# On the Monodromy and Period Map of the Winger Pencil

Yunpeng Zi

###### Abstract.

The sextic plane curves that are invariant under the standard action of the
icosahedral group on the projective plane make up a pencil of genus ten curves
(spanned by a sum of six lines and a three times a conic). This pencil was
first considered in a note by R. M. Winger in 1925 and is nowadays named after
him. We gave this a modern treatment and proved among other things that it
contains essentially every smooth genus ten curve with icosahedral symmetry.
We here consider the monodromy group and the period map naturally defined by
the icosahedral symmetry. We showed that this monodromy group is a subgroup of
finite index in $\operatorname{SL}_{2}(\mathds{Z}[\sqrt{5}])$ and the period
map brings the Winger pencil to a curve on the Hilbert modular surface
$\operatorname{SL}_{2}(\mathds{Z}[\sqrt{5}])/\mathds{H}^{2}$.

##  1\. Introduction

This is the last part of a series of paper concerning the Winger’s pencil and
also a continuation of the author’s Phd thesis. The Winger’s pencil
$\mathscr{W}_{\mathscr{B}}$ is a linear system of planer genus 10 curves with
$\mathcal{A}_{5}$-symmetry on each fiber which is studied in [11] by R.M
Winger. If $I$ is a complex 3-space endowed with a faithful
$\mathcal{A}_{5}$-action, it is defined as a hypersurface by the following
equation in the projective variety
$\mathds{P}(I)\times\mathscr{B}\cong\mathds{P}^{2}\times\mathds{P}^{1}$

(1) $g_{2}^{3}+tg_{6}=0$

Here $t\in\mathscr{B}$ be a parameter, $g_{2}$ and $g_{6}$ are two generators
of $\mathds{C}[I]^{\mathcal{A}}_{6}$ where $g_{2}$ is a polynomial of degree
two representing a smooth conic and $g_{6}$ is a polynomial of degree 6
representing the union of 6 lines. The singular members of this pencil appears
only at four points, they are $t=-1$ an irreducible curve with six nodes which
is also coming from identifying six pairs of double points on the Bring’s
curve, $t=0$ a smooth conic with multiple three, $t=\frac{27}{5}$ an
irreducible curve with ten node and the last $t=\infty$ the union of six
lines.

We have showed in [13] that with some modifications the new object ”Winger’s
family” parameterized all stable genus 10 curves with
$\mathcal{A}_{5}$-symmetry. It was showed in the same paper that for a smooth
member $C_{t}$ ($t\in\mathscr{B}^{\circ}$ a point in the smooth locus of
$\mathscr{W}\to\mathscr{B}$) of the Winger pencil, its space of holomorphic
forms $H^{0}(C,\omega_{C})$ is isomorphic to $V\oplus I\oplus
I^{\prime}=V\oplus E$ as a $\mathds{C}\mathscr{I}$-module where $V$ is the
permutation representation of dimension four and $I$ and $I^{\prime}$ are
three dimensional irreducible representations. This implies that
$H^{1}(C;\mathds{C})$ is isomorphic to $V^{\oplus 2}\oplus E^{\oplus 2}$.
Since both $V$ and $E$ are complexifications of irreducible
$\mathds{Q}\mathscr{I}$-modules $V_{\mathds{Q}}$ resp. $E_{\mathds{Q}}$ (which
are therefore self-dual), it follows that there exist a canonical isotypical
decomposition for $H_{1}(C;\mathds{Q})$

(2)
$H_{1}(C_{t};\mathds{Q})\cong(V_{\mathds{Q}}\otimes\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(V_{\mathds{Q}},H_{1}(C_{t},\mathds{Q})))\oplus(E_{\mathds{Q}}\otimes\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t},\mathds{Q})))$

with
$\dim_{\mathds{Q}}\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(V_{\mathds{Q}},H_{1}(C_{t},\mathds{Q}))=2$
and
$\dim_{\mathds{Q}(\sqrt{5})}\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t},\mathds{Q}))=2$.
We have proved in [6] that the (global) monodromy restricted to $V$-part has
image in $\operatorname{SL}_{2}(\mathds{Z})$, more explicitly it is the index
$8$ congruence subgroup $\Gamma_{1}(3)$ of
$\operatorname{SL}_{2}(\mathds{Z})$. Hence the period map
$p_{V}:\mathscr{B}^{\circ}\to\mathds{H}/\operatorname{SL}_{2}(\mathds{Z})$ is
a ramified finite morphism of degree 8.

In this paper, we will focus on the $E$-part of the decomposition (2). If
$E_{o}$ is a fixed integral form of the representation $E$ with endomorphism
ring $\mathcal{O}_{o}$, the monodromy group and period map related to $E_{o}$
is denoted by $\Gamma_{E_{o}}$ and $p_{E_{o}}$. We could also observe that
there exist an inner product on $E_{o}$ and a symplectic form on
$H_{1}(C_{t})$, the monodromy action will preserve these forms. This implies
that the monodromy group must be a subgroup of
$\operatorname{Sp}_{1}(\mathcal{O}_{o})\cong\operatorname{SL}_{2}(\mathcal{O}_{o})$.
The main theorems are the following:

######  Theorem 1.1.

The monodromy group $\Gamma_{E_{o}}$ is a subgroup of finite index in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$. In particular it is arithmetic.

And if $\mathscr{B}^{+}$ be the open subvariety of $\mathscr{B}$ obtained by
removing from $\mathscr{B}$ the three points representing nodal curves, we
have the following theorem about the period map.

######  Theorem 1.2.

The ’partial’ period map
$p_{E_{o}}:\mathscr{B}^{+}\to\Gamma_{E_{o}}/\mathds{H}^{2}\to\operatorname{SL}_{2}(\mathcal{O}_{o})/\mathds{H}^{2}$
has the property that the first arrow is open and the second map is finite.

Moreover with the help of computer program Magma, we could say a little more
property about the group $\Gamma_{E_{o}}$ namely

######  Theorem 1.3.

The monodromy group $\Gamma_{E_{o}}$ is of index two in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$.

The main tool of surveying the the monodromies and the periods are two models
of the genus 10 curve with an $\mathcal{A}_{5}$-symmetry. The first one which
we named it as $\Sigma$ is coming from the regular icosahedron with a natural
$\mathcal{A}_{5}$-action by removing in a $\mathcal{A}_{5}$-equivalent manner
a small triangle at each vertices and identifying the antipodal points on the
boundary. This is also the model that we used in [6]. The second which we call
it $\Pi$ is modified from the Euclidean realization of the Bring’s curve. This
realization is a regular polygon endowed with a $\mathcal{A}_{5}$-symmetry
namely the Great Dodecahedron. We will remove in a
$\mathcal{A}_{5}$-equivalent manner a small pentagram at each vertices and
identifying the antipodal points on the boundary. Each of the models give a
real one-dimensional family $\Sigma_{t}$ resp. $\Pi_{t}$ on the Winger pencil
such that they connect two different singular members of the Winger pencil.
Then instead of computing the local monodromies on the Winger pencil, we could
done it on the family $\Sigma_{t}$ or $\Pi_{t}$.

This paper is organized as following, we will introduce some basic lemmas and
fix notations after the introduction. And we will take the next two sections
devoting to introduce the details of the two models. We will use all these
information to determine the local monodromies in Section 4. A global
description to the monodromy group $\Gamma_{E_{o}}$ and period map $p_{E_{o}}$
will be given in the Section 5. And in the last section we will give the way
of computing the index in the last section. As we have talked above, this
computation is made by the computer program, the code for this computation has
uploaded to [12].

###  1.1. Acknowledgement

The author wants to thank Prof. Eduard Looijenga for his kind help and useful
discussion.

###  1.2. The Integral form of $E_{\mathds{Q}}$

Before we study this project in detail, let us introduce some properties of
$E$ the $6$-dimensional linear representation of $\mathcal{A}_{5}$. Let
$I_{\mathds{R}}$ be the Euclidean vector space with a faithful
$\mathcal{A}_{5}$-action and we will denote the image of
$\mathcal{A}_{5}\hookleftarrow\operatorname{GL}(I_{\mathds{R}})$ as
$\mathscr{I}$. The $\mathds{R}\mathscr{I}$-module $I_{\mathds{R}}$ is
irreducible, even its complexification $I$ is an irreducible
$\mathds{C}\mathscr{I}$-module, but $\mathscr{I}$ is not definable over
$\mathds{Q}$. If $I^{\prime}$ is obtained from $I$ by precomposing the
$\mathscr{I}$-action with an outer automorphism of $\mathscr{I}$, then
$E:=I\oplus I^{\prime}$ is as a representation naturally defined over
$\mathds{Q}$. And the character computation shows that it actually splits over
the field $\mathds{Q}[\sqrt{5}]$.

Let us take $V_{o}$ be the integral permutation representation of
$\mathcal{A}_{5}$ of rank 4 the same as the notations in [6]. Recall that if
we take $\mathds{Z}^{5}$ to be the free $\mathds{Z}$-module generated by
$\\{e_{i}\\}_{i=1}^{5}$ and $\mathcal{A}_{5}$ acts on the set of generators in
a natural way, the $\mathds{Z}\mathcal{A}_{5}$-module $V_{o}$ is defined by
the following exact sequence.

$0\to\mathds{Z}\to\mathds{Z}^{5}\to V_{o}\to 0$

This exact sequence gives an a surjective map
$\wedge^{2}\mathds{Z}^{5}\to\wedge^{2}V_{o}$ whose kernel is identified with
$\mathds{Z}^{5}\wedge(\sum_{i=1}^{5}e_{i})$, so that we have the exact
sequence of $\mathds{Z}\mathcal{A}_{5}$-modules

(3) $0\to V_{o}\to\wedge^{2}\mathds{Z}^{5}\to\wedge^{2}V_{o}\to 0$

It is clear to see that $\wedge^{2}V_{o}$ is an integral form of $E$ from the
character computations. We will always denote $E_{\mathds{Q}}$ to be the
vector space $\wedge^{2}V_{o}\otimes\mathds{Q}$. Using the similar notion of
[5] we will take $f_{i,j}$ as the image of $e_{i}\wedge e_{j}$ in
$\wedge^{2}V_{o}$. Let $\phi:\mathds{Z}^{5}\to\mathds{Z}$ be the morphism of
taking the coordinate sum. We will denote by $E_{o}$ the space generated by
the elements $(f_{i,j}+f_{j,k}+f_{k,i})$ for all $i,j,k\in\\{1,2,3,4,5\\}$.
Note that $E_{o}$ is the image of the $\mathds{Z}\mathscr{I}$-homomorphism

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Here $\iota_{\phi}$ is taking inner product with $\phi$. The following Lemma
is the Lemma 2.1 of [5].

######  Lemma 1.4.

Let $e:=\sum_{i}f_{i,i+1}\in\wedge^{2}V_{o}$. Then the $\mathscr{I}$-orbit of
$e$ is the union of a basis of $E_{o}$ and its antipodal. Hence the
$\mathds{Z}\mathscr{I}$-module $E_{o}$ is principal. Moreover there exists an
inner product

$s:E_{o}\times E_{o}\to\mathds{Z}$

for which this basis is orthogonal is $\mathscr{I}$-invariant.

######  Remark 1.5.

There is a simple observation that if we assume that
$(H,{\langle}-,-{\rangle})$ is a $\mathds{Z}\mathcal{A}_{5}$-module endowed
with a $\mathcal{A}_{5}$-invariant symplectic form ${\langle}-,-{\rangle}$,
the inner product $s$ and the symplectic form ${\langle}-,-{\rangle}$ gives a
symplectic form on $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H)$ in a
natural way, since $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H)$ is a
submodule of $\operatorname{Hom}(E_{o},H)$. This form is also symplectic form
and making $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H)$ a symplectic
$\mathds{Z}$-module.

Since $E_{\mathds{Q}}$ is not absolutely irreducible, there must have
endomorphisms which not a multiple of $\operatorname{Id}$. We will construct
one such example and show that it is defined over integers and generates the
endomorphism ring $\operatorname{End}_{\mathds{Z}\mathscr{I}}(E_{o})$. First
let us take the generators of $\mathcal{A}_{5}$ as $\sigma_{2}:=(1,5)(3,4)$,
$\sigma_{3}:=(2,5,3)$ and $\sigma_{5}=(1,2,3,4,5)$. It is clear that they
satisfies the relation that $\sigma_{2}\sigma_{3}\sigma_{5}=1$. Note that
$\sigma_{5}$ fixes $e$ and
$\sigma_{2}\sigma_{5}^{3}\sigma_{2}\sigma_{5}^{2}\sigma_{2}=(2,5)(3,4)$ maps
$e$ to $-e$. Then the Lemma 1.4 implies that the basis of $E_{o}$ is the
following

$\\{e,e_{0}:=\sigma_{2}(e),e_{1}:=\sigma_{5}(e_{0}),\cdots,e_{4}:=\sigma_{5}^{4}(e_{0})\\}$

Their relations are as following

$\displaystyle\sigma_{5}:$ $\displaystyle e_{0}\to e_{1}\to e_{2}\to e_{3}\to
e_{4}\to e_{0};\hbox{fixes $e$}$ $\displaystyle\sigma_{2}:$ $\displaystyle
e\leftrightarrow e_{0};e_{1}\leftrightarrow e_{4};e_{2}\leftrightarrow-
e_{2};e_{3}\leftrightarrow-e_{3};\hbox{fixes $(e+e_{0})$}$
$\displaystyle\sigma_{3}:$ $\displaystyle e\to e_{0}\to e_{1}\to e;e_{2}\to
e_{4}\to-e_{3}\to e_{2};\hbox{fixes $(e+e_{0}+e_{1})$}$

We will take the endomorphism $X\in\operatorname{End}(E_{o})$ as following

$\displaystyle X(e)$ $\displaystyle:=e_{0}+e_{1}+e_{2}+e_{3}+e_{4}$
$\displaystyle X(e_{0})$ $\displaystyle:=e+e_{1}-e_{2}-e_{3}+e_{4}$
$\displaystyle X(e_{1})$ $\displaystyle:=e+e_{0}+e_{2}-e_{3}-e_{4}$
$\displaystyle X(e_{2})$ $\displaystyle:=e-e_{0}+e_{1}+e_{3}-e_{4}$
$\displaystyle X(e_{3})$ $\displaystyle:=e-e_{0}-e_{1}+e_{2}+e_{4}$
$\displaystyle X(e_{4})$ $\displaystyle:=e+e_{0}-e_{1}-e_{2}+e_{3}$

It is clear to check that $\sigma_{i}X=X\sigma_{i}$ for all $i\in\\{2,3,5\\}$.
Hence it is a nontrivial element of
$\operatorname{End}_{\mathds{Z}\mathscr{I}}(E_{o})$ which is not a multiple by
an integer. Moreover $X$ satisfies the relation that
$X^{2}-5\operatorname{Id}=0$.

######  Proposition 1.6.

The endomorphism ring
$\mathcal{O}_{o}:=\operatorname{End}_{\mathds{Z}\mathscr{I}}(E_{o})$ is
generated by $X$ subjects to the relation $X^{2}-5\operatorname{Id}=0$. Hence
it is isomorphic to the quadratic algebraic integers $\mathds{Z}[\sqrt{5}]$.

###### Proof.

This is the Lemma 2.8 of [5]. ∎

From the Proposition 1.6, we get the following Corollaries. The first one 1.7
is a more explicit description of the endomorphism of $E_{\mathds{Q}}$ and the
second 1.8 concerns about the automorphism ring of $E_{o}$.

######  Corollary 1.7.

The endomorphism ring
$\mathcal{K}:=\operatorname{End}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}})$ of
$E_{\mathds{Q}}=E_{o}\otimes\mathds{Q}$ is generated by $X$ subjects to the
relation $X^{2}-5\operatorname{Id}=0$. Hence it is isomorphic to the quadratic
field extension $\mathds{Q}[\sqrt{5}]/\mathds{Q}$.

######  Corollary 1.8.

The automorphism group $\operatorname{Aut}_{\mathds{Z}\mathscr{I}}(E_{o})$ is
a multiplicative group cyclic of order two. More explicitly it is the group
$<\pm\operatorname{Id}>$.

###### Proof.

(Proof of Corollary 1.8) Let
$Y\in\operatorname{Aut}_{\mathds{Z}\mathscr{I}}(E_{o})$ be an automorphism of
$E_{o}$. Since $Y$ and $Y^{-1}$ are both endomorphisms of $E_{o}$, there exist
integers $a_{1}$, $b_{1}$,$a_{2}$ and $b_{1}$ such that
$Y=a_{1}X+b_{1}\operatorname{Id}$ and $Y^{-1}=a_{2}X+b_{2}\operatorname{Id}$.
From the Lemma 1.6, the relation $YY^{-1}=\operatorname{Id}$ implies that
$a_{1}=a_{2}=0$ and $b_{1}=b_{2}=\pm 1$. This finishes the proof. ∎

Despite $E_{o}$ there are other integral forms of $E_{\mathds{Q}}$. For
example we may take $E\subset E_{o}$ to be the subspace which has even
coefficients sum with respect to the basis in 1.4. It is a sublattice of index
two in $E_{o}$ and it was proved in the Lemma 2.8 of [5] that
$\mathcal{O}:=\operatorname{End}_{\mathds{Z}\mathscr{I}}(E)$ is isomorphic to
to the ring of algebraic integers in $\mathds{Q}[\sqrt{5}]$ i.e.
$\mathds{Z}[Y]/(Y^{2}-Y-1)$. Moreover there is an embedding
$\mathcal{O}_{o}\hookrightarrow\mathcal{O}$ given by $X\to(2Y-1)$.

