# On equations of fake projective planes with automorphism group of order 21

Lev Borisov  Department of Mathematics, Rutgers University, Piscataway, NJ
08854, USA<EMAIL_ADDRESS>
  *     * scAbstract. We study Dolgachev elliptic surfaces with a double and a triple fiber and find explicit equations for two new pairs of fake projective planes with $21$ automorphisms. This includes, in particular, the fake projective plane discovered by J. Keum.

scKeywords. Algebraic surfaces, fake projective planes

sc2020 Mathematics Subject Classification. 14J29

  *
cDecember 6, 2022Received by the Editors on September 21, 2021.
Accepted on April 19, 2023.

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

sce-mail<EMAIL_ADDRESS>
The author acknowledges the Office of Advanced Research Computing (OARC) at
Rutgers, The State University of New Jersey for providing access to the Amarel
cluster [Ama] and associated research computing resources that have
contributed to the results reported here.

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Families of (2,3) Dolgachev surfaces
  3. 3 Finding the special Dolgachev surface
  4. 4 Constructing and identifying the fake projective plane
  5. 5 Constructing the fake projective plane of Keum
  6. 6 Open problems
  7. Appendix

## 1\. Introduction

The theory of fake projective planes originated with the famous example of D.
Mumford [Mum79] of a surface of general type with the same Hodge numbers as
the usual projective plane $\mathbb{C}\mathbb{P}^{2}$. By its nature, the
construction did not yield any explicit equations of it. Over the subsequent
decades, work by multiple authors (see for instance [AK17, Ish88, Keu06,
Keu08, KK02, Kli03, PY07, PY10]) produced additional examples and general
results and culminated in the classification of all fake projective planes by
D. Cartwright and T. Steger [CS11, Car14]. These surfaces are classified as
free quotients of the complex $2$-dimensional ball $\mathbb
B^{2}=\\{(z_{1},z_{2}),|z_{1}|^{2}+|z_{2}|^{2}<1\\}$ by certain discrete
arithmetic subgroups. There are exactly $50$ conjugate pairs of such surfaces,
separated into $28$ classes. This extremely useful classification does not
lead to any polynomial equations either since there are no known methods for
constructing explicit automorphic forms for these groups.

Over the last several years, the author of this paper has been involved in
multiple collaborations with the goal of discovering explicit polynomial
equations that define fake projective planes and related surfaces; see [BK20,
BY20, BF20, BBF22]. This paper is a continuation of such efforts, originally
aimed at finding the equations of Mumford’s fake projective plane. While this
goal is still elusive, we find equations of the fake projective plane
constructed by J. Keum in [Keu06], which is commensurable to the Mumford’s
surface. We also find another interesting fake projective plane in the
process. As always, new approaches had to be developed for the case at hand.

It is not surprising that most of the currently computed fake projective
planes have nontrivial automorphism groups, as this provides some avenues for
exploration, and this paper continues the trend. According to the
classification of Cartwright and Steger, the maximum order of the automorphism
group of a fake projective plane is $21$. There are three conjugate pairs of
fake projective planes with automorphism group of this size, and in all three
cases the group is the semi-direct product of a normal subgroup $C_{7}$ and a
non-normal subgroup $C_{3}$. Specifically, here are the planes, with their
names in the CS classification and brief comments:

  * •

$(a=7$, $p=2$, $\emptyset$, $D_{3}2_{7})$ is the first example of the fake
projective plane for which explicit equations were found; see [BK20]. There
are two other conjugate pairs of fake projective planes in its class.

  * •

$(a=7$, $p=2$, $\\{7\\}$, $D_{3}2_{7})$ is the surface constructed by Keum in
[Keu06]. It has three more pairs of fake projective planes in its class,
including Mumford’s fake projective plane.

  * •

$(C20$, $p=2$, $\emptyset$, $D_{3}2_{7})$ is the last of the three surfaces;
it does not seem to be implicated in any other construction. There are no
other pairs in its class.

In all three cases, the quotient of $\mathbb{P}^{2}_{\operatorname{fake}}$ by
the subgroup $C_{7}$ of its automorphism group has a minimal resolution $Y$
with rather peculiar geometry; see [Keu06, Keu08]. The quotient has three
singular points of type $\frac{1}{3}(1,7)$, which are permuted by the residual
$C_{3}$-action of the automorphism group of
$\mathbb{P}^{2}_{\operatorname{fake}}$. The minimum resolution $Y$ of
$\mathbb{P}^{2}_{\operatorname{fake}}/C_{7}$ has three disjoint chains of
three lines with self-intersections $-3,-2,-2$ which we denote by

$S\relbar B\relbar C,\leavevmode\nobreak\ \leavevmode\nobreak\ S_{1}\relbar
B_{1}\relbar C_{1},\leavevmode\nobreak\ \leavevmode\nobreak\ S_{2}\relbar
B_{2}\relbar C_{2}.$

Here $\relbar$ indicates a transversal intersection point. In addition, $Y$ is
fibered over $\mathbb{C}\mathbb{P}^{1}$, with generic fibers of genus $1$, two
multiple fibers, three nodal fibers and one fiber of type $I_{9}$ (a ring of
nine $\mathbb{C}\mathbb{P}^{1}$ with self-intersection $(-2)$ each) with
components

$A\relbar B\relbar C\relbar A_{1}\relbar B_{1}\relbar C_{1}\relbar
A_{2}\relbar B_{2}\relbar C_{2}\relbar A.$

The residual automorphism group $C_{3}$ preserves the fibration structure and
acts by $A\to A_{1}\to A_{2}\to A$ and similarly for the $S$, $B$ and $C$
curves.

The multiplicities of the multiple fibers in the case of
$(a=7,p=2,\emptyset,D_{3}2_{7})$ are $2$ and $4$, and they are $2$ and $3$ in
the other two cases, which are the focus of this paper. In particular, in the
cases of interest, the curves $S$, $S_{1}$ and $S_{2}$ are $6$-sections of the
fibration. The two special fibers $3F_{2}$ and $2F_{3}$ have multiplicity $3$
and $2$, respectively. The reductions $F_{2}$ and $F_{3}$ are linearly
equivalent to $2F$ and $3F$ with $F=K_{Y}$, and the generic fiber is
equivalent to $6F$.

The first idea of this paper is to consider the ring

$\bigoplusop\displaylimits_{a,b\geq 0}H^{0}(Y,\mathcal{O}(aF+bS)).$

It has a double grading, and we can derive a formula for the graded dimension

$\sumop\displaylimits_{a,b\geq 0}\dim
H^{0}(Y,\mathcal{O}(aF+bS))\,t^{a}s^{b}=\frac{1+2st^{4}+2st^{5}+s^{2}t^{9}}{(1-t^{2})(1-t^{3})(1-s)(1-st^{3})}$

which is suggestive of a free module structure over the subring generated by
the variables $u_{0},u_{1},v_{1},v_{2}$ with weights
$(0,2),(0,3),(1,0),(1,3)$, respectively. The appropriate GIT quotient is a
birational model $Y_{0}$ of $Y$ that collapses all of the curves that
intersect $F$ and $S$ trivially. These are the curves
$C,B_{1},C_{1},B_{2},C_{2}$, and the image of the special fiber in $Y_{0}$
becomes

(1.1) $A\relbar B\relbar*\relbar A_{1}\relbar**\relbar A_{2}\relbar**\relbar
A,$

so that the intersection point of (the images of) $B$ and $A_{1}$ is a simple
node and the intersections of $A_{2}$ with both $A$ and $A_{1}$ are
$\frac{1}{3}(1,2)$ singularities. The construction of the new fake projective
planes then proceeds as follows.

  1. Step 1.

We construct a nine-parameter family of $(2,3)$-Dolgachev surfaces with a
rational $6$-section $S$. A general member of this family has twelve distinct
singular nodal fibers, in addition to the double and triple fibers. The
defining equations of the family are nine quadrics of weights $3\times(2,8)$,
$3\times(2,9)$ and $3\times(2,10)$ in the variables of weight

$(0,2),(0,3),(1,0),(1,3),(1,4),(1,4),(1,5),(1,5).$

The idea is to postulate the above free module structure and the weights of
the quadratic relations and to use the associativity conditions of the ring to
solve for the coefficients of the quadrics. It entails solving a system of
over $1600$ equations with $92$ unknowns, which is done by an ad hoc method
utilizing the Mathematica software system.

  2. Step 2.

We construct seven-, five- and two-parameter subfamilies with additional
conditions on the special fiber. Respectively, we require the special fiber to
contain a line, two disjoint lines, two disjoint lines with two nodes on one
and one node on the other. In particular, a generic element of the two-
parameter family has special fiber

$A\relbar B\relbar*\relbar A_{1}\relbar*\relbar A_{2}\relbar*\relbar A$

in the sense that the intersection points of $B$ with $A_{1}$ and $A_{2}$ with
$A$ and $A_{1}$ are nodes.

  3. Step 3.

We find a finite-field reduction of the surface $Y_{0}$ by looking through the
parameter choices over a finite field and checking whether the resulting
surfaces have worse-than-nodal singularities at two special points on the
curve $A_{2}$. The smallest prime for which we were able to find such surfaces
was $79$.

  4. Step 4.

We proceed by successively solving the (hard to write) conditions on being
more singular at the intersection points of $A$-curves for parameters modulo
powers of $79$. We then recognize the parameters as algebraic numbers and
construct $Y_{0}$ over a number field of degree $12$. We make a coordinate
change to realize $Y_{0}$ over the number field $\mathbb{Q}(\sqrt{-7})$.

  5. Step 5.

We study $Y_{0}$ to find its geometric features, such as the curves $S_{1}$
and $S_{2}$ and the birational action of $C_{3}$. We find the degree $7$
extension of the field of rational functions of $Y_{0}$ that gives
$\mathbb{P}^{2}_{\operatorname{fake}}$ and calculate the bicanonical linear
system of the latter. We then realize $\mathbb{P}^{2}_{\operatorname{fake}}$
as an intersection of $84$ cubic equations in $\mathbb{C}\mathbb{P}^{9}$,
following the blueprint of [BK20].

  6. Step 6.

We identify the fake projective plane as $(C20,p=2,\emptyset,D_{3}2_{7})$ by
exhibiting too many torsion line bundles for it to be
$(a=7,p=2,\\{7\\},D_{3}2_{7})$. The method is to find non-reduced
$C_{3}$-invariant elements of $|2K_{\mathbb{P}^{2}_{\operatorname{fake}}}|$
modulo a prime (this time it is $29$) by an exhaustive search and then lift
them to powers of the said prime and finally the algebraic numbers. We also
use one of the torsion line bundles to pick a more natural basis of
$H^{0}(\mathbb{P}^{2}_{\operatorname{fake}},2K)$ so that its equations have
smaller coefficients. Finally, we verify that it is indeed a fake projective
plane, as in [BK20].

  7. Step 7.

In Step 2, one actually finds _two different five-parameter families_ of
Dolgachev surfaces which contain two disjoint lines in the special fiber.
Unfortunately, for the second family we were unable to reduce the number of
parameters further by considering the condition of having nodes. However, we
are still able to go through Steps 3–5 in this case, by a brute force approach
to the finite-field search. By a process of elimination, we see that the new
pair of fake projective planes is the one constructed in [Keu06].

The paper is organized as follows. Section 2 contains the first two steps of
the construction. Section 3 contains Steps 3 and 4. Section 4 describes Steps
5 and 6. In Section 5 we discuss the last step. Finally, in Section 6 we talk
about the open problems associated with our construction. We also have an
appendix in which we put some equations that are too lengthy for the main body
of the paper. However, many of the key formulas are far too large to even be
included in the appendix. They are collected in [Bor22+] instead.

### Acknowledgments

This project depended heavily on the use of the Mathematica software system,
with certain steps performed in Magma, Macaulay2, PARI/GP and C; see [Mat,
Mag, Mac, Par]. The author thanks Sai Kee Yeung for a useful comment on the
first version of the paper.

## 2\. Families of (2,3) Dolgachev surfaces

We start by studying smooth projective surfaces $Y$ with a genus $1$ fibration
$Y\to\mathbb{C}\mathbb{P}^{1}$ with the following properties:

  * •

The class of the general fiber is $6F$, where $F$ is the canonical class of
$Y$. In particular, $F^{2}=0$.

  * •

There exist a double fiber $2F_{3}$ and a triple fiber $3F_{2}$. The classes
of $F_{2}$ and $F_{3}$ are $2F$ and $3F$, respectively.

  * •

There is a rational $6$-section $S$ with $SF=1$ and $S^{2}=-3$.

  * •

We have $p_{g}(Y)=q(Y)=0$.

Our motivation is that the minimal resolutions of the fake projective planes
we are interested in satisfy the above; see [Keu08].

