# Zariski multiples associated with quartic curves

Ichiro Shimada  Department of Mathematics, Graduate School of Science,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 JAPAN
<EMAIL_ADDRESS>
###### Abstract.

We investigate Zariski multiples of plane curves $Z_{1},\dots,Z_{N}$ such that
each $Z_{i}$ is a union of a smooth quartic curve, some of its bitangents, and
some of its $4$-tangent conics. We show that, for plane curves of this type,
the deformation types are equal to the homeomorphism types, and that the
number of deformation types grows as $O(d^{\,62})$ when the degree $d$ of the
plane curves tends to infinity.

###### 2020 Mathematics Subject Classification:

14H50, 14J26

This work was supported by JSPS KAKENHI Grant Number 20H01798 and 20K20879.

##  1\. Introduction

By a _plane curve_ , we mean a reduced, possibly reducible, complex projective
plane curve. We say that two plane curves $C$ and $C^{\prime}$ have the same
_combinatorial type_ if there exist tubular neighbourhoods $T(C)$ of $C$ and
$T(C^{\prime})$ of $C^{\prime}$ such that $(T(C),C)$ and
$(T(C^{\prime}),C^{\prime})$ are homeomorphic, whereas we say that $C$ and
$C^{\prime}$ have the same _homeomorphism type_ if
$(\mathord{\mathbb{P}}^{2},C)$ and $(\mathord{\mathbb{P}}^{2},C^{\prime})$ are
homeomorphic.

A _Zariski $N$-ple_ is a set of plane curves $\\{C_{1},\dots,C_{N}\\}$ such
that the curves $C_{i}$ have the same combinatorial type, but their
homeomorphism types are pairwise different. The notion of Zariski $N$-ples was
introduced by Artal-Bartolo [1] in reviving a classical example of
$6$-cuspidal curves of degree $6$ due to Zariski. Since then, this notion has
been studied by many people from various points of view. Some of the tools
that have been used in this investigation are: Alexander polynomials,
characteristic varieties, fundamental groups of complements, topological
invariants of branched coverings, and so on. See the survey paper [3].
Recently, Bannai et al. [4, 5, 6] have investigated Zariski $N$-ples such that
each member is a union of a smooth quartic curve and some of its bitangents.

In this paper, we introduce $4$-tangent conics of a smooth quartic curve, and
consider Zariski $N$-ples $Z_{1},\dots,Z_{N}$ such that each $Z_{i}$ is a
union of a smooth quartic curve, some of its bitangents, and some of its
$4$-tangent conics.

Let $Q$ be a smooth quartic curve. A _bitangent_ of $Q$ is a line whose
intersection multiplicity with $Q$ is even at each intersection point. It is
well known that every smooth quartic curve has exactly $28$ bitangents. We say
that a bitangent $\bar{l}$ of $Q$ is _ordinary_ if $\bar{l}$ is tangent to $Q$
at distinct $2$ points. A smooth conic
$\bar{c}\subset\mathord{\mathbb{P}}^{2}$ is called a _$4$ -tangent conic of
$Q$_ if $\bar{c}$ is tangent to $Q$ at $4$ distinct points. Every smooth
quartic curve has $63$ one-dimensional connected families of $4$-tangent
conics (see Theorem 4.1).

######  Definition 1.1.

Let $m$ and $n$ be non-negative integers such that $m\leq 28$. We say that a
plane curve $Z$ is a _$\mathord{\mathcal{Q}}^{(m,n)}$ -curve_ if $Z$ is of the
form

(1.1) $Z=Q+\bar{l}_{1}+\cdots+\bar{l}_{m}+\bar{c}_{1}+\cdots+\bar{c}_{n},$

where $Q$ is a smooth quartic curve, $\bar{l}_{1},\dots,\bar{l}_{m}$ are
distinct bitangents of $Q$, and $\bar{c}_{1},\dots,\bar{c}_{n}$ are distinct
$4$-tangent conics of $Q$, and they satisfy that

  1. (i)

the bitangents $\bar{l}_{1},\dots,\bar{l}_{m}$ are ordinary,

  2. (ii)

the intersection of any three of
$Q,\bar{l}_{1},\dots,\bar{l}_{m},\bar{c}_{1},\dots,\bar{c}_{n}$ is empty, and

  3. (iii)

the intersection of any two of
$\bar{l}_{1},\dots,\bar{l}_{m},\bar{c}_{1},\dots,\bar{c}_{n}$ is transverse.

It is obvious that any two $\mathord{\mathcal{Q}}^{(m,n)}$-curves have the
same combinatorial type. We construct a non-singular variety
$\mathord{\mathcal{Z}}^{(m,n)}$ parameterizing all
$\mathord{\mathcal{Q}}^{(m,n)}$-curves in Section 5.

######  Definition 1.2.

We say that two $\mathord{\mathcal{Q}}^{(m,n)}$-curves have the same
_deformation type_ if they belong to the same connected component of the
parameter space $\mathord{\mathcal{Z}}^{(m,n)}$.

It is obvious that $\mathord{\mathcal{Q}}^{(m,n)}$-curves of the same
deformation type have the same homeomorphism type. Our main results are the
following:

######  Theorem 1.3.

If two $\mathord{\mathcal{Q}}^{(m,n)}$-curves have the same homeomorphism
type, then they have the same deformation type.

We put

(1.2) $d^{(m,n)}:=\binom{28}{m}\cdot\binom{n+62}{62},$

which grows as $O(n^{62})$ when $n\to\infty$.

######  Theorem 1.4.

The number $N^{(m,n)}$ of deformation types of
$\mathord{\mathcal{Q}}^{(m,n)}$-curves satisfies

(1.3) $d^{(m,n)}/1451520\leq N^{(m,n)}\leq d^{(m,n)}.$

The main ingredient of the proof of these results is the monodromy argument of
Harris [9] (see Theorem 3.1). This argument converts the problem of
enumerating deformation types of $\mathord{\mathcal{Q}}^{(m,n)}$-curves to an
easy combinatorial problem of counting orbits of an action of the Weyl group
$W(E_{7})$ on a certain finite set. In Tables 1.3, 1.3, 1.3, we give a list of
$N^{(m,n)}$ for some $(m,n)$. See Section 5.2 for more detail.

$\begin{array}[]{c|cccccccccccccc}m&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\\
\hline\cr N&1&1&2&3&5&10&16&23&37&54&70&90&101&103\\\
G&1&1&2&3&5&9&16&23&37&54&70&90&101&103\end{array}$

Table 1.1. $N^{(m,0)}=N^{(28-m,0)}$

$\begin{array}[]{c|cccccccccc}n&1&2&3&4&5&6&7&8&9&10\\\ \hline\cr
N&1&3&9&30&112&501&2483&13791&81404&490750\\\
G&1&3&7&22&71&306&1585&9831&64790&425252\end{array}$

Table 1.2. $N^{(0,n)}$

$\begin{array}[]{c|cccccccccc}(m,n)&(1,1)&(1,2)&(2,1)&(1,3)&(2,2)&(3,1)&(1,4)&(2,3)&(3,2)&(4,1)\\\
\hline\cr N&2&8&4&33&23&9&162&132&66&20\\\
G&2&8&3&30&17&8&140&95&57&17\end{array}$

$\begin{array}[]{c|ccccccccccc}(m,n)&(1,5)&(2,4)&(3,3)&(4,2)&(5,1)&(1,6)&(2,5)&(3,4)&(4,3)&(5,2)&(6,1)\\\
\hline\cr N&901&889&508&190&45&5674&6503&4348&1854&531&103\\\
G&753&670&430&164&42&4829&5259&3812&1649&501&96\end{array}$

Table 1.3. $N^{(m,n)}$

To each $\mathord{\mathcal{Q}}^{(m,n)}$-curve $Z$, we associate a discrete
invariant $g(Z)$, which we call an _intersection graph_. This invariant is
similar to the _splitting graph_ defined in [19]. Note that each of the
bitangents $\bar{l}_{1},\dots,\bar{l}_{m}$ and a $4$-tangent conics
$\bar{c}_{1},\dots,\bar{c}_{n}$ of the smooth quartic curve $Q\subset Z$
splits by the double covering $Y\to\mathord{\mathbb{P}}^{2}$ branched along
$Q$. This data $g(Z)$ describes how the irreducible components of these pull-
backs intersect on $Y$. See Section 8 for the precise definition. In Tables,
we also present the number $G$ of non-isomorphic intersection graphs obtained
from $\mathord{\mathcal{Q}}^{(m,n)}$-curves. When $n=0$, the intersection
graph $g(Z)$ is the two-graph studied in [5], in which Bannai and Ohno studied
$\mathord{\mathcal{Q}}^{(m,0)}$-curves for $m\leq 6$, and enumerated their
homeomorphism types that can be distinguished by the two-graphs. See Sections
9.1 and 9.2 for the details.

