive product rule
$\vartheta_{P}\vartheta_{Q}=z^{\phi(P+Q)-\phi(P)-\phi(Q)}\vartheta_{+P+Q}$
being well-defined on the mirror. Thus one needs a corrected notion of this
product, which from symplectic geometry can be done by counting so-called
Maslov index two disks as described in [2]. These can be generated by gluing
on Maslov index zero disks onto standard Maslov index two disks. Rather than
trying to make this symplectic heuristic precise, we instead are motivated by
this picture as follows. If we stand well away from the singularities and
tropicalise these Maslov index zero disks they appear to be a collection of
lines passing out from the origin together with the information of their
class. This data is the inspiration for the definition below of a scattering
diagram on the base $B$.

######  Definition 3.1.

Let $B$ be an affine manifold with a single singularity, with $B$ homeomorphic
to $\mathbb{R}^{2}$ with singularity at the origin, so that
$B^{*}:=B\setminus\\{0\\}$ is an affine manifold. Let $\mathcal{M}$ be a
locally constant sheaf of Abelian groups on $B^{*}$ with a subsheaf of monoids
$\mathcal{M}^{+}\subset\mathcal{M}$ and equipped with a map
$r:\mathcal{M}\rightarrow\Lambda_{B}$. Let $\mathcal{J}$ be a sheaf of ideals
in $\mathcal{M}^{+}$ with stalk $\mathcal{J}_{x}$ maximal in
$\mathcal{M}^{+}_{x}$ for all $x\in B^{*}$. Let $\mathcal{R}$ denote the sheaf
of rings locally given by the completion of $k[\mathcal{M}^{+}]$ at
$\mathcal{J}$. A _scattering diagram with values in the pair
$(\mathcal{M},\mathcal{J})$ on $B$_ is a function $f$ which assigns to each
rational ray from the origin an section of the restriction of $\mathcal{R}$ to
the ray. We require the following properties of this function:

  * •

For each $\mathfrak{d}$ one has $f(\mathfrak{d})=1$ mod
$\mathcal{J}|_{\mathfrak{d}}$.

  * •

For each $n$ there are only finitely many $\mathfrak{d}$ for which
$f(\mathfrak{d})$ is not congruent to $1$ mod
$\mathcal{J}|_{\mathfrak{d}}^{n}$. These $\mathfrak{d}$ are called _walls_.

  * •

For each ray $\mathfrak{d}$ and for each monomial $z^{p}$ appearing in
$f(\mathfrak{d})$ one has $r(p)$ tangent to $\mathfrak{d}$. A line for which
$r(p)$ is a positive generator of $\mathfrak{d}$ for all $p$ with
$c_{p}\not=0$ is called an _incoming ray_. If instead $r(p)$ is a negative
generator of $\mathfrak{d}$ for all $p$ with $c_{p}\not=0$, it is called an
_outgoing ray_.

We denote such an object by the tuple $(B,f,\mathcal{M},\mathcal{J})$. We say
that $(B,f,\mathcal{M},\mathcal{J})$ is obtained from
$(B,f^{\prime},\mathcal{M},\mathcal{J})$ by _adding outgoing rays_ if for each
ray $\mathfrak{d}$ one can write
$f(\mathfrak{d})=f^{\prime}(\mathfrak{d})(1+\sum(c_{p}z^{p}))$ where for each
monoid element $p$ with $c_{p}\not=0$ the vector $-r(p)$ is a generator of
$\mathfrak{d}$.

In our case the sheaf of monoids $\mathcal{M}^{+}$ will be as given in
Definition 2.5, with the monoid $M$ being a finitely generated sharp submonoid
of $H_{2}(S,\mathbb{Z})$ containing $NE(S)$, the monoid generated by effective
curves on $S$. Being sharp means the only invertible element of $M$ is the
identity element, and so the maximal ideal is just the complement of the
identity. We introduce the choice of scattering diagram, the _canonical
scattering diagram_ , and then motivate and define each term appearing. A ray
$\mathfrak{d}=(av_{i}+bv_{i+1})\mathbb{R}^{\geq 0}$ specifies a blow up of $S$
given by refining the fan until $\mathfrak{d}$ is a one-cell and all the two-
cells are integrally isomorphic to $(\mathbb{R}^{\geq 0})^{2}$. The ray
$\mathfrak{d}$ then corresponds to a component $C$ in the inverse image of
$D$. To this ray then we assign the power series

$f(\mathfrak{d}):=\exp\left[\sum_{\beta}k_{\beta}N_{\beta}z^{\eta(\pi_{*}(\beta))-k_{\beta}m_{\mathfrak{d}}^{\prime}}\right],$

where $m_{\mathfrak{d}}^{\prime}$ is the unique lift of a primitive outward
pointing tangent vector $m_{\mathfrak{d}}$ to $\mathfrak{d}$ not contained in
the relative interior of $\mathcal{M}|_{\mathfrak{d}}$. The number $N_{\beta}$
counts the number of relative curves mapping to $(S,D)$ tangent to $C$ to
maximal order $k_{\beta}$ at a single point as outlined in [16] at the
beginning of section 3. By the Riemann-Roch formula this is of dimension

$dim\>S-3-K_{S}.\beta-k_{\beta}+1=0.$

By [5] this produces a Gromov-Witten invariant. This construction may also be
run using logarithmic Gromov-Witten invariants rather than these blow ups, see
[20] for the definition, and this will be explored in future work of Gross and
Siebert.

Conjecturally this encodes the glueing data for a generating set of the open
Gromov-Witten invariants. It should be possible to recreate the entire Gromov-
Witten theory of $S$, both open and closed from this data but this problem is
very difficult.

###  3.1 Broken lines

To encode the Maslov index two disks themselves we collapse them onto a
skeleton in the base of the SYZ fibration, having pushed all the singularities
to the origin. Then following [24] the disks appear as balanced tropical
curves together with a mark of the class that they lie in. The skeleta of
these disks are trivalent graphs with one leg ending at the origin and the
other two legs passing to infinity. The definition below defines how one of
these legs behave, the full picture only arising once we have discussed pairs
of pants and even then we suppress the third leg which by necessity ends at
origin.

######  Definition 3.2.

A _broken line from $v\in B(\mathbb{Z})$ to $P\in B$_ on a scattering diagram
$(B,f,R,J)$ is a choice of piecewise linear function $l:\mathbb{R}^{\leq
0}\rightarrow B$ and a map $m:\mathbb{R}^{\leq
0}\rightarrow\prod_{t\in\mathbb{R}^{\leq 0}}\mathcal{R}_{l(t)}$ such that the
following hold:

  * •

$l$ has only finitely many points where it is non-linear and these only occur
where $l$ maps into a rational ray of $B$.

  * •

$l(0)=P$.

  * •

$l(t)$ lies in the same cone as $v$ and is parallel to $v$ for all $t$
sufficiently negative.

  * •

$l$ does not map an interval to a ray through the origin.

  * •

$m(t)\in\mathcal{R}_{l(t)}$ for all $t$ and is a monomial in this ring,
written as $c_{t}z^{m_{t}}$. Further, on each domain of linearity of $l$,
$m(t)$ is given by a section of $\mathcal{R}$ pulled back to this domain of
linearity.

  * •

For $t$ very negative, $m_{t}\in\mathcal{M}^{+}_{l(t)}$ is the unique element
not lying in the interior of this monoid satisfying $r(m_{t})=v$.

  * •

$r(m(t_{0}))=-{\partial l\over\partial t}\big{|}_{t=t_{0}}$ wherever $l$ is
linear.

  * •

$m$ only changes at those points where $l$ is non-linear.

  * •

Let $t\in\mathbb{R}_{\leq 0}$ be a point where $l$ is non-linear. We write
$\partial(l_{+})$, $\partial(l_{-})$, $m_{+}$ and $m_{-}$ for the values of
$\partial l{/}\partial t$ and $m$ on either side of $t$. Suppose that $l(t)$
lies on a ray $\mathfrak{d}$ with primitive normal vector $n_{\mathfrak{d}}$,
negative on $\partial(l_{-})$. Then $m_{+}$ is a monomial term of
$m_{-}f_{\mathfrak{d}}^{\langle n_{\mathfrak{d}},\partial(l_{-})\rangle}$.