As implied in the introduction, we first compute(1)(1)(1)Technically, we could
just conjecture everything, with the justification for it being the final
outcome, but it is worth proving what we can. the graded dimension of the ring

$R=\bigoplusop\displaylimits_{a,b\geq 0}H^{0}(Y,\mathcal{O}(aF+bS))$

in a series of lemmas.

######  Lemma 2.1.

The graded dimension of $R_{0}=\bigoplusop\displaylimits_{a\geq
0}H^{0}(Y,\mathcal{O}(aF))$ is given by

$\sumop\displaylimits_{a\geq 0}t^{a}\dim
H^{0}(Y,\mathcal{O}(aF))=\frac{1}{(1-t^{2})(1-t^{3})}.$

###### Proof.

The ring $R_{0}$ is freely generated by the elements $u_{0}\in
H^{0}(Y,\mathcal{O}(2F))$ and $u_{1}\in H^{0}(Y,\mathcal{O}(3F))$ whose
divisors are $F_{2}$ and $F_{3}$, respectively. Indeed, these generate a
subring of $R_{0}$, and we will show that there are no other forms. The
subring for $a=0\mod 6$ is isomorphic to the homogeneous ring of the
$\mathbb{C}\mathbb{P}^{1}$ of the base of the fibration, so it is a polynomial
ring in $u_{0}^{3}$ and $u_{1}^{2}$. If $a$ is odd, then $\mathcal{O}(aF)$
restricts to a nontrivial bundle to $F_{3}$ (because the normal bundle of the
double fiber is nontrivial), so all its global sections vanish on $F_{3}$, and
the corresponding elements in $R_{0}$ are divisible by $u_{1}$. Similarly, if
$a$ is not divisible by $3$, then all global sections of $\mathcal{O}(aF)$
vanish on $F_{2}$, and the elements are divisible by $u_{0}$. Together, these
observations imply the result. ∎

######  Lemma 2.2.

The dimension of $H^{0}(Y,\mathcal{O}(aF+S))$ is $1$ for $a=0$ and is $a-1$
for $a>0$.

###### Proof.

For $0\leq a\leq 2$ we have $(aF+S)S<0$, so any section of $\mathcal{O}(aF+S)$
must vanish on $S$, and the statement follows from Lemma 2.1. For $a\geq 3$ we
have $\chi(aF+S)=\frac{1}{2}(aF+S)((a-1)F+S)+1=a-1$, so it suffices to show
that the invertible sheaf $\mathcal{O}(aF+S)$ has no higher cohomology. The
vanishing of $H^{2}(Y,\mathcal{O}(aF+S))$ for all integers $a$ is clear from
Serre duality and the fact that every effective divisor on $Y$ must have a
non-negative intersection with $F$. To see the vanishing of
$H^{1}(Y,\mathcal{O}(aF+S))$, we run induction on $a$. Specifically, consider
the long exact sequence.

$\displaystyle 0\longrightarrow H^{0}(Y,\mathcal{O}((a-2)F+S))\longrightarrow
H^{0}(Y,\mathcal{O}(aF+S))\longrightarrow
H^{0}\left(Y,i_{*}\mathcal{O}_{F_{2}}(aF+S)\right)$
$\displaystyle\longrightarrow H^{1}(Y,\mathcal{O}((a-2)F+S))\longrightarrow
H^{1}(Y,\mathcal{O}(aF+S))\longrightarrow
H^{1}\left(Y,i_{*}\mathcal{O}_{F_{2}}(aF+S)\right).$

Since $(aF+S)F_{2}=2$, the last term is $0$, so it suffices to prove that
$H^{1}(Y,\mathcal{O}(aF+S))=0$ for $a\in{1,2}$, which follows from our
computation of the global sections of these divisors. ∎

######  Lemma 2.3.

For $b\geq 1$ and $a\geq 3b$ the dimension of $H^{0}(Y,\mathcal{O}(aF+bS))$ is
equal to $\chi(\mathcal{O}(aF+bS))=$ $\frac{1}{2}(2ab-b-3b^{2}+2)$.

###### Proof.

We will prove it by induction on $b$ with the base case provided by Lemma 2.2.
As before, $\dim H^{2}(Y,\mathcal{O}(aF+bS))=\dim
H^{0}(Y,\mathcal{O}((1-a)F-bS))=0$, so the statement amounts to the relation
$\dim H^{1}(Y,\mathcal{O}(aF+bS))=0$. For the induction step, the short exact
sequence

$0\longrightarrow\mathcal{O}(aF+(b-1)S)\longrightarrow\mathcal{O}(aF+bS))\longrightarrow
i_{S*}\mathcal{O}(aF+bS)\longrightarrow 0$

leads to

$\longrightarrow H^{1}(Y,\mathcal{O}(aF+(b-1)S)\longrightarrow
H^{1}(Y,\mathcal{O}(aF+bS))\longrightarrow
H^{1}(S,i_{S}^{*}\mathcal{O}(aF+bS)).$

The terms on the left and on the right are zero by the induction hypothesis
and $(aF+bS)S=a-3b\geq 0$. ∎

The above lemmas allow us to compute the graded dimension of $R$.

######  Proposition 2.4.

$\sumop\displaylimits_{a,b\geq 0}\dim
H^{0}(Y,\mathcal{O}(aF+bS))\,s^{b}t^{a}=\frac{1+2st^{4}+2st^{5}+s^{2}t^{9}}{(1-t^{2})(1-t^{3})(1-s)(1-st^{3})}.$

###### Proof.

We write $\dim H^{0}(Y,\mathcal{O}(aF+bS))=c_{a,b}$ to simplify notation.
Since $(aF+bS)S<0$ implies that $S$ is a fixed component of $|aF+bS|$, we see
that for $a<3b$ there holds $c_{a,b}=c_{a,b-1}$. Therefore,
$\sumop\displaylimits_{a,b\geq 0}c_{a,b}t^{a}s^{b}$ equals

$\frac{s}{1-s}\sumop\displaylimits_{b\geq
0}\left(c_{3b,b}t^{3b}+c_{3b+1,b}t^{3b+1}+c_{3b+2,b}t^{3b+2}\right)s^{b}+\sumop\displaylimits_{b\geq
0}\sumop\displaylimits_{a\geq 3b}c_{a,b}t^{a}s^{b}.$

We then use Lemmas 2.1 and 2.3 to write the above as

$\displaystyle\frac{s}{1-s}(1+t^{2})+\frac{1}{(1-t^{2})(1-t^{3})}$
$\displaystyle\;+\sumop\displaylimits_{b\geq
1}s^{b}\left(\frac{s}{1-s}\left(\frac{1}{2}(2(3b)b-b-3b^{2}+2)t^{3b}+\frac{1}{2}(2(3b+1)b-b-3b^{2}+2)t^{3b+1}\right.\right.$
$\displaystyle\hphantom{\sumop\displaylimits_{b\geq
1}s^{b}\Big{(}\frac{s}{1-s}\frac{1}{2}}\;\;\;\;+\frac{1}{2}(2(3b+2)b-b-3b^{2}+2)t^{3b+2}\Big{)}+\sumop\displaylimits_{a\geq
3b}\frac{1}{2}(2ab-b-3b^{2}+2)t^{a}\Big{)},$

which is then easily computed by Mathematica.(2)(2)(2)It is, of course,
computable by hand, but this seems to be a fool’s errand given the subsequent
use of various software. ∎

As a consequence of Proposition 2.4, we conjecture that $R$ has the structure
of a rank $6$ graded free module over the ring

$\mathbb{C}[u_{0},u_{1},v_{1},v_{2}]$

generated by the sections $u_{0}$, $u_{1}$, $v_{1}$ and $v_{2}$ of
$\mathcal{O}(2F)$, $\mathcal{O}(3F)$, $\mathcal{O}(S)$ and
$\mathcal{O}(3F+S)$, respectively. Note that while $u_{0},u_{1},v_{1}$ are
defined uniquely up to scaling, the section $v_{2}$ can also be changed by
adding a scalar multiple of $u_{1}v_{1}$. We denote the generators of the
module at weights $(1,4)$ and $(1,5)$ by $v_{3},v_{4},v_{5},v_{6}$. We
conjecture that the generator at degree $(9,2)$ is equal to $v_{3}v_{5}$. It
can likely be proved that the ring $R$ is Gorenstein, which would then imply
that such a choice is possible, but we just take it as a sensible guess.

We will consider polynomial relations on $u_{0},\ldots,v_{6}$. By looking at
the graded dimension of $R$, we observe that these variables must satisfy
three linearly independent relations of degree $(2,8)$, three linearly
independent relations of degree $(2,9)$ and three relations of degree
$(2,10)$, which are linearly independent together with $u_{0}$ times the
$(2,8)$ relations. We will refer to these relations as quadrics (in $v_{i}$).
In general, genus $1$ curves of degree $6$ in $\mathbb{P}^{5}$ can be written
as an intersection of nine degree $2$ polynomials, and we expect that for most
values of $u_{0}$ and $u_{1}$, these quadrics describe the corresponding fiber
of $Y\to\mathbb{C}\mathbb{P}^{1}$.

We then set up the possible quadrics with undetermined coefficients, taking
care to undo multiple symmetries of the construction. For example, we make
sure that the coefficients of the $(2,8)$ quadrics in
$v_{3}^{2},v_{3}v_{4},v_{4}^{2}$ are the standard basis vectors, and similarly
for the $(2,9)$ and $(2,10)$ quadrics. We assume that $(v_{3},v_{4})$ and
$(v_{5},v_{6})$ are dual to each other in the socle pairing of $R/\langle
u_{0},u_{1},v_{1},v_{2}\rangle$. We can also make additional assumptions on
the coefficients in view of the possible changes of $v_{i}$ such as
$v_{2}\mapsto\alpha v_{2}+\beta u_{1}v_{1}$. The only such assumptions which
appear crucial to the success of the method are that the coefficients of the
first $(2,8)$ quadric at $v_{3}v_{1}u_{0}^{2}$ an $v_{4}v_{1}u_{0}^{2}$ are
$1$ and $0$, the coefficients of the second quadric at these monomials are
both $0$, and the coefficients of the third quadric are $0$ and $1$,
respectively. It can be shown that a generic collection of the quadrics can be
manipulated into this form by appropriate linear changes of $v_{3},v_{4}$
together with adding multiples of $u_{0}^{2}v_{1}$ to them. However, there are
six ways of doing so, which means that while we expect $Y_{0}$ to be defined
over a quadratic imaginary field, we cannot expect the coefficients of these
quadrics to be this simple.

Once the quadrics are written down, one can compute the multiplication table
for the generators of $R$ as a free module over
$\mathbb{C}[u_{0},u_{1},v_{1},v_{2}]$ and then set up the associativity
relations as equations on the coefficients of the quadrics; see [Bor22+,
DolgachevSurfaces.nb]. The associativity relations led to over $1600$
equations in $92$ variables. Fortunately, some of these equations were quite
simple, but still the task appeared daunting. We were able to use the
Mathematica “Solve” command to eventually reduce to nine free parameters. The
basic idea was to try to solve the easier equations first, with either byte
count or the number of terms used as measure of complexity. The drawback of
this technique is that one can at best hope to recover rational
parameterizations by a subset of the set of variables, but we were fortunate
in this case. The resulting equations are presented in Section A.1 in the
appendix. The variables $d_{i}$ are the parameters, and the variables
$u_{0},u_{1},v_{1},\ldots,v_{6}$ are the coordinates.

Having found the equations of a nine-parameter family, we then tried to impose
additional geometric conditions on it. We postulated without loss of
generality that the $I_{9}$ fiber of $Y$ occurs at $u_{0}^{3}=u_{1}^{2}$ (or
simply $u_{0}=u_{1}=1$ if one wants to dehomogenize). We know from the work of
Keum [Keu08] that possible intersections of $S$ with $(A,A_{1},A_{2})$ fall
into cases $(2,2,1)$ and $(1,1,3)$, so in either setting, the image of the
$I_{9}$ fiber in $Y_{0}$ should have two disjoint lines. We first solved for
the condition that the equations in Section A.1 vanish on one line, by
parameterizing the said line. This gave a seven-parameter family, and we used
its explicit description to put in a condition of having two such lines,
taking care not to get the degenerate cases where the lines intersect. Much
trial and error was involved, and even in the best case, the computations took
a long time. Due to an unfortunate mistake on the author’s part, some of the
intermediate steps were lost; however, the final five-parameter answer
survived and is explicitly written in [Bor22+, DolgachevSurfaces.nb].