The $4$-tangent conics of a smooth quartic curve $Q$ are related to the
$2$-torsion points of the Jacobian of $Q$ (see Remark 4.3). A similar idea
applied to plane cubic curves enabled us to construct in [15] certain
equisingular families of plane curves with many connected components. In [15],
it was also shown that these connected components cannot be distinguished by
the fundamental groups of the complements, because they are all ablelian. Then
it was shown in [8] and [18] that the homeomorphism types of distinct
connected components of these families can be distinguished by the invariant
called _linking numbers_.

The embedding topology of reducible plane curves whose irreducible components
are tangent to each other was also investigated by Artal Bartolo, Cogolludo,
and Tokunaga in [2] from the view point of dihedral covering of the plane
branched along the curve. In this case, the complement can have a non-abelian
fundamental group. See [2, Corollary 1].

It would be an interesting problem to study the fundamental groups of the
complements of $\mathord{\mathcal{Q}}^{(m,n)}$-curves, and their related
invariants such as linking numbers and/or (non-)existence of finite coverings
of the plane with prescribed Galois groups.

Via the cyclic covering of the plane of degree $4$ branched along a smooth
quartic curve, the geometry of $\mathord{\mathcal{Q}}^{(m,n)}$-curves is
related to the geometry of $K3$ surfaces. By considering the double covering
branched along a singular sextic curve and employing Torelli theorem for
complex $K3$ surfaces, we have investigated in [16] Zariski $N$-ples of plane
curves of degree $6$ with only simple singularities. We expect that a similar
idea can be applied to Zariski $N$-ples associated with singular quartic
curves.

This paper is organized as follows. In Section 2, we introduce cyclic
coverings $X_{u}\to Y_{u}\to\mathord{\mathbb{P}}^{2}$ branched over a smooth
quartic curve $Q_{u}$, and fix some notation. In Section 3, we recall the
result of Harris [9]. In Section 4, we construct the family of $4$-tangent
conics. In Section 5, we construct the space $\mathord{\mathcal{Z}}^{(m,n)}$
parameterizing all $\mathord{\mathcal{Q}}^{(m,n)}$-curves, and prove Theorems
1.3 and 1.4. In Section 6, we further study the family of $4$-tangent conics
in detail. The geometry of the $K3$ surface $X_{u}$ is closely investigated.
In Section 7, we study the configurations of lifts in $Y_{u}$ of bitangents
and $4$-tangent conics, and we define the intersection graph $g(Z)$ in Section
8. In Section 9, we examine some examples for small $m$ and $n$.

Acknowledgement. Thanks are due to Professor Shinzo Bannai and Professor
Taketo Shirane for discussions and comments. The author also thanks the
anonymous referee for his/her valuable comments on the first version of this
paper.

##  2\. Coverings of $\mathord{\mathbb{P}}^{2}$

For a positive integer $d$, we put

$\Gamma(d):=H^{0}(\mathord{\mathbb{P}}^{2},\mathord{\mathcal{O}}(d)).$

Let $\mathord{\mathcal{U}}$ denote the space of smooth quartic curves, which
is a Zariski open subset of $\mathord{\mathbb{P}}_{*}(\Gamma(4))$. Let $u$ be
a point of $\mathord{\mathcal{U}}$. We denote by
$Q_{u}\subset\mathord{\mathbb{P}}^{2}$ the smooth quartic curve corresponding
to the point $u$. We consider the following branched coverings:

$\gamma_{u}\colon
X_{u}\;\smash{\mathop{\;\longrightarrow\;}\limits^{\eta_{u}}}\;Y_{u}\;\smash{\mathop{\;\longrightarrow\;}\limits^{\pi_{u}}}\;\mathord{\mathbb{P}}^{2},$

where $\pi_{u}$ is the double covering of $\mathord{\mathbb{P}}^{2}$ branched
along $Q_{u}$, $\eta_{u}$ is the double covering of $Y_{u}$ branched along
$\pi_{u}^{-1}(Q_{u})$, and $\gamma_{u}=\pi_{u}\circ\eta_{u}$ is the cyclic
covering of degree $4$ of $\mathord{\mathbb{P}}^{2}$ branched along $Q_{u}$.
We put

$SY_{u}:=H^{2}(Y_{u},\mathord{\mathbb{Z}}),$

which is a unimodular lattice of rank $8$ with the cup-product
$\langle\phantom{i},\phantom{i}\rangle_{Y}$. Let $h_{u}\in SY_{u}$ be the
class of the pull-back of a line on $\mathord{\mathbb{P}}^{2}$ by $\pi_{u}$.
It is well known that $Y_{u}$ is a del Pezzo surface of degree $2$ with the
anti-canonical class $h_{u}$. (See [7, Chapters 6 and 8] about del Pezzo
surfaces.) On the other hand, the surface $X_{u}$ is a $K3$ surface. Let
$\langle\phantom{i},\phantom{i}\rangle_{X}$ denote the cup product of
$H^{2}(X_{u},\mathord{\mathbb{Z}})$, and let $\tilde{h}_{u}$ be the class
$\eta_{u}^{*}(h_{u})$. Then $\tilde{h}_{u}$ is an ample class of degree
$\langle\tilde{h}_{u},\tilde{h}_{u}\rangle_{X}=4$.

It is classically known that every smooth quartic curve $Q_{u}$ has exactly
$28$ bitangents. Moreover, if $u$ is general in $\mathord{\mathcal{U}}$, all
bitangents $l$ of $Q_{u}$ are ordinary, that is, $l$ is tangent to $Q_{u}$ at
distinct two points.

######  Definition 2.1.

A reduced conic $\bar{c}\subset\mathord{\mathbb{P}}^{2}$ is called a
_splitting conic_ of $Q_{u}$ if the intersection multiplicity of $Q_{u}$ and
$\bar{c}$ is even at each intersection point.

A smooth conic $\bar{c}$ is splitting if and only if
$\pi_{u}^{*}(\bar{c})\subset Y_{u}$ has two irreducible components. A singular
reduced conic $\bar{c}$ is splitting if and only if $\bar{c}$ is a union of
two distinct bitangents.

It is easy to see that a smooth conic $\bar{c}=\\{g=0\\}$ defined by
$g\in\Gamma(2)$ is a splitting conic of $Q_{u}=\\{\varphi=0\\}$ defined by
$\varphi\in\Gamma(4)$ if and only if there exist polynomials $f\in\Gamma(2)$
and $q\in\Gamma(2)$ such that $\varphi=fg+q^{2}$. By an easy dimension
counting, we see the following:

######  Lemma 2.2.

Suppose that $u$ is general in $\mathord{\mathcal{U}}$. Let
$\bar{c}\subset\mathord{\mathbb{P}}^{2}$ be a smooth splitting conic of
$Q_{u}$. Then the intersection multiplicities of $Q_{u}$ and $\bar{c}$ are
either $(2,2,2,2)$ or $(4,2,2)$. $\Box$

######  Definition 2.3.

A smooth splitting conic $\bar{c}\subset\mathord{\mathbb{P}}^{2}$ of $Q_{u}$
is called a _$4$ -tangent conic_ (resp. a _$3$ -tangent conic_) of $Q_{u}$ if
the intersection multiplicities of $Q_{u}$ and $\bar{c}$ are $(2,2,2,2)$
(resp. $(4,2,2)$).

The following is easy to verify. The results are summarized in Table 2.1.

$\begin{array}[]{l | l |l}\mathord{\mathbb{P}}^{2}&Y_{u}&X_{u}\\\ \hline\cr\bar{l}&\pi_{u}^{*}(\bar{l})=l+l^{\prime}&\gamma_{u}^{*}(\bar{l})=\tilde{l}+\tilde{l}^{\prime}\\\ &\langle l,l\rangle_{Y}=\langle l^{\prime},l^{\prime}\rangle_{Y}=-1&\langle\tilde{l},\tilde{l}\rangle_{X}=\langle\tilde{l}^{\prime},\tilde{l}^{\prime}\rangle_{X}=-2\\\ &\langle l,l^{\prime}\rangle_{Y}=2&\langle\tilde{l},\tilde{l}^{\prime}\rangle_{X}=4\\\ &\langle h_{u},l\rangle_{Y}=\langle h_{u},l^{\prime}\rangle_{Y}=1&\langle\tilde{h}_{u},\tilde{l}\rangle_{X}=\langle\tilde{h}_{u},\tilde{l}^{\prime}\rangle_{X}=2\\\ \hline\cr\bar{c}&\pi_{u}^{*}(\bar{c})=c+c^{\prime}&\gamma_{u}^{*}(\bar{c})=\tilde{c}+\tilde{c}^{\prime}\\\ &\langle c,c\rangle_{Y}=\langle c^{\prime},c^{\prime}\rangle_{Y}=0&\langle\tilde{c},\tilde{c}\rangle_{X}=\langle\tilde{c}^{\prime},\tilde{c}^{\prime}\rangle_{X}=0\\\ &\langle c,c^{\prime}\rangle_{Y}=4&\langle\tilde{c},\tilde{c}^{\prime}\rangle_{X}=8\\\ &\langle h_{u},c\rangle_{Y}=\langle h_{u},c^{\prime}\rangle_{Y}=2&\langle\tilde{h}_{u},\tilde{c}\rangle_{X}=\langle\tilde{h}_{u},\tilde{c}^{\prime}\rangle_{X}=4\end{array}$ Table 2.1. Pull-backs of bitangents and $4$-tangent conics

######  Proposition 2.4.

(1) Let $\bar{l}$ be an ordinary bitangent of $Q_{u}$. Then
$\pi_{u}^{*}(\bar{l})$ is a union of two smooth rational curves $l$ and
$l^{\prime}$ on $Y_{u}$ with self-intersection $-1$ that intersect at two
points transversely, and $\gamma_{u}^{*}(\bar{l})$ is a union of two smooth
rational curves $\tilde{l}$ and $\tilde{l}^{\prime}$ on $X_{u}$ with self-
intersection $-2$ that intersect at two points with intersection
multiplicities $(2,2)$.