###  1.3. Criterion of Generating a Lattice

As the end of this section, let us introduce some criterions about when a set
of elements become the generators of a lattice.

######  Lemma 1.9.

Let $(H,{\langle}-,-{\rangle})$ be a lattice of rank $n$ with bilinear form
${\langle}-,-{\rangle}$. Assume that $a_{1},\cdots,a_{n}$ be elements of $H$
such that they form a $\mathds{Q}$-basis of
$H_{\mathds{Q}}:=H\otimes\mathds{Q}$. If for every $n$ coprime integers
$\\{\alpha_{1},\cdots,\alpha_{n}\\}$ i.e.
$\gcd(\alpha_{1},\cdots,\alpha_{n})=1$, there exist
$y(\alpha_{1},\cdots,\alpha_{n})\in H$ such that

${\langle}\sum_{i=1}^{n}\alpha_{i}a_{i},y(\alpha_{1},\cdots,\alpha_{n}){\rangle}=1$

then $a_{1},\cdots,a_{n}$ generates $H$ over $\mathds{Z}$. Furthermore if we
assume $(H,{\langle}-,-{\rangle})$ is unimodular the converse also holds.

###### Proof.

(1) Let us assume that $a_{1},\cdots,a_{n}$ cannot generates $H$ over
$\mathds{Z}$ and denote the sublattice generated by $a_{1},\cdots,a_{n}$ by
$H^{\prime}$. Since $\\{a_{1},\cdots,a_{n}\\}$ is a $\mathds{Q}$-basis of
$\mathds{Q}$-vector space $H_{\mathds{Q}}$, $H^{\prime}$ must have the same
rank as $H$. There exist an element $x\in H$ but $x\notin H^{\prime}$
satisfying that for every positive integer $k>1$, $\frac{x}{k}\notin H$ and
there exist minimal positive integer $m\in\mathds{Z}$ such that $m\neq 1$ and
$mx\in H^{\prime}$. Hence there exist integers $\alpha_{1},\cdots,\alpha_{n}$
such that $mx=\sum_{i=1}^{n}\alpha_{i}a_{i}$. By the minimality of $x$ and
$m$, $\alpha_{1},\cdots,\alpha_{n}$ has the property that
$(\alpha_{1},\cdots,\alpha_{n})=1$. Hence there exist an element
$y(\alpha_{1},\cdots,\alpha_{n})\in H$, such that
${\langle}\sum_{i=1}^{n}\alpha_{i}a_{i},y(\alpha_{1},\cdots,\alpha_{n}){\rangle}=1=m{\langle}x,y(\alpha_{1},\cdots,\alpha_{n}){\rangle}$,
which is not possible.

(2) Let us assume that $a_{1},\cdots,a_{n}$ generates $H$ and
$(H,{\langle}-,-{\rangle})$ is unimodular. Let $\alpha_{1},\cdots,\alpha_{n}$
be $n$ integers satisfying $\gcd(\alpha_{1},\cdots,\alpha_{n})=1$. Hence there
exist integers $\beta_{i}$ such that $\sum_{i=1}^{n}\alpha_{i}\beta_{i}=1$.
Let us take $y(\alpha_{1},\cdots,\alpha_{n})\in H^{\vee}=H$ such that
${\langle}a_{i},y(\alpha_{1},\cdots,\alpha_{n}){\rangle}=\beta_{i}$. Hence we
have
${\langle}\sum_{i=1}^{n}\alpha_{i}a_{i},y(\alpha_{1},\cdots,\alpha_{n}){\rangle}=\sum_{i=1}^{n}\alpha_{i}\beta_{i}=1$.
∎

Using the similar argument, we can prove the following Lemma, which is
frequently used in the material below.

######  Lemma 1.10.

Let $(H,{\langle}-,-{\rangle})$ be an unimodular lattice of finite rank such
that $H$ is a $\mathds{Z}\mathcal{A}_{5}$-module and ${\langle}-,-{\rangle}$
is an $\mathcal{A}_{5}$-invariant bilinear form. Assume that
$\operatorname{Hom}_{\mathds{Z}\mathcal{A}_{5}}(E_{o},H)$ is free
$\mathds{Z}$-module of rank $n$ and $\phi_{1},\cdots,\phi_{n}$ are $n$
linearly inequivalent elements of
$\operatorname{Hom}_{\mathds{Z}\mathcal{A}_{5}}(E_{o},H)$ such that they form
a basis of $\mathds{Q}$-vector space
$\operatorname{Hom}_{\mathds{Q}\mathcal{A}_{5}}(E_{\mathds{Q}},H_{\mathds{Q}})$.
The elements $\phi_{1},\cdots,\phi_{n}$ generates
$\operatorname{Hom}_{\mathds{Z}\mathcal{A}_{5}}(E_{o}.H)$ over $\mathds{Z}$ if
and only if for every set of $n$ integers $\\{\alpha_{1},\cdots,\alpha_{n}\\}$
satisfying $\gcd(\alpha_{1},\cdots,\alpha_{n})=1$, there exist
$y(\alpha_{1},\cdots,\alpha_{n})\in H$ such that

${\langle}\sum_{i=1}^{n}\alpha_{i}\phi_{i}(e),y(\alpha_{1},\cdots,\alpha_{n}){\rangle}=1$

###### Proof.

The proof is similar as above. ∎

##  2\. Geometric Model from Icosahedron

We introduced a geometric model in Section 2 of [6] for a smooth fiber $C_{t}$
with $t\in\mathscr{B}^{\circ}$ and describe two stable degenerations in terms
of it. Here we will give a quick summary on this model without proof. Note
that we will always use $z$ to denote a 2-cell or a face of a polyhedron, $y$
to denote a 1-cell or an edge and $x$ to denote a 0-cell or a vertex beginning
from this section. For an oriented edge $y$, let $\textrm{in}(y)$ be its
initial point and $\textrm{tm}(y)$ be its terminal point.

Let us fix an oriented euclidean 3-space $I_{\mathds{R}}$ and a regular
dodecahedron $\tilde{\Sigma}\subset I_{\mathds{R}}$ centered at the origin.
Let $\iota$ be the antipodal map which is orientation reversing. Note that the
automorphism group $\mathscr{I}\subset\operatorname{SO}(I_{\mathds{R}})$ of
$\tilde{\Sigma}$ is isomorphic to $\mathcal{A}_{5}$, the alternating group of
five elements. Let $\hat{\Sigma}$ be obtained from the dodecahedron
$\tilde{\Sigma}$ by removing in a $\mathscr{I}$-invariant manner a small
regular triangle centered at each vertex of $\tilde{\Sigma}$ so that the faces
of $\hat{\Sigma}$ are oriented solid $10$-gons. We now identify opposite
points on the boundary of $\hat{\Sigma}$ and thus obtain a complex $\Sigma$
that is a closed oriented surface of genus $10$ endowed with an action of
$\mathscr{I}$ (See Figure 1).

Figure 1. Removing in a $\mathscr{I}$-invariant manner a small regular
triangle centered at each vertex of $\tilde{\Sigma}$

Recall that there exist two kinds of oriented 1-cells resp. 1-cycles on
$\Sigma$. The ones coming from the truncation is called _$1$ -cells resp.
1-cycles of truncation type_. The 1-cycles of truncation type are bijectively
indexed by the set $\mathscr{C}_{0}(\tilde{\Sigma})$ of vertices of
$\tilde{\Sigma}$, namely for each vertex $x\in\mathscr{C}_{0}(\tilde{\Sigma})$
the sum of three 1-cells of truncation type together with its counterclockwise
orientation is a 1-cycle of truncation type. This labeling is denoted by
$\delta_{x}$, the set of all 1-cells of truncation type is denoted by
$\Delta_{\textrm{trc}}$ and we have $\delta_{\iota x}=-\delta_{x}$. The ones
coming from the edges of $\tilde{\Sigma}$ is called _$1$ -cells resp. 1-cycles
of edge type_. The 1-cycles of edge type are indexed by
$\mathscr{C}_{1}(\tilde{\Sigma})$, namely for each oriented edges
$y\in\tilde{\Sigma}$ the division $(y-\iota y)$ is a 1-cycle of edge type.
This labeling is denoted by $\delta_{y}$ and the set of all 1-cells of edge
type is denoted by $\Delta_{\textrm{edge}}$ . Note that this labeling is not
bijective for we have the relation that $\delta_{-y}=\delta_{\iota
y}=-\delta_{y}$.

The polyhedron $\Sigma$ can be endowed with a natural complex structure
$J_{\tau}$ where $\tau$ is the length of the 1-cells of truncation type such
that the following proposition follows

######  Proposition 2.1.

(Proposition 3.4 of [6]) The Riemann surface $(\Sigma,J_{\tau})$ is the set of
complex points of a complex real algebraic curve. It has genus 10 and comes
with a faithful $\mathscr{I}$-action, hence is isomorphic to a member of the
Winger pencil. We thus have defined a continuous map
$\gamma:[0,1]\to\mathscr{B}$ which traverses the real interval
$[\infty,\frac{27}{5}]$ and which maps $(0,1)$ to $\mathscr{B}^{\circ}$ (and
so lands in the locus where $t$ is real and $>\frac{27}{5}$), such that the
pull-back of the Winger pencil yields the family constructed above. The
degenerations of $\Sigma$ into $\Sigma_{\textrm{edge}}$ resp.
$\Sigma_{\textrm{trc}}$ have $\Delta_{\textrm{edge}}$ resp.
$\Delta_{\textrm{trc}}$ as their sets of vanishing cycles.

Moreover it is clear to check that the intersection number of the above
1-cycles are as the following Lemma.

######  Lemma 2.2.

The intersection numbers of these $1$-cycles are as follows: any two loops of
the same type have intersection number zero and if
$x\in\mathscr{C}_{0}(\tilde{\Sigma})$ and
$y\in\mathscr{C}_{1}(\tilde{\Sigma})$, then
${\langle}\delta_{x},\delta_{y}{\rangle}=0$ unless $x$ lies on $y$ or on
$\iota y$, in which case ${\langle}\delta_{x},\delta_{y}{\rangle}\in\\{\pm
1\\}$ with the plus sign appearing if and only if $x$ is the end point of $y$.

###  2.1. Celluar Homology of $\Sigma$

Recall that the surface $\Sigma$ comes with a cellular decomposition. Hence
there exist a natural exact sequence i.e. the cellular decomposition of
$\Sigma$ enables us compute its homology as that of the combinatorial chain
complex

(4)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{C}_{2}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{\mathscr{C}_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{\mathscr{C}_{0}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Let us take $B_{i}(\Sigma):=\operatorname{im}{(\partial_{i+1})}$ and
$Z_{i}(\Sigma):=\ker(\partial_{i})$. Hence we have the following two exact
sequences where the second describes the first homology group. The first
describes the 1-boundaries namely the 1-boundaries of $\Sigma$ are generated
by the boundaries of 2-cells of $\Sigma$ and the sum of all 2-cells has no
boundary.

(5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker\partial_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{C}_{2}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{B_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{H_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

If we apply the left exact functor
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},-)$ to the above exact
sequences, we can get the long exact sequences

(7) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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17.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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17.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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95.24753pt\raise 6.46388pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 108.02844pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
108.02844pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
201.08807pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
201.08807pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Ext}^{1}_{\mathds{Z}\mathscr{I}}(E_{o},\mathds{Z})}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
(8) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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17.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
17.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.55963pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.55963pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 203.07413pt\raise 0.0pt\hbox{\hbox{\kern
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Note that $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathds{Z})$ must
be trivial. Unlike the $V$-case discussed in [6], the first term of exact
sequence (8) is nontrivial. This could be seen from the character computation
to the $\mathds{C}\mathscr{I}$-module
$\mathscr{C}_{2,\mathds{C}}(\Sigma):=\mathds{C}\otimes\mathscr{C}_{2}(\Sigma)$
which showed that $\mathscr{C}_{2,\mathds{C}}(\Sigma)$ is isomorphic to
$\mathds{C}\oplus W\oplus I\oplus I^{\prime}$ as
$\mathds{C}\mathscr{I}$-module, where $W$ is the $5$-dimensional irreducible
representation. Then the exact sequence (7) will help to give a explicit
description of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$. For the
term $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))$, we can
”divide” $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))$
into three parts, despite the one from the boundary, the other two parts are
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Sigma))$
and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Sigma))$.
We will discuss the last two separately. The main theorems of section is the
Proposition 2.8.

Let us first observe that the $\mathds{Z}$-module
$\mathscr{C}_{2}(\Sigma)/(\iota-1)$ is isomorphic to $E_{o}$ as
$\mathds{Z}\mathscr{I}$-module where the isomorphism is unique up to a sign
and if we choose a system $\mathscr{R}_{2}(\Sigma)$ of representatives of
$\iota$-symmetry on $\mathscr{C}_{2}(\Sigma)$, $\mathscr{R}_{2}(\Sigma)$ will
become the $\mathds{Z}$-basis of $E_{o}$ that we discussed in Lemma 1.4. Hence
we may fix one such isomorphism and let $e$ denote not only one of the
generators of $E_{o}$ but also a face in
$\mathscr{R}_{2}(\Sigma)\subset\mathscr{C}_{2}(\Sigma)$. Moreover for each
$z\in\mathscr{R}_{2}(\Sigma)$, let $h_{z}\in\mathcal{A}_{5}$ be a permutation
(which is not unique) such that $h_{z}z=-z$ and
$\mu_{5}\cong\operatorname{Stab}(z)\subset\mathscr{I}$ be its
$\mathscr{I}$-stabilizer.

To see this observation, recall that the $2$-cells i.e. the faces of $\Sigma$
can be canonically oriented clockwisely. Clearly they are bijectively indexed
by the faces of $\tilde{\Sigma}$. The set $\mathscr{C}_{2}(\Sigma)$ of
oriented 2-cells of $\Sigma$ admits both $\mathscr{I}$-symmetry and
$\iota$-symmetry, Note that the $\mathscr{I}$-symmetry keeps the orientation
and permutes the 12 faces, hence the stabilizer for each 2-cell is cyclic
order five. The $\iota$-symmetry commutes with $\mathscr{I}$-action and will
reverse the orientation. Hence it is clear that the $\mathds{Z}$-module
$\mathscr{C}_{2}(\Sigma)/(\iota-1)$ is isomorphic to $E_{o}$ as
$\mathds{Z}\mathscr{I}$-module. If we choose a system
$\mathscr{R}_{2}(\Sigma)$ of representatives of $\iota$-symmetry on
$\mathscr{C}_{2}(\Sigma)$, $\mathscr{R}_{2}(\Sigma)$ will become the
$\mathds{Z}$-basis of $E_{o}$ as we claimed above. From the Corollary 1.8,
such isomorphism is unique up to sign.

Let us first construct the morphisms $E_{o}\to\mathscr{C}_{2}(\Sigma)$ and
$E_{o}\to B_{1}(\Sigma)$. Let $e\in\mathscr{R}_{2}(\Sigma)$ be as above, the
elements $\delta_{\textrm{cell}}:=(e+\iota e)$ and
$\delta^{\prime}_{\textrm{cell}}:=\sum_{z\in\mathscr{R}_{2}(\Sigma)\backslash\\{e\\}}(z+\iota
z)$ are $\operatorname{Stab}(e)$-invariant and
$h_{e}\delta_{\textrm{cell}}=-\delta_{\textrm{cell}}$ resp.
$h_{e}\delta^{\prime}_{\textrm{cell}}=-\delta^{\prime}_{\textrm{cell}}$. Then
the elements $\delta_{\textrm{bound}}:=\partial\sigma_{\textrm{cell}}$ and
$\delta^{\prime}_{\textrm{bound}}:=\frac{1}{2}\partial(\delta_{\textrm{cell}}+\delta^{\prime}_{\textrm{cell}})$
are also $\operatorname{Stab}(e)$-invariant and signature reversal by $h_{e}$.
Since $\delta^{\prime}_{\textrm{bound}}$ is the boundary of
$\sum_{z\in\mathscr{R}_{2}(\Sigma)}z$, the two elements
$\delta_{\textrm{bound}}$ and $\delta^{\prime}_{\textrm{bound}}$ both lie in
$B_{1}(\Sigma)$. Therefore we may define the $\mathscr{I}$-morphisms
$E_{o}\to\mathscr{C}_{2}(\Sigma)$ by
$\sigma_{\textrm{cell}}:e\to\delta_{\textrm{cell}}$ resp.
$\sigma^{\prime}_{\textrm{cell}}:e\to\delta^{\prime}_{\textrm{cell}}$ and
$\mathscr{I}$-morphisms $E_{o}\to B_{1}(\Sigma)$ by
$\sigma_{\textrm{bound}}:e\to\delta_{\textrm{bound}}$ resp.
$\sigma^{\prime}_{\textrm{bound}}:e\to\delta^{\prime}_{\textrm{bound}}$.