There is more technical content in [16] exploring how to deform the Mumford
construction to produce a formal smoothing of the $n$-vertex. We do not
concern ourselves with those results at the moment since we now have enough to
define the multiplication rule on the $\vartheta$-functions. In [15] Gross,
Hacking and Siebert introduce an entirely abstract construction of the
$\vartheta$-functions.

The key data we will need is the count of pairs of pants, which are expressed
in terms of broken lines as pairs of broken lines $(l_{P},m_{P})$ and
$(l_{Q},m_{Q})$ from $P$ and $Q$ respectively to an irrational point near to
$R$ such that $(\partial l_{P}/\partial t)|_{t=0}+(\partial l_{Q}/\partial
t)_{t=0}=R$. Let $T_{P,Q\rightarrow R}$ denote the set of such pairs.

We have seen on page 1 how to define a product on the $\vartheta$-functions in
the toric case, so let us now study the general case. The outcome of our
conversation about Hochschild cohomology was that the ring of regular
functions on the mirror is the degree zero part of symplectic cohomology of
the original variety. Following [17] for each integral point $P$ in $B$
introduce a symbol $\vartheta_{P}$. As a $k$-vector space the ring
$QH(\check{W})$ is freely generated by the $\vartheta_{P}$. We take the
content of [16] Theorem 2.34 as the definition of the product of
$\vartheta_{P}$ and $\vartheta_{Q}$, so the product
$\vartheta_{P}\vartheta_{Q}$ is equal to

$\sum_{R}\sum_{((l_{P},m_{P}),(l_{Q},m_{Q}))\in T_{P,Q\rightarrow
R}}m_{P}(0)m_{Q}(0)\vartheta_{R}$

This is analogous to the approach of [15]. A key result of [16] is that for
fixed order $J^{n}$ and for a consistent scattering diagram in the sense of
[16], Definition 2.26, this does not depend on the choice of irrational point
near $R$. This consistency property will hold in particular for the canonical
scattering diagrams considered below. This sum need not terminate and indeed
will in general only produce a power series. However, in the case that $D$
supports an ample divisor, this power series will in fact be a polynomial. Our
goal in what follows is to limit the terms which can occur and describe a
generating basis of the $\vartheta$-functions.

###  3.2 The combinatorics of scattering diagrams

We now explain how to use these functions to embed the mirror into affine
space. At the moment it is not clear that the relations between the
$\vartheta$ functions should be algebraic. Let us explain how to exploit the
structure of a scattering diagram to prove algebraic convergence. To do this
we require an example to work with, for simplicity the del Pezzo surface
$dP_{5}$.

######  Example 3.3.

Figure 3.1 below gives a choice of boundary divisor, where circles denote
blown-up points and red lines denote components of $D$.

Figure 3.1:

From this we construct the base of the scattering diagram as in Figure 3.2,
where the grey section is glued isomorphically to the standard upper right
quadrant

$1$$2$$3$$4$$5$ Figure 3.2:

In this case one can construct the entire scattering diagram: it is finite and
only non-trivial on the one-cells. We will not do this since in future
examples the scattering diagram will be non-trivial on a dense set of rays.

Each point of $B$ defines a curve class as via $a\;v_{i}+b\;v_{i+1}\mapsto
aD_{i}+bD_{i+1}$. Define the piecewise linear function $E$ on $B$ given by
$-K_{S}\cdot(aD_{i}+bD_{i+1})$. From the Fano condition this defines a
strictly convex piecewise linear function on the space $B$ which is positive
everywhere but at the origin. The function $E$ extends to a linear function on
the tangent space at each point in the interior of a maximal cell. This allows
us to evaluate $E$ on tangent vectors to broken lines away from one-cells.

In fact, given a broken line, $E(\partial l/\partial t)$ is increasing as a
function of $t$. Indeed, at a point $t$ where $l$ is non-linear by definition
the tangent vector changes by a positive generator of the ray $\mathfrak{d}$,
and $E$ is positive on this generator. Similarly if $l$ crosses from one
maximal cell of $B$ to another then by convexity of $E$ the value of
$E(\partial l/\partial t)$ increases.

We use this as follows. Suppose that two broken lines $(l_{1},m_{1})$ and
$(l_{2}.m_{2})$ from $v_{1}$ and $v_{2}$ combine to form a pair of pants at
$P$. Then by definition we must have

$(\partial l_{1}/\partial t)|_{t=0}+(\partial l_{2}/\partial t)|_{t=0}+P=0.$

Since $E(P)\geq 0$, and $-E(v_{i})\leq E((\partial l_{i}/\partial t)|_{t=0})$,
we see that $E(v_{1})$ and $E(v_{2})$ control how many new maximal cells
$l_{1}$ and $l_{2}$ can enter, and how many times they can bend.

Let us begin by understanding what could possibly contribute to the product
$\vartheta_{(-1,0)}\vartheta_{(0,1)}$. Let us number the two-cells anti-
clockwise starting at the upper right cell as in Figure 3.2.

Let $l_{1}$ be a broken line coming from $(-1,0)$ and $l_{2}$ a broken line
from $(0,1)$. We have that $E((-1,0))=E((0,1))=1$. This immediately implies
that $0\leq E(P)\leq 2$. Further, if $E(P)=2-n$, then the total number of
bends and crossings of one-dimensional cells of $l_{1}$ and $l_{2}$ is $n$.
First note that $n=0$ does not occur: since $v_{1}$ and $v_{2}$ are in
different cones, at least one of the broken lines must cross a one-cell. If
$n=1$, then we cross one one-cell and further bends or crossings can occur.
There is then only one possibility, the obvious pair of pants as described in
Figure 3(a) contributing to $\vartheta_{(-1,1)}$. Finally let us study the
case $n=2$. In this case, $E(P)=0$, hence $P=0$, and we can choose a point
near the origin, say in the second two-cell. If $l_{2}$ enters the grey region
as detailed in Figure 3(c) it emerges moving in the direction $(-1,0)$ and so
has $E(\partial l/\partial t)=1$. Therefore in order to reach the second cell,
it must cross two more one-cells, and hence it can’t contribute to a pair of
pants. Thus $l_{2}$ must start in cell 2 and never leaves it.

A similar argument shows that $l_{1}$ can only pass from cell 3 to cell 2, and
either $l_{1}$ or $l_{2}$ bends at a ray $\mathbb{R}^{\geq 0}v$ with $E(v)=1$
and $v\in B(\mathbb{Z})$. There is only one choice for such a ray, namely the
one-cell separating cells 2 and 3. See Figure 3(b).

In general, there will only be a finite number of rays with bounded $E$ and
only a finite combination of bends. Thus in more complicated cases this
analysis can be implemented in a computer algebra package via the algorithm
outlined in Algorithm 1 and a non-optimised version written in Python is
available from me on request.

Algorithm 1 Theta relations algorithm

1:object AffineManifold

2:$\triangleright$ Define a data structure consisting of a fan $\Sigma$, a
piecewise linear function $\phi$ and a collection of integer points of
$\Sigma$, $\Sigma_{\mathbb{Z}}$.

3:object BrokenLine

4:$\triangleright$ Define a data structure consisting of an initial direction
$v$, a monomial, a tangent direction thought of as $\partial l/\partial
t\mid_{t=0}$ and a list of points $P_{i}$ generating rays along which the
broken line bent

5:function ProductFormula($l_{1}$, $l_{2}$, affinemanifold)

6:$\triangleright$ Define a function of two broken lines with tangent
directions $v_{1}$ and $v_{2}$ in an affine manifold, affinemanifold which
finds all possible terms appearing in the product
$\vartheta_{v_{1}}\vartheta_{v_{2}}$

7: int maxE := $\phi(v_{1})+\phi(v_{2})$

8: def endpoints := The set of points in $\Sigma_{\mathbb{Z}}$ such that
$\phi(P)\leq maxE$.