The general member of this five-parameter family has a double and a triple
fiber at $u_{1}=0$ and $u_{0}=0$, respectively, and a special fiber at
$u_{0}^{3}=u_{1}^{2}$ which has the following configuration of curves $A$,
$B$, $A_{1}$, $A_{2}$ and their intersection points $p_{1},\ldots,p_{4}$:

special fiberat
$u_{0}^{3}=u_{1}^{2}$$p_{3}$$p_{2}$$p_{1}$$p_{4}$$A$$B$$A_{2}$$A_{1}$

Here, $A_{2}$ and $B$ are the two disjoint lines in the special fiber, and $A$
and $A_{1}$ are both of degree $2$.

Our next step was to impose the conditions that $Y_{0}$ is singular at
$p_{1}$, $p_{3}$ and $p_{4}$, as would be expected by the geometry of the
surface. One immediate technical difficulty was that the points $p_{1}$ and
$p_{2}$ were not defined over the parameter space. However, we were able to
make a change of variables, similar to a rational parameterizations of a plane
conic, to resolve this difficulty. We also made a simple coordinate change to
simplify the formulas somewhat; see [Bor22+, DolgachevSurfaces.nb] for
details.

The conditions of being singular at each $p_{i}$ were computed as follows. We
looked at the Jacobian matrix of the quadrics at $p_{i}$ and computed its
minors. We then computed the greatest common factor of these minors, which
left us with large, but manageable expressions in the parameters
$t_{1},\ldots,t_{5}$. An ad hoc manipulation of the equations and the
parameters allowed us to replace $t_{i}$ with $s_{1},\ldots,s_{5}$ and then
solve for $s_{1}$ and $s_{5}$. The polynomial equation on the remaining
parameters $s_{2}$, $s_{3}$ and $s_{4}$ has several factors, with all but one
of them leading to undesirable degeneracies. We were left with a polynomial
with integer coefficients of degree $3$, $13$ and $12$ in $s_{2}$, $s_{3}$ and
$s_{4}$, respectively. It is about $130$ Kb long in its expanded form and thus
is not worth trying to write down explicitly in this paper; see [Bor22+,
DolgachevSurfaces.nb]. It appears that the resulting $2$-dimensional parameter
space is not rational, but it was simple enough for our purposes.

## 3\. Finding the special Dolgachev surface

This section describes how we found the surface $Y_{0}$. We expected that
$Y_{0}$ has $\frac{1}{3}(1,2)$ singularities at $p_{3}$ and $p_{4}$, and it
would be reasonable to try to encode these in terms of $s_{2}$, $s_{3}$ and
$s_{4}$ subject to the large polynomial equation. Unfortunately, this direct
approach was not computationally feasible, as equations ballooned to hundreds
of Mb in length. So we used the alternative method of finite-field reduction.
Specifically, we wanted to find a reduction of $Y_{0}$ modulo a prime $p$ and
then lift it first to $p$-adics and then to a number field.

We looked at various primes $p$ and parameters $(s_{2},s_{3},s_{4})\in\mathbb
F_{p}^{3}$, with $p$ chosen so that $\sqrt{-7}$ exists in $\mathbb F_{p}$ as
it appeared likely that we would need this in the field of definition. For
each triple of parameters we first checked whether it fits the polynomial
relation that defines the parameter space. Then we used Magma to compute the
structure of singularities at $p_{3}$ and $p_{4}$ (we have lengthy but
explicit formulas for $p_{i}$ in terms of the parameters). If both
singularities looked right, then we took note of the values of the parameters.
For one reason or another, we had to go to $p=79$ before any suitable examples
were found! However, at $p=79$ there were six possible solutions, see [Bor22+,
Search79]:

$(s_{2},s_{3},s_{4})\in\\{(14,47,52),(15,65,27),(19,32,14),(44,14,32),(58,27,65),(72,52,47)\\}.$

It was then an interesting challenge to lift these solutions to powers of
$79$. By solving for $v_{5}$ and $v_{6}$, one can reduce the problem to
codimension $2$. Then it is possible to encode the condition of having worse-
than-nodal singularity as a certain polynomial in terms of first and second
derivatives of the defining equations. Then we endeavored to have these
polynomials produce values that are zero modulo higher and higher powers of
$79$, as we are adjusting our parameters $s_{i}$ to a more accurate $p$-adic
approximation. Of course, we also have to keep the defining polynomial of the
family zero to the appropriate power of $79$. The resulting code is messy but
not particularly slow, and we were able to compute the values of $s_{i}$ up to
$79^{101}$ in a reasonable amount of time.

Once a $p$-adic approximation was found, it was then routine to find “simple”
algebraic numbers that give these parameters $s_{i}$. As in [BF20, BBF22], we
used a lattice reduction algorithm to find a small linear combination of
powers of $s_{i}$ modulo $79^{101}$ and $79^{101}$ itself. This suggested that
these are algebraic numbers of degree $12$. By looking at the standard
polynomial, we indeed found $12$ possible triples $(s_{2},s_{3},s_{4})$.
Perhaps not surprisingly, our finite-field search only picked up the six cases
that correspond to the simple roots of the reduction of the defining
polynomial of $s_{2}$

$\displaystyle
1048576+9633792s_{2}+47179776s_{2}^{2}+156022272s_{2}^{3}+376708864s_{2}^{4}$
$\displaystyle+693988960s_{2}^{5}+1003433368s_{2}^{6}+1148276192s_{2}^{7}+1023247890s_{2}^{8}$
$\displaystyle+681835980s_{2}^{9}+317640295s_{2}^{10}+91989513s_{2}^{11}+12492403s_{2}^{12}$

modulo $79$.

Since we expected that $Y_{0}$ can be defined over a quadratic field, we
wanted to find a coordinate change in $v_{i}$ to see it. Specifically, we
aimed to get the points $p_{i}$ to have simple coordinates. As it is a lot
faster in Mathematica, this was done numerically, and then the coefficients
were approximated by algebraic numbers. The resulting equations are listed in
Section A.2 of the appendix, where we use $w_{i}$ to denote new variables,
with $u_{0}$ and $u_{1}$ not affected by the coordinate change.

## 4\. Constructing and identifying the fake projective plane

This section describes how we found the fake projective plane (up to
conjugation) and identified it as $(C20,p=2,\emptyset,D_{3}2_{7})$. The
relevant calculations are contained in the Mathematica file SevenfoldCover.nb
and Magma files Torsion and CheckSmoothness, as well as Macaulay2 file
CheckFPP in [Bor22+].

We know that $S$ is cut out by $w_{1}=0$, but it takes a bit of effort to find
equations of $S_{1}$ and $S_{2}$. The first idea is that since
$(4F+S)S=4-3=1$, the sections $w_{3}$ and $w_{4}$ are linear on $S$. Thus,
points on $S$ can be parameterized by $t=\frac{w_{4}}{w_{3}}$. This is true
for any member of the $9$-dimensional family of Dolgachev surfaces; see
Section A.1. So we know both the equation and the parameterizations of $S$,
and we would like to do the same for $S_{1}$ and $S_{2}$.

It follows from intersection form considerations [Bor22+,
IntersectionForms.nb] that $S_{2}$ is cut out by an equation of bidegree
$(1,8)$ which also passes through $A$ and $A_{2}$ with multiplicities $1$ and
$3$, respectively. We pick multiple points on $A$ and $A_{2}$ and encode the
relevant conditions on the $(1,8)$ polynomial to find it to be

(4.1) $\displaystyle u_{0}^{4}$ $\displaystyle
w_{1}+\frac{1}{149}(-124-9\,{\rm
i}\sqrt{7})u_{0}u_{1}^{2}w_{1}+\frac{1}{8344}\left(-553+509\,{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{2}$
$\displaystyle+\frac{1}{149}\left(20-37\,{\rm
i}\sqrt{7}\right)u_{0}^{2}w_{3}+\frac{1}{596}\left(-155+175\,{\rm
i}\sqrt{7}\right)u_{0}^{2}w_{4}$
$\displaystyle+\frac{1}{8344}\left(3255+1093\,{\rm
i}\sqrt{7}\right)u_{1}w_{5}+\frac{1}{2086}\left(-763-369\,{\rm
i}\sqrt{7}\right)u_{1}w_{6}.$

We also needed to compute the parametric equation of $S_{2}$. After some of
the more straightforward approaches failed to finish, we did the following.
For a given value of the parameter $t$ on $S$ we can compute the corresponding
point $p(t)$ on $S$ and six points of $S_{2}$ in the same fiber of the genus
$1$ fibration. One of these points is the image $p_{1}(t)$ of $p(t)$ under the
predicted order $3$ birational automorphism of $Y_{0}$. If $t$ is an integer,
then this point on $S_{2}$ has to be defined over $\mathbb{Q}({\rm
i}\sqrt{7})$, which allows us to distinguish $p_{1}(t)$ from the five other
points. After the normalization $w_{1}=1$, the coordinates of $p_{1}(t)$ are
polynomial in $t$ of degree at most $5$, and we use several values of $t$ to
find their coefficients. The resulting parameterization is given in Section
A.3 in the appendix.

One could follow a similar approach to find $S_{1}$, which has a degree
$(1,10)$ equation vanish on it, but we chose to use the group law of the
elliptic fibers instead. Indeed, we know that for a given value of $t$ the
points $p(t)$, $p_{1}(t)$ and $p_{2}(t)$ satisfy $p_{1}(t)+p(t)=2p_{1}(t)$
under the group law of the fiber with any choice of origin. This allowed us to
find $p_{1}(t)$ and then compute the parameterization; see Section A.4. We
then found the degree $(1,10)$ polynomial

(4.2) $\displaystyle u_{0}^{5}$ $\displaystyle
w_{1}-u_{0}^{2}u_{1}^{2}w_{1}+\frac{1}{7112}\left(-1897-3\,{\rm
i}\sqrt{7}\right)u_{0}^{2}u_{1}w_{2}+\frac{1}{254}\left(-3+17\,{\rm
i}\sqrt{7}\right)u_{0}^{3}w_{3}$ $\displaystyle+\frac{1}{254}\left(39+33\,{\rm
i}\sqrt{7}\right)u_{1}^{2}w_{3}+\frac{1}{508}\left(-71-21\,{\rm
i}\sqrt{7}\right)u_{0}^{3}w_{4}+\frac{1}{254}\left(-71-21\,{\rm
i}\sqrt{7}\right)u_{1}^{2}w_{4}$
$\displaystyle+\frac{1}{7112}\left(6559+509\,{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{5}+\frac{1}{1778}\left(-1435-81\,{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{6}$

that vanishes on $S_{1}$.

We then computed the order $3$ birational automorphism by postulating its
general form and using the parameterization of $S$, $S_{1}$ and $S_{2}$ to
find the coefficients. It can be viewed as an order $3$ automorphism of a
genus $1$ curve defined over $\mathbb{C}(u_{0},u_{1})$. The formulas for the
automorphism are in Section A.5.

We were then ready to compute the equations of the sevenfold cover of $Y$. The
method was essentially the same as that of [BK20], but we briefly describe it
here for the benefit of the reader. We know that the sevenfold Galois cover
$\mathbb{P}^{2}_{\operatorname{fake}}\to Y$ is ramified at
$S,B,C,S_{1},B_{1},C_{1},S_{2},B_{2},C_{2}$. We consider the rational function

(4.3)
$\frac{f_{1,10}f_{1,8}^{2}}{u_{0}^{7}\left(u_{0}^{3}-u_{1}^{2}\right)^{2}w_{1}^{3}},$

where $f_{1,8}$ and $f_{1,10}$ are given in (4.1) and (4.2), respectively. Its
divisor is supported on the special curves, with all but the nine
$S,\ldots,C_{2}$ curves above having multiplicity divisible by $7$. So the
function field of $\mathbb{P}^{2}_{\operatorname{fake}}$ is obtained from that
of $Y$ by adding the seventh root of the function (4.3). We then compute ten
sections of the bicanonical linear system on
$\mathbb{P}^{2}_{\operatorname{fake}}$ by projecting them to $Y$ and realizing
them as sections of various line bundles on $Y_{0}$. Care is taken to have the
action of the automorphism group look nice in these coordinates. Specifically,
we have ten variables $P_{0},\ldots,P_{9}$ on which the order $7$ element acts
by

$g_{7}(P_{0},\ldots,P_{9})=\left(P_{0},\zeta_{7}P_{1},\zeta_{7}^{2}P_{2},\zeta_{7}^{4}P_{3},\zeta^{3}P_{4},\zeta^{6}P_{5},\zeta^{5}P_{6},\zeta^{3}P_{7},\zeta^{6}P_{8},\zeta^{5}P_{9}\right)$

and the order $3$ element acts by

$g_{3}(P_{0},\ldots,P_{9})=(P_{0},P_{2},P_{3},P_{1},P_{5},P_{6},P_{4},P_{8},P_{9},P_{7}).$

See [Bor22+, SevenfoldCover.nb] for more details.