(2) Let $\bar{c}$ be a $4$-tangent conic of $Q_{u}$. Then
$\pi_{u}^{*}(\bar{c})$ is a union of two smooth rational curves $c$ and
$c^{\prime}$ on $Y_{u}$ with self-intersection $0$ that intersect at four
points transversely, and $\gamma_{u}^{*}(\bar{c})$ is a union of two smooth
elliptic curves $\tilde{c}$ and $\tilde{c}^{\prime}$ on $X_{u}$ with self-
intersection $0$ that intersect at four points with intersection
multiplicities $(2,2,2,2)$. $\Box$

######  Definition 2.5.

A curve $l$ on $Y_{u}$ is called a _$Y$ -lift_ of a bitangent $\bar{l}$ of
$Q_{u}$ if $\pi_{u}$ maps $l$ to $\bar{l}$ isomorphically. We also say that a
curve $c$ on $Y_{u}$ is a _$Y$ -lift_ of a splitting conic $\bar{c}$ of
$Q_{u}$ if $\pi_{u}$ maps $c$ to $\bar{c}$ isomorphically.

##  3\. Monodromy

Let $u$ be a point of $\mathord{\mathcal{U}}$. It is well known that the
lattice $SY_{u}=H^{2}(Y_{u},\mathord{\mathbb{Z}})$ is isomorphic to the
lattice of rank $8$ whose Gram matrix is the diagonal matrix
$\mathord{\mathrm{diag}}(1,-1,\dots,-1)$, and that the orthogonal complement

$\Sigma_{u}:=(\mathord{\mathbb{Z}}h_{u}\hookrightarrow SY_{u})^{\perp}$

of the ample class $h_{u}$ in $SY_{u}$ is isomorphic to the negative-definite
root lattice of type $E_{7}$. The deck transformation

$\iota_{u}\colon Y_{u}\to Y_{u}$

of $\pi_{u}\colon Y_{u}\to\mathord{\mathbb{P}}^{2}$ acts on $\Sigma_{u}$ as
$-1$. Note that the group $\mathord{\mathrm{O}}(\Sigma_{u})$ of isometries of
$\Sigma_{u}$ is equal to the Weyl group $W(E_{7})$, which is of order
$2903040$. Hence there exists an injective homomorphism

(3.1)
$\mathord{\mathrm{O}}(SY_{u},h_{u}):=\\{\,{g\in\mathord{\mathrm{O}}(SY_{u})}\,|\,{h_{u}^{g}=h_{u}}\,\\}\;\;\hookrightarrow\;\;W(E_{7}).$

It is easy to check that the action on $\Sigma_{u}$ of each of the standard
generators of $W(E_{7})$ lifts to an isometry of $SY_{u}$ that fixes $h_{u}$.
Hence the homomorphism (3.1) is in fact an isomorphism. The family of lattices
$\\{\,{SY_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$ forms a locally constant
system

$\mathrm{\mathcal{SY}}\to\mathord{\mathcal{U}}.$

Let $b$ be a general point of $\mathord{\mathcal{U}}$, which will serve as a
base point of $\mathord{\mathcal{U}}$. The monodromy action of
$\pi_{1}(\mathord{\mathcal{U}},b)$ on the lattice $SY_{b}$ preserves $h_{b}\in
SY_{b}$.

######  Theorem 3.1 (Harris [9]).

The monodromy homomorphism

(3.2)
$\pi_{1}(\mathord{\mathcal{U}},b)\to\mathord{\mathrm{O}}(SY_{b},h_{b})\cong
W(E_{7})$

is surjective.

The original statement in [9] is not on the monodromy action on the lattice
$SY_{b}$, but on the Galois group $W(E_{7})/\\{\pm
1\\}\cong\mathord{\mathrm{GO}}_{6}(\mathord{\mathbb{F}}_{2})$ of bitangents of
$Q_{b}$. Moreover the proof in [9] is via the proof of a similar result on
cubic surfaces with $E_{7}$ replaced by $E_{6}$. Hence we give a direct and
simple proof of Theorem 3.1 below.

For the proof, we prepare some more notation, which will be used throughout
this paper. We denote by $\overline{L}_{u}$ the set of bitangents of $Q_{u}$,
and $L_{u}$ the set of $Y$-lifts of bitangents of $Q_{u}$. Let
$\Sigma_{u}^{\vee}$ denote the dual lattice of $\Sigma_{u}$. By identifying
$l\in L_{u}$ with its class $[l]\in SY_{u}$, we have an identification

$\displaystyle L_{u}$ $\displaystyle\;=\;$ $\displaystyle\\{\,{v\in
SY_{u}}\,|\,{\langle v,h_{u}\rangle_{Y}=1,\;\langle v,v\rangle_{Y}=-1}\,\\}$
$\displaystyle\;\cong\;$
$\displaystyle\\{\,{v\in\Sigma_{u}^{\vee}}\,|\,{\langle
v,v\rangle_{Y}=-3/2}\,\\},$

where the second bijection is obtained by the orthogonal projection
$SY_{u}\to\Sigma_{u}^{\vee}$. We put
$\overline{SY}_{u}:=SY_{u}/\langle\iota_{u}\rangle$, and consider the
commutative diagram

(3.7) $\displaystyle\begin{array}[]{ccc}L_{u}&\hookrightarrow&SY_{u}\\\
\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow\\\
\overline{L}_{u}&\hookrightarrow&\overline{SY}_{u},\end{array}$

where vertical arrows are quotient maps by the involution $\iota_{u}$. Since
the action of $\pi_{1}(\mathord{\mathcal{U}},b)$ on $SY_{b}$ commutes with
$\iota_{b}$, we have a monodromy action of $\pi_{1}(\mathord{\mathcal{U}},b)$
on $\overline{SY}_{b}$. Thus we obtain a diagram

(3.11)
$\displaystyle\begin{array}[]{ccc}\mathord{\mathcal{L}}&\hookrightarrow&\mathrm{\mathcal{SY}}\\\
\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow\\\
\overline{\mathord{\mathcal{L}}}&\hookrightarrow&\overline{\mathrm{\mathcal{SY}}}\end{array}$

of locally constant systems over $\mathord{\mathcal{U}}$ parameterizing the
diagram (3.7) over $\mathord{\mathcal{U}}$, where vertical arrows are quotient
maps by the family of involutions

$\iota_{\hskip
0.3pt\mathord{\mathcal{U}}}:=\\{\,{\iota_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}.$

Note that $\overline{\mathord{\mathcal{L}}}$ is the space parameterizing all
bitangents of smooth quartic curves.

###### Proof of Theorem 3.1.

Let $L^{\\{7\\}}_{u}$ (resp. $L^{[7]}_{u}$) be the set of non-ordered
$7$-tuples $\\{l_{1},\dots,l_{7}\\}$ (resp. ordered $7$-tuples
$[l_{1},\dots,l_{7}]$) of elements $l_{1},\dots,l_{7}\in L_{u}$ such that
$\langle l_{i},l_{j}\rangle_{Y}=0$ for $i\neq j$. By (3), we can enumerate all
elements of $L^{\\{7\\}}_{u}$. It turns out that $|L^{\\{7\\}}_{u}|=576$, and
hence

(3.12) $|L^{[7]}_{u}|=576\cdot 7!=2903040=|W(E_{7})|.$

(See also Remark 6.6.) For $7$-tuples $\lambda=[l_{1},\dots,l_{7}]$ and
$\lambda^{\prime}=[l^{\prime}_{1},\dots,l^{\prime}_{7}]$ in $L^{[7]}_{u}$,
there exists a unique isometry
$g_{\lambda,\lambda^{\prime}}\in\mathord{\mathrm{O}}(SY_{u}\otimes\mathord{\mathbb{Q}})$
such that

$g_{\lambda,\lambda^{\prime}}(h_{u})=h_{u},\quad
g_{\lambda,\lambda^{\prime}}(l_{i})=l^{\prime}_{i}\quad(i=1,\dots,7).$

It is enough show that, when $u=b$, these elements
$g_{\lambda,\lambda^{\prime}}$ are contained in the image of the monodromy
(3.2). Indeed, by (3.12), this claim implies that these isometries
$g_{\lambda,\lambda^{\prime}}$ constitute the whole group
$\mathord{\mathrm{O}}(SY_{b},h_{b})\cong W(E_{7})$. To prove this claim, it is
enough to show that $\pi_{1}(\mathord{\mathcal{U}},b)$ acts on $L^{[7]}_{b}$
transitively by the monodromy, or equivalently, to show that the total space
$\mathord{\mathcal{L}}^{[7]}$ of the locally constant system

$\mathord{\mathcal{L}}^{[7]}\to\mathord{\mathcal{U}}$

obtained from the family
$\\{\,{L^{[7]}_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$ is connected.