9: for P in endpoints do

10: if $l_{1}$ and $l_{2}$ could pass through P and $v_{1}+v_{2}+P=0$ then

11: print the data of how $l_{1}$ and $l_{2}$ have bent

12: if $l_{1}$ could pass through $P$ then

13: def newline := $l_{1}$ but now also scattering off of $\mathbb{R}^{\geq
0}P$ as well

14: ProductFormula (newline, $l_{2}$)

15: if $l_{2}$ could pass through $P$ then

16: def newline := $l_{2}$ but now also scattering off of $\mathbb{R}^{\geq
0}P$ as well

17: ProductFormula ($l_{1}$, newline)

18:def affinemanifold := AffineManifold( …)

19:$\triangleright$ Define the base of the appropriate scattering diagram

20:def $L_{1}$ := BrokenLine ( …)

21:$\triangleright$ Define $L_{1}$ to be a broken line, not scattering off of
any rays with initial direction $v_{1}$

22:def $L_{2}$ := BrokenLine ( …)

23:$\triangleright$ Define $L_{2}$ to be a broken line, not scattering off of
any rays with initial direction $v_{2}$

24:ProductFormula ($L_{1}$, $L_{2}$, affinemanifold)

(a)

(b)

(c)

Figure 3.3:

Doing this systematically for each pair
$\vartheta_{v_{i-1}}\vartheta_{v_{i+1}}$ we can find schematic relations for
the products which highlight which relative invariants we need to find.
Filling out the relations for $dP_{5}$ we find the following relations.

$\displaystyle\vartheta_{(1,0)}\vartheta_{(-1,1)}$
$\displaystyle=g_{1}\vartheta_{(0,1)}+f_{1}$ (3.1)
$\displaystyle\vartheta_{(0,1)}\vartheta_{(-1,0)}$
$\displaystyle=g_{2}\vartheta_{(-1,1)}+f_{2}$ (3.2)
$\displaystyle\vartheta_{(-1,1)}\vartheta_{(0,-1)}$
$\displaystyle=g_{3}\vartheta_{(-1,0)}+f_{3}$ (3.3)
$\displaystyle\vartheta_{(-1,0)}\vartheta_{(1,0)}$
$\displaystyle=g_{4}\vartheta_{(0,-1)}+f_{4}$ (3.4)
$\displaystyle\vartheta_{(0,-1)}\vartheta_{(0,1)}$
$\displaystyle=g_{5}\vartheta_{(1,0)}+f_{5}$ (3.5)

The terms $f_{i}$ count curves meeting $D_{i}$ at a single point with tangency
order one together with some monomials coming from broken lines looping around
the origin.

As in the work of Hulya Arguz [1] the easy case is when the boundary consists
of at least four components and the corresponding algebra is a quadratic
algebra. The three component case is tractable but to compute the two
component case $dP_{2}$ it is best to blow up the two intersection points of
the two divisors. This refines the dual complex but keeps the interior
$S\setminus D$ invariant. As a consequence, this changes the mirror family
because the base $\operatorname{Spec}\;k[P]$ increases in dimension. However,
the mirror family to the original surface can be found by restricting to a
subfamily, see [16], Section 6.2 for details. In keeping with our previous
notation we denote the exceptional curves over these by $E_{8}$ and $E_{9}$.
The choice of dual complex now becomes:

we re-label the theta functions by the corresponding cohomology class, so
$\vartheta_{D_{i}}$ for $\vartheta_{i}$, further we write $\vartheta_{nD_{i}}$
for the function $\vartheta_{nv_{i}}$. With these labels we can apply the
techniques of Example 3.3 to show that the possible terms appearing in the
relations between the theta functions become:

$\displaystyle\vartheta_{C}\vartheta_{L}$
$\displaystyle=r_{1}\vartheta_{E_{8}}+r_{2}\vartheta_{E_{9}}+r_{3}+c_{1}(E_{8})$
(3.6) $\displaystyle\vartheta_{E_{8}}\vartheta_{E_{9}}$
$\displaystyle=r_{4}\vartheta_{3L}+\vartheta_{2L}c_{1}(L)+\vartheta_{L}c_{2}(L)+c_{3}(C)+r_{5}$
$\displaystyle\quad+\vartheta_{C}c_{2}(C)+\vartheta_{2C}c_{1}(C)+r_{6}\vartheta_{3C}$
(3.7) $\displaystyle\vartheta_{C}^{3}$
$\displaystyle=\vartheta_{3C}+c_{1}(L)\vartheta_{C}+r_{7}+c_{1}(E_{9})+r_{8}+r_{9}\vartheta_{E_{8}}+r_{10}\vartheta_{E_{9}}$
(3.8) $\displaystyle\vartheta_{L}^{3}$
$\displaystyle=\vartheta_{3L}+c_{1}(C)\vartheta_{L}+r_{11}+c_{1}(E_{8})+r_{12}+r_{13}\vartheta_{E_{9}}+r_{14}\vartheta_{E_{8}}$
(3.9) $\displaystyle\vartheta_{C}^{2}$
$\displaystyle=\vartheta_{2C}+r_{15}\vartheta_{C}+c_{1}(L)$ (3.10)
$\displaystyle\vartheta_{L}^{2}$
$\displaystyle=\vartheta_{2L}+r_{16}\vartheta_{L}+c_{1}(C)$ (3.11)

where the functions $r_{i}$ come from the combinatorics of broken lines which
don’t bend. The monomials which they carry come from the fact that the values
of $\phi(\partial l/\partial t)$ are different at the start and end of the
broken lines. The functions $c_{n}(D)$ are counts of broken lines bending due
to curves tangent to a class $D$ at a single point to order $n$. It is this
finite list of open Gromov-Witten invariants which we must calculate. To
calculate these invariants we have two options, either one could directly
compute the moduli space and find the degree of the virtual fundamental class
or one can use properties of the scattering diagram. Since we have not yet
explained the role monodromy plays in the scattering diagram we will do the
latter since it forms an important tool in the theory.

##  4 Explicit Gromov-Witten counts and Monodromy

We cannot apply the Mumford construction globally in the non-toric case since
the function $\phi$ is not global but we can apply it locally to each face. To
do so if $\tau$ is a one- or two-dimensional face of $\Delta_{(S,D)}$, we
write $\Delta_{(S,D)}/\tau$ for the localised fan at $\tau$, i.e., the fan
given by adding $\mathbb{R}\tau$ to each cone containing $\tau$ and removing
cones of $\Delta_{(S,D)}$ not containing $\tau$. Then after choosing a
representative for $\phi$, $\phi_{\tau}$, we have a piecewise linear function
single-valued on $\Delta_{(S,D)}/\tau$ and one can consider the graph
$\Gamma(\Delta_{(S,D)}/\tau,\phi_{\tau})$. If $\tau$ lies as the one-cell
separating two two-cells, $\sigma_{+}$ and $\sigma_{-}$, let
$\phi_{\sigma_{\pm}}$ be the linear extensions of $\phi_{\tau}$ restricted to
the maximal cones $\sigma_{\pm}$. Then we have canonical inclusions of graphs
and an equality

$\Gamma(\Delta_{(S,D)}/\tau,\phi_{\tau})=\Gamma(\Delta_{(S,D)}/\sigma_{+},\phi_{\sigma_{+}})\cap\Gamma(\Delta_{(S,D)}/\sigma_{-},\phi_{\sigma_{-}})$
(4.1)

All the monoids $\Gamma(\Delta_{(S,D)}/\rho,\phi_{\rho})$ have actions by $M$,
giving rings $k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]$ with $k[M]$-algebra
structures, and different choices of representative for $\phi$ lead to
canonically isomorphic $k[M]$-algebras. Choose a monomial ideal $I\subset M$
such that $A_{I}:=k[M]/I$ is Artinian, and write

$k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]_{I}:=k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]\otimes_{k[M]}A_{I}.$

Let

$U_{\rho}=\operatorname{Spec}\;k[\Gamma(\Delta_{(S,D)}/\rho),\phi_{\rho}]_{I}.$

Then the relation (4.1) leads to natural open inclusions
$U_{\sigma_{\pm}}\subset U_{\tau}$, which gives us gluing data for a scheme.
The paper [16] gives explicit formulae for the rings involved and proves that
this gluing data does indeed satisfy the descent condition. This is the strict
analogue of the Mumford degeneration, but does not contain any data about the
instanton corrections. In particular, with the above gluing, it is not obvious
that the glued scheme will carry enough global functions to be embedded into
an affine variety. For example, a naive approach to finding global functions
is to attempt to extend monomial functions from $U_{\rho}$, but this
frequently doesn’t work precisely because of monodromy. The authors of [16]
correct the gluing using automorphisms of the rings
$k[\Gamma(\Delta_{(S,D)}/\sigma,\phi_{\sigma})]_{I}$ for the two-cells
$\sigma$ and taking a limit over all $I$.