Having constructed a fake projective plane with $21$ automorphisms, we want to
try to identify it in accordance with the Cartwright–Steger classification.
Since the Dolgachev surface it is built from has a double and a triple fiber,
there are two possibilities: Keum’s surface $(a=7,p=2,\\{7\\},D_{3}2_{7})$ and
$(C20,p=2,\emptyset,D_{3}2_{7})$. In the former case, the torsion in the
Picard group is $C_{2}^{3}$, with the automorphism group $G$ acting
transitively on seven nontrivial elements. Therefore, for every cyclic
subgroup of order $3$ in $G$ there should be a unique $G$-invariant torsion
element on $\mathbb{P}^{2}_{\operatorname{fake}}$. This naturally lead to us
trying to construct such torsion classes.

If $L$ is a torsion class in the Picard group of a fake projective plane, then
$H^{0}(\mathbb{P}^{2}_{\operatorname{fake}},K+L)$ is $1$-dimensional; see for
example [GKS23]. If $L$ is furthermore a $2$-torsion element, then the square
of the corresponding section is in
$H^{0}(\mathbb{P}^{2}_{\operatorname{fake}},2K)$. In the other direction,
$2$-torsion elements of the Picard group can be constructed from sections of
$2K$ which give non-reduced curves. The additional condition of $C_{3}$
invariance means that we should consider sections

(4.4)
$a_{0}P_{0}+a_{1}(P_{1}+P_{2}+P_{3})+a_{2}(P_{4}+P_{5}+P_{6})+a_{3}(P_{7}+P_{8}+P_{9})$

up to scaling.

We looked at the finite-field reduction of
$\mathbb{P}^{2}_{\operatorname{fake}}$ to look for such $a_{i}$. Specifically,
$p=29$ was the smallest prime that both had $\sqrt{-7}$ in it and had the
expected Hilbert polynomial of the reduction of
$\mathbb{P}^{2}_{\operatorname{fake}}$. We then ran through all possible
$a_{i}$ with $a_{0}=1$ and used Magma to check if the resulting scheme is non-
reduced; see [Bor22+, Torsion]. We got three solutions, which was already a
likely indicator that the $\mathbb{P}^{2}_{\operatorname{fake}}$ is not
$(a=7,p=2,\\{7\\},D_{3}2_{7})$ but rather is $(C20,p=2,\emptyset,D_{3}2_{7})$,
which has a larger torsion subgroup $C_{2}^{6}$ in its Picard group. However,
we needed to ascertain it by lifting to characteristic zero.

The lifting procedure was pretty typical of such approach: we lifted the
$a_{i}$ in (4.4) to modulo $p^{k}$ for large enough $k$ and then guessed the
algebraic numbers that could give these reductions. It is worth mentioning how
exactly we did the lifting. We found via Magma several points on the non-
reduced linear cut of the reduction of $\mathbb{P}^{2}_{\operatorname{fake}}$
modulo $p$; then at each point we found two linearly independent tangent
vectors in $\mathbb{P}^{9}(\mathbb F_{29})$ which are orthogonal to the
gradients of all $84$ cubic polynomials and the linear polynomial (4.4). We
lifted these vectors to small integers and then were successively adjusting
them so that the aforementioned orthogonality held modulo higher powers of
$p$. At each stage we modified the points, the tangent vectors and the
polynomial (4.4) to the next power of $p$; this amounted to solving a system
of linear equations modulo $p$, which was not time-consuming. Afterwards, we
identified the corresponding algebraic numbers.

One of the solutions was the linear polynomial

(4.5) $\displaystyle P_{0}+\tfrac{1}{2}\left(1+\,{\rm
i}\sqrt{7}\right)(P_{1}+P_{2}+P_{3})+\left(-122+2\,{\rm
i}\sqrt{7}\right)(P_{4}+P_{5}+P_{6})$
$\displaystyle+\tfrac{1}{7}\left(84-4\,{\rm
i}\sqrt{7}\right)(P_{7}+P_{8}+P_{9}),$

and the other two were defined over a degree $4$ number field. For each of
these equations we then verified that the resulting cuts are non-
reduced.(3)(3)(3)This verification was only done numerically rather than
symbolically, but it is sufficiently convincing for our purposes. An
interested reader is welcome to try their hand at verifying it symbolically.
We note that the resulting $2$-torsion line bundles are not $C_{7}$-invariant,
so we have established at least $21$ nontrivial $2$-torsion elements of the
Picard group, which indicates that our surface is (up to complex conjugation)
$(C20,p=2,\emptyset,D_{3}2_{7})$.

Last but not least, we used the description of the above non-reduced linear
cut to find a more pleasant basis for
$H^{0}(\mathbb{P}^{2}_{\operatorname{fake}},2K)$; see [Bor22+,
SevenfoldCover.nb]. Namely, we used a coordinate change from $P_{i}$ to
$Q_{i}$ in which the aforementioned non-reduced cut (4.5) becomes

$Q_{0}+Q_{1}+Q_{2}+Q_{3}+Q_{7}+Q_{8}+Q_{9}.$

This allowed us to find a model of it with smaller coefficients of the $84$
equations, which we recorded in [Bor22+, EqsFPPwithrrQ]. We then went through
the verification procedure developed in [BK20] to make sure that the scheme
cut out by the $84$ cubic relations in $10$ variables is indeed a fake
projective plane in its bicanonical embedding; see [Bor22+, CheckFPP,
CheckSmoothness]. The smoothness calculation was performed by looking at three
specific minors which are nonzero at the $C_{7}$ invariant points of
$\mathbb{P}^{2}_{\operatorname{fake}}$. Each minor took a few hours to compute
on our hardware.

## 5\. Constructing the fake projective plane of Keum

While looking for Dolgachev surfaces with two disjoint lines, we have found a
five-parameter subfamily of Dolgachev surfaces with two disjoint lines in the
special fiber with the intersections of $S$ with $A$, $A_{1}$ and $A_{2}$
given by $(1,1,3)$. This makes it is provably different from the family in
Step 2. This family was noticeably harder to work with. It was not even
entirely straightforward to find the intersection points of the components of
the special fiber. In particular, the conditions of being singular at three of
these points were far too complicated to simplify. The details are in [Bor22+,
SecondFamily.nb]

To find a member of this family with additional singularities (1.1), we used a
brute force approach. Namely, we ran a search over the parameter space
$\mathbb F_{p}^{5}$ for small $p$, computed the intersection points of the
components of the special fiber and checked to see if the tangent space at the
three points of interest had the correct dimension. This was too time-
consuming in Mathematica, so we used lower-level languages. We first tried
PARI/GP and then eventually C; see [Bor22+, finitefieldnodes.c]. Clearly, this
is very easy to parallelize, and we ran multiple computations at a time on the
Amarel cluster [Ama]. This allowed us to reduce the set of possible parameters
to roughly $p^{2}$. For each of these we used Mathematica to check if there
are worse-than-nodal singularities at the two expected points.

The first successful prime was $p=53$. We proceeded to lift the solution to
$p$-adics, as described in Section 3. We computed the sevenfold cover along
the lines of Section 4; see [Bor22+, FindingKeumFPP.nb]. We do not write down
the parametric equations or the $C_{3}$ symmetry, but do list the equations in
Section A.6 for the record; they are also available at [Bor22+,
QuadricsWforKeum]. The field extension needed to construct the cyclic cover
was obtained by adding the seventh root of

$\displaystyle\left(4u_{0}^{4}w_{1}-4u_{0}u_{1}^{2}w_{1}+\left(18-6{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{2}-\left(3-{\rm
i}\sqrt{7}\right)u_{0}^{2}w_{3}+\left(5+3{\rm
i}\sqrt{7}\right)u_{0}^{2}w_{4}\right.$
$\displaystyle\phantom{(}+\left.\left(-1-3{\rm
i}\sqrt{7}\right)u_{1}w_{5}+\left(-19-7{\rm
i}\sqrt{7}\right)u_{1}w_{6}\right)\;\times$
$\displaystyle\left(-66u_{0}^{2}u_{1}w_{2}+\left(8+4{\rm
i}\sqrt{7}\right)u_{0}^{3}w_{3}+\left(-2+10{\rm
i}\sqrt{7}\right)u_{1}^{2}w_{3}+\left(-31+{\rm
i}\sqrt{7}\right)u_{0}^{3}w_{4}\right.$
$\displaystyle\phantom{(}+\left(-34-6{\rm
i}\sqrt{7}\right)u_{1}^{2}w_{4}+\left.\left(-17-3{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{5}+\left(142-6{\rm
i}\sqrt{7}\right)u_{0}u_{1}w_{6}\right)^{2}\big{/}\left(u_{0}^{14}w_{1}^{3}\right)$

We verified that the resulting equations give a fake projective plane by the
same method. By elimination, it must be the one constructed by Keum in
[Keu06]. We did not try to simplify the equations of the Keum’s fake
projective plane using the torsion elements in its Picard group, even though
it must be possible to do. The plan is to have an undergraduate researcher to
try to use our approach to simplify equations of this and other known fake
projective planes.

## 6\. Open problems

The construction of this paper raises several interesting questions.

One can try to apply our approach to computing order $2$ elements of the
Picard group of the fake projective plane $(C20,p=2,\emptyset,D_{3}2_{7})$ to
other fake projective planes. It is important to have at least a $C_{3}$
symmetry; otherwise the finite-field search may take too long. One can even
try to do it with $C_{3}$-invariant elements of higher order by looking for
reducible linear cuts in the bicanonical embedding, although it is less clear
how to do the lifting in this case.

One can try to find a nicer set of variables so that the equations of the two
special Dolgachev surfaces $Y_{0}$ are simplified.

It is hoped that the knowledge of Keum’s fake projective plane will allow one
to find explicit equations of Mumford’s fake projective plane. This appears to
boil down to finding a degree $7$ non-Galois cover of the quotient of
$\mathbb{P}^{2}_{\rm Keum}$ by its $C_{3}$ automorphism group, with the Galois
group of the corresponding splitting field being $\operatorname{PSL}(2,7)$.
Perhaps the key to this is understanding such covers for the fibers of the map
$Y\to\mathbb{P}^{1}$. In fact, the triple fiber leads one to look for such
degree $7$ covers $C\to E$ of a genus $7$ curve $C$ which are ramified over
two explicitly known points on an explicitly known genus $1$ curve $E$. This
is currently work in progress.