Let $\lambda=[l_{1},\dots,l_{7}]$ be a point of $\mathord{\mathcal{L}}^{[7]}$
over $u\in\mathord{\mathcal{U}}$. Contracting the $(-1)$-curves
$l_{1},\dots,l_{7}$, we obtain a birational morphism

$\mathord{\mathrm{bl}}_{\lambda}\colon Y_{u}\to\mathord{\mathbf{P}}_{\lambda}$

to a projective plane $\mathord{\mathbf{P}}_{\lambda}$. We put
$\beta_{\lambda}:=[\mathord{\mathrm{bl}}_{\lambda}(l_{1}),\dots,\mathord{\mathrm{bl}}_{\lambda}(l_{7})]$.
Conversely, we fix a projective plane $\mathord{\mathbf{P}}$, and let
$\mathord{\mathbf{P}}^{[7]}$ denote the set of ordered $7$-tuples
$[p_{1},\dots,p_{7}]$ of distinct points of $\mathord{\mathbf{P}}$. For a
general point $\beta=[p_{1},\dots,p_{7}]$ of $\mathord{\mathbf{P}}^{[7]}$, let

$\mathord{\mathrm{bl}}^{\beta}\colon Y^{\beta}\to\mathord{\mathbf{P}}$

be the blowing-up at the points $p_{1},\dots,p_{7}$. Then $Y^{\beta}$ is a del
Pezzo surface of degree $2$, and the complete linear system of the anti-
canonical divisor on $Y^{\beta}$ gives a double covering
$Y^{\beta}\to\mathord{\mathbb{P}}^{2}$ branched along a smooth quartic curve
$Q^{\beta}$ such that each of the $7$ exceptional curves over
$p_{1},\dots,p_{7}$ is a $Y$-lift of a bitangent of $Q^{\beta}$. Hence there
exist a point $\lambda\in\mathord{\mathcal{L}}^{[7]}$ and an isomorphism
$\mathord{\mathbf{P}}_{\lambda}\cong\mathord{\mathbf{P}}$ that maps
$\beta_{\lambda}$ to $\beta$.

We put

$\mathord{\mathcal{I}}:=\left\\{\;{(\lambda,\gamma,\beta)}\;\left|\;{\;\;\parbox{199.16928pt}{$\lambda\in\mathord{\mathcal{L}}^{[7]}$,
$\beta\in\mathord{\mathbf{P}}^{[7]}$, and $\gamma$ is an isomorphism
$\mathord{\mathbf{P}}_{\lambda}\cong\mathord{\mathbf{P}}$ that maps
$\beta_{\lambda}$ to $\beta$}}\;\right.\right\\}.$

Then the projection $\mathord{\mathcal{I}}\to\mathord{\mathcal{L}}^{[7]}$ is
surjective with fibers isomorphic to
$\mathord{\mathrm{PGL}}_{3}(\mathord{\mathbb{C}})$, whereas the projection
$\mathord{\mathcal{I}}\to\mathord{\mathbf{P}}^{[7]}$ is dominant with fibers
isomorphic to $\mathord{\mathrm{PGL}}_{3}(\mathord{\mathbb{C}})$. Since
$\mathord{\mathbf{P}}^{[7]}$ and
$\mathord{\mathrm{PGL}}_{3}(\mathord{\mathbb{C}})$ are connected, we see that
$\mathord{\mathcal{L}}^{[7]}$ is connected. $\Box$

##  4\. Family of $4$-tangent conics

In this section, we construct a space $\overline{\mathord{\mathcal{C}}}$
parameterizing all $4$-tangent conics of smooth quartic curves.

Let $\overline{C}_{u}$ denote the set of $4$-tangent conics of $Q_{u}$, and
let $C_{u}$ be the set of $Y$-lifts of $4$-tangent conics of $Q_{u}$. We put

(4.1) $F_{u}:=\\{\,{v\in SY_{u}}\,|\,{\langle v,h_{u}\rangle_{Y}=2,\langle
v,v\rangle_{Y}=0}\,\\}\;\cong\;\\{\,{v\in\Sigma_{u}}\,|\,{\langle
v,v\rangle_{Y}=-2}\,\\},$

where the second bijection is given by the orthogonal projection
$SY_{u}\to\Sigma_{u}^{\vee}$. As was shown in Table 2.1, we have $[c]\in
F_{u}$ for any $c\in C_{u}$. Note that $|F_{u}|=126$, the number of roots of
the root lattice $\Sigma_{u}$ of type $E_{7}$. We put

$\overline{F}_{u}:=F_{u}/\langle\iota_{u}\rangle\;\;\subset\;\;\overline{SY}_{u}.$

Then we have a commutative diagram

(4.2)
$\begin{array}[]{ccccc}C_{u}&\;\smash{\mathop{\;\longrightarrow\;}\limits^{\Phi_{u}}}&F_{u}&\hookrightarrow&SY_{u}\\\
\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow\\\
\overline{C}_{u}&\;\smash{\mathop{\;\longrightarrow\;}\limits^{\overline{\Phi}_{u}}}&\overline{F}_{u}&\hookrightarrow&\overline{SY}_{u}\end{array}$

where $\Phi_{u}\colon C_{u}\to F_{u}$ is given by $c\mapsto[c]\in SY_{u}$, and
the vertical arrows are quotient by the involution $\iota_{u}\colon Y_{u}\to
Y_{u}$. We have locally constant systems
$\mathord{\mathcal{F}}\to\mathord{\mathcal{U}}$ and
$\overline{\mathord{\mathcal{F}}}\to\mathord{\mathcal{U}}$ obtained from the
families $\\{\,{F_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$ and
$\\{\,{\overline{F}_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$.

######  Theorem 4.1.

There exists a commutative diagram

(4.3)
$\begin{array}[]{ccccc}\mathord{\mathcal{C}}&\;\smash{\mathop{\;\longrightarrow\;}\limits^{\Phi_{\hskip
0.3pt\mathord{\mathcal{U}}}}}&\mathord{\mathcal{F}}&\hookrightarrow&\mathrm{\mathcal{SY}}\\\
\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow&&\phantom{\Big{\downarrow}}\hskip-8.0pt\downarrow\\\
\overline{\mathord{\mathcal{C}}}&\;\smash{\mathop{\;\longrightarrow\;}\limits^{\overline{\Phi}_{\hskip
0.3pt\mathord{\mathcal{U}}}}}&\overline{\mathord{\mathcal{F}}}&\hookrightarrow&\overline{\mathrm{\mathcal{SY}}}\end{array}$

of morphisms over $\mathord{\mathcal{U}}$ that parameterizes the diagrams
(4.2) over $\mathord{\mathcal{U}}$. The morphisms $\Phi_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\mathord{\mathcal{C}}\to\mathord{\mathcal{F}}$
and $\overline{\Phi}_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\overline{\mathord{\mathcal{C}}}\to\overline{\mathord{\mathcal{F}}}$
are smooth and surjective, and every fiber of them is a Zariski open subset of
$\mathord{\mathbb{P}}^{1}$.

For the proof, we use the double covering $\eta_{u}\colon X_{u}\to Y_{u}$ of
$Y_{u}$ by the $K3$ surface $X_{u}$. We consider the Néron-Severi lattice

$S\hskip-0.9ptX_{u}:=H^{2}(X_{u},\mathord{\mathbb{Z}})\cap H^{1,1}(X_{u})$

with the intersection form $\langle\,,\,\rangle_{X}$. Then $\eta_{u}$ induces
a embedding of the lattice

$\eta_{u}^{*}\colon SY_{u}(2)\hookrightarrow S\hskip-0.9ptX_{u},$

where $SY_{u}(2)$ is the lattice obtained from $SY_{u}$ by multiplying the
intersection form by $2$. Let $j_{u}\colon X_{u}\to X_{u}$ be a generator of
the cyclic group
$\operatorname{\mathrm{Gal}}\nolimits(X_{u}/\mathord{\mathbb{P}}^{2})$ of
order $4$. Then $\eta_{u}\colon X_{u}\to Y_{u}$ is the quotient morphism by
$j_{u}^{2}$. Hence $j_{u}^{2}$ acts on the image of $\eta_{u}^{*}\colon
SY_{u}(2)\hookrightarrow S\hskip-0.9ptX_{u}$ trivially.