Suppose that we have a scattering diagram $(f,\Delta_{(S,D)},R,J)$ and fix a
thickening ideal $I$. Let $\sigma$ be a two-cell with boundary one-cells
$\tau_{+}$, $\tau_{-}$. Choose points $P_{+},P_{-}\in\sigma$ very close to
$\tau_{+}$, $\tau_{-}$ respectively. Choose a path $\gamma$ connecting $P_{+}$
to $P_{-}$. Such a path selects a primitive normal vector $n_{\mathfrak{d}}$
to each ray through the origin in $\sigma$, pairing negatively with the
tangent vector of the path. By assumption there are only finitely many rays
$\mathfrak{d}$ passing through the origin such that $f(\mathfrak{d})$ is not 1
mod $I$. To each such ray take $\theta_{\mathfrak{d}}$ to be the automorphism
defined on a toric monomial $z^{p}$,
$p\in\Gamma(\Delta_{(S,D)}/\sigma,\phi_{\sigma})$ by

$\theta_{\mathfrak{d}}(z^{p})=z^{p}f(\mathfrak{d})^{\langle
n_{\mathfrak{d}},r(p)\rangle}$

Now we define the automorphism $\theta_{\sigma}$ to be the ordered composition
of the $\theta_{\mathfrak{d}}$ as one moves along $\gamma$. This yields an
automorphism of $U_{\sigma}$. In addition, the definition of $U_{\tau}$ is
modified slightly using $f(\tau)$, see [16], Equation (2.7). Thus we obtain a
modification of the $U_{\tau}$’s and a modification of the gluing inclusions
$U_{\tau_{+}}\supset U_{\sigma}\subset U_{\tau_{-}}$. This gives a new glued
scheme which depends on the scattering diagram and the ideal $I$. Consistent
scattering diagrams then have the property that the associated rings have a
natural basis of regular functions
$\\{\vartheta_{p}\,|\,p\in\Delta_{(S,D)}(\mathbb{Z})\\}$ which yield an
embedding in an affine scheme over $\operatorname{Spec}\;A_{I}$. One of the
main theorems of [16], namely Theorem 3.8, states the canonical scattering
diagram is consistent. Importantly there is an algorithmic way of computing
consistent scattering diagrams in certain situations, and a way of reducing
the calculation of the canonical scattering diagram to these situations.

So for the moment, consider a case where $\Delta_{(S,D)}$ is in fact affine
isomorphic to $\mathbb{R}^{2}$ and $\phi$ is single-valued. Then the sheaf
$\mathcal{M}^{+}$ extends across $0$, with the stalk at $0$ being
$\Gamma(\Delta_{(S,D)},\phi)$. In this construction $\mathcal{R}_{0}$ is the
completion of $k[\mathcal{M}^{+}_{0}]$ at its maximal monomial ideal. All of
the automorphisms $\theta_{\sigma}$ act on $\mathcal{R}_{0}$. Our construction
so far has not attempted to incorporate the monoid ring $\mathcal{R}_{0}$.
Take a cyclic ordering of the two-cells $\sigma_{1},\ldots\sigma_{n}$ and let
$\theta$ be the composite
$\theta_{\sigma_{n}}\circ\ldots\circ\theta_{\sigma_{1}}$ considered as a map
$\mathcal{R}_{0}\rightarrow\mathcal{R}_{0}$. Then the scattering diagram is
consistent if the map $\theta$ is the identity, see [7] and [16], Theorem
3.30. Let us give an example calculation.

######  Example 4.1.

Any toric variety relative to its toric boundary has trivial canonical
scattering diagram. Therefore we will find no interesting examples here. Let
us give an example motivated instead by $dP_{5}$: we will see shortly the
connection.

Our initial scattering diagram is $\mathbb{R}^{2}$ with two incoming rays
$(1+z^{(1,0)})$ and $(1+z^{(0,1)})$:

Let us calculate the composition $\theta$ along a circle anti-clockwise around
the origin starting at $(-1,-2)$ acting on $z^{(1,0)}$ and $z^{(0,1)}$. The
first wall we meet is $\mathfrak{d}_{1}=\mathbb{R}_{\geq 0}(1,0)$. Applying
the above definition of the automorphism $\theta$:

$\theta_{\mathfrak{d}_{1}}(z^{(1,0)})=z^{(1,0)}(1+z^{(1,0)})^{\langle(1,0),(0,-1)\rangle}=z^{(1,0)}$
$\theta_{\mathfrak{d}_{1}}(z^{(0,1)})=z^{(0,1)}(1+z^{(1,0)})^{\langle(0,1),(0,-1)\rangle}=z^{(0,1)}(1+z^{(1,0)})^{-1}$

Continuing along our path we meet the wall $\mathfrak{d}_{2}=\mathbb{R}_{\geq
0}(0,1)$. Here $\theta_{\mathfrak{d}_{2}}$ is given by:

$\theta_{\mathfrak{d}_{2}}(z^{(1,0)})=z^{(1,0)}(1+z^{(0,1)})^{\langle(1,0),(1,0)\rangle}=z^{(1,0)}(1+z^{(0,1)})$
$\theta_{\mathfrak{d}_{2}}(z^{(0,1)})=z^{(0,1)}(1+z^{(0,1)})^{\langle(0,1),(1,0)\rangle}=z^{(0,1)}$

Therefore the composition $\theta$ sends $z^{(1,0)}$ to
$z^{(1,0)}(1+z^{(0,1)})$ and $z^{(0,1)}$ to
$z^{(0,1)}(1+z^{(1,0)}+z^{(1,1)})^{-1}$. To first order this looks as if we
are missing outgoing rays with monomials $(1+z^{(1,0)})$ and $(1+z^{(0,1)})$.
Let us add those in and then repeat this calculation.

Taking again a path anti-clockwise starting at $(-1,-2)$ we first meet the
wall $\mathfrak{d}_{3}=\mathbb{R}_{\geq 0}(0,-1)$. The associated
$\theta_{\mathfrak{d}_{3}}$ acts by:

$\theta_{\mathfrak{d}_{3}}(z^{(1,0)})=z^{(1,0)}(1+z^{(0,1)})^{\langle(1,0),(-1,0)\rangle}=z^{(1,0)}(1+z^{(0,1)})^{-1}$
$\theta_{\mathfrak{d}_{3}}(z^{(0,1)})=z^{(0,1)}(1+z^{(0,1)})^{\langle(0,1),(-1,0)\rangle}=z^{(0,1)}$

The next wall is $\mathfrak{d}_{1}$, which again acts as before. The third
wall we meet is $\mathfrak{d}_{2}$, again acting as before. The final wall we
encounter is $\mathfrak{d}_{4}=\mathbb{R}_{\geq 0}(-1,0)$, where the
associated $\theta$ is given by:

$\theta_{\mathfrak{d}_{4}}(z^{(1,0)})=z^{(1,0)}(1+z^{(1,0)})^{\langle(1,0),(0,1)\rangle}=z^{(1,0)}$
$\theta_{\mathfrak{d}_{4}}(z^{(0,1)})=z^{(0,1)}(1+z^{(1,0)})^{\langle(0,1),(0,1)\rangle}=z^{(0,1)}(1+z^{(1,0)})$

One easily calculates the composite of all of these and one finds that to
second order the composite acts by $z^{(1,0)}\mapsto
z^{(1,0)}(1+z^{(1,1)}+\ldots)$ so it appears that we are missing an outgoing
ray carrying monomial $(1+z^{(1,1)})$. Introducing such a ray to obtain a new
scattering diagram

Now one calculates $\theta$ is the identity, so we have finally constructed a
consistent scattering diagram.

The term-by-term nature of the above construction seems to suggest that in the
background lurks a general procedure. The Kontsevich-Soibelman lemma of [22]
proves that from any scattering diagram embedded in $\mathbb{R}^{2}$ (i.e, one
with no singularities at the origin) one can construct a consistent scattering
diagram by adding in only outgoing rays. In particular one can perform this
correction order-by-order as in our example making it computable. This is
explicitly described in the following lemma.