######  Remark 2.3.

We claim that
$\sigma^{\prime}_{\textrm{bound}}\in\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1})$
is an element not coming from the image of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$.
If we assume the contrary that
$\sigma^{\prime}_{\textrm{bound}}\in\operatorname{im}\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$,
the $\operatorname{Stab}(e)$-invariance and $(h_{e}+1)$-invariance of
$\sigma^{\prime}_{\textrm{bound}}(e)$ implies that
$\sigma^{\prime}_{\textrm{bound}}$ must be the image of the following
morphisms

$e\to\frac{1}{2}\sum_{z\in\mathscr{R}_{2}(\Sigma)}(z+\iota z)$

However it is clear that
$\frac{1}{2}\sum_{z\in\mathscr{R}_{2}(\Sigma)}(z+\iota z)$ is not an element
of $\mathscr{C}_{2}(\Sigma)$. This is a contradiction!

We have proved in [6] that $Z_{1}(\Sigma)$ contains two disjoint factors
$Z_{\textrm{trc}}(\Sigma)$ and $Z_{\textrm{edge}}(\Sigma)$. We will deal with
the two factors separately and begin with the $E_{o}$-copy in
$Z_{\textrm{trc}}(\Sigma)$. Recall that the 2-cell $e$ is a 10-gon which is
modified from $\tilde{e}$, a regular pentagon on $\tilde{\Sigma}$. And
$\tilde{e}$ has five vertices, the set of theses vertices is a
$\operatorname{Stab}{(e)}$-orbit. On the other hand, let
$V_{e}\subset\mathscr{C}_{0}(\Sigma)$ be the set of terminal points of
oriented edges $y\in\mathscr{C}_{1}(\tilde{\Sigma})$ who are not parallel to
$\tilde{e}$ and have initial points on $\tilde{e}$ . This is a 5-element set
and is a single $\operatorname{Stab}(e)$-orbit in
$\mathscr{C}_{0}(\tilde{\Pi})$. Note that despite the five vertices on
$\tilde{e}$ and five vertices on $\iota\tilde{e}$, there are ten vertices of
$\tilde{\Sigma}$ that do not lie on neither $\tilde{e}$ nor $\iota\tilde{e}$.
They are the points of $V_{e}$ and $\iota V_{e}$. Hence we may take the
5-elements sum $\delta_{\textrm{trc}}:=\sum_{x\in\tilde{e}}\delta_{x}$ and
$\delta^{\prime}_{\textrm{trc}}:=\sum_{x\in V_{e}}\delta_{x}$. They are
$\operatorname{Stab}(e)$-invariant and satisfying that
$h_{e}\delta_{\textrm{trc}}=-\delta_{\textrm{trc}}$
resp.$h_{e}\delta^{\prime}_{\textrm{trc}}=-\delta^{\prime}_{\textrm{trc}}$.
Therefore they define two $\mathscr{I}$-equivariant homomorphism
$\sigma_{\textrm{trc}}$ resp. $\sigma^{\prime}_{\textrm{trc}}$ from $E_{o}$ to
$Z_{\textrm{trc}}(\Sigma)$ by taking
$\sigma_{\textrm{trc}}(e):=\delta_{\textrm{trc}}$ resp.
$\sigma^{\prime}_{\textrm{trc}}(e):=\delta^{\prime}_{\textrm{trc}}$.

For the 2-cell $e$, the following two subsets of
$\mathscr{C}_{1}(\tilde{\Sigma})$ have special interest to us

$\displaystyle E_{e}$ $\displaystyle:=$
$\displaystyle\\{y\in\mathscr{C}_{1}(\tilde{\Sigma})$
$\displaystyle:\textrm{in}(y)\in e,\ \textrm{tm}(y)\notin e$
$\displaystyle\\}$ $\displaystyle E^{\prime}_{e}$ $\displaystyle:=$
$\displaystyle\\{y\in\mathscr{C}_{1}(\tilde{\Sigma})$
$\displaystyle:\textrm{in}(y)\in V_{e},\ \textrm{tm}(y)\in\iota V_{e}$
$\displaystyle\\}$

Note that $E_{e}$ is a 5-elements set which is
$\operatorname{Stab}{(e)}$-invariant and $E^{\prime}_{e}$ is a $10$-elements
set which is $\operatorname{Stab}(e)\times\iota$-invariant. Each of the set
defines a 1-cycle of edge type namely $\delta_{\textrm{edge}}:=\sum_{y\in
E_{e}}\delta_{y}$ and $\delta^{\prime}_{\textrm{edge}}:=\frac{1}{2}\sum_{y\in
E^{\prime}_{e}}\delta_{y}$. From the construction both of the elements are
$\operatorname{Stab}(e)$ stable and signature reversal by $h_{e}$. Hence the
$\mathscr{I}$-orbit of $\delta_{\textrm{edge}}$ resp.
$\delta^{\prime}_{\textrm{edge}}$ has 12 elements and comes into 6 antipodal
pairs. Therefore each of the two elements defines an equivariant homomorphism
$\sigma_{\textrm{edge}}:E_{o}\to Z_{\textrm{edge}}(\Sigma)$ resp.
$\sigma^{\prime}_{\textrm{edge}}:E_{o}\to Z_{\textrm{edge}}(\Sigma)$ with
$\sigma_{\textrm{edge}}(e)=\delta_{\textrm{edge}}$ resp.
$\sigma^{\prime}_{\textrm{edge}}(e)=\delta^{\prime}_{\textrm{edge}}$.

######  Remark 2.4.

Let us consider the following sum in $Z_{\textrm{edge}}(\Sigma)$

$\sum_{y\subset e\hbox{\ and oriented by $e$}}\delta_{y}$

It is $\operatorname{Stab}(e)$-invariant. However from the properties of
$\tilde{\Sigma}$, this element is $h_{e}$-invariant. Moreover the sum over the
$\mathscr{I}$-orbit of this elment is zero. This is because each closed loop
of edge type appears twice in this sum with opposite orientation. Hence this
is a copy of $W_{o}$, the 5-dimensional permutation representation of
$\mathcal{A}_{5}$.

######  Proposition 2.5.

The $\mathds{Z}$-modules
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Sigma))$
and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Sigma))$
are both free $\mathds{Z}$-modules of rank two. Moreover they are both
$\mathcal{O}_{o}$-modules where

  1. (1)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$ is
isomorphic to $\mathcal{O}_{o}$ a free $\mathcal{O}_{o}$-module of rank one
with

(9) $\displaystyle X\sigma_{\textrm{cell}}$ $\displaystyle=$
$\displaystyle\sigma^{\prime}_{\textrm{cell}}$ $\displaystyle
X\sigma^{\prime}_{\textrm{cell}}$ $\displaystyle=$ $\displaystyle
5\sigma_{\textrm{cell}}$

  2. (2)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$ is
isomorphic to $\mathcal{O}$ with

(10) $\displaystyle X\sigma_{\textrm{bound}}$ $\displaystyle=$
$\displaystyle-\sigma_{\textrm{bound}}+2\sigma^{\prime}_{\textrm{bound}}$
$\displaystyle X\sigma^{\prime}_{\textrm{bound}}$ $\displaystyle=$
$\displaystyle 2\sigma_{\textrm{bound}}+\sigma^{\prime}_{\textrm{bound}}$

Hence it contains the image of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$ as
a submodule of index two,

  3. (3)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Sigma))$
is a free $\mathcal{O}_{o}$-module of rank one with

(11) $\displaystyle X\sigma_{\textrm{trc}}$ $\displaystyle=$ $\displaystyle
2\sigma_{\textrm{trc}}+\sigma^{\prime}_{\textrm{trc}}$ $\displaystyle
X\sigma^{\prime}_{\textrm{trc}}$ $\displaystyle=$
$\displaystyle\sigma_{\textrm{trc}}-2\sigma^{\prime}_{\textrm{trc}}$

  4. (4)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Sigma))$
is isomorphic to $\mathcal{O}$ with

(12) $\displaystyle X\sigma_{\textrm{edge}}$ $\displaystyle=$
$\displaystyle\sigma_{\textrm{edge}}+2\sigma^{\prime}_{\textrm{edge}}$
$\displaystyle X\sigma^{\prime}_{\textrm{edge}}$ $\displaystyle=$
$\displaystyle 2\sigma_{\textrm{edge}}-\sigma^{\prime}_{\textrm{edge}}$

###### Proof.

Since $E_{o}$ is principal $\mathds{Z}\mathscr{I}$-module and
$\mathscr{C}_{2}(\Sigma)$, $B_{1}(\Sigma)$, $Z_{\textrm{edge}}(\Sigma)$ and
$Z_{\textrm{trc}}(\Sigma)$ are free $\mathds{Z}$-module,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Sigma))$
and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Sigma))$
are both free $\mathds{Z}$-modules. The rank are clear from the
$\mathds{Q}$-dimension of their $\mathds{Q}$-extension.

(Claim 1 and Claim 2): It is clear from the computation that the Equations (9)
and (10) hold. Hence
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$ is
isomorphic to the module $\mathcal{O}_{o}[\sigma_{\textrm{cell}}]$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$ is
isomorphic to $\mathcal{O}$ and the image of the module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Sigma))$ is
contained in $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$
as a submodule of index at least two. We claim that
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$ is generated
by $\sigma_{\textrm{bound}}$ and $\sigma^{\prime}_{\textrm{bound}}$ as
$\mathds{Z}$-module. Then all the assertions in the proposition implies from
this claim. Let us assume the contrary that there exist a map
$v\in\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$, such
that $v$ is not generated by $\sigma_{\textrm{bound}}$ and
$\sigma^{\prime}_{\textrm{bound}}$ over $\mathds{Z}$. However we can find
rational numbers $a$ and $a^{\prime}$ with at least one of them is not an
integer, such that
$v=a\sigma_{\textrm{bound}}+a^{\prime}\sigma^{\prime}_{\textrm{bound}}$, since
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$ is rank two
and $\sigma_{\textrm{bound}}$ and $\sigma^{\prime}_{\textrm{bound}}$ are not
linearly equivalent. Then by counting the 1-cells of edge type on $\Sigma$, we
find that both $a$ and $a^{\prime}$ are integers, which is a contradiction.

(Claim 3 and Claim 4): The direct computation shows that the Equations (12)
and (11) hold. Then from the construction and Remark 2.4,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Sigma))$
is isomorphic to $\mathcal{O}_{o}[\sigma_{\textrm{trc}}]$ and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Sigma))$
is isomorphic to $\mathcal{O}$. ∎

The following Lemma gave the intersection number between the class defined
above and the vanishing cycles of the two degenerations. Without loss of
generality, Let $\mathscr{R}_{0}(\tilde{\Sigma})$ be a systems of
representatives of $\iota$-symmetry on $\mathscr{C}_{0}(\tilde{\Sigma})$
consists of the vertices of $e$ and the elements of $V_{e}$. And let
$\mathscr{R}_{1}(\tilde{\Sigma})$ be a system of representatives of
$\iota\times(-1)$-action on $\mathscr{C}_{1}(\tilde{\Sigma})$ such that each
element $y$ has initial point in $\mathscr{R}_{0}(\tilde{\Sigma})$. Therefore
the quantity of $\mathscr{R}_{0}(\tilde{\Sigma})$ is 10 and
$\mathscr{R}_{1}(\tilde{\Sigma})$ is 15.

######  Lemma 2.6.

Let $e$, $\Delta_{\textrm{edge}}$, $\Delta_{\textrm{trc}}$,
$\sigma_{\textrm{trc}}$, $\sigma^{\prime}_{\textrm{trc}}$,
$\sigma_{\textrm{edge}}$ and $\sigma^{\prime}_{\textrm{edge}}$ be as defined
above. Then the class $[\sigma_{\textrm{edge}}(e)]$ and
$[\sigma^{\prime}_{\textrm{edge}}(e)]$ has zero intersection number with the
elements of $\Delta_{\textrm{edge}}$ resp. $[\sigma_{\textrm{trc}}(e)]$ and
$[\sigma^{\prime}_{\textrm{trc}}(e)]$ has zero intersection number with the
elements of $\Delta_{\textrm{trc}}$, whereas for
$x\in\mathscr{R}_{0}(\tilde{\Sigma})$ resp.
$y\in\mathscr{R}_{1}(\tilde{\Sigma})$,

$\displaystyle{\langle}[\sigma_{\textrm{edge}}(e)],[\delta_{x}]{\rangle}=\begin{cases}1&\text{if\
}x\in e,\\\ -1,&\text{if\ }x\in V_{e}.\\\ \end{cases}$
$\displaystyle{\langle}[\sigma^{\prime}_{\textrm{edge}}(e)],[\delta_{x}]{\rangle}=\begin{cases}0&\text{if
}x\in e,\\\ 2,&\text{if }x\in V_{e}.\end{cases}$
$\displaystyle{\langle}[\sigma_{\textrm{trc}}(e)],[\delta_{y}]{\rangle}=\begin{cases}-1&\text{if
}\textrm{in}(y)\in e\hbox{ and }\textrm{tm}(y)\in V_{e},\\\
0,&\text{otherwise.}\\\ \end{cases}$
$\displaystyle{\langle}[\sigma^{\prime}_{\textrm{trc}}(e)],[\delta_{y}]{\rangle}=\begin{cases}1&\text{if
}\textrm{in}(y)\in e\hbox{ and }\textrm{tm}(y)\in V_{e},\\\ -2&\text{if
}\textrm{in}(y)\in V_{e}\hbox{ and }\textrm{tm}(y)\in\iota V_{e},\\\
0,&\text{otherwise.}\\\ \end{cases}$

###### Proof.

This is clear from the definitions (see also Figure 1). ∎

######  Lemma 2.7.

The element
$(\delta^{\prime}_{\textrm{bound}}+\delta^{\prime}_{\textrm{edge}}+\delta_{\textrm{trc}}+\delta^{\prime}_{\textrm{trc}})$
is divisible by 2 in $Z_{1}(\Sigma)$. In particular the class
$[\delta^{\prime}_{\textrm{edge}}+\delta_{\textrm{trc}}+\delta^{\prime}_{\textrm{trc}}]$
is a boundary in $H_{1}(\Sigma,\mathds{Z}/2)$.

###### Proof.

Let us $T_{e}$ to be the set of oriented 1-cells of truncation type on
$\Sigma$ who are the intersection of $z\in\mathscr{R}_{2}(\Sigma)$ and
$\delta_{x}$ with $z\neq e$ and $x\in\iota V_{e}$ and the orientation inherits
from $z$. It is clear to check that
$(\delta^{\prime}_{\textrm{bound}}+\delta^{\prime}_{\textrm{edge}}+\delta_{\textrm{trc}}+\delta^{\prime}_{\textrm{trc}})$
is a closed 1-cycle with even coefficients i.e. $2(\sum_{\textrm{in}(y)\in
e,\textrm{tm}(y)\in V_{e}}y+\sum_{y\in T_{e}}y)$. ∎

We are going to prove the following Proposition which describes the
$\mathds{Z}\mathscr{I}$-module structure of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))$.

######  Proposition 2.8.

Let us take the morphism $\sigma$ to be
$\sigma:=\frac{1}{2}(\sigma^{\prime}_{\textrm{bound}}+\sigma^{\prime}_{\textrm{edge}}+\sigma_{\textrm{trc}}+\sigma^{\prime}_{\textrm{trc}})$.
In the exact sequence (8) the cokernal of the map

(13)
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))/\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))$

is the free abelian group generated by
$\sigma_{\textrm{trc}}$,$\sigma^{\prime}_{\textrm{trc}}$,
$\sigma_{\textrm{edge}}$ and $\sigma$. The cokernel of $p_{*}$ is trivial.

###### Proof.

We claim that the $\mathds{Z}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))$ is free
generated by $\sigma_{\textrm{bound}}$, $\sigma^{\prime}_{\textrm{bound}}$,
$\sigma_{\textrm{trc}}$, $\sigma^{\prime}_{\textrm{trc}}$,
$\sigma_{\textrm{edge}}$ and $\sigma$. It is clear that the
$\mathds{Q}$-dimension of
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},\mathds{Q}\otimes
Z_{1}(\Sigma))$ is 6 and the 6 morphisms are not linearly equivalent. Hence if
the claim didn’t holds, there exist rational numbers
$a_{\textrm{bound}},a^{\prime}_{\textrm{bound}},\cdots,a_{\textrm{edge}}$ and
$a$ with at least one of them is not an integer such that the following there
linear combinations $a_{\textrm{bound}}\sigma_{\textrm{bound}}+\cdots+a\sigma$
lie in $Z_{1}(\Sigma)$ The integrality of the coefficients of 1-cells of $\Pi$
implies that
$a_{\textrm{bound}},a^{\prime}_{\textrm{bound}},\cdots,a_{\textrm{edge}}$ and
$a$ are both integers. This is a contradiction!