###### Proof of Theorem 4.1.

Note that the family of involutions $\iota_{\hskip
0.3pt\mathord{\mathcal{U}}}=\\{\,{\iota_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$
acts on $\mathord{\mathcal{F}}$ over $\mathord{\mathcal{U}}$ without fixed
points. Hence, if the parameterizing space $\Phi_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\mathord{\mathcal{C}}\to\mathord{\mathcal{F}}$
of $\Phi_{u}\colon C_{u}\to F_{u}$ is constructed, then
$\overline{\Phi}_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\overline{\mathord{\mathcal{C}}}\to\overline{\mathord{\mathcal{F}}}$
is constructed as a quotient of $\Phi_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\mathord{\mathcal{C}}\to\mathord{\mathcal{F}}$
by $\iota_{\hskip 0.3pt\mathord{\mathcal{U}}}$.

Let $u$ be an arbitrary point of $\mathord{\mathcal{U}}$, and let $v$ be an
element of $F_{u}\subset SY_{u}$. We put $\tilde{v}:=\eta_{u}^{*}(v)\in
S\hskip-0.9ptX_{u}$. We can easily confirm that there exist exactly $6$ pairs
$\\{l_{i},l^{\prime}_{i}\\}$ ($i=1,\dots,6$) of $Y$-lifts of bitangents of
$Q_{u}$ such that $\langle l_{i},l_{i}^{\prime}\rangle_{Y}=1$ and
$v=[l_{i}]+[l_{i}^{\prime}]$, and that these $12$ curves
$l_{1},l_{1}^{\prime},\dots,l_{6},l_{6}^{\prime}$ are distinct. Hence the
complete linear system on the $K3$ surface $X_{u}$ corresponding to
$\tilde{v}\in S\hskip-0.9ptX_{u}$ has no fixed components. The class
$\tilde{v}$ is primitive in $S\hskip-0.9ptX_{u}$ with
$\langle\tilde{v},\tilde{v}\rangle_{X}=0$ and
$\langle\tilde{h}_{u},\tilde{v}\rangle_{X}=4$. Therefore there exists an
elliptic fibration on $X_{u}$ such that the class of a fiber is equal to
$\tilde{v}$. We denote this elliptic fibration by $\phi_{v}\colon
X_{u}\to\mathord{\mathbb{P}}^{1}$ .

If $c\in C_{u}$, then $\eta_{u}^{*}(c)$ is an elliptic curve, and hence
$\eta_{u}^{*}(c)$ is a smooth fiber of an elliptic fibration of
$\phi_{v}\colon X_{u}\to\mathord{\mathbb{P}}^{1}$, where $v=[c]$. Conversely,
suppose that $(u,v)\in\mathord{\mathcal{F}}$, and let $f$ be a smooth fiber of
the elliptic fibration $\phi_{v}\colon X_{u}\to\mathord{\mathbb{P}}^{1}$. We
denote by $a\subset\mathord{\mathbb{P}}^{2}$ the plane curve $\gamma_{u}(f)$
with the reduced structure. Let $d$ be the degree of $a$, and $\delta$ the
mapping degree of $\gamma_{u}|f\colon f\to a$. Since
$\langle\tilde{h}_{u},f\rangle_{X}=4$ and $\gamma_{u}\colon
X_{u}\to\mathord{\mathbb{P}}^{2}$ is Galois, we have $(d,\delta)=(1,4)$,
$(2,2)$, or $(4,1)$. If $(d,\delta)=(1,4)$, then $f=\gamma_{u}^{-1}(a)$ is
invariant under the action of
$\operatorname{\mathrm{Gal}}\nolimits(X_{u}/\mathord{\mathbb{P}}^{2})=\langle
j_{u}\rangle$, and hence the class $[f]\in S\hskip-0.9ptX_{u}$ is a non-zero
multiple of $\tilde{h}_{u}$, which contradicts $\langle f,f\rangle_{X}=0$. If
$(d,\delta)=(4,1)$, then $f$, $j_{u}(f)$, $j_{u}^{2}(f)$, $j_{u}^{3}(f)$ are
distinct curves that intersect over the points of $a\cap Q_{u}$. On the other
hand, since
$[f]=\eta_{u}^{*}(v)\in\operatorname{\mathrm{Im}}\nolimits\eta_{u}^{*}$, we
have $j_{u}^{*2}([f])=[f]$. This contradicts $\langle f,f\rangle_{X}=0$. Hence
$(d,\delta)=(2,2)$, and we see that $a$ is a smooth splitting conic. Note that
$a$ is $4$-tangent, because otherwise $f$ would be singular. Thus we have
proved that $c\mapsto\eta_{u}^{*}(c)$ gives a bijection from $C_{u}$ to the
union of the sets of smooth fibers of elliptic fibrations $\phi_{v}\colon
X_{u}\to\mathord{\mathbb{P}}^{1}$, where $v$ runs through $F_{u}$.

Let $\mathord{\mathcal{X}}\to\mathord{\mathcal{U}}$ be the universal family of
$\\{\,{X_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$, and let
$\pi_{\mathord{\mathcal{F}}}\colon\mathord{\mathcal{F}}\times_{\mathord{\mathcal{U}}}\mathord{\mathcal{X}}\to\mathord{\mathcal{F}}$
be the pull-back of $\mathord{\mathcal{X}}\to\mathord{\mathcal{U}}$ by
$\mathord{\mathcal{F}}\to\mathord{\mathcal{U}}$. Let $\mathord{\mathcal{M}}$
be a line bundle on
$\mathord{\mathcal{F}}\times_{\mathord{\mathcal{U}}}\mathord{\mathcal{X}}$
such that the class $[\mathord{\mathcal{M}}|X_{u}]\in S\hskip-0.9ptX_{u}$ of
the line bundle $\mathord{\mathcal{M}}|X_{u}$ on
$\pi_{\mathord{\mathcal{F}}}^{-1}(u,v)=X_{u}$ is equal to $v\in F_{u}$. Then
$\pi_{\mathord{\mathcal{F}}*}\mathord{\mathcal{M}}\to\mathord{\mathcal{F}}$ is
a vector bundle of rank $2$. The fiber over $(u,v)\in\mathord{\mathcal{F}}$ of
the $\mathord{\mathbb{P}}^{1}$-bundle
$\mathord{\mathbb{P}}_{*}(\pi_{\mathord{\mathcal{F}}*}\mathord{\mathcal{M}})\to\mathord{\mathcal{F}}$
is identified with the base curve of the elliptic fibration $\phi_{v}\colon
X_{u}\to\mathord{\mathbb{P}}^{1}$. We can construct $\mathord{\mathcal{C}}$ as
the open subset of
$\mathord{\mathbb{P}}_{*}(\pi_{\mathord{\mathcal{F}}*}\mathord{\mathcal{M}})$
consisting of non-critical points of $\phi_{v}\colon
X_{u}\to\mathord{\mathbb{P}}^{1}$. $\Box$

The non-singular varieties $\mathord{\mathcal{C}}$ and
$\overline{\mathord{\mathcal{C}}}$ parameterize all pairs $(u,c)$ and
$(u,\bar{c})$, respectively, where $u\in\mathord{\mathcal{U}}$ and $c\in
C_{u}$, $\bar{c}\in\overline{C}_{u}$. Since $\Phi_{u}$ and
$\overline{\Phi}_{u}$ have connected fibers, we can regard $F_{u}$ as the set
of connected families of $Y$-lifts of $4$-tangent conics of $Q_{u}$, and
$\overline{F}_{u}$ as the set of connected families of $4$-tangent conics. The
following observation obtained in the proof of Theorem 4.1 will be used in the
next section.

######  Proposition 4.2.

Every connected family $[c]\in F_{u}$ of $Y$-lifts of splitting conics is a
pencil with no base points. $\Box$

######  Remark 4.3.

A line section $\Lambda_{u}$ of $Q_{u}\subset\mathord{\mathbb{P}}^{2}$ is a
canonical class of the genus $3$ curve $Q_{u}$. Let
$\mathord{\mathrm{Pic}}^{0}(Q_{u})$ be the Picard group of line bundles of
degree $0$ of $Q_{u}$, and let $\mathord{\mathrm{Pic}}^{0}(Q_{u})[2]$ be the
subgroup of $2$-torsion points of $\mathord{\mathrm{Pic}}^{0}(Q_{u})$. For a
$4$-tangent conic $\bar{c}$ of $Q_{u}$, let $\Theta_{u}(\bar{c})$ be the
reduced part of the divisor $\bar{c}\cap Q_{u}$ of $Q_{u}$. Then the class of
the divisor $\Theta_{u}(\bar{c})-\Lambda_{u}$ of degree $0$ is a point of
$\mathord{\mathrm{Pic}}^{0}(Q_{u})[2]-\\{0\\}$, and this correspondence gives
a bijection
$\overline{F}_{u}\cong\mathord{\mathrm{Pic}}^{0}(Q_{u})[2]-\\{0\\}$.