######  Lemma 4.2 (The Kontsevich-Soibelman lemma).

Let $(B,f,R,J)$ be a scattering diagram with $B\cong\mathbb{R}^{2}$ as affine
manifolds. Let $P$ be a point of $B$ contained inside a two-cell $\sigma$. The
ordered composite of the functions $\theta$ along a path $\gamma$ around the
origin with $\gamma(0)=\gamma(1)=P$ gives rise to an automorphism
$\theta_{\gamma}$ of the stalk of sheaf $\mathcal{R}$ at zero. There exists
another scattering diagram $(B,f^{\prime},R,J)$ such that the corresponding
automorphism $\theta_{\gamma}$ is the identity and such that for each ray
$\mathfrak{d}$ one has

$f^{\prime}(\mathfrak{d})=f(\mathfrak{d})(1+\sum a_{i}z^{m_{i}})$

where the $z^{m_{i}}$ are all outgoing.

A key result of Gross, Hacking and Keel, Theorem 3.25 of [16] explores how to
flatten the base of the scattering diagram by using _toric models_. A toric
model of $(S,D)$ is a choice of birational morphism
$\phi:(S,D)\rightarrow(S^{\prime},D^{\prime})$ obtained by blowing down curves
in the interior of $S$ meeting $D$ in precisely one point such that
$S^{\prime}$ is toric and $D^{\prime}$ the toric boundary. This restricts to
an isomorphism $D\cong D^{\prime}$ and so induces a map of dual complexes
which “flattens” the singularities in the sense that the ensuing dual
intersection complex is isomorphic to $\mathbb{R}^{2}$ as an affine manifold.
In the philosophy of Gross Hacking and Keel it pushes the singularities out to
infinity. We reincorporate the singularities by instead including them into
the scattering diagram as rays carrying expressions
$1+z^{[E_{i}]+\phi(v_{i})}$ where $E_{i}$ is an exceptional curve meeting
$D_{i}$ and $v_{i}$ is the primitive generator of the ray in
$\Delta_{(S^{\prime},D^{\prime})}$ corresponding to $D_{i}$. We make this
construction explicit in the example in the next section. There is a canonical
morphism $\nu:\Delta_{(S,D)}\rightarrow\Delta_{(S^{\prime},D)}$ and in Lemmas
3.27 and 3.28 of [16] the authors prove that this induces an isomorphism of
sheaves $\mathcal{R}(S,D)\cong\mathcal{R}(S^{\prime},D)$ together with a
bijection of broken lines between $\Delta_{(S,D)}$ and
$\Delta_{(S^{\prime},D)}$. To be precise we have the following theorem of
[16]:

######  Theorem 4.3.

Let $(S,D)$ be a Looijenga pair with $(S^{\prime},D)$ a toric model. Then the
complex $\Delta_{S^{\prime},D}$ is isomorphic as an affine manifold to
$\mathbb{R}^{2}$. Let
$(\mathbb{R}^{2},f_{S^{\prime}},\underline{M},\underline{M}\setminus 0)$ be
the scattering diagram with values in $A_{1}(S)$ on
$\Delta_{(S^{\prime},D^{\prime})}$ whose value along $\mathbb{R}_{\geq
0}v_{i}$ is the product of incoming rays
$\prod_{j}(1+z^{[E_{i,j}]+\phi(v_{i})})$ and outgoing rays
$\prod_{j}(1+z^{-[E_{i,j}]+\phi(v_{i})})^{-1}$ where the $E_{i,j}$ are the
blown-down curves meeting $D_{i}$. Since this satisfies the assumptions of the
Kontsevich-Soibelman lemma, we let $\bar{f}_{S^{\prime}}$ be the associated
consistent diagram.

Let $\bar{f}_{S}$ be the canonical scattering diagram on $\Delta_{(S,D)}$.
Locally $\Delta_{(S,D)}$ and $\Delta_{(S^{\prime},D^{\prime})}$ are isomorphic
and we can evaluate $f_{S^{\prime}}$, $\bar{f}_{S}$ and $\bar{f}_{S^{\prime}}$
on the same line $\mathfrak{d}$. Then one has an equality

$\bar{f}_{S^{\prime}}(\mathfrak{d})=\bar{f}_{S}(\mathfrak{d})f_{S^{\prime}}(\mathfrak{d}).$

this has the effect of replacing the base of the scattering diagram by an
affine manifold isomorphic to $\mathbb{R}^{2}$ and swapping the directions of
rays $(1+z^{[E_{i,j}]+\phi(v_{i})})$ from outgoing to incoming. This is seen
explicitly in the example in the next section.

After possibly blowing up some double points of $D$, we can then blow down
appropriate curves to construct toric models, see [16], Proposition 1.3. This
allows one to calculate the original scattering diagram by applying the
Kontsevich-Soibelman lemma to $f_{S^{\prime}}$. An implementation of this
algorithm in Sage is available alongside this paper.

##  5 Explicit formulae

Let us practice constructing toric models in the cases we are interested in.

######  Example 5.1.

Recall the dual intersection complex for $dP_{5}$:

This is certainly not affine isomorphic to $\mathbb{R}^{2}$. The boundary has
five components, and this suggests that we should attempt to find a toric
model isomorphic to $dP_{7}$, since there is a choice of boundary divisor
there also consisting of five components. A model for $dP_{7}$ together with
its toric boundary is contained below. The red lines are a triangle in
$\mathbb{P}^{2}$, the red circles are blown up to obtain $dP_{7}$, and the
black points are further blown up to obtain $dP_{5}$ with $dP_{7}$ as its
toric model.

12

The corresponding dual complex is given by

In particular the self intersection of every component of the boundary for
$dP_{5}$ was -1, whereas two of the components of the boundary of $dP_{7}$
have self intersection 0.

We now incorporate this into the construction of the scattering diagram.
Initially after making a choice of ordering we have have the scattering
diagram:

$1+z^{E_{1}+\phi(v_{1})}$$1+z^{E_{2}+\phi(v_{2})}$

and we saw in our motivation for the Kontsevich-Soibelman lemma that this
produces a consistent diagram

$1+z^{E_{1}+\phi(v_{1})}$$1+z^{E_{1}+\phi(v_{1})}$$1+z^{E_{1}+\phi(v_{1})}$$+z^{E_{2}+\phi(v_{2})}$$1+z^{E_{2}+\phi(v_{2})}$$1+z^{E_{2}+\phi(v_{2})}$

which agrees up to changing the direction of the incoming rays and pushing
forwards with the scattering diagram found in [16] Figure 1.1. This is exactly
as predicted by the Kontsevich-Soibelman Lemma.

The attached implementation is optimised in three ways. Firstly one only is
interested in terms of order $n$, so we may reduce all the series modulo terms
of higher order and use efficient power calculations to do so. Secondly this
can be distributed over multiple cores. Thirdly each ray stores a list of all
powers already calculated so as to minimise repeat calculations. I would
appreciate any input on how to significantly speed up this algorithm.

Since by the previous section only a finite collection of classes need be
calculated we may terminate the calculation once those coefficients have been
obtained and substitute them into the formulae (3.6)-(3.11) we found using the
techniques in section 2.2.2.

Recall that the Picard group of $dP_{d}$ is generated by a hyperplane class
$H$ and the class of the exceptional curves $E_{1},\ldots E_{9-d}$. By using
the attached implementation of the Kontsevich-Soibelman lemma one can read off
the desired coefficients and the resulting families are contained in Table
5.1.