## Appendix

### A.1. Equations of the nine-parameter family of Dolgachev surfaces

$\displaystyle\bullet$
$\displaystyle\;d_{2}d_{5}d_{9}u_{0}u_{1}^{2}v_{1}^{2}+d_{2}d_{3}d_{9}^{2}u_{0}u_{1}^{2}v_{1}^{2}-d_{2}d_{6}d_{9}^{2}u_{0}u_{1}^{2}v_{1}^{2}+d_{3}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{2}d_{4}d_{9}u_{0}u_{1}v_{1}v_{2}$
$\displaystyle\;+d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{2}+2d_{2}d_{3}d_{9}^{2}u_{0}u_{1}v_{1}v_{2}-2d_{2}d_{6}d_{9}^{2}u_{0}u_{1}v_{1}v_{2}+d_{3}d_{9}u_{0}v_{2}^{2}+d_{2}d_{4}d_{9}u_{0}v_{2}^{2}$
$\displaystyle\;+d_{2}d_{3}d_{9}^{2}u_{0}v_{2}^{2}-d_{2}d_{6}d_{9}^{2}u_{0}v_{2}^{2}+d_{2}u_{0}^{2}v_{1}v_{3}+d_{2}v_{3}^{2}-d_{2}d_{9}u_{1}v_{1}v_{5}-v_{2}v_{5}-d_{2}d_{9}v_{2}v_{5}$
$\displaystyle\;+d_{2}d_{9}u_{1}v_{1}v_{6}+d_{2}d_{9}v_{2}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;-d_{2}d_{5}d_{9}u_{0}u_{1}^{2}v_{1}^{2}-d_{2}d_{3}d_{8}d_{9}u_{0}u_{1}^{2}v_{1}^{2}+d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}^{2}v_{1}^{2}-d_{3}d_{8}u_{0}u_{1}v_{1}v_{2}-d_{2}d_{4}d_{9}u_{0}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{2}-d_{2}d_{3}d_{7}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{2}d_{6}d_{7}d_{9}u_{0}u_{1}v_{1}v_{2}-d_{2}d_{3}d_{8}d_{9}u_{0}u_{1}v_{1}v_{2}$
$\displaystyle\;+d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}v_{1}v_{2}-d_{3}d_{7}u_{0}v_{2}^{2}-d_{2}d_{4}d_{9}u_{0}v_{2}^{2}-d_{2}d_{3}d_{7}d_{9}u_{0}v_{2}^{2}+d_{2}d_{6}d_{7}d_{9}u_{0}v_{2}^{2}$
$\displaystyle\;+d_{2}v_{3}v_{4}+d_{2}d_{9}u_{1}v_{1}v_{5}+d_{2}d_{9}v_{2}v_{5}-d_{2}d_{9}u_{1}v_{1}v_{6}-d_{2}d_{9}v_{2}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;d_{2}d_{5}d_{9}u_{0}u_{1}^{2}v_{1}^{2}+d_{2}d_{3}d_{8}d_{9}u_{0}u_{1}^{2}v_{1}^{2}-d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}^{2}v_{1}^{2}+d_{1}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{2}d_{4}d_{9}u_{0}u_{1}v_{1}v_{2}$
$\displaystyle\;+d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{2}d_{3}d_{7}d_{9}u_{0}u_{1}v_{1}v_{2}-d_{2}d_{6}d_{7}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{2}d_{3}d_{8}d_{9}u_{0}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}v_{1}v_{2}+d_{1}d_{9}u_{0}v_{2}^{2}+d_{2}d_{4}d_{9}u_{0}v_{2}^{2}+d_{2}d_{3}d_{7}d_{9}u_{0}v_{2}^{2}-d_{2}d_{6}d_{7}d_{9}u_{0}v_{2}^{2}$
$\displaystyle\;+d_{2}u_{0}^{2}v_{1}v_{4}+d_{2}v_{4}^{2}-d_{2}d_{9}u_{1}v_{1}v_{5}-d_{2}d_{9}v_{2}v_{5}+d_{2}d_{8}u_{1}v_{1}v_{6}+d_{2}d_{7}v_{2}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;d_{3}u_{0}^{3}v_{1}v_{2}+d_{2}u_{1}^{2}v_{1}v_{2}+d_{2}u_{1}v_{2}^{2}+d_{2}d_{3}d_{9}u_{0}u_{1}v_{1}v_{4}-d_{2}d_{6}d_{9}u_{0}u_{1}v_{1}v_{4}+d_{3}u_{0}v_{2}v_{4}$
$\displaystyle\;+d_{2}d_{3}d_{9}u_{0}v_{2}v_{4}-d_{2}d_{6}d_{9}u_{0}v_{2}v_{4}+d_{2}v_{3}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;-d_{2}d_{5}d_{9}u_{0}^{3}u_{1}v_{1}^{2}+d_{2}d_{6}d_{8}d_{9}u_{0}^{3}u_{1}v_{1}^{2}-d_{2}d_{3}d_{9}^{2}u_{0}^{3}u_{1}v_{1}^{2}+d_{2}^{2}d_{8}d_{9}u_{1}^{3}v_{1}^{2}-d_{2}^{2}d_{9}^{2}u_{1}^{3}v_{1}^{2}$
$\displaystyle\;-d_{2}d_{4}d_{9}u_{0}^{3}v_{1}v_{2}+d_{2}d_{6}d_{7}d_{9}u_{0}^{3}v_{1}v_{2}-d_{2}d_{3}d_{9}^{2}u_{0}^{3}v_{1}v_{2}+d_{2}d_{8}u_{1}^{2}v_{1}v_{2}+d_{2}^{2}d_{7}d_{9}u_{1}^{2}v_{1}v_{2}$
$\displaystyle\;+d_{2}^{2}d_{8}d_{9}u_{1}^{2}v_{1}v_{2}-2d_{2}^{2}d_{9}^{2}u_{1}^{2}v_{1}v_{2}+d_{2}d_{7}u_{1}v_{2}^{2}+d_{2}^{2}d_{7}d_{9}u_{1}v_{2}^{2}-d_{2}^{2}d_{9}^{2}u_{1}v_{2}^{2}$
$\displaystyle\;-d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{3}+d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}v_{1}v_{3}-d_{1}d_{2}d_{9}^{2}u_{0}u_{1}v_{1}v_{3}-d_{1}d_{9}u_{0}v_{2}v_{3}$
$\displaystyle\;-d_{2}d_{4}d_{9}u_{0}v_{2}v_{3}+d_{2}d_{6}d_{7}d_{9}u_{0}v_{2}v_{3}-d_{1}d_{2}d_{9}^{2}u_{0}v_{2}v_{3}-d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{4}$
$\displaystyle\;-d_{2}d_{4}d_{9}u_{0}v_{2}v_{4}+d_{2}d_{9}u_{0}^{2}v_{1}v_{5}+d_{2}d_{9}v_{3}v_{5}-d_{2}d_{9}u_{0}^{2}v_{1}v_{6}-d_{2}d_{9}v_{4}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;-d_{3}d_{8}u_{0}^{3}u_{1}v_{1}^{2}+d_{3}d_{9}u_{0}^{3}u_{1}v_{1}^{2}-d_{2}d_{8}u_{1}^{3}v_{1}^{2}+d_{2}d_{9}u_{1}^{3}v_{1}^{2}-d_{3}d_{7}u_{0}^{3}v_{1}v_{2}+d_{3}d_{9}u_{0}^{3}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{7}u_{1}^{2}v_{1}v_{2}-d_{2}d_{8}u_{1}^{2}v_{1}v_{2}+2d_{2}d_{9}u_{1}^{2}v_{1}v_{2}-d_{2}d_{7}u_{1}v_{2}^{2}+d_{2}d_{9}u_{1}v_{2}^{2}-d_{3}d_{8}u_{0}u_{1}v_{1}v_{3}$
$\displaystyle\;+d_{1}d_{9}u_{0}u_{1}v_{1}v_{3}-d_{3}d_{7}u_{0}v_{2}v_{3}+d_{1}d_{9}u_{0}v_{2}v_{3}+v_{4}v_{5}$
$\displaystyle\bullet$
$\displaystyle\;d_{2}d_{3}d_{9}u_{0}^{5}v_{1}^{2}+d_{2}^{2}d_{9}u_{0}^{2}u_{1}^{2}v_{1}^{2}-d_{2}d_{3}d_{5}d_{8}d_{9}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{1}d_{2}d_{5}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}^{2}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{3}d_{5}d_{7}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}-d_{2}d_{3}d_{4}d_{8}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}+d_{1}d_{2}d_{4}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{1}d_{2}d_{5}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{3}d_{4}d_{7}d_{9}u_{0}^{2}v_{2}^{2}+d_{1}d_{2}d_{4}d_{9}^{2}u_{0}^{2}v_{2}^{2}+d_{1}d_{2}d_{9}u_{0}^{3}v_{1}v_{3}+d_{2}^{2}d_{8}u_{1}^{2}v_{1}v_{3}+d_{2}^{2}d_{7}u_{1}v_{2}v_{3}$
$\displaystyle\;+d_{2}d_{3}d_{9}u_{0}^{3}v_{1}v_{4}+d_{2}^{2}d_{9}u_{1}^{2}v_{1}v_{4}+d_{2}^{2}d_{9}u_{1}v_{2}v_{4}-d_{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{5}+d_{2}d_{6}d_{8}d_{9}u_{0}u_{1}v_{1}v_{5}$
$\displaystyle\;-d_{1}d_{2}d_{9}^{2}u_{0}u_{1}v_{1}v_{5}-d_{1}d_{9}u_{0}v_{2}v_{5}-d_{2}d_{4}d_{9}u_{0}v_{2}v_{5}+d_{2}d_{6}d_{7}d_{9}u_{0}v_{2}v_{5}-d_{1}d_{2}d_{9}^{2}u_{0}v_{2}v_{5}$
$\displaystyle\;+d_{2}d_{9}v_{5}^{2}-d_{2}d_{3}d_{8}d_{9}u_{0}u_{1}v_{1}v_{6}+d_{1}d_{2}d_{9}^{2}u_{0}u_{1}v_{1}v_{6}-d_{2}d_{3}d_{7}d_{9}u_{0}v_{2}v_{6}+d_{1}d_{2}d_{9}^{2}u_{0}v_{2}v_{6}$
$\displaystyle\bullet$
$\displaystyle\;d_{2}d_{3}u_{0}^{5}v_{1}^{2}+d_{2}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}d_{3}^{2}d_{8}d_{9}u_{0}^{2}u_{1}^{2}v_{1}^{2}-d_{2}d_{3}d_{6}d_{8}d_{9}u_{0}^{2}u_{1}^{2}v_{1}^{2}-d_{1}d_{2}d_{3}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}$
$\displaystyle\;+d_{1}d_{2}d_{6}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{3}^{2}d_{8}u_{0}^{2}u_{1}v_{1}v_{2}-d_{1}d_{3}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}d_{3}^{2}d_{7}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{3}d_{6}d_{7}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}d_{3}^{2}d_{8}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}-d_{2}d_{3}d_{6}d_{8}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}-2d_{1}d_{2}d_{3}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;+2d_{1}d_{2}d_{6}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{3}^{2}d_{7}u_{0}^{2}v_{2}^{2}-d_{1}d_{3}d_{9}u_{0}^{2}v_{2}^{2}+d_{2}d_{3}^{2}d_{7}d_{9}u_{0}^{2}v_{2}^{2}-d_{2}d_{3}d_{6}d_{7}d_{9}u_{0}^{2}v_{2}^{2}$
$\displaystyle\;-d_{1}d_{2}d_{3}d_{9}^{2}u_{0}^{2}v_{2}^{2}+d_{1}d_{2}d_{6}d_{9}^{2}u_{0}^{2}v_{2}^{2}+d_{2}d_{3}u_{0}^{3}v_{1}v_{3}+d_{2}^{2}u_{1}^{2}v_{1}v_{3}+d_{2}^{2}u_{1}v_{2}v_{3}$
$\displaystyle\;+d_{2}d_{3}u_{0}^{3}v_{1}v_{4}+d_{2}^{2}u_{1}^{2}v_{1}v_{4}+d_{2}^{2}u_{1}v_{2}v_{4}+d_{2}v_{5}v_{6},$
$\displaystyle\bullet$
$\displaystyle\;d_{2}^{2}d_{3}d_{9}u_{0}^{5}v_{1}^{2}+d_{2}^{3}d_{9}u_{0}^{2}u_{1}^{2}v_{1}^{2}-d_{2}^{2}d_{3}d_{5}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}^{2}d_{5}d_{6}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}^{2}d_{3}d_{6}d_{8}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}$
$\displaystyle\;-d_{2}^{2}d_{6}^{2}d_{8}d_{9}^{2}u_{0}^{2}u_{1}^{2}v_{1}^{2}-d_{1}d_{2}^{2}d_{3}d_{9}^{3}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{1}d_{2}^{2}d_{6}d_{9}^{3}u_{0}^{2}u_{1}^{2}v_{1}^{2}+d_{2}^{3}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}d_{3}d_{5}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}d_{3}d_{6}d_{8}d_{9}u_{0}^{2}u_{1}v_{1}v_{2}-2d_{1}d_{2}d_{3}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}-d_{2}^{2}d_{3}d_{4}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}^{2}d_{3}d_{5}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{1}d_{2}d_{6}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}^{2}d_{4}d_{6}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}^{2}d_{5}d_{6}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;+d_{2}^{2}d_{3}d_{6}d_{7}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}-d_{2}^{2}d_{6}^{2}d_{7}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}+d_{2}^{2}d_{3}d_{6}d_{8}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}$
$\displaystyle\;-d_{2}^{2}d_{6}^{2}d_{8}d_{9}^{2}u_{0}^{2}u_{1}v_{1}v_{2}-2d_{1}d_{2}^{2}d_{3}d_{9}^{3}u_{0}^{2}u_{1}v_{1}v_{2}+2d_{1}d_{2}^{2}d_{6}d_{9}^{3}u_{0}^{2}u_{1}v_{1}v_{2}-d_{1}d_{3}d_{9}u_{0}^{2}v_{2}^{2}$
$\displaystyle\;+-d_{2}d_{3}d_{4}d_{9}u_{0}^{2}v_{2}^{2}+d_{2}d_{3}d_{6}d_{7}d_{9}u_{0}^{2}v_{2}^{2}-2d_{1}d_{2}d_{3}d_{9}^{2}u_{0}^{2}v_{2}^{2}-d_{2}^{2}d_{3}d_{4}d_{9}^{2}u_{0}^{2}v_{2}^{2}+d_{1}d_{2}d_{6}d_{9}^{2}u_{0}^{2}v_{2}^{2}$
$\displaystyle\;+d_{2}^{2}d_{4}d_{6}d_{9}^{2}u_{0}^{2}v_{2}^{2}+d_{2}^{2}d_{3}d_{6}d_{7}d_{9}^{2}u_{0}^{2}v_{2}^{2}-d_{2}^{2}d_{6}^{2}d_{7}d_{9}^{2}u_{0}^{2}v_{2}^{2}-d_{1}d_{2}^{2}d_{3}d_{9}^{3}u_{0}^{2}v_{2}^{2}+d_{1}d_{2}^{2}d_{6}d_{9}^{3}u_{0}^{2}v_{2}^{2}$
$\displaystyle\;+d_{2}^{2}d_{3}d_{9}u_{0}^{3}v_{1}v_{3}+d_{2}^{3}d_{9}u_{1}^{2}v_{1}v_{3}+d_{2}^{3}d_{9}u_{1}v_{2}v_{3}+d_{2}^{2}d_{6}d_{9}u_{0}^{3}v_{1}v_{4}+d_{2}^{3}d_{9}u_{1}^{2}v_{1}v_{4}+d_{2}^{2}u_{1}v_{2}v_{4}$
$\displaystyle\;+d_{2}^{3}d_{9}u_{1}v_{2}v_{4}+d_{2}^{2}d_{3}d_{9}^{2}u_{0}u_{1}v_{1}v_{5}-d_{2}^{2}d_{6}d_{9}^{2}u_{0}u_{1}v_{1}v_{5}+d_{2}d_{3}d_{9}u_{0}v_{2}v_{5}+d_{2}^{2}d_{3}d_{9}^{2}u_{0}v_{2}v_{5}$
$\displaystyle\;-d_{2}^{2}d_{6}d_{9}^{2}u_{0}v_{2}v_{5}+d_{2}^{2}d_{5}d_{9}u_{0}u_{1}v_{1}v_{6}+d_{2}^{2}d_{4}d_{9}u_{0}v_{2}v_{6}+d_{2}^{2}d_{9}v_{6}^{2}$