It is clear that from the Exact Sequence (8), the map

(14)
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))/\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))\to\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma,\mathds{Z}))$

is injective. The image of $\sigma_{\textrm{bound}}$ and
$\sigma^{\prime}_{\textrm{bound}}$ vanishes. Hence we only need to prove that
the image of $\sigma_{\textrm{trc}}$, $\sigma^{\prime}_{\textrm{trc}}$,
$\sigma_{\textrm{edge}}$ and $\sigma$ generates the $\mathds{Z}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma,\mathds{Z}))$.
We could check that the dimension of
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{o}\otimes\mathds{Q},H_{1}(\Sigma,\mathds{Q}))$
is four. And the image of $\sigma_{\textrm{trc}}$,
$\sigma^{\prime}_{\textrm{trc}}$, $\sigma_{\textrm{edge}}$ and $\sigma$ are
not linearly equivalent.Therefore we only need to check the conditions in
Lemma 1.10.

Now let $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\alpha_{4}$ be distinct
integers such that $\gcd(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=1$. The
morphism $s$ is taking as
$s:=\alpha_{1}\sigma_{\textrm{trc}}+\cdots+\alpha_{4}\sigma$. Let us assume
$x_{1}$ and $x_{2}$ be elements of $\mathscr{R}_{0}(\tilde{\Sigma})$ such that
$x_{1}\in e$ and $x_{2}\in V_{e}$, $y_{1}$ and $y_{2}$ be the elements in
$\mathscr{R}_{1}(\tilde{\Sigma})$ such that $\textrm{in}(y_{1})\in e$,
$\textrm{tm}(y_{1})\in V_{e}$ and $\textrm{in}(y_{1})\in V_{e}$,
$\textrm{tm}(y_{1})\in\iota V_{e}$. Recall that we have discussed in Remark
3.2 of [6] that there exist a system
$\\{\gamma_{x}\\}_{x\in\mathscr{R}_{0}(\tilde{\Sigma})}$ of simple closed
loops which is not necessarily $\mathscr{I}$-invariant such that
$\\{[\gamma_{x}],[\delta_{x}]\\}_{x\in\mathscr{C}_{0}(\tilde{\Sigma})}$ become
a basis for $H_{1}(\Sigma)$. And we have
${\langle}[\gamma_{x}],[\delta_{x}]{\rangle}$ equals 1 if and only if
$x=x^{\prime}$, otherwise it is always 0. Hence by the Lemma 2.6, we have the
following intersection numbers

$\displaystyle{\langle}[s(e)],[\delta_{x_{1}}]{\rangle}$ $\displaystyle=$
$\displaystyle\alpha_{3}$
$\displaystyle{\langle}[s(e)],[\delta_{x_{2}}]{\rangle}$ $\displaystyle=$
$\displaystyle-\alpha_{3}+\alpha_{4}$
$\displaystyle{\langle}[s(e)],[\delta_{y_{1}}]{\rangle}$ $\displaystyle=$
$\displaystyle\alpha_{2}-\alpha_{1}$
$\displaystyle{\langle}[s(e)],[\delta_{y_{2}}]{\rangle}$ $\displaystyle=$
$\displaystyle\alpha_{4}-2\alpha_{2}$
$\displaystyle{\langle}[s(e)],[\gamma_{x_{1}}]{\rangle}$ $\displaystyle=$
$\displaystyle\alpha_{1}+N(\alpha_{3},\alpha_{4})$

Here $N(\alpha_{3},\alpha_{4})$ is a integral linear combination of
$\alpha_{3}$ and $\alpha_{4}$. We could check that the five numbers
$\alpha_{3}$, $-\alpha_{3}+\alpha_{4}$, $\alpha_{2}-\alpha_{1}$,
$\alpha_{4}-2\alpha_{2}$ and $\alpha_{1}+N(\alpha_{3},\alpha_{4})$ are
coprime. Hence there exist coprime numbers $\\{\beta_{i}\\}_{i=1}^{5}$ such
that
$\beta_{1}\alpha_{3}+\beta_{2}(-\alpha_{3}+\alpha_{4})+\beta_{3}(\alpha_{2}-\alpha_{1})+\beta_{4}(\alpha_{4}-2\alpha_{2})+\beta_{5}(\alpha_{1}+N(\alpha_{3},\alpha_{4}))=1$.
Then let us take $y(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ to be the
combination
$\beta_{1}[\delta_{x_{1}}]+\beta_{2}[\delta_{x_{2}}]+\beta_{3}[\delta_{y_{1}}]+\beta_{4}[\delta_{y_{2}}]+\beta_{5}[\gamma_{x_{1}}]$.
It is clear that
${\langle}[s(e)],[y(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})]{\rangle}=1$.
Hence the Lemma 1.10 above implies the second assertion. This finishes the
proof. ∎

######  Corollary 2.9.

Let $p$ is the natural map $Z_{1}(\Sigma)\to H_{1}(\Sigma)$ as above, the
$\mathscr{I}$-equivariant morphism $U_{\Sigma}$ and $V_{\Sigma}$ in
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))$ be defined
as $U_{\Sigma}:=p\circ\sigma_{\textrm{trc}}$ and $V_{\Sigma}:=p\circ\sigma$.
The $\mathcal{O}_{o}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))$ is a free
$\mathcal{O}_{o}$-module of rank two with generators $U_{\Sigma}$ and
$V_{\Sigma}$.

###### Proof.

We have proved in Proposition 2.8, the $\mathds{Z}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))$ is freely
generated by the image of $\sigma_{\textrm{trc}}$,
$\sigma^{\prime}_{\textrm{trc}}$, $\sigma^{\prime}_{\textrm{edge}}$ and
$\sigma$. Since the images of $\sigma_{\textrm{bound}}$ and
$\sigma^{\prime}_{\textrm{bound}}$ vanishes in $H_{1}(\Sigma)$, we have
$V_{\Sigma}=\frac{1}{2}(p\circ\sigma^{\prime}_{\textrm{edge}}+p\circ\sigma_{\textrm{trc}}+p\circ\sigma^{\prime}_{\textrm{trc}})$.
Combined with the Equations (11) and (12), we have the following

(15) $\displaystyle p\circ\sigma^{\prime}_{\textrm{trc}}$ $\displaystyle=$
$\displaystyle(X-2)U_{\Sigma}$ $\displaystyle p\circ\sigma_{\textrm{edge}}$
$\displaystyle=$ $\displaystyle-2U_{\Sigma}+(X+1)V_{\Sigma}$ $\displaystyle
p\circ\sigma^{\prime}_{\textrm{edge}}$ $\displaystyle=$
$\displaystyle(1-X)U_{\Sigma}+2V_{\Sigma}$

Moreover it is clear to check that $U_{\Sigma}$ and $V_{\Sigma}$ are not
linearly equivalent over $\mathcal{O}_{o}$. These facts imply the Corollary. ∎

##  3\. The Geometric Model from the Bring’s Curve

In this section we will introduce a geometric model for the smooth fiber
$C_{t}$ where $t\in\mathscr{B}^{\circ}$ constructed form the geometric model
of the Bring’s Curve. We will describe two stable degeneration given by this
model and give explicit descriptions of the vanishing cycle of the
degeneration. At beginning we will introduce some properties of the Bring’s
curve that is required below. For more detailed introduction to the Bring’s
curve, we refer to the survey paper [3] by H. Braden and L. Disney-Hogg.

###  3.1. Preliminaries on the Bring’s Curve

Let us begin with a regular icosahedron in $I_{\mathds{R}}$ centered at the
origin where $I_{\mathds{R}}$ is a fixed Euclidean 3-space with
$\mathcal{A}_{5}$-symmetry as the section 1.2. It has 12 vertices, 30 edges
and 20 faces. Moreover the antipodal map $\iota$ is well-defined as above.
Then a great dodecahedron $\tilde{\Pi}$ has the same edges and vertices as the
icosahedron above. However its faces are replaced by inscribed planar regular
pentagons that connects 5 coplanar vertices. Hence the number of faces is 12.
Every face $z$ has a unique parallel face which is by construction $\iota z$.
Note that if $z$ and $z^{\prime}$ are two different faces that are not
parallel, they will intersect on the edges or the vertices and no where else.
If we denote the $i$-th cells of $\tilde{\Pi}$ by
$\mathscr{C}_{i}(\tilde{\Pi})$ with $i=0,1,2$ similar as the last section, the
numbers of each set is $12$, $30$ and $12$. Hence the Euler formula gives

$\sharp\mathscr{C}_{0}(\tilde{\Pi})-\sharp\mathscr{C}_{1}(\tilde{\Pi})+\sharp\mathscr{C}_{2}(\tilde{\Pi})=-6$

which is the Euler characteristic of a genus 4 surface. It is actually a
complex algebraic curve of genus 4 with at least $\mathcal{A}_{5}$-symmetry,
for the flat structure on each face can glue together making it a locally flat
surface with the vertices as the singularities. Also remember that it was
proved in [8] that Bring’s curve is the only non-hyperelliptic genus 4 curve
with $\mathcal{A}_{5}$-symmetry. In other words $\tilde{\Pi}$ with the complex
structure above is isomorphic to the Bring’s curve.

######  Remark 3.1.

Let us consider the map projection away from the origin to the circumscribed
icosahedron. We could endow a flat structure on each face of the icosahedron
and make it a Riemann surface isomorphic to $\mathds{P}^{1}$. Then this map
will become a ramified triple cover branched at the vertices of the
icosahedron. The ramification index is 2 at the vertices of $\tilde{\Pi}$ and
1 at the center of the faces. We can check that these data make the Riemann-
Hurwitz formula holds. Hence this map realized the Bring’s curve as the
branched triple cover of $\mathds{P}^{1}$ which is not
$\mathscr{I}$-equivariant. This gives another way of making $\tilde{\Pi}$ a
complex algebraic curve by pulling back the complex structure on
$\mathds{P}^{1}$ through this triple cover.

###  3.2. The Geometric Model $\Pi$ and its Degenerations

Figure 2. Removing in an $\mathscr{I}$-equivariant manner a small regular
penta-pyramid at each vertex of $\tilde{\Pi}$

Next we will modify the model $\tilde{\Pi}$ of the Bring’s curve to get a
model $\Pi$ of genus 10 curve and we will introduce two degenerations coming
from this model. Let $\hat{\Pi}$ be obtained from $\tilde{\Pi}$ by removing in
an $\mathscr{I}$-equivariant manner a small regular penta-pyramid at each
vertex $x\in\mathscr{C}_{0}(\tilde{\Pi})$. Then the boundary of $\hat{\Pi}$
consists of 12 disjoint closed loops, each of them is a regular pentagram
centered at $x$. Note that this operation is the same as following: for each
face $z\in\mathscr{C}_{2}(\tilde{\Pi})$, we will remove a small isosceles
triangles in $\operatorname{Stab}{(z)}$-manner at each vertex of $z$. Hence
each face of $\hat{\Pi}$ is a decagon. We will identify the opposite points on
the boundary of $\hat{\Pi}$ and thus obtained a complex $\Pi$ that is a closed
oriented surface of genus 10 endowed with an $\mathscr{I}$-symmetry. It is
clear that $\Pi$ has a structure of cellular decomposition: the set of 2-cells
of $\Pi$ consists of 12 decagons and is indexed by
$\mathscr{C}_{2}(\tilde{\Pi})$ i.e. the 2-cells of the great dodecahedron. The
set of 0-cells of $\Pi$ are represented by the antipodal pairs of vertices of
$\hat{\Pi}$ and are naturally indexed by the oriented 1-cells of
$\tilde{\Pi}$. The set of 1-cells of $\Pi$ consists two disjoint subset: those
lie on the edge of $\tilde{\Pi}$ (hence called of edge type and denoted as
$\mathscr{C}_{\textrm{edge}}({\Pi})$) and those come from the boundary of
$\hat{\Pi}$ and they form 6 antipodal pairs of pentagrams (hence called of
truncation type and denoted as $\mathscr{C}_{\textrm{trc}}({\Pi})$).

######  Proposition 3.2.

The action of $\mathscr{I}$ on the cells of $\Pi$ is as following:

  1. (1)

the action of $\mathscr{I}$ on the set $\mathscr{C}_{0}({\Pi})$ of 0-cells of
${\Pi}$ is transitive, each 0-cell has a stabilizer cyclic of order five,

  2. (2)

the set $\mathscr{C}_{1}({\Pi})$ of oriented 1-cells of ${\Pi}$ consists of
two regular orbits $\mathscr{C}_{\textrm{edge}}({\Pi})$ and
$\mathscr{C}_{\textrm{trc}}({\Pi})$,

  3. (3)

the action of $\mathscr{I}$ on the set $\mathscr{C}^{+}_{2}(\Pi)$ of
canonically oriented 2-cells is transitive, the stabilizer of each such cell
is cyclic of order five.

An oriented 1-cell of $\Pi$ is part of a unique loop consisting of oriented
cells of the same type. We will analysis this in detail. A loop of truncation
type consists of 5 oriented 1-cells of that type and each oriented 1-cell of
truncation type appears in a unique such loop. They are indexed by the set
$\mathscr{C}_{0}(\tilde{\Pi})$: every vertex $x$ of $\tilde{\Pi}$ lies in the
center of a solid regular pentagram whose interior is removed to form
$\hat{\Pi}$. The boundary of this pentagram together with its counterclockwise
orientation is a sum $\theta_{x}$ of five oriented 1-cells of truncation type.
Moreover we have that $\theta_{x}=-\theta_{\iota x}$. We will call the closed
loop constructed above loops of the truncation type. There are 12 such closed
loops and the $\mathscr{I}$-action permutes them transitively. The stabilizer
of each closed loop of truncation type is cyclic of order 5. We will denote
the set of such twelve 1-cycles by $\Theta_{\textrm{trc}}$.

A loop of edge type is the sum of 1-cells of that type and its image under
$-\iota$. They are bijectively indexed by the set
$\mathscr{C}_{1}(\tilde{\Pi})$and we will denote the oriented loop of edge
type defined by $y\in\mathscr{C}_{1}(\tilde{\Pi})$ by $\theta_{y}$. It is
clear that we have $\theta_{\iota y}=-\theta_{y}$ and
$\theta_{-y}=-\theta_{y}$. The set of such 1-cycles, which we will denote it
by $\Theta_{\textrm{edge}}$, is an $\mathscr{I}$-orbit consists of 30
elements. In other words, the $\mathscr{I}$-stabilizer of each 1-cycle of edge
type is cyclic of order two. The following Lemma gives the intersection number
between the two kinds of 1-cycles and is straightforward to check.

######  Lemma 3.3.

The intersection numbers of these $1$-cycles are as follows: any two loops of
the same type have intersection number zero and if
$x\in\mathscr{C}_{0}(\tilde{\Pi})$ and $y\in\mathscr{C}_{1}(\tilde{\Pi})$,
then ${\langle}\theta_{x},\theta_{y}{\rangle}=0$ unless $x$ lies on $y$ or on
$\iota y$, in which case ${\langle}\theta_{x},\theta_{y}{\rangle}\in\\{\pm
1\\}$ with the plus sign appearing if and only if $x$ is the end point of $y$.

Let us describe two kinds of degenerations realized by the model $\Pi$, they
are both genus 10 nodal curves with $\mathscr{I}$-symmetry. They will have the
properties that one has $\Theta_{\textrm{edge}}$ as vanishing cycles and the
other one has $\Theta_{\textrm{trc}}$ as vanishing cycles.

First let us note that there exist a one-parameter family of piecewise
Euclidean structure on $\hat{\Pi}$. It is given as following: let us assume
the length of the 1-cells of the edge type is $\tau>0$ and the length of the
1-cells of the truncation type is $1-\tau$. This determines $\hat{\Pi}$ as a
metric space. It is clear that this metric is piecewise Euclidean and
invariant under both $\mathscr{I}$ and $\iota$-symmetry. It defines a
conformal structure $I_{\tau}$ first on $\Pi$ except the vertices and then it
can be extended across these vertices. The given orientation on $\Pi$ makes
this conformal structure an $\mathscr{I}$-invariant complex structure.

If $\tau$ tends to 1, we got a complex structure on the singular surface
$\Pi_{\textrm{trc}}$ that is obtained from $\Pi$ by contracting each 1-cycle
of truncation type into a point. It is clear that this singular surface can
also be obtained by identifying the 6 pairs in $\mathscr{C}_{0}(\tilde{\Pi})$.
The complex structure $I_{1}$ makes it a singular curve isomorphic to
$C_{-1}$. Similarly, if $\tau$ tends to $0$, the length of 1-cycles of edge
type will tend to 0. We got a complex structure on the singular surface
$\Pi_{\textrm{edge}}$ that is obtained from $\Pi$ by contracting each 1-cycle
of edge type into a point. The complex structure $I_{0}$ makes it a singular
curve isomorphic to $C_{\infty}$. Summarize above, we got the following
Proposition.

######  Proposition 3.4.

The Riemann surface $(\Pi,I_{\tau})$ is the set of complex points of a complex
real algebraic curve. It has genus 10 and comes with a faithful
$\mathscr{I}$-action, hence is isomorphic to a member of the Winger pencil. We
thus have defined a continuous map $\gamma_{\Pi}:[0,1]\to\mathscr{B}$ which
transverse $(0,1)$ to $\mathscr{B}^{\circ}$ and $\tau=1$ to
$t=-1\in\mathscr{B}$ resp. $\tau=0$ to $t=\infty\in\mathscr{B}$ such that the
pull-back of the Winger pencil yields the family constructed above. The
degenerations of $(\Pi,I_{\tau})$ into $\Pi_{\textrm{trc}}$ resp.
$\Pi_{\textrm{edge}}$ have $\Theta_{\textrm{trc}}$ resp.
$\Theta_{\textrm{edge}}$ as its set of vanishing cycles.