Surface | Equations | Super-potential
---|---|---
$dP_{5}$ | $\vartheta_{1}\vartheta_{3}=z^{[D_{2}]}\vartheta_{2}+z^{[D_{4}]+[D_{5}]}$ | $\sum\vartheta_{i}$
| $\vartheta_{2}\vartheta_{4}=z^{[D_{3}]}\vartheta_{3}+z^{[D_{1}]+[D_{4}]}$ |
| $\vartheta_{3}\vartheta_{5}=z^{[D_{4}]}\vartheta_{4}+z^{[D_{1}]+[D_{2}]}$ |
| $\vartheta_{4}\vartheta_{1}=z^{[D_{5}]}\vartheta_{5}+z^{[D_{2}]+[D_{3}]}$ |
| $\vartheta_{5}\vartheta_{2}=z^{[D_{1}]}\vartheta_{1}+z^{[D_{3}]+[D_{4}]}$ |
$dP_{4}$ | $\vartheta_{1}\vartheta_{3}=z^{[D_{2}]}\vartheta_{2}+z^{[D_{4}]}\vartheta_{4}+$ | $\sum\vartheta_{i}$
|  $z^{H-E_{1}}+z^{2H-E_{1}-E_{2}-E_{3}-E_{5}}+z^{2H-E_{1}-E_{2}-E_{4}-E_{5}}$ |
| $\vartheta_{2}\vartheta_{4}=z^{[D_{1}]}\vartheta_{1}+z^{[D_{3}]}\vartheta_{3}+$ |
|  $z^{H-E_{3}}+z^{H-E_{4}}+z^{2H-E_{2}-E_{3}-E_{4}-E_{5}}$ |
$dP_{3}$ | $\vartheta_{1}\vartheta_{2}\vartheta_{3}=z^{[D_{1}]}\vartheta_{1}^{2}+z^{[D_{2}]}\vartheta_{2}^{2}+z^{[D_{3}]}\vartheta_{3}^{2}+$ | $\sum\vartheta_{i}$
|  $z^{[D_{1}]}\vartheta_{1}(z^{E_{1}}+z^{E_{2}}+z^{H-E_{3}-E_{5}}+z^{H-E_{3}-E_{6}}+$ |
|  $z^{H-E_{4}-E_{5}}+z^{H-E_{4}-E_{6}})+z^{[D_{2}]}\vartheta_{2}(z^{E_{3}}+z^{E_{4}}+$ |
|  $z^{H-E_{1}-E_{5}}+z^{H-E_{1}-E_{6}}+z^{H-E_{2}-E_{5}}+z^{H-E_{2}-E_{6}})+$ |
|  $z^{[D_{3}]}\vartheta_{3}(z^{E_{5}}+z^{E_{6}}+z^{H-E_{1}-E_{3}}+z^{H-E_{1}-E_{4}}$ |
|  $z^{H-E_{2}-E_{3}}+z^{H-E_{2}-E_{4}})+z^{H}+$ |
|  $(z^{2H}+z^{4H-\sum E_{i}})\alpha+$ |
|  $z^{3H-\sum E_{i}}(z^{E_{2}-E_{1}}+z^{E_{1}-E_{2}}+z^{E_{4}-E_{3}}+z^{E_{3}-E_{4}}+$ |
|  $z^{E_{6}-E_{5}}+z^{E_{6}-E_{5}})+z^{5H-2\sum E_{i}}-4z^{[D]}$ |
| $\alpha=(z^{-E_{1}}+z^{-E_{2}})(z^{-E_{3}}+z^{-E_{4}})(z^{-E_{5}}+z^{-E_{6}})$ |
Table 5.1: Explicit equations for mirror families

The expression for the mirror family to $dP_{3}$ agrees with that found in
[16], Example 6.13. To express the formulae for $dP_{2}$ requires some
additional notation. In this case we had to blow up two additional points
which lay on the intersection of the chosen conic $C$ and line $L$. Let
$E_{8}$, $E_{9}$ be the exceptional divisors lying over the intersections
points of $C$ and $L$. Let us relabel the $\vartheta_{i}$ by the cohomology
class of the corresponding boundary divisor, e.g., writing $\vartheta_{D_{i}}$
for $\vartheta_{i}$. There is an action of $C_{2}$ exchanging $E_{1}$ and
$E_{2}$, an action of $S_{5}$ permuting $E_{3}$ through $E_{7}$ and an action
of $C_{2}$ exchanging $E_{8}$ and $E_{9}$. When we write $\prod(1+z^{hH+\sum
a_{i}E_{i}})$ we mean to take the product of $1+z^{h[H]+\sum a_{i}[E_{i}]}$
over elements of the orbit under the action of $C_{2}\times S_{5}$. In
particular we choose here representatives with $a_{1}\geq a_{2}$ and $a_{3}$
through $a_{7}$ decreasing. For instance $\prod(1+z^{H+E_{3}})$ would be the
product
$(1+z^{H+E_{3}})(1+z^{H+E_{4}})(1+z^{H+E_{5}})(1+z^{H+E_{6}})(1+z^{H+E_{7}})$.
Searching through the output of our implementation of the Kontsevich-Soibelman
lemma we are able to find the coefficients we were missing in the equations
(3.6) to (3.11) for the mirror family to $dP_{2}$ relative to our choice of
boundary. After calculating these we find that the mirror family is given by

$\displaystyle\vartheta_{C}\vartheta_{L}$
$\displaystyle=z^{E_{8}}\vartheta_{E_{8}}+z^{E_{9}}\vartheta_{E_{9}}+3z^{C+L+2E_{8}+2E_{9}}+B_{1}(E_{8})z^{E_{8}}$
$\displaystyle\vartheta_{E_{8}}\vartheta_{E_{9}}$
$\displaystyle=\vartheta_{3L}z^{L}+\vartheta_{2L}B_{1}(L)z^{L}+\vartheta_{L}B_{2}(L)z^{L}$
$\displaystyle\quad+z^{C}B_{3}(C)+3z^{2C+2L+3E_{8}+3E_{9}}+\vartheta_{C}B_{2}(C)z^{C}$
$\displaystyle\quad+\vartheta_{2C}B_{1}(C)z^{C}+\vartheta_{3C}z^{C}$
$\displaystyle\vartheta_{C}^{3}$
$\displaystyle=\vartheta_{3C}+3B_{1}(L)z^{L+E_{8}+E_{9}}\vartheta_{C}+6z^{C+2L+3E_{8}+3E_{9}}+6B_{1}(E_{9})$
$\displaystyle\quad
z^{L+E_{8}+2E_{9}}+3z^{L+2E_{8}+E_{9}}\vartheta_{E_{8}}+3z^{L+E_{8}+2E_{9}}\vartheta_{E_{9}}$
$\displaystyle\vartheta_{L}^{3}$
$\displaystyle=\vartheta_{3L}+3B_{1}(C)z^{C+E_{8}+E_{9}}\vartheta_{L}+6z^{2C+L+3E_{8}+3E_{9}}+6B_{1}(E_{8})$
$\displaystyle\quad
z^{C+2E_{8}+E_{9}}+3z^{C+E_{8}+2E_{9}}\vartheta_{E_{9}}+3z^{C+2E_{8}+E_{9}}\vartheta_{E_{8}}$
$\displaystyle\vartheta_{C}^{2}$
$\displaystyle=\vartheta_{2C}+2\vartheta_{L}z^{L+E_{8}+E_{9}}+2B_{1}(L)z^{L+E_{8}+E_{9}}$
$\displaystyle\vartheta_{L}^{2}$
$\displaystyle=\vartheta_{2L}+2\vartheta_{C}z^{C+E_{8}+E_{9}}+2B_{1}(C)z^{C+E_{8}+E_{9}}$

where the coefficients $B_{k}(E)$ are terms in the scattering diagram related
to the relative invariants of curves meeting the curve $E$ in one point with
tangency order $k$. The term $B_{i}(-)$ is the coefficient of $t^{i}$ in the
product $\prod N_{k}(-,t^{k})$, where the $N_{k}(-,t)$ are given by the
following formulae:

$\displaystyle N_{3}(C,t)$
$\displaystyle=\prod(1+9tz^{2H-E_{1}-E_{2}-E_{3}})\prod(1+9tz^{3H-2E_{1}-E_{2}-E_{3}-E_{4}-E_{5}})\prod(1+9tz^{4H-2E_{1}-2E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}})$
$\displaystyle\prod(1+9tz^{4H-3E_{1}-E_{2}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})\prod(1+72tz^{4H-2E_{1}-2E_{2}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})$
$\displaystyle\prod(1+9tz^{5H-3E_{1}-2E_{2}-2E_{3}-2E_{4}-E_{5}-E_{6}-E_{7}})\prod(1+9tz^{6H-3E_{1}-3E_{2}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-E_{7}})$
$\displaystyle N_{2}(C,t)$
$\displaystyle=\prod(1+4tz^{H-E_{1}})\prod(1+4tz^{2H-E_{1}-E_{2}-E_{3}-E_{4}})\prod(1+4tz^{3H-2E_{1}-E_{2}-E_{3}-E_{4}-E_{5}-E_{6}})$
$\displaystyle\prod(1+4tz^{4H-2E_{1}-2E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})$
$\displaystyle N_{1}(C,t)$
$\displaystyle=\prod(1+tz^{E_{3}})\prod(1+tz^{H-E_{1}-E_{3}})\prod(1+tz^{2H-E_{1}-E_{2}-E_{3}-E_{4}-E_{5}})$
$\displaystyle\prod(1+tz^{3H-2E_{1}-2E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}})$
$\displaystyle N_{3}(L,t)$
$\displaystyle=\prod(1+9tz^{3H-2E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})\prod(1+9tz^{4H-E_{1}-2E_{3}-2E_{4}-2E_{5}-E_{6}-E_{7}})$
$\displaystyle\prod(1+9tz^{5H-E_{1}-E_{2}-3E_{3}-2E_{4}-2E_{5}-2E_{6}-E_{7}})\prod(1+72tz^{5H-E_{1}-E_{2}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-2E_{7}})$
$\displaystyle\prod(1+9tz^{5H-2E_{1}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-2E_{7}})\prod(1+9tz^{6H-2E_{1}-E_{2}-3E_{3}-3E_{4}-2E_{5}-2E_{6}-2E_{7}})$
$\displaystyle\prod(1+9tz^{7H-2E_{1}-2E_{2}-3E_{3}-3E_{4}-3E_{5}-3E_{6}-2E_{7}})$
$\displaystyle N_{2}(L,t)$
$\displaystyle=\prod(1+4tz^{2H-E_{3}-E_{4}-E_{5}-E_{6}})\prod(1+4tz^{3H-E_{1}-2E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})$
$\displaystyle\prod(1+4tz^{4H-E_{1}-E_{2}-2E_{3}-2E_{4}-2E_{5}-E_{6}-E_{7}})\prod(1+4tz^{5H-2E_{1}-E_{2}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-2E_{7}})$
$\displaystyle N_{1}(L,t)$
$\displaystyle=\prod(1+tz^{E_{1}})\prod(1+tz^{H-E_{3}-E_{4}})\prod(1+tz^{2H-E_{1}-E_{3}-E_{4}-E_{5}-E_{6}})$
$\displaystyle\prod(1+tz^{3H-E_{1}-E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}-E_{7}})$
$\displaystyle N_{1}(E_{8},t)$
$\displaystyle=\prod(1+tz^{H-E_{3}-E_{8}})\prod(1+tz^{2H-E_{1}-E_{3}-E_{4}-E_{5}-E_{8}})\prod(1+9tz^{3H-E_{1}-E_{2}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}-E_{8}})$
$\displaystyle\prod(1+tz^{3H-E_{1}-E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}-E_{8}})\prod(1+tz^{3H-2E_{1}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}-E_{8}})$
$\displaystyle\prod(1+tz^{4H-2E_{1}-E_{2}-2E_{3}-2E_{4}-E_{5}-E_{6}-E_{7}-E_{8}})\prod(1+tz^{5H-2E_{1}-2E_{2}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-E_{7}-E_{8}})$
$\displaystyle N_{1}(E_{9},t)$
$\displaystyle=\prod(1+tz^{H-E_{3}-E_{9}})\prod(1+tz^{2H-E_{1}-E_{3}-E_{4}-E_{5}-E_{9}})\prod(1+9tz^{3H-E_{1}-E_{2}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}-E_{9}})$
$\displaystyle\prod(1+tz^{3H-E_{1}-E_{2}-2E_{3}-E_{4}-E_{5}-E_{6}-E_{9}})\prod(1+tz^{3H-2E_{1}-E_{3}-E_{4}-E_{5}-E_{6}-E_{7}-E_{9}})$
$\displaystyle\prod(1+tz^{4H-2E_{1}-E_{2}-2E_{3}-2E_{4}-E_{5}-E_{6}-E_{7}-E_{9}})\prod(1+tz^{5H-2E_{1}-2E_{2}-2E_{3}-2E_{4}-2E_{5}-2E_{6}-E_{7}-E_{9}})$

After setting all of the monomials equal to one we find that the values of
$B_{i}(-)$ reduce to $6561$, $459$, $27$, $6561$, $459$, $27$, $81$ and $81$
respectively and this fibre of the family is smooth, we will use this in
calculations later. We expect that each fibre of this family is a double cover
of the plane. This can be seen by projecting onto $\vartheta_{C}$ and
$\vartheta_{L}$, then the equations are entirely symmetric under exchanging
these two variables, noting of course that $z^{C}B_{3}(C)=z^{L}B_{3}(L)$.

Unfortunately this is a deformation of the four-vertex, not of the two-vertex.
This stems from our decision to blow up two additional points at the start,
and so the family we actually want is the restriction to the locus
$z^{E_{8}}=z^{E_{9}}=1$. Setting $z^{C}=z^{L}=0$ we see that coordinates on
the central fibre are $\vartheta_{C},\vartheta_{L}$ and
$\vartheta_{E_{8}}-\vartheta_{E_{9}}$. Eliminating the extra variables we
obtain an equation for the family:

$\displaystyle(\vartheta_{E_{8}}-\vartheta_{E_{9}})^{2}$
$\displaystyle=\vartheta_{C}^{2}\vartheta_{L}^{2}-4\vartheta_{C}^{3}z^{C}-4\vartheta_{L}^{3}z^{L}+18\vartheta_{C}\vartheta_{L}z^{C+L}-27z^{2C+2L}-4\vartheta_{L}^{2}z^{L}B_{1}(L)-4\vartheta_{C}^{2}z^{C}B_{1}(C)$
$\displaystyle\quad+20\vartheta_{C}z^{C+L}B_{1}(L)+20\vartheta_{L}z^{C+L}B_{1}(C)+16z^{C+L}B_{1}(L)B_{1}(C)-4\vartheta_{L}z^{L}B_{2}(L)$
$\displaystyle\quad-4\vartheta_{C}z^{C}B_{2}(C)-2\vartheta_{C}\vartheta_{L}B_{1}(E_{8})+30z^{C+L}B_{1}(E_{8})-4z^{C}B_{3}(C)-B_{1}(E_{8})^{2}$
(5.1)

Now we see clearly the leading order terms
$\vartheta_{C}^{2}\vartheta_{L}^{2}=(\vartheta_{E_{8}}-\vartheta_{E_{9}})^{2}$,
giving rise to the two-vertex over the origin. The appropriate super-potential
remains $\vartheta_{C}+\vartheta_{L}$.

One could ask if there is a better way to present the data hiding in the
scattering diagram. In the cases of $dP_{5}$ and $dP_{4}$ the canonical
scattering diagram is finite and can be computed, as was done in Example 3.7
of [16] for one of these. By an observation of Gross, Hacking and Keel the
entire scattering diagram for $dP_{3}$ is generated by the series associated
to a single ray under the birational automorphism group of this surface,
generated by a finite collection of classes. This breaks down in lower
degrees, as far as I am aware there is no known closed form expression for the
scattering diagram associated to the final surface.

There are however more elegant ways to describe terms arising in the equation,
as was seen in Example 6.13 of [16]. Following these ideas we can describe the
terms arising in equation 5.1. To do this we write $N_{i}(-,t)^{(j)}$ for the
coefficient of $t^{j}$ in $N_{i}(-,t)$. Then we find the following birational
descriptions:

Term | Description
---|---
$N_{1}(C,t)^{(1)}z^{C}$ | The sum of $\frac{1}{16}z^{\pi^{*}(H-E)}$ over all contractions to $dP_{8}$ obtained by contracting
| one $-1$-curve meeting $C$ and five meeting $L$, none of which meet
$\pi^{*}E$ or $\pi^{*}H$.
$N_{1}(L,t)^{(1)}z^{L}$ | The sum of $\frac{1}{16}z^{\pi^{*}(H-E)}$ over all contractions to $dP_{8}$ obtained by contracting
| one $-1$-curve meeting $L$ and five meeting $C$, none of which meet
$\pi^{*}E$ or $\pi^{*}H$.
$N_{1}(C,t)^{(2)}z^{C}$ | The sum of $z^{\pi^{*}H}$ over all contractions to $\mathbb{P}^{2}$ obtained by blowing down
| two $-1$-curves meeting $C$ and five meeting $L$.
$N_{1}(L,t)^{(2)}z^{L}$ | The sum of $z^{\pi^{*}H}$ over all contractions to $\mathbb{P}^{2}$ obtained by blowing down
| two $-1$-curves meeting $L$ and five meeting $C$.