### A.2. Equations of $\boldsymbol{Y_{0}}$ in the case of
$\boldsymbol{(C20,p=2,\emptyset,D_{3}2_{7})}$

$\displaystyle\bullet$ $\displaystyle\;\frac{1}{32}(-63+259\,{\rm
i}\sqrt{7})u_{0}^{4}w_{1}^{2}+\frac{1}{32}(63-259\,{\rm
i}\sqrt{7})u_{0}u_{1}^{2}w_{1}^{2}+\frac{1}{64}(21+31\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{32}(-1-3\,{\rm
i}\sqrt{7})u_{0}w_{2}^{2}+\frac{1}{16}(133-17\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+w_{3}^{2}+\frac{1}{32}(-217+21\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{64}(77-57\,{\rm
i}\sqrt{7})u_{1}w_{1}w_{5}+\frac{1}{32}(-67+23\,{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{16}(-49+13\,{\rm i}\sqrt{7})u_{1}w_{1}w_{6}$
$\displaystyle+\frac{1}{8}(9-5\,{\rm i}\sqrt{7})w_{2}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{4}(7+30\,{\rm
i}\sqrt{7})u_{0}^{4}w_{1}^{2}+\frac{1}{4}(-7-30\,{\rm
i}\sqrt{7})u_{0}u_{1}^{2}w_{1}^{2}+\frac{1}{32}(11+9\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{56}(-7-5\,{\rm
i}\sqrt{7})u_{0}w_{2}^{2}+\frac{1}{4}(23-4\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+\frac{1}{16}(-77+9\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}+w_{3}w_{4}$
$\displaystyle+\frac{1}{32}(51-7\,{\rm
i}\sqrt{7})u_{1}w_{1}w_{5}+\frac{1}{56}(-77+41\,{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{8}(-23+3\,{\rm i}\sqrt{7})u_{1}w_{1}w_{6}$
$\displaystyle+\frac{1}{14}(7-9\,{\rm i}\sqrt{7})w_{2}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{64}(275+471\,{\rm
i}\sqrt{7})u_{0}^{4}w_{1}^{2}+\frac{1}{64}(-275-471\,{\rm
i}\sqrt{7})u_{0}u_{1}^{2}w_{1}^{2}+\frac{1}{448}(161-11\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{28}(-7-2\,{\rm
i}\sqrt{7})u_{0}w_{2}^{2}+\frac{1}{64}(229+17\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+\frac{1}{4}(-10-3\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}+w_{4}^{2}$
$\displaystyle+\frac{1}{112}(175-31\,{\rm
i}\sqrt{7})u_{1}w_{1}w_{5}+\frac{1}{28}(-21+22\,{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{14}(-42+11\,{\rm i}\sqrt{7})u_{1}w_{1}w_{6}$
$\displaystyle-\frac{5}{7}\,{\rm i}\sqrt{7}w_{2}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{1472}(2107-567\,{\rm
i}\sqrt{7})u_{0}^{3}u_{1}w_{1}^{2}+\frac{1}{1472}(-2107+567\,{\rm
i}\sqrt{7})u_{1}^{3}w_{1}^{2}+\frac{1}{1472}(-973+617\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}$ $\displaystyle+\frac{1}{736}(427-607\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}+\frac{1}{736}(-177+73\,{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{16}(-63+47\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}$ $\displaystyle+\frac{1}{184}(254-29\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}+\frac{1}{184}(861-497\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{736}(-1071+147\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}$ $\displaystyle+\frac{1}{64}(161-61\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}+\frac{1}{368}(-1169+413\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+w_{3}w_{6}+\frac{1}{92}(-63-13\,{\rm
i}\sqrt{7})w_{4}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{736}(2457-735\,{\rm
i}\sqrt{7})u_{0}^{3}u_{1}w_{1}^{2}+\frac{1}{736}(-2457+735\,{\rm
i}\sqrt{7})u_{1}^{3}w_{1}^{2}+\frac{1}{368}(-413+201\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}$ $\displaystyle+\frac{1}{92}(119-70\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}+\frac{1}{736}(-257+139\,{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{8}(-49+23\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}$ $\displaystyle+\frac{1}{736}(1565-167\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}+\frac{1}{92}(651-203\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{184}(-399+71\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}$ $\displaystyle+\frac{1}{16}(105-37\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}+w_{3}w_{5}+\frac{1}{92}(-707+171\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+\frac{1}{23}(-14-8\,{\rm i}\sqrt{7})w_{4}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{5888}(4249-171\,{\rm
i}\sqrt{7})u_{0}^{3}u_{1}w_{1}^{2}+\frac{1}{5888}(-4249+171\,{\rm
i}\sqrt{7})u_{1}^{3}w_{1}^{2}+\frac{1}{2944}(-1551+589\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}$ $\displaystyle+\frac{1}{368}(371-41\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}+\frac{1}{5888}(-903+565\,{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{128}(-399+77\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}$ $\displaystyle+\frac{1}{5888}(6223-653\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}+\frac{1}{184}(707-33\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{736}(-765+151\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}$ $\displaystyle+\frac{1}{16}(57-27\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}+w_{4}w_{5}+\frac{1}{368}(-1755+433\,{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+\frac{1}{184}(-159-35\,{\rm
i}\sqrt{7})w_{4}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{32}(245+217\,{\rm
i}\sqrt{7})u_{0}^{5}w_{1}^{2}+\frac{1}{32}(-245-217\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}^{2}w_{1}^{2}+\frac{1}{256}(287-157\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{256}(-71-11\,{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{32}(119-21\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{256}(-7+181\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{256}(-121-53\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{8}(-49-7\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{32}(119-21\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{32}(-7+5\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{32}(-343+17\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{32}(33+29\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}+w_{5}^{2}$ $\displaystyle+\frac{1}{16}(133+25\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{16}(-17-13\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{32}(245+217\,{\rm
i}\sqrt{7})u_{0}^{5}w_{1}^{2}+\frac{1}{32}(-245-217\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}^{2}w_{1}^{2}+\frac{1}{256}(287-157\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{256}(-71-11\,{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{32}(119-21\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{256}(-7+181\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{256}(-121-53\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{8}(-49-7\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{32}(119-21\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{32}(-7+5\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{32}(-343+17\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{32}(33+29\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}+w_{5}^{2}$ $\displaystyle+\frac{1}{16}(133+25\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{16}(-17-13\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{16}(98+119\,{\rm
i}\sqrt{7})u_{0}^{5}w_{1}^{2}+\frac{1}{16}(-98-119\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}^{2}w_{1}^{2}+\frac{1}{32}(28-29\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{64}(-27+\,{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{32}(133+77\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{64}(-49+19\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{64}(11-17\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{32}(-147-105\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{8}(28-7\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{8}(-7+2\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{64}(-651-17\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{16}(19+13\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}$ $\displaystyle+\frac{1}{8}(56+21\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{16}(-17-13\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}+w_{5}w_{6},$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{128}(735+1057\,{\rm
i}\sqrt{7})u_{0}^{5}w_{1}^{2}+\frac{1}{128}(-735-1057\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}^{2}w_{1}^{2}+\frac{1}{256}(273-333\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{32}(-19+2\,{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{32}(56+175\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{64}(-63+13\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{64}(59-21\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{128}(49-749\,{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{64}(161-77\,{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{64}(-105+21\,{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{256}(-2471-405\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{8}(9+7\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}$ $\displaystyle+\frac{1}{64}(315+273\,{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}\frac{1}{16}(-13-15\,{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}+w_{6}^{2}$

### A.3. Parametric equation of $\boldsymbol{S_{2}}$ in $\boldsymbol{Y_{0}}$
in the case of $\boldsymbol{(C20,p=2,\emptyset,D_{3}2_{7})}$

$\displaystyle u_{0}$ $\displaystyle=\tfrac{1}{2}\big{(}5+9\,{\rm
i}\sqrt{7}-16(1+\,{\rm i}\sqrt{7})t+(13+7\,{\rm i}\sqrt{7})t^{2}\big{)}$
$\displaystyle u_{1}$ $\displaystyle=\tfrac{1}{2}\big{(}-47+29\,{\rm
i}\sqrt{7}+84(1-\,{\rm i}\sqrt{7})t+(-14+86\,{\rm
i}\sqrt{7})t^{2}+(-21-31\,{\rm i}\sqrt{7})t^{3}\big{)}$ $\displaystyle w_{1}$
$\displaystyle=1,$ $\displaystyle w_{2}$
$\displaystyle=\tfrac{1}{2}\big{(}3121-403\,{\rm i}\sqrt{7}+(-9820+1052\,{\rm
i}\sqrt{7})t+4(2545-221\,{\rm i}\sqrt{7})t^{2}+(-3473+237\,{\rm
i}\sqrt{7})t^{3}\big{)}$ $\displaystyle w_{3}$
$\displaystyle=\tfrac{1}{2}\big{(}3505-2963\,{\rm
i}\sqrt{7}+8(-2259+1391\,{\rm i}\sqrt{7})t+16(2039-955\,{\rm
i}\sqrt{7})t^{2}+(-24950+9054\,{\rm i}\sqrt{7})t^{3}$
$\displaystyle\hphantom{=\;}+7(985-277\,{\rm i}\sqrt{7})t^{4}\big{)}$
$\displaystyle w_{4}$ $\displaystyle=4(-1+t)\big{(}-108+316\,{\rm
i}\sqrt{7}+32(25-27\,{\rm i}\sqrt{7})t+(-1239+731\,{\rm
i}\sqrt{7})t^{2}+8(67-23\,{\rm i}\sqrt{7})t^{3}\big{)}$ $\displaystyle w_{5}$
$\displaystyle=\tfrac{1}{2}\big{(}2265-2539\,{\rm
i}\sqrt{7}+8(-2941+1523\,{\rm i}\sqrt{7})t+736(99-29\,{\rm i}\sqrt{7})t^{2}$
$\displaystyle\hphantom{=;}+(-99624+16264\,{\rm
i}\sqrt{7})t^{3}+48(1323-95\,{\rm i}\sqrt{7})t^{4}+(-15479-5\,{\rm
i}\sqrt{7})t^{5}\big{)}$ $\displaystyle w_{6}$
$\displaystyle=\tfrac{1}{2}(-1+t)\big{(}4720+336\,{\rm
i}\sqrt{7}-32(313+43\,{\rm i}\sqrt{7})t+(-952-40\,{\rm i}\sqrt{7})t^{2}$
$\displaystyle\hphantom{=;}+8(1571+393\,{\rm
i}\sqrt{7})t^{3}+(-6325-2047\,{\rm i}\sqrt{7})t^{4}\big{)}$

### A.4. Parametric equation of $\boldsymbol{S_{1}}$ in $\boldsymbol{Y_{0}}$
in the case of $\boldsymbol{(C20,p=2,\emptyset,D_{3}2_{7})}$