######  Remark 3.5.

The polygon $\Pi_{\textrm{edge}}$ has interesting properties itself. If we
treat each $\theta_{x}$ as a solid regular pentagon not as a closed loop, the
resulting polygon is named as a dodecadodecahedron. This polygon has 24 faces,
12 of them are regular pentagons and 12 of them are regular pentagrams, 60
edges and 30 vertices, giving the Euler characteristic $\chi=-6$. It was
showed in [10] that this is also an Euclidean realization for the Bring’s
curve.

###  3.3. Cellular Homology of $\Pi$

The geometric model $\Pi$ admits a cellular structure which enables us to
compute its homology as the homology of the combinatorial chain complex

(16)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{C}_{2}(\Pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{2}}$$\textstyle{\mathscr{C}_{1}(\Pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{\mathscr{C}_{0}(\Pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that the middle term $\mathscr{C}_{1}(\Pi)$ admits a direct sum
decomposition namely
$\mathscr{C}_{1}(\Pi)=\mathscr{C}_{\textrm{edge}}\oplus\mathscr{C}_{\textrm{trc}}$.
Similar as above we will denote the set of $i$-cycles as
$Z_{i}(\Pi):=\ker(\partial_{i})$ and $i$-boundaries as
$B_{i}(\Pi):=\operatorname{im}(\partial_{i+1})$. Let us apply the functor
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}(E_{o},-)}$ to the Exact Sequence
(5) and (6) with $\Sigma$ replaced by $\Pi$. For the first one we have the
long exact sequence

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It is from the construction of $\Pi$ that $\ker\partial_{2}$ is isomorphic to
trivial representation of $\mathscr{I}$. Hence the first term vanishes. We
could see from the character computation that
$\mathscr{C}_{2}(\Pi)\otimes\mathds{C}$ is isomorphic to
$\operatorname{Id}\oplus W\oplus I\oplus I^{\prime}$. Hence the second and the
third terms are nontrivial. We will see below that one of them is isomorphic
to $\mathcal{O}_{o}$ and another is isomorphic to $\mathcal{O}$ as
$\mathcal{O}_{o}$-modules and the quotient of them is finite but nonzero. For
the second we have the following

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3.0pt}}}}}}\ignorespaces{\hbox{\kern 203.62973pt\raise 0.0pt\hbox{\hbox{\kern
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We will introduce four elements of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Pi))$ with two of
them have image in the $\mathds{Z}$-module spanned by $\Theta_{\textrm{edge}}$
(which we will denote it as $Z_{\textrm{edge}}(\Pi)$) denoted by
$\pi_{\textrm{edge}}$ and the other two $\pi_{\textrm{trc}}$ and
$\pi^{\prime}_{\textrm{trc}}$ in $\mathds{Z}$-module spanned by
$\Theta_{\textrm{trc}}$ (which we will denote it as $Z_{\textrm{trc}}(\Pi)$).

Let us first observe that the $\mathds{Z}\mathscr{I}$-module
$\mathscr{C}_{0}(\tilde{\Pi})/(\iota+1)$ is isomorphic to $E_{o}$ where this
isomorphism is unique up to a sign and there exist a system of representatives
$\mathscr{R}_{0}(\tilde{\Pi})$ of $\iota$-action on
$\mathscr{C}_{0}(\tilde{\Pi})$ such that this isomorphism will identify this
system with the basis $\\{e,e_{0},\cdots,e_{4}\\}$ of $E_{o}$. To see this
observation recall that $\mathscr{C}_{0}(\tilde{\Pi})$ consists of 12 vertices
where the $\mathscr{I}$-symmetry permutes them making the set one
$\mathscr{I}$-orbit. The map $\iota$ commutes with this
$\mathscr{I}$-symmetry. Moreover the uniqueness comes from the Lemma 1.8. We
will take $e$ represent not only the element in the basis of $E_{o}$ but also
a vertex in $\mathscr{R}_{0}(\tilde{\Pi})$. Finally for each
$x\in\mathscr{R}_{0}(\tilde{\Pi})$, let us fix an element
$h_{x}\in\mathscr{I}$ cyclic of order two, which is not necessarily unique,
such that $h_{x}x=-x$.

Let us begin with the modules
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Pi))$ and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$. For arbitrary
vertex $x\in\mathscr{R}_{0}(\tilde{\Pi})$, there exist two parallel planer
pentagons such that the vertices of $\tilde{\Pi}$ despite $x$ and $\iota x$
lie on one of them. We will denote the two planer pentagon together with its
counterclockwise orientation by $z_{x}$ resp. $z_{\iota x}$. Clearly each of
them associates to a oriented 2-cell in $\mathscr{C}_{2}(\Pi)$ in a natural
way, we will denote the two 2-cells in the same symbols. It is clear to check
that $z_{\iota x}=z_{h_{x}x}=h_{x}z_{x}=-\iota z_{x}$.

From this observation there are two elements in $\mathscr{C}_{2}(\Pi)$ that
draw our attention namley $\theta_{\textrm{cell}}:=z_{e}+\iota z_{e}$ and
$\theta^{\prime}_{\textrm{cell}}:=\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})\backslash\\{e\\}}(z_{x}+\iota
z_{x})$. They are $\operatorname{Stab}(e)$-invariant and satisfies the
property that $h_{e}\theta_{\textrm{cell}}=-\theta_{\textrm{cell}}$
resp.$h_{e}\theta^{\prime}_{\textrm{cell}}=-\theta^{\prime}_{\textrm{cell}}$.
The two elements give two $\mathscr{I}$-equivariant morphisms
$\pi_{\textrm{cell}}$ resp. $\pi^{\prime}_{\textrm{cell}}$ of
$E_{o}\to\mathscr{C}_{2}(\Pi)$, namely $e\to\theta_{\textrm{cell}}$ resp.
$e\to\theta^{\prime}_{\textrm{cell}}$.

The boundaries of $\theta_{\textrm{cell}}$ resp.
$\theta^{\prime}_{\textrm{cell}}$ satisfies the relation that
$\partial_{2}\theta^{\prime}_{\textrm{cell}}=\partial_{2}\theta_{\textrm{cell}}+2\partial_{2}(-\iota
z_{e}+\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})\backslash\\{e\\}}z_{x})$. Let us
take the two elements $\theta_{\textrm{bound}}$ resp.
$\theta^{\prime}_{\textrm{bound}}$ as
$\theta_{\textrm{bound}}:=\partial_{2}\theta_{\textrm{cell}}$ and
$\theta^{\prime}_{\textrm{bound}}:=\partial_{2}(-\iota
z_{e}+\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})\backslash\\{e\\}}z_{x})$. Since
the element $\theta^{\prime}_{\textrm{bound}}$ is the boundary of
$\frac{1}{2}(\theta^{\prime}_{\textrm{cell}}-\theta_{\textrm{cell}})$, it has
to be $\operatorname{Stab}(e)$-equivariant and satisfy the relation
$h_{e}\theta^{\prime}_{\textrm{bound}}=-\theta^{\prime}_{\textrm{bound}}$. It
is the same for $\theta_{\textrm{bound}}$. Hence we have two
$\mathscr{I}$-equivariant morphisms in
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$ i.e.
$\pi_{\textrm{bound}}$ resp. $\pi^{\prime}_{\textrm{bound}}$ given as
$e\to\theta_{\textrm{bound}}$ resp. $e\to\theta^{\prime}_{\textrm{bound}}$.

######  Remark 3.6.

Observe that the element $(-\iota
z_{e}+\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})\backslash\\{e\\}}z_{x})\in\mathscr{C}_{2}(\Pi)$
is $\operatorname{Stab}(e)$-invariant. However instead of inverting its
signature, the element $h_{e}$ will fix it. These facts implies that
$\pi^{\prime}_{\textrm{bound}}$ is an element of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$ but it does not
lie in the image of $\partial_{2}$.

Next let us consider the $\mathds{Z}\mathscr{I}$-modules
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Pi))$ and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Pi))$.
For the $Z_{\textrm{trc}}(\Pi)$-part, let us take
$\theta_{\textrm{trc}}=\theta_{e}$ and
$\theta^{\prime}_{\textrm{trc}}:=\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})\backslash\\{e\\}}\theta_{x}$.
It is clear that they are $\operatorname{Stab}(e)$-invariant and signature
reversal by $h_{e}$. Therefore we may define the map $E_{o}\to
Z_{\textrm{trc}}$ in an $\mathscr{I}$-equivariant manner $\pi_{\textrm{trc}}$
to be the morphism as $e\to\theta_{e}$ resp.
$\pi^{\prime}_{\textrm{trc}}:e\to\theta^{\prime}_{\textrm{trc}}$.

For the $Z_{\textrm{edge}}(\Pi)$-part, observe that for each vertex
$x\in\mathscr{R}_{0}(\tilde{\Pi})$ there exist five oriented 1-cells
$y\in\mathscr{C}_{1}(\tilde{\Pi})$ such that they have $x$ as common initial
point. Besides of these edges and their $\iota$-dual, the oriented edges that
don’t parallel to $z_{e}$ is the 10-element subset
$\\{y\in\mathscr{C}_{1}(\tilde{\Pi}):\textrm{in}(y)\in z_{e},\textrm{tm}(y)\in
z_{\iota_{e}}\\}$ of $\mathscr{C}_{1}(\tilde{\Pi})$ that admits the symmetry
of $\operatorname{Stab}(e)\times(-\iota)$. This set consists of two
$\operatorname{Stab}(e)$-orbit and $-\iota$ exchanges the two orbits. From
these observations, let us take
$\theta_{\textrm{edge}}:=\sum_{y\in\mathscr{C}_{1}(\tilde{\Pi}),\textrm{in}(y)=e}\theta_{y}$
and
$\theta^{\prime}_{\textrm{edge}}:=\frac{1}{2}\sum_{y\in\mathscr{C}_{1}(\tilde{\Pi}),\textrm{in}(y)\in
z_{e},\textrm{tm}(y)\in z_{\iota e}}\theta_{y}$. Since the 1-cycle of edge
type satisfies the relation that $\theta_{-\iota y}=\theta_{y}$, the element
$\theta^{\prime}_{\textrm{edge}}$ lies in $Z_{\textrm{edge}}(\Pi)$. Moreover
the two elements are both $\operatorname{Stab}{(e)}$-invariant and signature
reversal by $h_{e}\in\mathscr{I}$. Hence we can define the morphism
$\pi_{\textrm{edge}}$ resp. $\pi^{\prime}_{\textrm{edge}}$ to be the
$\mathscr{I}$-equivariant map $e\to\theta_{\textrm{edge}}$ resp.
$e\to\theta^{\prime}_{\textrm{edge}}$.

######  Remark 3.7.

The element $\sum_{y\subset\partial z_{e}}\theta_{y}$, where $y$ is naturally
oriented such that it is the same as the boundary of $z_{e}$, is also
$\operatorname{Stab}(e)$-invariant. However $h_{e}$ will fix this element.
Hence it cannot give a morphism from $E_{o}$ to $Z_{\textrm{edge}}(\Pi)$.

We have the following Propositions.

######  Proposition 3.8.

The $\mathds{Z}$-modules
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Pi))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$,
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Pi))$ and
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Pi))$ are
both free $\mathds{Z}$ of rank two. Moreover they are both
$\mathcal{O}_{o}$-modules where

  1. (1)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Pi))$ is a
free $\mathcal{O}_{o}$-module of rank one with

(19) $\displaystyle X\pi_{\textrm{cell}}$ $\displaystyle=$
$\displaystyle\pi^{\prime}_{\textrm{cell}}$ $\displaystyle
X\pi^{\prime}_{\textrm{cell}}$ $\displaystyle=$ $\displaystyle
5\pi_{\textrm{cell}}$

  2. (2)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$ is isomorphic
to $\mathcal{O}$ as $\mathcal{O}_{o}$-modules with

(20) $\displaystyle X\pi_{\textrm{bound}}$ $\displaystyle=$
$\displaystyle\pi_{\textrm{bound}}+2\pi^{\prime}_{\textrm{bound}}$
$\displaystyle X\pi^{\prime}_{\textrm{bound}}$ $\displaystyle=$ $\displaystyle
2\pi_{\textrm{bound}}-\pi^{\prime}_{\textrm{bound}}$

Moreover it contains the image of
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\mathscr{C}_{2}(\Pi))$ as a
submodule of index two.

  3. (3)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{trc}}(\Pi))$ is a
free $\mathcal{O}_{o}$-module of rank one with

(21) $\displaystyle X\pi_{\textrm{trc}}$
$\displaystyle=\pi^{\prime}_{\textrm{trc}}$ $\displaystyle
X\pi^{\prime}_{\textrm{trc}}$ $\displaystyle=5\pi_{\textrm{trc}}$

  4. (4)

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{\textrm{edge}}(\Pi))$ is
isomorphic to $\mathcal{O}$ as $\mathcal{O}_{o}$-module with

(22) $\displaystyle X\pi_{\textrm{edge}}$ $\displaystyle=$
$\displaystyle-\pi_{\textrm{edge}}+2\pi^{\prime}_{\textrm{edge}}$
$\displaystyle X\pi^{\prime}_{\textrm{edge}}$ $\displaystyle=$ $\displaystyle
2\pi_{\textrm{edge}}+\pi^{\prime}_{\textrm{edge}}$

###### Proof.

The proof for this Proposition is similar to the proof of Proposition 2.5. The
Equations 19, 20, 21 and 22 come from direct computation. And the other claims
come from counting on the coefficients of 1-cells. ∎

######  Lemma 3.9.

The element
$(\theta^{\prime}_{\textrm{bound}}+\theta_{\textrm{trc}}-\theta^{\prime}_{\textrm{trc}}+\theta^{\prime}_{\textrm{edge}})$
is divisible by two in $Z_{1}(\Pi)$. In particular, the class
$[\theta_{\textrm{trc}}-\theta^{\prime}_{\textrm{trc}}+\theta^{\prime}_{\textrm{edge}}]$
is a boundary in $H_{1}(\Pi,\mathds{Z}/2)$.

###### Proof.

This is a direct computation. ∎

######  Proposition 3.10.

Let us take the morphism $\pi$ to be
$\pi:=\frac{1}{2}(\pi^{\prime}_{\textrm{bound}}+\pi_{\textrm{trc}}-\pi^{\prime}_{\textrm{trc}}+\pi^{\prime}_{\textrm{edge}})$.
Then in the Exact Sequence (18), the cokernel of the map $i_{*}$

$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Pi))/\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Pi))$

is free abelian group generated by $\pi_{\textrm{trc}}$,
$\pi^{\prime}_{\textrm{trc}}$,$\pi_{\textrm{edge}}$ and $\pi$. The cokernel of
$p_{*}$ is trivial.

Before we give the proof of the Proposition 3.10, let us given the
intersection numbers of some cycle class.

######  Proposition 3.11.

Let $e$, $\Theta_{\textrm{edge}}$, $\Theta_{\textrm{trc}}$,
$\pi_{\textrm{edge}}$, $\pi^{\prime}_{\textrm{edge}}$, $\pi_{\textrm{trc}}$
and $\pi^{\prime}_{\textrm{trc}}$ be defined as before. Then the classes
$[\pi_{\textrm{edge}}(e)]$ and $[\pi^{\prime}_{\textrm{edge}}(e)]$ resp.
$[\pi_{\textrm{trc}}(e)]$ and $[\pi^{\prime}_{\textrm{trc}}(e)]$ has zero
intersection number with the elements in $\Theta_{\textrm{edge}}$ resp.
$\Theta_{\textrm{trc}}$. Meanwhile for $x\in\mathscr{R}_{0}(\tilde{\Pi})$ and
$y\in\mathscr{C}_{1}(\tilde{\Pi})$ with
$\textrm{in}(y)\in\mathscr{R}_{0}(\tilde{\Pi})$, we have

$\displaystyle{\langle}[\pi_{\textrm{edge}}(e)],[\theta_{x}]{\rangle}=\begin{cases}5&\text{if
}x=e,\\\ -1,&\text{if\ }x\neq e\\\ \end{cases}$
$\displaystyle{\langle}[\pi^{\prime}_{\textrm{edge}}(e)],[\theta_{x}]{\rangle}=\begin{cases}0&\text{if
}x=e,\\\ 2,&\text{if }x\neq e.\end{cases}$
$\displaystyle{\langle}[\pi_{\textrm{trc}}(e)],[\theta_{y}]{\rangle}=\begin{cases}-1&\text{if
}\textrm{in}(y)=e,\\\ 0,&\text{if }\text{otherwise.}\\\ \end{cases}$
$\displaystyle{\langle}[\pi^{\prime}_{\textrm{trc}}(e)],[\theta_{y}]{\rangle}=\begin{cases}1&\text{if
}\textrm{in}(y)=e,\\\ -2&\text{if }\textrm{in}(y)\in z_{e}\text{\ and\
}\textrm{tm}(y)\in z_{\iota e},\\\ 0,&\text{if }\text{otherwise.}\\\
\end{cases}$

###### Proof.

This is a direct compute from the model $\Pi$. ∎

We have seen on the model of $\Sigma$, each $\delta_{x}$ with
$x\in\mathscr{C}_{0}(\tilde{\Sigma})$ admits a ”dual” class such that they
span $H_{1}(\Sigma)$ together. The similar construction can be made for the
model $\Pi$. However they will only span a primitive sublattice of
$H_{1}(\Pi)$.