##  5\. Proof of the main results

In this section, we construct the space $\mathord{\mathcal{Z}}^{(m,n)}$
parameterizing all $\mathord{\mathcal{Q}}^{(m,n)}$-curves, and prove Theorems
1.3 and 1.4.

###  5.1. Deformation types

We fix some notation. For a set $A$, let $S^{k}(A)$ denote the symmetric
product $A^{k}/\mathord{\mathfrak{S}}_{k}$, where $A^{k}=A\times\dots\times A$
($k$ times), and let $S_{0}^{k}(A)$ denote the complement in $S^{k}(A)$ of the
image of the big diagonal in $A^{k}$.

For a morphism $\mathord{\mathcal{A}}\to\mathord{\mathcal{U}}$, let
$\mathord{\mathcal{S}}^{k}(\mathord{\mathcal{A}})$ denote the symmetric
product $\mathord{\mathcal{A}}^{k}/\mathord{\mathfrak{S}}_{k}$, where
$\mathord{\mathcal{A}}^{k}:=\mathord{\mathcal{A}}\times_{\mathord{\mathcal{U}}}\dots\times_{\mathord{\mathcal{U}}}\mathord{\mathcal{A}}$
($k$ times), and let $\mathord{\mathcal{S}}_{0}^{k}(\mathord{\mathcal{A}})$
denote the complement in $\mathord{\mathcal{S}}^{k}(\mathord{\mathcal{A}})$ of
the image of the big diagonal in $\mathord{\mathcal{A}}^{k}$. Note that, if
$\mathord{\mathcal{A}}$ is smooth over $\mathord{\mathcal{U}}$ with relative
dimension $1$, then $\mathord{\mathcal{S}}^{k}(\mathord{\mathcal{A}})$ is
smooth over $\mathord{\mathcal{U}}$ with relative dimension $k$.

Recall that $\overline{\mathord{\mathcal{L}}}\to\mathord{\mathcal{U}}$ and
$\overline{\mathord{\mathcal{C}}}\to\mathord{\mathcal{U}}$ are the spaces
parameterizing all bitangents and all $4$-tangent conics of smooth quartic
curves, respectively. We put

$\mathord{\mathcal{Z}}^{\prime(m,n)}:=\mathord{\mathcal{S}}_{0}^{m}(\overline{\mathord{\mathcal{L}}})\times_{\mathord{\mathcal{U}}}\mathord{\mathcal{S}}_{0}^{n}(\overline{\mathord{\mathcal{C}}}),$

which is the space parameterizing all curves $Z^{\prime}$, where $Z^{\prime}$
is a union of a smooth quartic curve $Q$, $m$ distinct bitangents of $Q$, and
$n$ distinct $4$-tangent conics of $Q$. Now we can construct the parameter
space

$\varpi\colon\mathord{\mathcal{Z}}^{(m,n)}\to\mathord{\mathcal{U}}$

of $\mathord{\mathcal{Q}}^{(m,n)}$-curves as the open subvariety of
$\mathord{\mathcal{Z}}^{\prime(m,n)}$ consisting of points corresponding to
plane curves $Z^{\prime}$ satisfying conditions (i), (ii), (iii) in Definition
1.1. For a point $\zeta\in\mathord{\mathcal{Z}}^{(m,n)}$, we denote by
$Z_{\zeta}$ the $\mathord{\mathcal{Q}}^{(m,n)}$-curve corresponding to
$\zeta$.

For $u\in\mathord{\mathcal{U}}$, we put

$P^{(m,n)}_{u}:=S_{0}^{m}(\overline{L}_{u})\times
S^{n}(\overline{F}_{u})\;\;\subset\;\;S_{0}^{m}(\overline{SY}_{u})\times
S^{n}(\overline{SY}_{u}).$

The size of $P^{(m,n)}_{u}$ is equal to $d^{(m,n)}$ defined by (1.2). Then we
obtain a finite étale covering

$\rho\colon\mathord{\mathcal{P}}^{(m,n)}:=\mathord{\mathcal{S}}_{0}^{m}(\overline{\mathord{\mathcal{L}}})\times_{\mathord{\mathcal{U}}}\mathord{\mathcal{S}}^{n}(\overline{\mathord{\mathcal{F}}})\to\mathord{\mathcal{U}}$

of degree $d^{(m,n)}$ parameterizing the family
$\\{\,{P^{(m,n)}_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$. Using
$\overline{\Phi}_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\overline{\mathord{\mathcal{C}}}\to\overline{\mathord{\mathcal{F}}}$
in Theorem 4.1, we have a morphism
$\theta^{\prime}\colon\mathord{\mathcal{Z}}^{\prime(m,n)}\to\mathord{\mathcal{P}}^{(m,n)}$.
Restricting $\theta^{\prime}$ to the open subvariety
$\mathord{\mathcal{Z}}^{(m,n)}\subset\mathord{\mathcal{Z}}^{\prime(m,n)}$, we
obtain a morphism

$\theta\colon\mathord{\mathcal{Z}}^{(m,n)}\to\mathord{\mathcal{P}}^{(m,n)},$

which maps $\zeta\in\mathord{\mathcal{Z}}^{(m,n)}$ to

(5.1)
$\theta(\zeta):=(\\{\bar{l}_{1},\dots,\bar{l}_{m}\\},[[\bar{c}_{1}],\dots,[\bar{c}_{n}]])\;\in\;P^{(m,n)}_{\varpi(\zeta)},$

where $Z_{\zeta}$ has the irreducible components as in (5.4). Thus we obtain
the following commutative diagram.

$\begin{array}[]{ccccc}\mathord{\mathcal{Z}}^{(m,n)}&&\raise
2.0pt\hbox{$\;\smash{\mathop{\;\longrightarrow\;}\limits^{\theta}}\;$}&&\mathord{\mathcal{P}}^{(m,n)}\\\
&\hskip-10.0pt\hbox to0.0pt{\hss\tiny$\varpi$}\searrow&&\swarrow\hbox
to0.0pt{\tiny$\rho$\hss}\hskip-10.0pt\\\ &&\mathord{\mathcal{U}}&&\end{array}$

To investigate the image of $\theta$, we put

$\mathord{\mathcal{U}}^{\prime}:=\left\\{\;\;{u\in\mathord{\mathcal{U}}}\;\;\left|\;\;\vbox{\hbox{\parbox{227.62204pt}{every
bitangent of $Q_{u}$ is ordinary, and their union has only ordinary nodes as
its singularities}}}\;\;\right.\right\\},$

which is a Zariski open dense subset of $\mathord{\mathcal{U}}$.

######  Lemma 5.1.

The morphism
$\theta\colon\mathord{\mathcal{Z}}^{(m,n)}\to\mathord{\mathcal{P}}^{(m,n)}$ is
smooth with each non-empty fiber being an irreducible variety of dimension
$n$. The image of $\theta$ contains
$\rho^{-1}(\mathord{\mathcal{U}}^{\prime})$. In particular, the morphism
$\theta$ is dominant.

###### Proof.

Since $\overline{\Phi}_{\hskip
0.3pt\mathord{\mathcal{U}}}\colon\overline{\mathord{\mathcal{C}}}\to\overline{\mathord{\mathcal{F}}}$
is smooth and surjective with each fiber being a Zariski open subset of
$\mathord{\mathbb{P}}^{1}$, the morphism
$\theta^{\prime}\colon\mathord{\mathcal{Z}}^{\prime(m,n)}\to\mathord{\mathcal{P}}^{(m,n)}$
is smooth and surjective with each fiber being an irreducible variety of
dimension $n$. Suppose that $u\in\mathord{\mathcal{U}}^{\prime}$, and let
$p:=(\\{\bar{l}_{1},\dots,\bar{l}_{m}\\},[[\bar{c}_{1}],\dots,[\bar{c}_{n}]])$
be a point of $P^{(m,n)}_{u}$. By Proposition 4.2 and Bertini’s theorem, if we
choose each $4$-tangent conic $\bar{c}_{j}^{\prime}$ in the connected family
$[\bar{c}_{j}]\in\overline{F}_{u}$ generally, the curve
$Q_{u}+\sum\bar{l}_{i}+\sum\bar{c}_{j}^{\prime}$ satisfies conditions (ii) and
(iii) in Definition 1.1. Hence
$\theta^{-1}(p)=\theta^{\prime-1}(p)\cap\mathord{\mathcal{Z}}^{(m,n)}$ is non-
empty. $\Box$