$\displaystyle u_{0}$ $\displaystyle=\frac{1}{2}\big{(}5+9\,{\rm
i}\sqrt{7}-16(1+\,{\rm i}\sqrt{7})t+(13+7\,{\rm i}\sqrt{7})t^{2}\big{)}$
$\displaystyle u_{1}$ $\displaystyle=\frac{1}{2}\big{(}-47+29\,{\rm
i}\sqrt{7}+84(1-\,{\rm i}\sqrt{7})t+(-14+86\,{\rm
i}\sqrt{7})t^{2}+(-21-31\,{\rm i}\sqrt{7})t^{3}\big{)}$ $\displaystyle w_{1}$
$\displaystyle=1$ $\displaystyle w_{2}$
$\displaystyle=\frac{1}{2}(1049-379\,{\rm i}\sqrt{7})+(-908+564\,{\rm
i}\sqrt{7})t+(357-529\,{\rm i}\sqrt{7})t^{2}+2(15+77\,{\rm i}\sqrt{7})t^{3}$
$\displaystyle w_{3}$ $\displaystyle=\frac{1}{2}\big{(}-(1271-363\,{\rm
i}\sqrt{7})+16(193+37\,{\rm i}\sqrt{7})t+(-5446-530\,{\rm
i}\sqrt{7})t^{2}+4096t^{3}+\frac{1}{4}(-4417+477\,{\rm
i}\sqrt{7})t^{4}\big{)}$ $\displaystyle w_{4}$ $\displaystyle=-1105-77\,{\rm
i}\sqrt{7}+16(261+7\,{\rm i}\sqrt{7})t+(-6096+208\,{\rm
i}\sqrt{7})t^{2}+4(997-113\,{\rm i}\sqrt{7})t^{3}+(-965+209\,{\rm
i}\sqrt{7})t^{4}$ $\displaystyle w_{5}$
$\displaystyle=2(-1+t)\big{(}-112+48\,{\rm i}\sqrt{7}+8(71+11\,{\rm
i}\sqrt{7})t+(-944-400\,{\rm i}\sqrt{7})t^{2}+64(13+7\,{\rm
i}\sqrt{7})t^{3}+(-369-179\,{\rm i}\sqrt{7})t^{4}\big{)}$ $\displaystyle
w_{6}$ $\displaystyle=\frac{1}{4}(-1+t)\big{(}16(75+\,{\rm
i}\sqrt{7})+64(21+47\,{\rm i}\sqrt{7})t+(-10696-6232\,{\rm
i}\sqrt{7})t^{2}+48(275+89\,{\rm i}\sqrt{7})t^{3}$
$\displaystyle\hphantom{=;}+(-5197-967\,{\rm i}\sqrt{7})t^{4}\big{)}$

### A.5. (Birational) automorphism of order 3 of $\boldsymbol{Y_{0}}$ for
$\boldsymbol{(C20,p=2,\emptyset,D_{3}2_{7})}$

$\displaystyle u_{0}$ $\displaystyle\longmapsto u_{0}$ $\displaystyle u_{1}$
$\displaystyle\longmapsto u_{1}$ $\displaystyle w_{1}$
$\displaystyle\longmapsto\frac{1}{1792(u_{0}^{3}-u_{1}^{2})}\big{(}7(383-29\,{\rm
i}\sqrt{7})u_{0}^{4}w_{1}+u_{0}u_{1}(7(-331+\,{\rm
i}\sqrt{7})u_{1}w_{1}+(-91+177\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+7u_{0}^{2}(w_{3}-99\,{\rm
i}\sqrt{7}w_{3}+40(-1+3\,{\rm i}\sqrt{7})w_{4})+16u_{1}((77+17\,{\rm
i}\sqrt{7})w_{5}+(-77-25\,{\rm i}\sqrt{7})w_{6})\big{)}$ $\displaystyle w_{2}$
$\displaystyle\longmapsto\frac{1}{256(u_{0}^{3}-u_{1}^{2})}\big{(}u_{0}^{4}((2471-405\,{\rm
i}\sqrt{7})u_{1}w_{1}+16(3+7\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+u_{0}u_{1}^{2}((-2387+281\,{\rm
i}\sqrt{7})u_{1}w_{1}+(-325+47\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+u_{0}^{2}u_{1}((1961-603\,{\rm
i}\sqrt{7})w_{3}+8(-301+71\,{\rm i}\sqrt{7})w_{4})+128u_{0}^{3}((-1+\,{\rm
i}\sqrt{7})w_{5}$ $\displaystyle\hphantom{\longmapsto+}+(6-2\,{\rm
i}\sqrt{7})w_{6})+16u_{1}^{2}((19+15\,{\rm i}\sqrt{7})w_{5}+(-19-7\,{\rm
i}\sqrt{7})w_{6})\big{)}$ $\displaystyle w_{3}$
$\displaystyle\longmapsto\frac{1}{256(u_{0}^{3}-u_{1}^{2})}\big{(}(287+131\,{\rm
i}\sqrt{7})u_{0}^{6}w_{1}+32u_{1}^{3}((35+11\,{\rm
i}\sqrt{7})u_{1}w_{1}+(-5-\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+u_{0}^{3}u_{1}((-1547-447\,{\rm
i}\sqrt{7})u_{1}w_{1}+(259+87\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+16u_{0}u_{1}^{2}((-45+7\,{\rm
i}\sqrt{7})w_{3}+8(7-\,{\rm i}\sqrt{7})w_{4})+u_{0}^{4}((673-579\,{\rm
i}\sqrt{7})w_{3}$ $\displaystyle\hphantom{\longmapsto+}+8(-133+79\,{\rm
i}\sqrt{7})w_{4})+16u_{0}^{2}u_{1}((11+7\,{\rm
i}\sqrt{7})w_{5}+5(w_{6}-3\,{\rm i}\sqrt{7}w_{6}))\big{)}$ $\displaystyle
w_{4}$
$\displaystyle\longmapsto\frac{1}{896(u_{0}^{3}-u_{1}^{2})}\big{(}14(39+43\,{\rm
i}\sqrt{7})u_{0}^{6}w_{1}+14u_{1}^{3}((291+55\,{\rm
i}\sqrt{7})u_{1}w_{1}+3(-9-5\,{\rm i}\sqrt{7})w_{2})$
$\displaystyle\hphantom{\longmapsto+}+u_{0}^{3}u_{1}(56(-89-21\,{\rm
i}\sqrt{7})u_{1}w_{1}+(693+257\,{\rm
i}\sqrt{7})w_{2})+14u_{0}^{4}((185-139\,{\rm i}\sqrt{7})w_{3}$
$\displaystyle\hphantom{\longmapsto+}+8(-33+19\,{\rm
i}\sqrt{7})w_{4})+7u_{0}u_{1}^{2}((-467+153\,{\rm
i}\sqrt{7})w_{3}+8(71-21\,{\rm i}\sqrt{7})w_{4})$
$\displaystyle\hphantom{\longmapsto+}+16u_{0}^{2}u_{1}((7+11\,{\rm
i}\sqrt{7})w_{5}+(21-31\,{\rm i}\sqrt{7})w_{6})\big{)}$ $\displaystyle w_{5}$
$\displaystyle\longmapsto\frac{1}{256}\big{(}(77+57\,{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}+16u_{1}((-17-13\,{\rm i}\sqrt{7})w_{3}+16\,{\rm
i}\sqrt{7}w_{4})$
$\displaystyle\hphantom{\longmapsto+}+\frac{32}{u_{0}^{3}-u_{1}^{2}}u_{0}^{3}(u_{0}^{2}(w_{2}+\,{\rm
i}\sqrt{7}w_{2})+(-3-7\,{\rm i}\sqrt{7})u_{1}w_{3}+8\,{\rm
i}\sqrt{7}u_{1}w_{4}$ $\displaystyle\hphantom{\longmapsto+}+(6+2\,{\rm
i}\sqrt{7})u_{0}w_{5}+(-4-4\,{\rm i}\sqrt{7})u_{0}w_{6})+u_{0}((11-65\,{\rm
i}\sqrt{7})u_{0}w_{2}$ $\displaystyle\hphantom{\longmapsto+}+16((-3+9\,{\rm
i}\sqrt{7})w_{5}+(15-13\,{\rm i}\sqrt{7})w_{6}))\big{)}$ $\displaystyle w_{6}$
$\displaystyle\longmapsto\frac{1}{32}\big{(}u_{0}^{2}((7+11\,{\rm
i}\sqrt{7})u_{1}w_{1}+(-5-5\,{\rm i}\sqrt{7})w_{2})+4u_{1}(-4(4+\,{\rm
i}\sqrt{7})w_{3}+(7+5\,{\rm i}\sqrt{7})w_{4})$
$\displaystyle\hphantom{\longmapsto+}+6(7+5\,{\rm
i}\sqrt{7})u_{0}w_{5}+8(-3-5\,{\rm i}\sqrt{7})u_{0}w_{6}\big{)}$

### A.6. Equations of $\boldsymbol{Y_{0}}$ in the case of Keum’s fake
projective plane

$\displaystyle\bullet$ $\displaystyle\;\frac{1}{4096}(4439-677{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}w_{1}^{2}+\frac{1}{256}(-117-417{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}+\frac{1}{16}(-27-9{\rm
i}\sqrt{7})u_{0}w_{2}^{2}$ $\displaystyle+\frac{1}{128}(-193+3{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+w_{3}^{2}+\frac{1}{256}(119+315{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}+\frac{1}{32}(-15-7{\rm
i}\sqrt{7})u_{1}w_{1}w_{5}$ $\displaystyle+\frac{1}{16}(-3+15{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{32}(7+43{\rm
i}\sqrt{7})u_{1}w_{1}w_{6}+\frac{1}{8}(21-9{\rm i}\sqrt{7})w_{2}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{2048}(169+165{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}w_{1}^{2}+\frac{1}{128}(9+21{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{16}(-9-27{\rm
i}\sqrt{7})u_{0}w_{2}^{2}+\frac{1}{64}(5+{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+\frac{1}{128}(-331+33{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}+w_{3}w_{4}$
$\displaystyle+\frac{1}{8}u_{1}w_{1}w_{5}+\frac{1}{16}(-33-3{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{16}(-27-7{\rm
i}\sqrt{7})u_{1}w_{1}w_{6}+\frac{1}{8}(45+15{\rm i}\sqrt{7})w_{2}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{1024}(-57+11{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}w_{1}^{2}+\frac{1}{64}(-39-3{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{2}+\frac{1}{16}(45-9{\rm
i}\sqrt{7})u_{0}w_{2}^{2}$ $\displaystyle+\frac{1}{32}(-3+{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{3}+\frac{1}{64}(-39-27{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{4}+w_{4}^{2}+\frac{1}{16}(-1+{\rm
i}\sqrt{7})u_{1}w_{1}w_{5}$ $\displaystyle+\frac{1}{16}(-3-9{\rm
i}\sqrt{7})w_{2}w_{5}+\frac{1}{8}(9-3{\rm
i}\sqrt{7})u_{1}w_{1}w_{6}+\frac{1}{8}(-27+15{\rm i}\sqrt{7})w_{2}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{4096}(-161+67{\rm
i}\sqrt{7})u_{0}^{3}u_{1}w_{1}^{2}+\frac{1}{4096}(161-67{\rm
i}\sqrt{7})u_{1}^{3}w_{1}^{2}+\frac{1}{256}(93+9{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}$ $\displaystyle+\frac{1}{512}(-63+93{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}+\frac{1}{128}(-81-45{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{256}((31+3{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}$ $\displaystyle+\frac{1}{64}(-33+3{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}+\frac{1}{512}(-233+27{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{128}(51-57{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}$ $\displaystyle+\frac{1}{64}(11-{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}+\frac{1}{64}(-69-17{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+w_{3}w_{6}+\frac{1}{4}(3+3{\rm
i}\sqrt{7})w_{4}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{4096}(-1663+285{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{1}w_{1}^{2}+\frac{1}{256}(-477-9{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}$ $\displaystyle+\frac{1}{512}(-561+243{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}+\frac{1}{32}(9-27{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{256}(121+53{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}$ $\displaystyle+\frac{1}{16}(-3+9{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}+\frac{1}{512}(-1911-251{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{32}(21-63{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}$ $\displaystyle+\frac{1}{64}(-29+23{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}+w_{3}w_{5}+\frac{1}{64}(203-65{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+\frac{1}{4}(7+11{\rm i}\sqrt{7})w_{4}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{256}(-13+7{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{1}w_{1}^{2}+\frac{1}{8}((-3-3{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{2}+\frac{1}{256}(147+39{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{2}$ $\displaystyle+\frac{1}{64}(-99+9{\rm
i}\sqrt{7})u_{1}w_{2}^{2}+\frac{1}{128}(-9-5{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{3}+\frac{1}{32}(-3+9{\rm
i}\sqrt{7})u_{0}w_{2}w_{3}$ $\displaystyle+\frac{1}{256}(-17-109{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{4}+\frac{1}{64}(105-27{\rm
i}\sqrt{7})u_{0}w_{2}w_{4}+\frac{1}{32}(-5-{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{5}$
$\displaystyle+w_{4}w_{5}+\frac{1}{32}(19+7{\rm
i}\sqrt{7})u_{0}^{2}w_{1}w_{6}+\frac{1}{2}(-3+{\rm i}\sqrt{7})w_{4}w_{6}$
$\displaystyle\bullet$ $\displaystyle\;\frac{1}{32768}(5579+3199{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}^{2}w_{1}^{2}+\frac{1}{4096}(12009-1563{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$
$\displaystyle+\frac{1}{2048}(-3897+3339{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{4096}(-1267-7{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}$ $\displaystyle+\frac{1}{4096}(317-1175{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}+\frac{1}{1024}(843+1023{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}$ $\displaystyle+\frac{1}{4096}(49-1267{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{1024}(1267+7{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}+\frac{1}{2048}(-8589+903{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}$ $\displaystyle+\frac{1}{512}(141-135{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{256}(387+87{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}+w_{5}^{2}$
$\displaystyle+\frac{1}{512}(-1043+345{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{256}(147-729{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{32768}(623+275{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}^{2}w_{1}^{2}+\frac{1}{4096}(1227-129{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$
$\displaystyle+\frac{1}{2048}(-1359+909{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{4096}(315+47{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{4096}(-221-137{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{1024}(573+105{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{4096}(-329+315{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{256}(31+3{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{2048}(-1851+465{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{512}(-5-17{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{256}(-39+21{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}$ $\displaystyle+\frac{1}{512}((-213+95{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{256}(57-267{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}+w_{5}w_{6}$ $\displaystyle\bullet$
$\displaystyle\;\frac{1}{32768}(623+275{\rm
i}\sqrt{7})(u_{0}^{3}-u_{1}^{2})u_{0}^{2}w_{1}^{2}+\frac{1}{4096}(1227-129{\rm
i}\sqrt{7})u_{0}^{2}u_{1}w_{1}w_{2}$ $\displaystyle+\frac{1}{2048}(135+459{\rm
i}\sqrt{7})u_{0}^{2}w_{2}^{2}+\frac{1}{4096}(161-67{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{3}+\frac{1}{4096}(-67-23{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{3}$ $\displaystyle+\frac{1}{1024}(171-33{\rm
i}\sqrt{7})u_{1}w_{2}w_{3}+\frac{1}{4096}(147+295{\rm
i}\sqrt{7})u_{0}^{3}w_{1}w_{4}+\frac{1}{1024}(5+17{\rm
i}\sqrt{7})u_{1}^{2}w_{1}w_{4}$ $\displaystyle+\frac{1}{2048}(-141+135{\rm
i}\sqrt{7})u_{1}w_{2}w_{4}+\frac{1}{512}(-5-17{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{5}+\frac{1}{256}(27+15{\rm
i}\sqrt{7})u_{0}w_{2}w_{5}$ $\displaystyle+\frac{1}{512}(-213+95{\rm
i}\sqrt{7})u_{0}u_{1}w_{1}w_{6}+\frac{1}{256}(-309-129{\rm
i}\sqrt{7})u_{0}w_{2}w_{6}+w_{6}^{2}$