######  Proposition 3.12.

For each vertex $x\in\mathscr{R}_{0}{(\tilde{\Pi})}$, there exist a 1-cycle
$[\varepsilon_{x}]$ such that
${\langle}[\varepsilon_{x}],\theta_{x^{\prime}}{\rangle}=1$ if and only if
$x^{\prime}=x$ and otherwise it is $0$ for all
$x^{\prime}\in\mathscr{R}_{0}(\tilde{\Pi})$. In particular, if $x\neq e$ we
could require additional conditions for $\varepsilon_{x}$ such that

  1. (1)

${\langle}[\pi_{\textrm{edge}}(e)],[\varepsilon_{x}]{\rangle}=0$ and

  2. (2)

${\langle}[\pi^{\prime}_{\textrm{edge}}(e)],[\varepsilon_{x}]{\rangle}=-1$.

###### Proof.

The proof for the first claim is clear. The construction above made the $\Pi$
a genus 10 Riemann surface and each of the six loops of truncation type
represents a generators of $\pi_{1}(\Sigma)$ which is canonical. Hence their
exist ”dual” class $[\varepsilon_{x}]$ such that
${\langle}[\varepsilon_{x}],\theta_{x^{\prime}}{\rangle}=1$ if and only if
$x^{\prime}=x$.

Figure 3. The intersection graph of 2-cells on $\tilde{\Pi}$. The vertices
represents the 2-cells on $\tilde{\Pi}$ and two vertices are joined by an edge
if the corresponding 2-cells intersect through an edge of $\tilde{\Pi}$.

The proof for the second assertion is a direct construction. Observe that the
intersection graph of 2-cells on $\tilde{\Pi}$ is as Figure 3 where the
vertices in the graph represent the 2-cells of $\tilde{\Pi}$ and two vertices
are connected by an edge if and only if the 2-cells of $\tilde{\Pi}$ they
represented intersect at a 1-cell of $\tilde{\Pi}$. Hence if we starting from
the 2-cell $z_{e}$, we can reach $\iota z_{e}$ by crossing at least three
2-cells. From this observation, let $a\in\mathscr{R}_{0}(\tilde{\Pi})$ be any
fixed vertex which is not $e$. Let us choose a point $p_{a}$ lies on both
$z_{e}$ and $\theta_{a}$. We also choose a small open band $\hat{U}_{x}$ of
$\theta_{\iota x}$ in $\hat{\Pi}$ for each $x\in\mathscr{C}_{0}(\tilde{\Pi})$.
We will let the path $\hat{\varepsilon}_{a}$ start from $p_{a}$ walk through
the ”shortest” path mentioned above while avoiding $\cup_{x\neq\pm a}U_{x}$
and ending in $\iota p_{a}$. Note that if $y$ is a 1-cell of edge type of
$\hat{\Pi}$, $\hat{\varepsilon}_{a}$ intersects with $y$ only if $y$ has
boundary point one on $z_{e}$ and another one on $z_{\iota e}$. The
intersection point can be modified to lie in $U_{\iota a}$ and multiplicity is
one. Hence from construction that the image $\varepsilon_{a}$ of
$\hat{\varepsilon}_{a}$ in $\Pi$ will be a closed loop and the intersection
numbers are as listed. ∎

###### Proof.

(Proof of the Proposition 3.10) The proof for this Proposition is similar to
the proof of Proposition 2.8. We claim that the $\mathds{Z}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Pi))$ is free
generated by $\pi_{\textrm{bound}}$, $\pi^{\prime}_{\textrm{bound}}$,
$\pi_{\textrm{trc}}$, $\pi^{\prime}_{\textrm{trc}}$, $\pi_{\textrm{edge}}$ and
$\pi$. This is showed by counting on the coefficients of 1-cells on $\Pi$.

For the second part, we need to prove that the image of $\pi_{\textrm{edge}}$,
$\pi^{\prime}_{\textrm{edge}}$, $\pi_{\textrm{trc}}$, $\pi$ generates the
module $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Pi))$. This is
done by checking the conditions in Lemma 1.10. ∎

######  Corollary 3.13.

Let $p$ is the natural map $Z_{1}(\Pi)\to H_{1}(\Pi)$ as above, the
$\mathscr{I}$-equivariant morphism $U_{\Pi}$ and $V_{\Pi}$ in
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Pi))$ be defined as
$U_{\Pi}:=p\circ\pi_{\textrm{trc}}$ and $V_{\Pi}:=p\circ\pi$. The
$\mathcal{O}_{o}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Pi))$ is free
$\mathcal{O}_{o}$-module of rank two with generators $U_{\Pi}$ and $V_{\Pi}$.

###### Proof.

We have proved in Proposition 2.8, the $\mathds{Z}$-module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))$ is freely
generated by the image of $\pi_{\textrm{trc}}$, $\pi^{\prime}_{\textrm{trc}}$,
$\pi_{\textrm{edge}}$ and $\pi$. Since the images of $\pi_{\textrm{bound}}$
and $\pi^{\prime}_{\textrm{bound}}$ vanishes in $H_{1}(\Pi)$, we have
$V_{\Pi}=\frac{1}{2}(p\circ\pi^{\prime}_{\textrm{edge}}+p\circ\pi_{\textrm{trc}}-p\circ\pi^{\prime}_{\textrm{trc}})$.
Combined with the Equations (21) and (22), we have the following

(23) $\displaystyle p\circ\pi^{\prime}_{\textrm{trc}}$ $\displaystyle=$
$\displaystyle XU_{\Pi}$ $\displaystyle p\circ\pi_{\textrm{edge}}$
$\displaystyle=$ $\displaystyle-(X-3)U_{\Pi}+(X-1)V_{\Pi}$ $\displaystyle
p\circ\sigma^{\prime}_{\textrm{edge}}$ $\displaystyle=$
$\displaystyle(X-1)U_{\Pi}+2V_{\Pi}$

Moreover it is clear to check that $U_{\Pi}$ and $V_{\Pi}$ are not linearly
equivalent over $\mathcal{O}_{o}$. These facts imply the Corollary. ∎

######  Remark 3.14.

(Other Models of the Bring’s Curve) In the article [7], G. Riera and R.
Rodriguez introduced a hyperbolic model $\tilde{\Pi}^{\textrm{hyp}}$ of the
Bring’s curve. This model also appears with great importance in [4]. It is a
non-euclidean 20-gon lie on the Poincare’s disk with the edges identified as
in the Figure 1. It is known that the polygon’s vertices fall into three
equivalence classes $P_{1}$, $P_{2}$ and $P_{3}$ which is marked in the Figure
1 and the genus of the curve is 4. The 20-gon can be tessellated by 240
triangles (or 120 double triangles) with interior angles $\frac{\pi}{5}$,
$\frac{\pi}{4}$ and $\frac{\pi}{2}$ which is named as a (2,4,5)-triangle.
Hence it is clear to see that the tessellation of the 20-gon $\tilde{\Lambda}$
has 112 vertices which coming with three types.

  1. (1)

The intersection of 4 (2,4,5)-triangles, the total number is 60,

  2. (2)

The intersection of 8 (2,4,5)-triangles, the total number is 30,

  3. (3)

The intersection of 10 (2,4,5)-triangles, the total number is 24.

The $\mathcal{S}_{5}$-symmetry is given by permuting the double triangles.
There are two kinds of regular hyperbolic pentagons on
$\tilde{\Pi}^{\textrm{hyp}}$. Twenty-four of them, which we call
$\frac{\pi}{2}$-pentagon, has inner angle $\frac{\pi}{2}$ which centered at
the points of the third type and their vertices are always the points of the
second type. Another twenty-four of them, which we call
$\frac{2\pi}{5}$-pentagon, has inner angle $\frac{2\pi}{5}$. Both centers and
vertices are the points of the third type and the the midpoints of edges are
of the second type.

The hyperbolic model admits a Euclidean realization namely the great
dodecahedron. The realization map is constructed by mapping 2-cells of
$\tilde{\Pi}$ to the $\frac{2\pi}{5}$-pentagons in an
$\mathscr{I}$-equivariant way. The points of the third type are divided into
two disjoint 12-elements-sets, one is the images of vertices of $\tilde{\Pi}$
and another one is the images of barycenters of the faces. The $\iota$-map is
induced from $\tilde{\Pi}$ in a natural way. We could remove in an
$\mathscr{I}$-equivariant manner a small regular $\frac{\pi}{2}$-pentagon at
each ”vertices of $\tilde{\Pi}$”. And identifying $\iota$ to get a model of
genus 10 $\mathscr{I}$-curves. The advantage for this model is that we could
see clearly the $\mathcal{S}_{5}$-symmetry on the Bring’s curve. Note that the
orientations of $\frac{\pi}{4}$ at the points of second type are only
hyperbolic automorphisms and they are not euclidean.

Figure 4. The edges of the 20-gon are identified as below. The points of
second type are marked with red point and the points of third type are marked
with green plus. The Figure is modified from the Figure 2 in [4]

##  4\. Local Monodromy on the $E$-part

Let us recall some basic ideas that we used in [6] and which is also useful in
here. Recall that on $\Sigma$ resp. $\Pi$ we defined a family of complex
structures $J_{\tau}$ resp.$I_{\tau}$ with $\tau\in(0,1)$ resp. $\tau\in(0,1)$
which defined a path $\gamma_{\Sigma}:(0,1)\to\mathscr{B}^{\circ}$ in the base
of the Winger pencil traversing the positive interval $(\infty,\frac{27}{5})$
resp. a path $\gamma_{\Pi}:(0,1)\to\mathscr{B}^{\circ}$ in the base of the
Winger pencil traversing the positive interval $(-1,\infty)$. This path had a
continuous extension to $[0,1]$ resp.$[0,1]$ that gave rise to the stable
degenerations $\Sigma_{\textrm{edge}}$ (for $\gamma_{\Sigma}(0)=\infty$) and
$\Sigma_{\textrm{trc}}$ (for $\gamma_{\Sigma}(1)=\frac{27}{5})$
resp.$\Pi_{\textrm{edge}}$ (for $\gamma_{\Pi}(0)=\infty$) and
$\Pi_{\textrm{trc}}$ (for $\gamma_{\Pi}(1)=-1)$. We will determine the
monodromies of these degenerations. When determining the local monodromies
given by $\Sigma$, it is convenient to regard $\gamma_{\Sigma}|(0,1)$ as a
base point for $\mathscr{B}^{\circ}$ and denote the fundamental group of
$\mathscr{B}^{\circ}$ with this base point by $\pi_{\Sigma}$. Similarly when
determining the local monodromies given by $\Pi$, we will regard
$\gamma_{\Pi}|(0,1)$ as a base point for $\mathscr{B}^{\circ}$. In this case
we denote the fundamental group of $\mathscr{B}^{\circ}$ with this base point
by $\pi_{\Pi}$. They are parts of the monodromy representation of $\pi$ on
$H_{1}(\Sigma)$. Clearly that $\pi_{\Sigma}$ and $\pi_{\Pi}$ are conjugate to
each other. Hence We will denote this group by $\pi$ if there is no
ambiguities.

Let $t\in\mathscr{B}^{\circ}$, recall that we have an isotropic decomposition

(24)
$H_{1}(C_{t};\mathds{Q})\cong(V_{\mathds{Q}}\otimes\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(V_{\mathds{Q}},H_{1}(C_{t},\mathds{Q})))\oplus(E_{\mathds{Q}}\otimes\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t},\mathds{Q})))$

Since the monodromy action will preserve this decomposition, we have a
monodromy representation of $\pi$ on both
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(V_{\mathds{Q}},H_{1}(C_{t};\mathds{Q}))$
and
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t};\mathds{Q}))$.
We have already determined the first type, together with an integral version
of it $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(V_{o},H_{1}(C_{t}))$ in [6].
Here we will focus on the second type and its integral version i.e.
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(C_{t}))$. This
integral global monodromy representation will be denoted by $\rho_{E_{o}}$.
The space
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t},\mathds{Q}))$
is of dimension four over $\mathds{Q}$ since $E_{\mathds{Q}}$ admits non-
trivial endomorphisms. However it will be of dimension two if treated as
$\mathcal{K}$-vector space, where $\mathcal{K}$ is the endomorphism field of
$E_{\mathds{Q}}$ defined in Corollary 1.7. As we observed in Remark 1.5 that
the symplectic form on $H_{1}(C_{t};\mathds{Q})$ and the inner product on
$E_{\mathds{Q}}$ give rise to a symplectic form on
$\operatorname{Hom}_{\mathds{Q}\mathscr{I}}(E_{\mathds{Q}},H_{1}(C_{t};\mathds{Q}))$.
The monodromies should keep this symplectic form, hence $\rho_{E_{o}}$ takes
its values in
$\operatorname{Sp}(1,\mathcal{O}_{o})\cong\operatorname{SL}_{2}(\mathcal{O}_{o})$.

Now let $C_{s}$ represents a singular member of the Winger pencil and
$U_{s}\subset\mathscr{B}$ a small disk-like neighborhood of $s$ (so that
$C_{s}\subset\mathscr{W}_{U_{s}}$ is a homotopy equivalence), we will
determine $\rho_{E_{o}}$ locally for the degenerations
$\Sigma_{\textrm{trc}}$, $\Sigma_{\textrm{edge}}$, $\Pi_{\textrm{trc}}$,
$\Pi_{\textrm{edge}}$ and do a local discussion for degeneration of $3K$ in
this section. If we choose $\alpha_{s}$ where
$s\in\\{\frac{27}{5},0,-1,\infty\\}$ be a simple closed loop around $s$ only,
the local fundamental groups is isomorphic to $\mathds{Z}$ with generator
represented by $\alpha_{s}$ in these local cases. Hence the local monodromy
around $s$ is determined by it value on $[\alpha_{s}]$. The requirements of
$U_{s}$ implies that for any $t\in U-\\{s\\}$ the natural map $H_{1}(C_{t})\to
H_{1}(\mathscr{W}_{U})\cong H_{1}(C_{s})$ is onto. So if $L$ denotes the
kernel, then we get the short exact sequence

(25) $0\to L\to H_{1}(C_{t})\to H_{1}(C_{s})\to 0$

In case $C_{s}$ has only nodal singularities, $L$ is an
$\mathscr{I}$-invariant isotropic primitive sublattice generated by the
vanishing cycles. The monodromies will preserve this exact sequence and acts
non-trivially only on the middle term. If we denote the set of vanishing
cycles by $\Delta$ and $x$ be a class in $H_{1}(C_{t})$, then the monodromies
of $[\alpha_{s}]$ is given by the following well-known Picard-Lefschetz
formula

(26) $\rho_{s}(\alpha_{s})(x)-x=\sum_{l\in\Delta/\\{\pm
1\\}}{\langle}x,l{\rangle}l$

These are the basic tools we will use in this section.

###  4.1. The Monodromies of the Degenerations of $\Sigma$

In this section, we will determine the local monodromies at the end points of
$\gamma_{\Sigma}$. We have proved in Corollary 2.9 that
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))$ is free
generated by $U_{\Sigma}$ and $V_{\Sigma}$ as $\mathcal{O}_{o}$-module. So it
is natural to express the local monodromies in terms of these generators in
this section. We will denote the local monodromies by
$\rho_{\Sigma,\textrm{trc}}$ and $\rho_{\Sigma,\textrm{edge}}$ respectively
and we will brief $p\circ\sigma_{\textrm{trc}}$ to $\sigma_{\textrm{trc}}$
which is the same for other symbols, since we will always work on
$H_{1}(\Sigma)$ in this section. The Theorem 4.1 below will give the local
monodromy in each case.

######  Theorem 4.1.

The monodromy $\rho_{\Sigma,\textrm{trc}}$ fixes $U_{\Sigma}$ and brings
$V_{\Sigma}$ to $(X-2)U_{\Sigma}+V$. The monodromy
$\rho_{\Sigma,\textrm{edge}}$ brings $U_{\Sigma}$ to
$3U_{\Sigma}-(X+1)V_{\Sigma}$ and brings $V_{\Sigma}$ to
$(X-1)U_{\Sigma}-V_{\Sigma}$.

Let us recall some facts about the vanishing cycles $L_{\textrm{trc}}$ resp.
$L_{\textrm{edge}}$ we discussed above before we give the proof of Theorem
4.1. Let $G_{\Sigma,\textrm{edge}}$ be the dual intersection graph of
$\Sigma_{\textrm{edge}}$. It has six vertices and every two vertices are
joined by an edge. Hence in this case we get the complete graph with six
vertices, i.e. a graph of type $K_{6}$. If $\hat{\Sigma}_{\textrm{edge}}$ is
the normalization of the singular curve $\Sigma_{\textrm{edge}}$, the set of
connected components of $\hat{\Sigma}_{\textrm{edge}}$ is denoted by
$\mathscr{L}$, then it has $6$ elements and $\mathscr{I}$ acts on it by
permutations. There is a natural homotopy class of maps
$\Sigma_{\textrm{edge}}\to G_{\Sigma,\textrm{edge}}$ which induces an
isomorphism $H_{1}(\Sigma_{\textrm{edge}})\to
H_{1}(G_{\Sigma,\textrm{edge}})$. Recall that
$H_{1}(G_{\Sigma,\textrm{edge}})$ is free of rank 10, so that the kernel
$L_{\Sigma,\textrm{edge}}$ of $H_{1}(\Sigma)\to H_{1}(\Sigma_{\textrm{edge}})$
is in fact a primitive Lagrangian sublattice. The intersection product then
identifies $L_{\Sigma,\textrm{edge}}$ with the dual of
$H_{1}(G_{\Sigma,\textrm{edge}})$ so that the short exact sequence (25)
becomes the following

(27)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{\Sigma,\textrm{edge}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{L_{\Sigma,\textrm{edge}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We have proved in [6] the following Lemma.