Term | Description
---|---
$N_{2}(C,t)^{(1)}z^{C}$ | The sum of $4z^{\pi^{*}\text{-}K_{\bar{S}}}$ over all contractions of a single $-1$-curve meeting $L$
| and not meeting $C$.
$N_{2}(L,t)^{(1)}z^{C}$ | The sum of $4z^{\pi^{*}\text{-}K_{\bar{S}}}$ over all contractions of a single $-1$-curve meeting $C$
| and not meeting $L$.
$N_{1}(E_{8},t)^{(1)}z^{E_{8}}$ | The sum of $z^{\pi^{*}(H-E)}$ over all contractions to $dP_{8}$ obtained by contracting
| four $-1$-curves meeting one of $C$ or $L$ and three meeting the other.

The term $\frac{1}{16}$ in $N_{1}(C,t)^{(1)}z^{C}$ arises because given one of
these classes $\alpha$ there are sixteen $-1$-curves (each of degree at most
three) meeting $\alpha$ and $C$ but not meeting $L$. Fixing any one of these
produces a unique contraction contributing the desired monomial. In other
words a subgroup of the birational automorphism group of the surface
stabilises the class $\alpha$. Finally we observe by direct computation that
$z^{C}N_{3}(C,t)^{(1)}=9z^{C+L}N_{1}(E_{8},t)^{(1)}-9z^{-2K_{S}}$. This gives
a birational interpretation of every term appearing in equation 5.1.

##  6 Comparisons and evidence

We begin this section by comparing the predictions of [4] for the mirror
families. These predictions state the mirror to $dP_{n}$ ought to be a
compactification of a fibration of the two-torus
$W:\mathbb{G}_{m}^{2}\rightarrow\mathbb{C}$ with an $I_{n}$ fibre at infinity.
Here $W$ is the super-potential. For example, in the case of $dP_{5}$ we can
take equations:

$x_{i+1}x_{i-1}=x_{i}+1$

defining a surface inside $\mathbb{A}^{5}$ with $W=\sum x_{i}$. It turns out
this is already a partial compactification of the mirror to $dP_{5}$. To
further compactify, we take the projective closure of the above surface,
taking coordinates $Z_{1},\ldots,Z_{5}$ and $T$ to obtain equations

$Z_{i+1}Z_{i-1}=Z_{i}T+T^{2}.$

Then $T$, $W$ can be viewed as sections of ${\mathcal{O}}(1)$ on this surface,
hence giving a pencil. After resolving the base-points of this pencil, one
obtains the desired compactification. The fibre at infinity therefore has
equations

$Z_{i-1}Z_{i+1}=0$

for $i\in\\{1,2,3,4,5\\}$ inside $\mathbb{P}^{4}$, hence defining a cycle of
five planes. This is exactly as predicted by [4]. For the other del Pezzo
surfaces $dP_{n}$ with $n\geq 4$, the same shape of the mirror as defined by
quadratic equations confirms that the fibre at infinity will be an $I_{n}$
fibre. More careful considerations with the equations show the same for $n=3$
and $2$.

There are two other tests available to us to determine whether the families
described above are indeed the mirror families. The first we implement here,
whilst the second is of wider interest.

###  6.1 Dimension of the Jacobian ring

The rank of the quantum cohomology is equal to the rank of the regular
cohomology, and therefore we already know the expected rank of all of the
Jacobian rings in each case we have constructed. If $S$ is a del Pezzo surface
of degree $d$ then the rank of the middle homology is $10-d$, generated by a
hyperplane and the exceptional curves. Therefore the rank of the quantum
cohomology should be $12-d$. Now let $(\check{V},\check{W})$ be a smooth fibre
of the family such that $\check{W}$ has isolated critical points. The rank of
the Jacobian ring of $\check{W}$ should then be equal to $12-d$. This is the
weakest test since it tells us no enumerative information about the surface.
This rank is at least sensitive to deforming the equations defining the
mirror.

Let us attempt this calculation for the case of the mirror to $dP_{2}$. First
recall that the super-potential on Landau-Ginzburg model is the partial sum
$x_{C}+x_{L}$. If one took the full sum one would construct the mirror to the
non-Fano double blow up of $dP_{2}$, and we do not believe that this mirror
construction should apply.

######  Example 6.1.

Recall that the Jacobian ring of a potential
$W:\mathbb{C}^{2}\rightarrow\mathbb{C}$ is the quotient

$\mathbb{C}[u_{1},u_{2}]/\langle\partial\check{W}/\partial u_{i}\rangle$

We are not quite in this situation and so we need to be more careful. The
Jacobian ring of $(S,W:S\rightarrow\mathbb{C})$ is the function ring of the
critical points of $W$. In our case $S$ is defined by two equations in
$\mathbb{A}^{4}$, $f_{1}$ and $f_{2}$. Therefore we are interested in the
points of $S$ where $\nabla W$ is contained in the span of $\nabla f_{1}$ and
$\nabla f_{2}$. This asks that all the $3\times 3$ minors of the following
matrix vanish

$M=\begin{bmatrix}\partial f_{1}/\partial\vartheta_{C}&\partial
f_{1}/\partial\vartheta_{L}&\partial f_{1}/\partial\vartheta_{8}&\partial
f_{1}/\partial\vartheta_{9}\\\ \partial f_{2}/\partial\vartheta_{C}&\partial
f_{2}/\partial\vartheta_{L}&\partial f_{2}/\partial\vartheta_{8}&\partial
f_{2}/\partial\vartheta_{9}\\\ \partial W/\partial\vartheta_{C}&\partial
W/\partial\vartheta_{L}&\partial W/\partial\vartheta_{8}&\partial
W/\partial\vartheta_{9}\\\ \end{bmatrix}$

This can be calculated using Sage and generating elements of the defining
ideal are:

$(\vartheta_{9}+2)\vartheta_{C}+3\vartheta_{C}^{2}-(\vartheta_{9}+2)\vartheta_{L}-3\vartheta_{L}^{2}+58\vartheta_{C}-58\vartheta_{L},(\vartheta_{8}+2)\vartheta_{C}+3\vartheta_{C}^{2}-(\vartheta_{8}+2)\vartheta_{L}-3\vartheta_{L}^{2}+58\vartheta_{C}-58\vartheta_{L},\\\
-\vartheta_{8}+\vartheta_{9},-\vartheta_{8}+\vartheta_{9},\vartheta_{C}\vartheta_{L}-\vartheta_{8}-\vartheta_{9}-84,\\\
-\vartheta_{C}^{3}-\vartheta_{L}^{3}+\vartheta_{8}\vartheta_{9}-27\vartheta_{C}^{2}+4\vartheta_{C}\vartheta_{L}-27\vartheta_{L}^{2}+2\vartheta_{8}+2\vartheta_{9}-324\vartheta_{C}-324\vartheta_{L}-3000$
(6.1)

Sage allows us to calculate a $k$-basis of the quotient ring and we find that
the quotient is 10-dimensional as a vector space, exactly as hoped for.

###  6.2 Structure of the Jacobian ring

In favourable situations we may calculate both structures, the quantum
cohomology and the Jacobian ring. The quantum cohomology for high degree del
Pezzo surfaces was calculated in [12]. In section 4 they produce a list of
relevant curve classes and in section 5 describe the product as a sum over
those classes.

The question is then what the conjectured isomorphism is. For a divisor $D$ it
was suggested in [13] that this map should send $D$ to the derivative of $W$
with respect to the log vector field $z^{D}\partial z^{D}$. This of course
leaves the question of where a point should be sent. This can be computed by
computing products on both sides, for instance once we know where a hyperplane
is sent we may compute its square to find the class of a point, subtracting
any quantum corrections.

To do this calculation explicitly would be un-enlightening. Rather it would be
an interesting project to attempt to automate this construction. Sage has the
necessary computational tools to carry this out. In future work I hope to
perform the same analysis of the quantum cohomology as was done in [12], but
for the degree two del Pezzo surface. The already know examples would provide
a fertile collection of examples with which to demonstrate the algorithm,
whilst simultaneously allowing us to demonstrate that our construction here is
correct.

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