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# On Controlled DeEntanglement for Natural Language Processing

###### Abstract

Latest addition to the toolbox of human species is Artificial
Intelligence(AI). Thus far, AI has made significant progress in low stake low
risk scenarios such as playing Go and we are currently in a transition toward
medium stake scenarios such as Visual Dialog. In my thesis††Presented at
Doctoral Consortium, Interspeech 2019, I argue that we need to incorporate
controlled de-entanglement as first class object to succeed in this
transition. I present mathematical analysis from information theory to show
that employing stochasticity leads to controlled de-entanglement of relevant
factors of variation at various levels. Based on this, I highlight results
from initial experiments that depict efficacy of the proposed framework. I
conclude this writeup by a roadmap of experiments that show the applicability
of this framework to scalability, flexibility and interpretibility.

##  1 Motivation of Research

In order to bring technology closer to humans it is important to provide
mechanisms for them to interact with the same. Applications like Speech
Recognition, Speech Synthesis, Visual Question Answering and Machine
Translation to name a few have been shown to be massively useful in this
context. Such Language Technologies are both rich in terms of quantity and
diversity of tasks as well as inherent complexity due to the presence of human
element. Hence they are the epitomy of medium risk medium stake applications
forming a bridge between low stake applications such as playing games, spam
filtering and high stake applications such as climate control, health related
applications. Consequently, we can employ such applications as sanity tests in
gauging the ability of AI - the latest addition to human toolbox - in
fulfilling its potential to impact our lives. However, there are three broad
problems that challenge this promise - (1) Scalability \- These technologies
are currently only accessible in a handful number of languages around the
planet. In order to have a meaningful impact it is imperative that such
technologies need to at least exist in many more languages. (2) Flexibility \-
Although deep learning based systems outperform their shallow learning
counterparts in terms of quality, they still pale away in terms of flexibility
and controllability. (3) Interpretibility \- Almost every deep learning system
today is akin to a black box: we can neither interpret nor justify predictions
by most of these models. In this thesis, I argue that employing a slightly
different framework of learning - one with controlled de-entanglement of
relevant factors of variation as a first class object - has the potential to
address all the three issues simultaneously and henceforth rapidly accelerate
progress in language technologies.

###  1.1 Controlled DeEntanglement

Let us consider a typical deep learning architecture such as AlexNet[1]. It is
characterized by a series of convolutional layers (feature extraction module)
followed by a pooling layer and a SoftMax layer(classification module). Note
that while I mention AlexNet as an example, this abstraction can be extended
to most sequence to sequence architectures with encoder as feature extraction
module and decoder as the classification module[2] across modalities and
tasks. It can be shown that the pooling layer acts as information
bottleneck[3] module in such architectures. I point out[4] that in case of
conventional Seq2Seq architectures deployed today, attention plays the role of
information bottleneck module regulating the amount of information being
utilized by the decoder. In [4, 5] I show that this module controls
optimization in encoder decoder models leading to (1) Disentanglement of
Causal Factors of variation in the data distribution (2) Marginalization of
nuisance factors of variation from the input distribution. In case of models
that employ stochasticity, two more effects can be observed : (a) Posterior
collapse or Degeneration due to powerful decoders and (b) Loss of output
fidelity due to finite capacity decoders. In current architectures,
marginalization and disentanglement are realized implicitly and often lead to
(a) and (b) when deployed in practise. I posit that designing learning
paradigms such that we explicitly control de-entanglement of relevant factors
of variation while marginalizing the nuisance factors of variation leads to
massive improvements. I refer to this as controlled DeEntanglement. Such an
approach, I claim, leads to further advantages in the context of both
generative processes - generation of novel content and discriminative
processes - robustness of such models to noise and attacks.

##  2 Key Issues identified and/or addressed

The most popular approach to obtain disentanglement in neural models is by
employing stochastic random variables. This approach provides flexibility to
jointly train the latent representations as well as the downstream network. It
has been observed that the latent representations resemble disentangled
representations under certain conditions [6, 7, 8, 9]. Note that although
obtaining such degenerate representations is typical, it is not the only
manifestation of disentanglement: it also manifests as continuous
representations[10] and other abstract phenomena(e.g. grounding). I argue that
explicitly controlling what and how much gets de-entangled [7] is better than
implicit disentanglement as is followed today[11]. I identify four ways to
computationally control de-entanglement in encoder decoder models, by
employing (1) suitable priors (2) additional adversarial or multi task
objectives (3) an alternative formulation of probability density estimation
and (4) a different objective of divergence. In my thesis, I present
experimental findings from various tasks to show that the proposed framework
has the ability to address all the three problems mentioned in section 1:
scalability via priors based controlled de-entanglement, flexibility via
priors, adversarial training and interpretibility via alternative formulation
of density estimation and objective of divergence.

###  2.1 Scalability - Discovery of Acoustic Units and Styles

A major bottleneck in the progress of many data-intensive language processing
tasks such as speech recognition and synthesis is scalability to new languages
and domains. Building such technologies for unwritten or under-resourced
languages is often not feasible due to lack of annotated data or other
expensive resources. I posit that there are two distinct categorizations that
pose challenges in terms of scalability: (1) Unwritten languages and low
resource scenarios and (2) Code switching and other non native speech
phenomena.

####  2.1.1 Unwritten Languages

Let us consider building speech technology for unwritten or under-resourced
languages. A fundamental resource required to build speech technology stack in
such languages is phonetic lexicon: something that translates acoustic input
to textual representation. Having such a lexicon - even if noisy and
incomplete - can help bootstrap speech recognition and synthesis models which
in turn enable other applications such as key word spotting. Therefore I
proposed to employ controlled de-entanglement for unsupervised acoustic unit
discovery in the context of our submission to ZeroSpeech Challenge 2019 [12].
We make an observation that articulatory information about speech production
presents a discrete set of independent constraints. For instance, manner and
place of articulation are two articulatory dimensions characterized by
discrete sets(labial vs dental, etc). Based on this, we condition the prior
space to conform to articulatory conditions by using a bank of discrete prior
distributions.

####  2.1.2 Code switching and other non native Speech phenomena

Speech has both continuous as well as discrete priors: The generative process
of speech assumes a Gaussian prior distribution which is continuous in nature.
However, the language which is also present in the utterance can be
approximated to be sampled from a discrete prior distribution. Exact
manifestation of this in linguistics can be at different levels: phonemes,
words, syllables, sub word units, etc. Based on this insight, I show[4] that
incorporating priors help encode language independent information thereby
facilitating synthesis of code mixed content. In addition to basing priors on
knowledge about characteristics, I believe that it is also possible to base
them on discovered patterns. In [13], we have discovered several code
switching styles based on [14]. I plan to incorporate priors based on this
style information to help speech recognition models targeted at decoding code
switched speech.

###  2.2 Flexibility - Global Control in Deep Generative Models

We humans exhibit explicit global as well as fine grained control over how we
deliver information. This enables us communicate more effectively in a
conversation. The goal is to build AI models that can mimic this behavior. I
have thus far worked on image captioning in the context of global control and
emphatic text to speech in the context of fine grained control.

####  2.2.1 Targeted Image Captioning

In the context of image captioning, an interesting observation is that both
the involved modalities - textual even though primarily symbolic and visual
even though primarily spatial - are characterized by distinct discrete and
continuous factors of variation. For instance, distinct objects or entities
would intuitively perhaps be better represented by discrete variables, while
their spatial location and relationships between them might be represented by
continuous variables. Therefore, we split the latent prior space[5] used for
approximating the posterior distribution into continuous and discrete
counterparts. Pressurizing the model to encode such prior information into the
latent space provides us the flexibility to control the generative process by
pinging different latent states during inference.

####  2.2.2 Emphatic Text to Speech

I am interested in investigating approaches to incorporate automatically
derivable information from speech into the model architecture for better
modeling and controlling prosody. In a sample implementation111Samples can be
found here: https://t.ly/NExyy, I have attempted disentangling heuristics
about tonal information to accomplish local control. I first quantize
fundamental frequency($F_{0}$) - a feature highly correlated with prosody -
into multiple bins. I then incorporate the resultant discretized ordinal
information at the phone level into our speech generation architecture in two
different scenarios: (1) Explicit labeling scenario where we input the
quantized $F_{0}$ values alongside the phoneme embeddings as input to the
encoder and (2) Implicit incorporation as an additional task to predict these
quantized $F_{0}$s at the output of encoder. The model was optimized using
ordinal triplet divergence [15]. We show that our approach generates
appropriate emphasis at word level and significantly outperforms AuToBI in
terms of flexibility.

###  2.3 Justification

Language is inherently composite in nature. Hence it is vital to model the
abstract relationships so as to capture the unseen compositions of seen
concepts at test time. However, accomplishing this is a deceptively non
trivial task and might lead to models learning just surface level
associations[16, 17, 18]. I present a case that flexible generative models
provide additional information to improve performance in such tasks. Further,
I hypothesize that when optimized using either a disjoint learning mechanism
or a different divergence function, such models can also act as justifying
modules for the task at hand. To ground this argument, I present two example
applications that employ flexible models mentioned in the previous
subsections: (1) Visual Question Answering System(VQA) that receivers
additional information in the form of targeted captions. I propose to use
Reward Augmented Maximum Likelihood[19] to generate and integrate captions in
the framework of Visual Question Answering. I show that tying the reward
function to length of the generated caption forces the model to encode most
relevant information thereby acting as justification to the selected answer.
(2) Based on similar insights, I propose to apply[19] to obtain and integrate
speech recognition transcripts in the context of Acoustic Topic Identification
System.

##  3 Acknowledgements

I sincerely thank Alan W Black for his invaluable guidance towards my
dissertation. I thank AFRL for funding my research and student volunteers for
taking part in the various subjective evaluations. I am very grateful to my
collaborators for providing me an opportunity to work with them and the
reviewers for useful feedback at various stages in my research career.

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