######  Lemma 4.2.

The natural homotopy class of maps $\Sigma_{\textrm{edge}}\to
G_{\Sigma,\textrm{edge}}$ induces an isomorphism on $H^{1}$ and the map which
assigns to the ordered distinct pair $(l,l^{\prime})$ in $\mathscr{L}$ the
$1$-cocycle on $G_{\Sigma,\textrm{edge}}$ spanned by the vertices defined by
$l$ and $l^{\prime}$ induces an $\mathscr{I}$-equivariant isomorphism
$\wedge^{2}W_{0}\cong H^{1}G_{\Sigma,\textrm{edge}}$. If we call that
$H^{1}(G_{\Sigma,\textrm{edge}})$ is naturally identified with the vanishing
homology of the degeneration $\Sigma$ into $\Sigma_{\textrm{edge}}$, then this
isomorphism identifies the set $\Delta_{\Sigma,\textrm{trc}}$ of vanishing
cycles with the set of unordered distinct pairs in $\mathscr{L}$. Dually,
$L_{\Sigma,\textrm{edge}}^{\vee}=H_{1}(\Sigma_{\textrm{edge}})$ is as a
$\mathds{Z}\mathscr{I}$-module isomorphic to $\wedge^{2}W_{o}^{\vee}$.

From this Lemma the short exact sequence (27) becomes the following sequence
of $\mathds{Z}\mathscr{I}$-modules.

(28)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\wedge^{2}W_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Sigma,\textrm{edge}}}$$\textstyle{H_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\wedge^{2}W_{o})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{Z_{1}(\Sigma)/B_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Note that $\wedge^{2}W_{o}$ has a single generator as a
$\mathds{Z}\mathscr{I}$-module, for example $\bar{l}\wedge\bar{l^{\prime}}$
with $l,l^{\prime}$ distinct. We have an $\mathscr{I}$-isomorphism
$\iota_{\textrm{edge}}:\wedge^{2}W_{o}\to Z_{1}(\Sigma)$ which sends
$\bar{l}\wedge\bar{l^{\prime}}$ to the element in $Z_{\textrm{edge}}(\Sigma)$
with the same stabilizer. Let us apply the left exact functor
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\cdot)$ to the short exact
sequence (28) and combine it with the exact sequence (6)

(29)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\wedge^{2}W_{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Sigma,\textrm{edge},*}}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\wedge^{2}W_{o}^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ext}_{\mathds{Z}\mathscr{I}}(E_{o},\wedge^{2}W_{o})}$$\textstyle{\operatorname{coker}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

By Proposition 2.8, the vertical arrow

$\operatorname{coker}(i)=\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},Z_{1}(\Sigma))/\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},B_{1}(\Sigma))\to\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(V_{o},H_{1}(\Sigma))$

is an isomorphism.

Likewise at the other end: if $G_{\Sigma,\textrm{trc}}$ is the dual
intersection graph of $\Sigma_{\textrm{trc}}$, then the kernel of
$H_{1}(\Sigma)\to H_{1}(\Sigma_{\textrm{trc}})\cong
H_{1}({\Sigma,\textrm{trc}})$ is the primitive Lagrangian sublattice
$L_{\Sigma,\textrm{trc}}$ we introduced earlier and we get a similar short
exact sequence and a similar description of the associated monodromy
$\rho_{\Sigma,\textrm{trc}}$ in terms of $\Delta_{\Sigma,\textrm{trc}}$.

The short exact sequence (25) becomes the following short exact sequence of
$\mathds{Z}\mathscr{I}$-modules

(30)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{\Sigma,\textrm{trc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Sigma,\textrm{trc}}}$$\textstyle{{H_{1}(\Sigma,\mathds{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{L_{\Sigma,\textrm{trc}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{Z_{1}(\Sigma)/B_{1}(\Sigma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here $j_{\Sigma,\textrm{trc}}$ is the obvious map. Applying the left exact
functor $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\cdot)$ to the short
exact sequence (30) and combine it with the exact sequence (6)

(31)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},L_{\Sigma,\textrm{trc}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Sigma,\textrm{trc}\
*}}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Sigma))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},L_{\Sigma,\textrm{trc}}^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ext}_{\mathds{Z}\mathscr{I}}(E_{o},L_{\Sigma,\textrm{trc}})}$$\textstyle{\operatorname{coker}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The vertical arrow is an isomorphism same as above.

###### Proof.

(Proof of Theorem 4.1)

By the Proposition 2.1 the images of $\sigma_{\textrm{edge}}$ and
$\sigma^{\prime}_{\textrm{edge}}$ lie in $L_{\Sigma,\textrm{edge}}$ and the
image of $\sigma_{\textrm{trc}}$ and $\sigma^{\prime}_{\textrm{trc}}$ lie in
$L_{\Sigma,\textrm{trc}}$. Hence the monodromy $\rho_{\Sigma,\textrm{trc}}$
fixes $U_{\Sigma}=\sigma_{\textrm{trc}}$ and $\sigma^{\prime}_{\textrm{trc}}$,
while $\rho_{\Sigma,\textrm{edge}}$ fixes $\sigma_{\textrm{edge}}$ and
$\sigma^{\prime}_{\textrm{edge}}$.

By the Picard-Lefschetz formula

$\rho_{\Sigma,\textrm{trc}}(V_{\Sigma}(e))-(V_{\Sigma}(e))=\sum_{\delta\in\Delta_{\Sigma,\textrm{trc}}/\\{\pm
1\\}}{\langle}[V_{\Sigma}(e)],\delta{\rangle}\delta=\frac{1}{2}\sum_{x\in\mathscr{R}_{0}(\tilde{\Sigma})}{\langle}[\sigma^{\prime}_{\textrm{edge}}(e)],\delta_{x}{\rangle}\delta_{x}$

Let
$x\in\mathscr{R}_{2}(\tilde{\Sigma})\subset\mathscr{C}_{2}(\tilde{\Sigma})$,
from the Lemma 2.6
${\langle}[\sigma^{\prime}_{\textrm{edge}}(e)],\delta_{x}{\rangle}$ equals 2
if $x\notin e$, otherwise it is $0$. Hence
$\sum_{x\in\mathscr{R}_{0}(\Sigma)}{\langle}[\sigma^{\prime}_{\textrm{edge}}(e)],\delta_{x}{\rangle}\delta_{x}$
equals $\delta^{\prime}_{\textrm{trc}}$. Hence we have
$\rho_{\Sigma,\textrm{trc}}(V_{\Sigma})=V_{\Sigma}+\sigma^{\prime}_{\textrm{trc}}=(X-2)U_{\Sigma}+V_{\Sigma}$
from Equation (15).

Similarly, for $\rho_{\Sigma,\textrm{edge}}$ we have

$\displaystyle\rho_{\Sigma,\textrm{edge}}(U_{\Sigma}(e))-U_{\Sigma}(e)$
$\displaystyle=$
$\displaystyle\sum_{\delta\in\Delta_{\Sigma,\textrm{edge}}/\\{\pm
1\\}}{\langle}[U_{\Sigma}(e)],\delta{\rangle}\delta$ $\displaystyle=$
$\displaystyle\sum_{y\in\mathscr{R}_{1}(\tilde{\Sigma})}{\langle}[\sigma_{\textrm{trc}}(e)],\delta_{y}{\rangle}\delta_{y}$
$\displaystyle\rho_{\Sigma,\textrm{edge}}(V_{\Sigma}(e))-V_{\Sigma}(e)$
$\displaystyle=$
$\displaystyle\sum_{\delta\in\Delta_{\Sigma,\textrm{edge}}/\\{\pm
1\\}}{\langle}[V_{\Sigma}(e)],\delta{\rangle}\delta$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{y\in\mathscr{R}_{1}(\tilde{\Sigma})}{\langle}[\sigma_{\textrm{trc}}(e)]+[\sigma^{\prime}_{\textrm{trc}}(e)],\delta_{y}{\rangle}\delta_{y}$

From the Lemma 2.6, ${\langle}[\sigma_{\textrm{trc}}(e)],\delta_{y}{\rangle}$
equals $-1$ if $\textrm{in}(y)\in e$ but $\textrm{tm}(y)\notin e$ otherwise it
equals 0. And
${\langle}[\sigma^{\prime}_{\textrm{trc}}(e)],\delta_{y}{\rangle}$ equals $1$
if $\textrm{in}(y)\in e$ but $\textrm{tm}(y)\notin e$, $-2$ if
$\textrm{in}(y)\notin e$ and $0$ in other cases. Hence we have
$\sum_{y\in\mathscr{R}_{1}(\tilde{\Sigma})}{\langle}[\sigma_{\textrm{trc}}(e)],\delta_{y}{\rangle}\delta_{y}$
is $-\delta_{\textrm{edge}}$,
$\sum_{y\in\mathscr{R}_{1}(\tilde{\Sigma})}{\langle}[\sigma^{\prime}_{\textrm{trc}}(e)],\delta_{y}{\rangle}\delta_{y}$
is $\delta_{\textrm{edge}}-2\delta^{\prime}_{\textrm{edge}}$. Therefore we
have
$\rho_{\Sigma,\textrm{edge}}(U_{\Sigma})=U_{\Sigma}-\sigma_{\textrm{edge}}=3U_{\Sigma}-(X+1)V_{\Sigma}$
and
$\rho_{\Sigma,\textrm{edge}}(V_{\Sigma})=V_{\Sigma}-\sigma^{\prime}_{\textrm{edge}}=(X-1)U_{\Sigma}-V_{\Sigma}$.
This finishes the proof. ∎

###  4.2. The Monodromies of the Degenerations of $\Pi$

In this section, we will determine the local monodromies at the end points of
$\gamma_{\Pi}$. The Theorem 4.3 below will give the local monodromy in each
case. Recall that by Corollary 3.13, the module
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Pi))$ is freely
generated by $U_{\Pi}$ and $V_{\Pi}$ as $\mathcal{O}_{o}$-module. So we will
express the monodromies $\rho_{\Pi,\textrm{edge}}$ and
$\rho_{\Pi,\textrm{trc}}$ in terms of these generators when computing the
local monodromies defined by $\Pi$.

######  Theorem 4.3.

The monodromy $\rho_{\Pi,\textrm{trc}}$ fixes $U_{\Pi}$ and takes $V_{\Pi}$ to
$XU_{\Pi}+V_{\Pi}$. The monodromy $\rho_{\Pi,\textrm{edge}}$ brings $U_{\Pi}$
to $(X-2)U_{\Pi}-(X-1)V_{\Pi}$ and $V_{\Pi}$ to $(2X-4)U_{\Pi}+(4-X)V_{\Pi}$.

It is clear that the dual intersection graph $G_{\Pi,\textrm{edge}}$ of
$\Pi_{\textrm{edge}}$ is the same as $G_{\Sigma,\textrm{edge}}$.Therefore we
have the same results as the exact sequences (27), (28) and (6) with $\Sigma$
replaced by $\Pi$. There is some difference at the other end: if
$G_{\Pi,\textrm{trc}}$ is the dual intersection graph of $\Pi_{\textrm{trc}}$,
$G_{\Pi,\textrm{trc}}$ has only one vertex and six edges with the vertex
marked with 4. In this case the kernel $L_{\Pi,\textrm{trc}}$ of
$H_{1}(\Pi)\to H_{1}(\Pi_{\textrm{trc}})$ in the exact sequence (25) is
generated by 6 elements i,e the vanishing cycles which denote their collection
by $\Delta_{\Pi,\textrm{trc}}$. Then it is a primitive isotropic sublattice
$L_{\Pi,\textrm{trc}}$ of rank six which is not Lagrangian. Hence the exact
sequence (25) will become the following.

(32)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{\Pi,\textrm{trc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{1}(\Pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{H_{1}(C_{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

However we have a similar description of the associated monodromy
$\rho_{\Pi,\textrm{trc}}$ in terms of $\Delta_{\Pi,\textrm{trc}}$.

(33)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{\Pi,\textrm{trc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Pi,\textrm{trc}}}$$\textstyle{{H_{1}(\Pi,\mathds{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{H_{1}(C_{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{Z_{1}(\Pi)/B_{1}(\Pi)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here $j_{\Pi,\textrm{trc}}$ is the obvious map. Applying the left exact
functor $\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},\cdot)$ to the short
exact sequence (33) and combine it with the exact sequence (18)

(34)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},L_{\Pi,\textrm{trc}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{\Pi,\textrm{trc}\
*}}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(\Pi))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$$\textstyle{\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(C_{-1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ext}_{\mathds{Z}\mathscr{I}}(E_{o},L_{\Sigma,\textrm{trc}})}$$\textstyle{\operatorname{coker}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The vertical arrow is an isomorphism same as above.

###### Proof.

(Proof of Theorem 4.3) The proof is similar to the proof of Theorem 4.1. By
the Proposition 3.4 the image of $\pi_{\textrm{edge}}$ and
$\pi^{\prime}_{\textrm{edge}}$ lie in $L_{\Pi,\textrm{edge}}$ and the image of
$\pi_{\textrm{trc}}$ and $\pi^{\prime}_{\textrm{trc}}$ lie in
$L_{\Pi,\textrm{trc}}$. Hence the monodromy $\rho_{\Pi,\textrm{edge}}$ fixes
$\pi_{\textrm{edge}}$ and $\pi^{\prime}_{\textrm{edge}}$, while the monodromy
$\rho_{\Pi,\textrm{trc}}$ fixes $U_{\Pi}=\pi_{\textrm{trc}}$ and
$\pi^{\prime}_{\textrm{trc}}$.

Recall that we take $\mathscr{R}_{i}(\tilde{\Pi})$ be systems of
representatives of $\iota$-symmetry on $\mathscr{C}_{i}(\tilde{\Pi})$. By the
Picard-Lefschetz formula

$\rho_{\Pi,\textrm{trc}}(V_{\Pi}(e))-V_{\Pi}(e)=\sum_{\theta\in\Delta_{\Pi,\textrm{trc}}/\\{\pm
1\\}}{\langle}[V_{\Pi}],\theta{\rangle}\theta=\frac{1}{2}\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})}{\langle}[\pi^{\prime}_{\textrm{edge}}(e)],\theta_{x}{\rangle}\theta_{x}$

Let $x\in\mathscr{R}_{0}(\tilde{\Pi})\subset\mathscr{C}_{0}(\tilde{\Pi})$,
from the Lemma 3.12
${\langle}[\pi^{\prime}_{\textrm{edge}}(e)],\theta_{x}{\rangle}$ equals $2$ if
$x\neq e$, otherwise it is $0$. Hence the sum
$\sum_{x\in\mathscr{R}_{0}(\tilde{\Pi})}{\langle}[\pi^{\prime}_{\textrm{edge}}(e)],\theta_{x}{\rangle}\theta_{x}$
equals $2\theta^{\prime}_{\textrm{trc}}$. Therefore we have
$\rho_{\Pi,\textrm{trc}}(V_{\Pi}(e))=V_{\Pi}+\pi^{\prime}_{\textrm{trc}}=XU_{\Pi}+V_{\Pi}$
from Equation (23).