###### Proof of Theorem 1.4.

By Lemma 5.1, the connected components of $\mathord{\mathcal{Z}}^{(m,n)}$ are
in bijective correspondence with the connected components of
$\mathord{\mathcal{P}}^{(m,n)}$, and hence with the
$\pi_{1}(\mathord{\mathcal{U}},b)$-orbits in $P^{(m,n)}_{b}$. By Theorem 3.1,
the number $N^{(m,n)}$ of $\pi_{1}(\mathord{\mathcal{U}},b)$-orbits in
$P^{(m,n)}_{b}$ satisfies (1.3), because $|W(E_{7})/\\{\pm 1\\}|=1451520$.
$\Box$

###  5.2. Computation of $N^{(m,n)}$

Recall that $\Sigma_{b}$ is a negative-definite root lattice of type $E_{7}$.
Let $\Sigma$ be the negative-definite root lattice of type $E_{7}$ with the
standard basis, and let $\Sigma^{\vee}$ be its dual. According to (3) and
(4.1), we define the subsets

$\overline{L}:=\\{\,{v\in\Sigma^{\vee}}\,|\,{\langle
v,v\rangle=-3/2}\,\\}/\\{\pm 1\\}$

of $\overline{\Sigma}^{\vee}:=\Sigma^{\vee}/\\{\pm 1\\}$, and

$\overline{F}:=\\{\,{v\in\Sigma}\,|\,{\langle v,v\rangle=-2}\,\\}/\\{\pm 1\\}$

of $\overline{\Sigma}:=\Sigma/\\{\pm 1\\}$. We then put

$P^{(m,n)}:=S_{0}^{m}(\overline{L})\times S^{n}(\overline{F}).$

The group $W(E_{7})$ is generated by seven standard reflections. The
permutations on $\overline{L}$ and on $\overline{F}$ induced by these
generators are easily calculated. Hence the permutations on $P^{(m,n)}$
induced by these generators are also calculated. Thus we can compute the orbit
decomposition of $P^{(m,n)}$ by $W(E_{7})$, and obtain the number $N^{(m,n)}$
of deformation types of $\mathord{\mathcal{Q}}^{(m,n)}$-curves.

######  Example 5.2.

The size $d^{(4,0)}$ of $P^{(4,0)}$ is $20475$. The group $W(E_{7})$
decomposes this set into three orbits of sizes $315,5040,15120$. Hence
$N^{(4,0)}=3$.

######  Example 5.3.

The size $d^{(0,4)}$ of $P^{(0,4)}$ is $720720$. The group $W(E_{7})$
decomposes this set into $30$ orbits as follows:

$\displaystyle 720720=63+945\times 3+1008\times 2+1890+2016+3780\times
2+5040\times 2+10080+$ $\displaystyle+11340+15120\times 5+22680+30240\times
5+45360\times 2+90720+120960\times 2.$

Hence $N^{(0,4)}=30$.

######  Example 5.4.

The size $d^{(2,2)}$ of $P^{(2,2)}$ is $762048$. The group $W(E_{7})$
decomposes this set into $23$ orbits as follows:

$\displaystyle 762048=378+1890+3780\times 3+6048+7560\times 2+12096\times
2+15120+22680+$ $\displaystyle+30240\times 3+45360\times 2+60480\times
4+120960\times 2.$

Hence $N^{(2,2)}=23$.

######  Remark 5.5.

For the computation, we used GAP [20], which is good at computations of
permutation groups.

###  5.3. Real quartic curves

######  Definition 5.6.

Note that
$H^{2}(\mathord{\mathbb{P}}^{2},\mathord{\mathbb{Z}})\cong\mathord{\mathbb{Z}}$
has a canonical generator, that is, the class of a line. Let $C$ and
$C^{\prime}$ be plane curves with the same homeomorphism type. A homeomorphism
$\sigma\colon(\mathord{\mathbb{P}}^{2},C)\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},C^{\prime})$
is said to be _orientation-preserving_ (resp.  _orientation-reversing_) if the
action of $\sigma$ on $H^{2}(\mathord{\mathbb{P}}^{2},\mathord{\mathbb{Z}})$
is the identity (resp. the multiplication by $-1$).

######  Example 5.7.

Suppose that $\mathord{\mathcal{Q}}^{(m,n)}$-curves $Z_{\zeta}$ and
$Z_{\zeta^{\prime}}$ are of the same deformation type. Let $\alpha\colon
I\to\mathord{\mathcal{Z}}^{(m,n)}$ be a path from $\zeta$ to $\zeta^{\prime}$,
where $I:=[0,1]\subset\mathord{\mathbb{R}}$. By the parallel transport along
$\alpha$, we obtain a homeomorphism
$\alpha_{*}\colon(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\zeta^{\prime}})$.
It is obvious that $\alpha_{*}$ is orientation-preserving.

######  Proposition 5.8.

Every $\mathord{\mathcal{Q}}^{(m,n)}$-curve $Z_{\zeta}$ admits an orientation-
reversing self-homeomorphism
$(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\zeta})$.

For the proof of Proposition 5.8, we use a classical result on real quartic
curves. We give a structure of the $\mathord{\mathbb{R}}$-scheme to
$\mathord{\mathbb{P}}^{2}$. We denote by $\Gamma_{\mathord{\mathbb{R}}}(d)$
the space of homogeneous polynomials of degree $d$ on
$\mathord{\mathbb{P}}^{2}$ with real coefficients, and consider the real
projective space $\mathord{\mathbb{P}}_{*}(\Gamma_{\mathord{\mathbb{R}}}(4))$
as a closed subset of $\mathord{\mathbb{P}}_{*}(\Gamma(4))$. We then put

$\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}}}:=\mathord{\mathcal{U}}\cap\mathord{\mathbb{P}}_{*}(\Gamma_{\mathord{\mathbb{R}}}(4)).$

The topological types of smooth real quartic curves are classified by Zeuthen
and Klein, and the result is summarized in [12, Theorem 1.7]. Using this
result, we obtain the following:

######  Theorem 5.9 (Zeuthen (1873) and Klein (1876)).

There exists a unique connected component
$\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$ of
$\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}}}$ consisting of points
$u\in\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}}}$ such that the real plane
curve $Q_{u}(\mathord{\mathbb{R}})$ is a union of $4$ ovals. If
$u\in\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$, the ovals in
$Q_{u}(\mathord{\mathbb{R}})$ are pairwise non-nested, and every bitangent of
$Q_{u}$ is defined over $\mathord{\mathbb{R}}$. $\Box$

######  Remark 5.10.

For beautiful pictures of real plane quartic curves with real $28$ bitangents,
see [11] and [14, Section 10.5]. These pictures are in fact defined over
$\mathord{\mathbb{Q}}$, and were obtained by the theory of Mordell-Weil
lattices.

For an algebraic variety $V$ defined over $\mathord{\mathbb{R}}$, we denote by
$H^{*}(V,\mathord{\mathbb{Z}})$ the cohomology ring of the topological space
$V(\mathord{\mathbb{C}})$ of $\mathord{\mathbb{C}}$-valued points of $V$, by

$\tau_{V}\colon
V(\mathord{\mathbb{C}})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}V(\mathord{\mathbb{C}})$

the self-homeomorphism of $V(\mathord{\mathbb{C}})$ obtained by the complex
conjugation, and by $V_{\mathord{\mathbb{C}}}$ the variety
$V\otimes_{\mathord{\mathbb{R}}}\mathord{\mathbb{C}}$ defined over
$\mathord{\mathbb{C}}$. Let $S$ be an algebraic surface defined over
$\mathord{\mathbb{R}}$, and let $C$ be a reduced irreducible curve on
$S_{\mathord{\mathbb{C}}}$. Then there exists a unique reduced irreducible
curve $C^{\prime}$ on $S_{\mathord{\mathbb{C}}}$ such that $\tau_{S}$ induces
an orientation-reversing homeomorphism
$C(\mathord{\mathbb{C}})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}C^{\prime}(\mathord{\mathbb{C}})$.
We denote this curve $C^{\prime}$ by $\tau_{S}[C]$. Then we have

$[\tau_{S}[C]]=-\tau_{S}^{*}([C])$

in $H^{2}(S,\mathord{\mathbb{Z}})$. If $C$ is also defined over
$\mathord{\mathbb{R}}$, then $\tau_{S}[C]=C$, and hence
$\tau_{S}^{*}([C])=-[C]$. If $H^{2}(S,\mathord{\mathbb{Z}})$ is generated by
classes of curves defined over $\mathord{\mathbb{R}}$, then $\tau_{S}$ acts on
$H^{2}(S,\mathord{\mathbb{Z}})$ as the multiplication by $-1$, and hence, for
any curve $C$ (not necessarily defined over $\mathord{\mathbb{R}}$), we have
$[\tau_{S}[C]]=[C]$ in $H^{2}(S,\mathord{\mathbb{Z}})$.

######  Lemma 5.11.

Let $r$ be a point of $\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$. If
$\bar{c}$ is a $4$-tangent conic of $Q_{r}$, then the $4$-tangent conic
$\tau_{\mathord{\mathbb{P}}^{2}}[\bar{c}]$ of
$\tau_{\mathord{\mathbb{P}}^{2}}[Q_{r}]=Q_{r}$ is in the same connected family
as $\bar{c}$.