Similarly, for $\rho_{\Pi,\textrm{edge}}$ we have

$\displaystyle\rho_{\Pi,\textrm{edge}}(U_{\Pi}(e))-U_{\Pi}(e)$
$\displaystyle=$
$\displaystyle\sum_{\theta\in\Delta_{\Pi,\textrm{edge}}/\\{\pm
1\\}}{\langle}[U_{\Pi}(e)],\theta{\rangle}\theta$ $\displaystyle=$
$\displaystyle\sum_{y\in\mathscr{R}_{1}(\tilde{\Pi})}{\langle}[\pi_{\textrm{trc}}(e)],\theta_{y}{\rangle}\theta_{y}$
$\displaystyle\rho_{\Pi,\textrm{edge}}(V_{\Pi}(e))-V_{\Pi}(e)$
$\displaystyle=$
$\displaystyle\sum_{\theta\in\Delta_{\Pi,\textrm{edge}}/\\{\pm
1\\}}{\langle}[V_{\Pi}(e)],\theta{\rangle}\theta$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{y\in\mathscr{R}_{1}(\tilde{\Pi})}{\langle}[\pi_{\textrm{trc}}(e)]-[\pi^{\prime}_{\textrm{trc}}(e)],\theta_{y}{\rangle}\theta_{y}$

From the Lemma 3.12, ${\langle}[\pi_{\textrm{trc}}(e)],\theta_{y}{\rangle}$
equals $-1$ if the initial point of $y$ is $e$ otherwise it equals 0. And
${\langle}[\pi^{\prime}_{\textrm{trc}}(e)],\theta_{y}{\rangle}$ equals $1$ if
the initial point of $y$ is $e$, $2$ if the initial point of $y$ lies on
$P_{e}$ while the terminal point of $y$ lies on $P_{\iota e}$ and $0$ in other
cases. Hence we have
$\sum_{y\in\mathscr{R}_{1}(\tilde{\Pi})}{\langle}[\pi_{\textrm{trc}}(e)],\theta_{y}{\rangle}\theta_{y}$
equals to $-\theta_{\textrm{edge}}$ and
$\sum_{y\in\mathscr{R}_{1}(\tilde{\Pi})}{\langle}[\pi^{\prime}_{\textrm{trc}}(e)],\theta_{y}{\rangle}\theta_{y}$
equals to $\theta_{\textrm{edge}}-2\theta^{\prime}_{\textrm{edge}}$. Therefore
we have the monodromies
$\rho_{\Pi,\textrm{edge}}(U_{\Pi})=U_{\Pi}-\pi_{\textrm{edge}}=(X-2)U_{\Pi}-(X-1)V_{\Pi}$
and
$\rho_{\Pi,\textrm{edge}}(V_{\Pi})=V_{\Pi}-\pi_{\textrm{edge}}+\pi^{\prime}_{\textrm{edge}}=(2X-4)U_{\Pi}+(4-X)V_{\Pi}$.
This finishes the proof. ∎

###  4.3. The Local Monodromies Near the Triple Conic

We claim that the monodromy around $s=0$ is of order three. Remember that
$C_{0}$ is the unstable curve $3K$, where $K$ is an $\mathscr{I}$-invariant
(smooth) conic. Let ${U_{0}}\subset\mathscr{B}$ be an open disk centered at
$s=0$ of radius $<\frac{27}{5}$. We proved in [13] that by doing a base change
over ${U_{0}}$ of order 3 (with Galois group $\mu_{3}$), given by
$\hat{t}\in\hat{U}_{0}\to t=\hat{t}^{3}\in{U_{0}}$, the pull back of
$\mathscr{W}_{U_{0}}/{U_{0}}$ can be modified over the central fiber $C_{0}$
only to make it a smooth family $\hat{\mathscr{W}}_{\hat{U}_{0}}/\hat{U}_{0}$
which still retains the $\mu_{3}$-action. The central fiber is then a smooth
curve $\hat{C}_{0}$ with an action of $\mathscr{I}\times\mu_{3}$ whose
$\mu_{3}$-orbit space gives $K$. This implies that the monodromy of the
original family around $0$ (which is a priori only given as an isotopy class
of diffeomorphisms of a nearby smooth fiber) can be represented by the action
of a generator $\phi\in\mu_{3}$ on $\hat{C}_{0}$ (which indeed commutes with
the $\mathscr{I}$-action on $C_{0}$).

######  Corollary 4.4.

Let $t\in{U_{0}}\backslash\\{0\\}\subset\mathscr{B}$, the monodromy
automorphism $\rho_{0}$ acts on
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(C_{t}))$ with order
three.

###### Proof.

This comes from the Corollary 4.8 of [6]. ∎

##  5\. Global Monodromy and Period Map on the $E$-part

We will determine the global monodromy group and the period map in this
section. Let us take $\psi$ and $\psi^{\prime}$ be the two naturally defined
embeddings of $\mathcal{K}\hookrightarrow\mathds{R}$. It is clear that $\psi$
and $\psi^{\prime}$ induce two different embeddings
$\operatorname{SL}_{2}(\mathcal{O}_{o})\hookrightarrow\operatorname{SL}_{2}(\mathds{R})$
which we still denote they by $\psi$ and $\psi^{\prime}$. Hence the map
$(\psi,\psi^{\prime})$ will embed the group
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ into
$\operatorname{SL}_{2}(\mathds{R})$ with the diagonal isomorphic to
$\operatorname{SL}_{2}(\mathds{Z})$. This could also be described as follows:
there exist a Galois involution $\varphi$ of
$\operatorname{SL}_{2}(\mathcal{O}_{o})$ which will exchange the image of the
two embeddings in
$\operatorname{SL}_{2}(\mathds{R})\times\operatorname{SL}_{2}(\mathds{R})$,
the fixed points are the group $\operatorname{SL}_{2}(\mathds{Z})$. Moreover
we could observe that $\operatorname{SL}_{2}(\mathcal{O}_{o})$ acts faithfully
and discontinuously on $\mathds{H}^{2}$ through this embedding. The quotient
$\operatorname{SL}_{2}(\mathcal{O}_{o})/\mathds{H}^{2}$ is then a algebraic
surface called Hilbert’s modular surface. We will need the Theorem 4.6 in [5]
listed below which is also a special case of the main theorem of [1].

######  Theorem 5.1.

Let $K$ be a real quadratic number field, $\mathcal{O}_{K}$ its ring of
integers and $\Omega<K$ a lattice. Let
$\Lambda<\operatorname{SL}_{2}(\mathcal{O}_{K})$ be the subgroup generated by
matrix of the form $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ with $c\neq 0$,
together with the set of matrices

$\\{\begin{pmatrix}1,\omega\\\ 0,1\end{pmatrix}:\omega\in\Omega\\}$

If $\psi,\ \psi^{\prime}:K\to\mathds{R}$ are the two real embeddings of $K$,
then the associated embedding
$\operatorname{SL}_{2}(\mathcal{O}_{K})\to\operatorname{SL}_{2}(\mathds{R})\times\operatorname{SL}_{2}(\mathds{R})$
maps $\Lambda$ onto a lattice in
$\operatorname{SL}_{2}(\mathds{R})\times\operatorname{SL}_{2}(\mathds{R})$. In
particular $\Lambda$ has finite index in
$\operatorname{SL}_{2}(\mathcal{O}_{K})$.

Since the two models $\Sigma$ and $\Pi$ gives the same singular fiber at the
”edge” ends, it is clear that $\rho_{\Sigma,\textrm{edge}}$ and
$\rho_{\Pi,\textrm{edge}}$ should conjugate to each other by a transformation
in
$\operatorname{Sp}_{1}(\mathcal{O}_{o})\cong\operatorname{SL}_{2}(\mathcal{O}_{o})$.
It is clear to check that if we take the linear transformation $P$ as
Equations (35), we will have
$\rho_{\Pi,\textrm{edge}}=P^{-1}\rho_{\Sigma,\textrm{edge}}P$.

(35) $\displaystyle PU_{\Pi}$ $\displaystyle=$ $\displaystyle-V_{\Sigma}$
$\displaystyle PV_{\Pi}$ $\displaystyle=$ $\displaystyle
U_{\Sigma}-V_{\Sigma}$

From this observation, we will take the basis $(U:=U_{\Sigma},V:=V_{\Sigma})$
as a basis for
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(C_{t}))$. First we
give a direct computation to the Corollary 4.4 that we proved in the last
section.

######  Corollary 5.2.

The monodromy $\rho_{0}$ action on
$\operatorname{Hom}_{\mathds{Z}\mathscr{I}}(E_{o},H_{1}(C_{t}))$ is cyclic of
order three. More explicitly, it brings $U$ to $-U+V$ and $V$ to $-U$.

###### Proof.

Since the smooth locus $\mathscr{B}^{\circ}$ is obtained by removing four
points from $\mathds{P}^{1}$ we will have the following equation

$\rho_{0}^{-1}=\rho_{\Sigma,\textrm{trc}}\rho_{\Sigma,\textrm{edge}}P^{-1}\rho_{\Pi,\textrm{trc}}P$

Then the computation shows that $\rho_{0}^{-1}$ is given as following:

$\displaystyle\rho_{0}^{-1}(U)$ $\displaystyle=$ $\displaystyle-V$
$\displaystyle\rho_{0}^{-1}(V)$ $\displaystyle=$ $\displaystyle U-V$

Hence it is cyclic of order three. ∎

######  Theorem 5.3.

The monodromy group $\Gamma_{E_{o}}$ is a subgroup of finite index in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$. In particular it is arithmetic.

###### Proof.

In order to show this theorem we only need to check the conditions in Theorem
5.1. From the above observation that $\Gamma_{E_{o}}$ is generated by the
following three generators

(36) $\rho_{\Sigma,\textrm{trc}}=\begin{pmatrix}1&X-2\\\ 0&1\end{pmatrix},\
\rho_{\Sigma,\textrm{edge}}=\begin{pmatrix}3&X-1\\\ -(1+X)&-1\end{pmatrix},\
P^{-1}\rho_{\Pi,\textrm{trc}}P=\begin{pmatrix}1&0\\\ -X&1\end{pmatrix}$

It is clear that $\Gamma_{E}$ is generated by matrix of the form
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ with $c\neq 0$ and upper triangular
matrices. Hence $\Gamma_{E_{o}}$ is a finite index subgroup of
$\operatorname{SL}_{2}(\mathcal{O}_{o})$. ∎

Finally summarize all the facts about the monodromy, we could determine the
’partial’ period map. Let $\mathscr{B}^{+}$ be the open subvariety of
$\mathscr{B}$ obtained by removing from $\mathscr{B}$ the three points
representing nodal curves.

######  Theorem 5.4.

The ’partial’ period map
$p_{E_{o}}:\mathscr{B}^{+}\to\Gamma_{E_{o}}/\mathds{H}^{2}\to\operatorname{SL}_{2}(\mathcal{O}_{o})/\mathds{H}^{2}$
has the property that the first arrow is open and the second map is finite.

##  6\. Computation for the Index of $\Gamma_{E_{o}}$ in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$

Recall that $\mathcal{O}_{o}=\mathds{Z}[X]/(X^{2}-5)$ is the endomorphisms
ring of $\mathds{Z}\mathcal{A}_{5}$-module $E_{o}$ and
$\mathcal{O}=\mathds{Z}[Y]/(Y^{2}-Y-1)$ is isomorphic to the ring of integers
in the algebraic field $\mathds{Q}[\sqrt{5}]$. Their relations are as
following: the natural map given by quotient $2\mathcal{O}$ gives the exact
sequence

$\mathcal{O}\to\mathcal{O}/2\mathcal{O}\to 1$

It has the properties that the last term $\mathcal{O}/2\mathcal{O}$ is
isomorphic to $\mathds{F}_{4}$ and $\mathcal{O}_{o}$ is the pullback of
$\mathds{F}_{2}\subset\mathds{F}_{4}$. The similar properties also holds if we
consider special linear groups with entries in $\mathcal{O}$ and
$\mathcal{O}_{o}$. We have the following exact sequence of the groups

(37)
$\operatorname{SL}_{2}(\mathcal{O})\to\operatorname{SL}_{2}(\mathcal{O}/2\mathcal{O})\cong\operatorname{SL}_{2}(\mathds{F}_{4})\to
1$

The subgroup $\operatorname{SL}_{2}(\mathcal{O}_{o})$ is the pullback of
$\operatorname{SL}_{2}(\mathds{F}_{2})\subset\operatorname{SL}_{2}(\mathds{F}_{4})$.
From these facts, we have the following proposition

######  Proposition 6.1.

The Exact Sequence 37 induced an one-to-one correspondence of sets of left
cosets

$\operatorname{SL}_{2}(\mathcal{O})/\operatorname{SL}_{2}(\mathcal{O}_{o})\to\operatorname{SL}_{2}(\mathds{F}_{4})/\operatorname{SL}_{2}(\mathds{F}_{2})$

In particular $\operatorname{SL}_{2}(\mathcal{O}_{o})$ has index 10 in
$\operatorname{SL}_{2}(\mathcal{O})$.

###### Proof.

It is clear to check that this map is well-defined, surjective and injective.
The index comes from the facts that $\operatorname{SL}_{2}(\mathds{F}_{4})$ is
isomorphic to $\mathcal{A}_{5}$ and $\operatorname{SL}_{2}(\mathds{F}_{2})$ is
isomorphic to $\mathcal{S}_{3}$. ∎

Besides we have an explicit description for the generating set of
$\operatorname{SL}_{2}(\mathcal{O})$ which we will consider the following
matrices in $\operatorname{SL}_{2}(\mathcal{O})$:

$\displaystyle A_{0}=-\operatorname{Id},\ $ $\displaystyle
A_{1}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\ $ $\displaystyle
A_{2}=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ $\displaystyle
A_{3}=\begin{pmatrix}X&0\\\ 0&X-1\end{pmatrix},\ $ $\displaystyle
A_{4}=\begin{pmatrix}1&X\\\ 0&1\end{pmatrix}$

The following Proposition is the Corollary 2.3 in [9] which showed that
$\operatorname{SL}_{2}(\mathcal{O})$ is generated by all the $A_{i}$s together
with $-\operatorname{Id}$ subject to some relations.

######  Proposition 6.2.

The group $\operatorname{SL}_{2}(\mathcal{O})$ is generated by
$A_{0},\cdots,A_{4}$ subject to the following relations

$C_{0}=A_{0}^{2}$, |  |
---|---|---
$C_{1}=[A_{0},A_{1}]$, |  |  $C_{2}=[A_{0},A_{2}]$,
$C_{3}=[A_{0},A_{3}]$, |  |  $C_{4}=[A_{0},A_{4}]$,
$R_{1}=A_{0}A_{1}^{2}$, |  |  $R_{2}=(A_{1}A_{2})^{3}$,
$R_{3}=A_{0}(A_{1}A_{3})^{2}$, |  |  $R_{4}=[A_{2},A_{4}]$,
$R_{5}=A_{3}A_{2}A_{3}^{-1}(A_{2}A_{4})^{-1}$, |  |  $R_{6}=A_{3}A_{4}A_{3}^{-1}(A_{2}A_{4}^{2})^{-1}$,
$R_{7}=A_{0}A_{1}A_{4}A_{1}(A_{2}A_{4}^{-1}A_{1}A_{4}^{-1}A_{3})^{-1}$, |  |

######  Theorem 6.3.

The monodromy group $\Gamma_{E_{o}}$ is a subgroup of index 20 in
$\operatorname{SL}_{2}(\mathcal{O})$. Hence it has index 2 in
$\operatorname{SL}_{2}(\mathcal{O}_{o})$.

###### Proof.

We first consider the index of $\Gamma_{E_{o}}$ in
$\operatorname{SL}_{2}(\mathcal{O})$. It is clear to check that
$\rho_{\Sigma,\textrm{trc}}=A_{4}^{2}A_{2}^{-3}$,
$\rho_{\Sigma,\textrm{edge}}=A_{4}^{-2}A_{2}^{2}A_{1}A_{4}^{-2}A_{1}$ and
$P^{-1}\rho_{\Pi,\textrm{trc}}P=-A_{1}A_{4}^{2}A_{2}^{-1}A_{1}$. Using the
Index function in the computer program Magma [2], we could compute that the
index $[\operatorname{SL}_{2}(\mathcal{O}):\Gamma_{E_{o}}]$ is 20. The Program
has been uploaded to [12]. From the Proposition 6.1 the index
$[\operatorname{SL}_{2}(\mathcal{O}):\operatorname{SL}_{2}(\mathcal{O}_{o})]$
is 10, hence the index
$[\operatorname{SL}_{2}(\mathcal{O}_{o}):\Gamma_{E_{o}}]$ is 2. ∎

## References

  * [1] Y. Benoist and H. Oh.  Discreteness criterion for subgroups of products of SL(2).  Transformation Groups, 15(3):503–515, 2010.
  * [2] W. Bosma, J. Cannon, and C. Playoust.  The Magma algebra system I: The user language.  Journal of Symbolic Computation, 24(3-4):235–265, 1997.
  * [3] H.W. Braden and L. Disney-Hogg.  Bring’s curve: Old and new.  arXiv preprint arXiv:2208.13692, 2022.
  * [4] H.W. Braden and T.P. Northover.  Bring’s curve: its period matrix and the vector of Riemann constants.  SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 8:065, 2012.
  * [5] B. Farb and E. Looijenga.  Arithmeticity of the monodromy of the Wiman–Edge pencil.  Annales de l’Institut Fourier, 71(4):1325–1361, 2021.
  * [6] E. Looijenga and Y. Zi.  Monodromy and period map of the Winger pencil.  arXiv preprint arXiv:2109.01810, 2021.
  * [7] G. Riera and R. Rodriguez.  The period matrix of Bring’s curve.  Pacific Journal of Mathematics, 154(1):179–200, 1992.
  * [8] C. Shramov and I. Cheltsov.  Cremona Groups and Icosahedron.  2015\.
  * [9] M. Stover.  Geometry of the Wiman–Edge monodromy.  Journal of Topology and Analysis, pages 1–29, 2021.
  * [10] M. Weber.  Kepler’s small stellated dodecahedron as a Riemann surface.  Pacific journal of mathematics, 220(1):167–182, 2005.
  * [11] R.M Winger.  On the invariants of the ternary icosahedral group.  Mathematische Annalen, 93(1):210–216, 1925.
  * [12] Y. Zi.  https://github.com/ZYPThu/Magma_Code_MonoE/blob/main/Winger.m.
  * [13] Y. Zi.  Geometry of the Winger pencil.  European Journal of Mathematics, pages 1074–1101, 2021.