###### Proof.

Note that, for $\varphi\in\Gamma_{\mathord{\mathbb{R}}}(4)$ and
$x\in\mathord{\mathbb{P}}^{2}(\mathord{\mathbb{R}})$, the sign of $\varphi(x)$
is well-defined, because $\lambda^{4}>0$ for any
$\lambda\in\mathord{\mathbb{R}}^{\times}$. We choose a defining equation
$\varphi\in\Gamma_{\mathord{\mathbb{R}}}(4)$ of $Q_{r}$ in such a way that
$\varphi(x)>0$ for any point $x$ of
$\mathord{\mathbb{P}}^{2}(\mathord{\mathbb{R}})$ in the outside of the ovals
of $Q_{r}(\mathord{\mathbb{R}})$. We let $Y_{r}$ be defined over
$\mathord{\mathbb{R}}$ by $w^{2}=\varphi$, and consider the self-homeomorphism
$\tau_{Y}\colon
Y_{r}(\mathord{\mathbb{C}})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}Y_{r}(\mathord{\mathbb{C}})$
given by the complex conjugation. For any bitangent $\bar{l}$ of $Q_{r}$, each
of its $Y$-lifts $l$ satisfies $\tau_{Y}[l]=l$, because $\varphi(x)\geq 0$ for
any point $x$ of $\bar{l}(\mathord{\mathbb{R}})$. Since the classes of these
curves $l$ span $SY_{r}=H^{2}(Y_{r},\mathord{\mathbb{Z}})$, we see that
$\tau_{Y}$ acts on $SY_{r}$ as the multiplication by $-1$. Therefore, for any
curve $C$ on $Y_{r}$, we have $[\tau_{Y}[C]]=[C]$. In particular, if $c\subset
Y_{r}$ is a $Y$-lift of $\bar{c}$, then $\tau_{Y}[c]$ is a $Y$-lift of the
$4$-tangent conic $\tau_{\mathord{\mathbb{P}}^{2}}[\bar{c}]$. Then
$[\tau_{Y}[c]]=[c]$ in $SY_{r}$ implies that
$\tau_{\mathord{\mathbb{P}}^{2}}[\bar{c}]$ and $\bar{c}$ belong to the same
connected family of $4$-tangent conics. $\Box$

###### Proof of Proposition 5.8.

Since $\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$ is open in
$\mathord{\mathbb{P}}_{*}(\Gamma_{\mathord{\mathbb{R}}}(4))$, it follows that
$\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$ is Zariski dense in
$\mathord{\mathcal{U}}$, and hence there exists a point
$r\in\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}\cap\mathord{\mathcal{U}}^{\prime}$.
By Lemma 5.1, we see that $\varpi^{-1}(r)$ intersects every connected
component of $\mathord{\mathcal{Z}}^{(m,n)}$. Let $\xi$ be a point of
$\varpi^{-1}(r)$ that belongs to the same connected component as $\zeta$, and
let

$Z_{\xi}=Q_{r}+\bar{l}_{1}^{\prime}+\cdots+\bar{l}_{m}^{\prime}+\bar{c}_{1}^{\prime}+\cdots+\bar{c}_{n}^{\prime}$

be the decomposition of $Z_{\xi}$ into irreducible components. Remark that
$Q_{r}$ and all of its bitangents are defined over $\mathord{\mathbb{R}}$ by
the definition of $\mathord{\mathcal{U}}_{\,\mathord{\mathbb{R}},4}$ (see
Theorem 5.9). We choose a path $\alpha\colon
I\to\mathord{\mathcal{Z}}^{(m,n)}$ from $\zeta$ to $\xi$. Then we obtain an
orientation-preserving homeomorphism

$\alpha_{*}\colon(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\xi}).$

For simplicity, we write $\tau$ instead of $\tau_{\mathord{\mathbb{P}}^{2}}$.
We have an orientation-reversing homeomorphism

$\tau\colon(\mathord{\mathbb{P}}^{2},Z_{\xi})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},\tau[Z_{\xi}])$

obtained by the complex conjugation. Since

$\tau[Z_{\xi}]=Q_{r}+\bar{l}_{1}^{\prime}+\cdots+\bar{l}_{m}^{\prime}+\tau[\bar{c}_{1}^{\prime}]+\cdots+\tau[\bar{c}_{n}^{\prime}],$

and, for $j=1,\dots,n$, the $4$-tangent conic $\tau[\bar{c}_{j}^{\prime}]$ of
$\tau[Q_{r}]=Q_{r}$ belongs to the same connected family as
$\bar{c}_{j}^{\prime}$ by Lemma 5.11, we see that
$\mathord{\mathcal{Q}}^{(m,n)}$-curves $\tau[Z_{\xi}]$ and $Z_{\xi}$ have the
same deformation type, and we have an orientation-preserving homeomorphism

$\beta_{*}\colon(\mathord{\mathbb{P}}^{2},\tau[Z_{\xi}])\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\xi}).$

Composing $\alpha_{*}$, $\tau$, $\beta_{*}$ and $\alpha_{*}^{-1}$, we obtain
an orientation-reversing self-homeomorphism of
$(\mathord{\mathbb{P}}^{2},Z_{\zeta})$. $\Box$

###  5.4. Homeomorphism types

Let $\zeta$ be a point of $\mathord{\mathcal{Z}}^{(m,n)}$ such that
$Z_{\zeta}$ has the decomposition

$Z=Q+\bar{l}_{1}+\cdots+\bar{l}_{m}+\bar{c}_{1}+\cdots+\bar{c}_{n}.$

We consider another point $\zeta^{\prime}\in\mathord{\mathcal{Z}}^{(m,n)}$
with the decomposition

$Z_{\zeta^{\prime}}=Q_{u^{\prime}}+\bar{l}^{\prime}_{1}+\cdots+\bar{l}^{\prime}_{m}+\bar{c}^{\prime}_{1}+\cdots+\bar{c}^{\prime}_{n}.$

Recall that the involution $\iota_{u}$ of $Y_{u}$ acts on the orthogonal
complement $\Sigma_{u}$ of $h_{u}\in SY_{u}$ as $-1$. Let $g\colon
SY_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{u^{\prime}}$ be an
isometryof lattices. Suppose that $g$ maps $h_{u}$ to $h_{u^{\prime}}$. Then
we have $g\circ\iota_{u}=\iota_{u^{\prime}}\circ g$. Moreover, by definitions
(3) and (4.1), the isometry $g$ maps $L_{u}$ to $L_{u^{\prime}}$ and $F_{u}$
to $F_{u^{\prime}}$. Therefore $g$ induces a bijection
$P^{(m,n)}_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}P^{(m,n)}_{u^{\prime}}$.

Theorem 1.3 is an immediate consequence of the following:

######  Theorem 5.12.

The following are equivalent:

  1. (i)

$\zeta$ and $\zeta^{\prime}$ belong to the same connected component of
$\mathord{\mathcal{Z}}^{(m,n)}$,

  2. (ii)

$\theta(\zeta)$ and $\theta(\zeta^{\prime})$ belong to the same connected
component of $\mathord{\mathcal{P}}^{(m,n)}$,

  3. (iii)

there exists an isometry $g\colon
SY_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{u^{\prime}}$ of
lattices that maps $h_{u}$ to $h_{u^{\prime}}$ and such that the induced
bijection
$P^{(m,n)}_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}P^{(m,n)}_{u^{\prime}}$
maps $\theta(\zeta)$ to $\theta(\zeta^{\prime})$, and

  4. (iv)

there exists a homeomorphism
$(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\zeta^{\prime}})$.

###### Proof.

By Proposition 5.8, condition ${\rm(iv)}$ is equivalent to the following:

  * ${\rm(iv)}^{\prime}$

there exists an orientation-preserving homeomorphism
$(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\zeta^{\prime}})$.

We will show that ${\rm(i)}$, ${\rm(ii)}$, ${\rm(iii)}$ and
${\rm(iv)}^{\prime}$ are equivalent. The implication
${\rm(i)}\Longleftrightarrow{\rm(ii)}$ follows from Lemma 5.1, and
${\rm(i)}\Longrightarrow{\rm(iv)^{\prime}}$ follows from Example 5.7.

We show ${\rm(iv)^{\prime}}\Longrightarrow{\rm(iii)}$. Suppose that
$\sigma\colon(\mathord{\mathbb{P}}^{2},Z_{\zeta})\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}(\mathord{\mathbb{P}}^{2},Z_{\zeta^{\prime}})$
is an orientation-preserving homeomorphism. We can assume, after renumbering
the curves, that $\sigma$ induces homeomorphisms
$\bar{l}_{i}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}\bar{l}_{i}^{\prime}$
and
$\bar{c}_{j}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}\bar{c}_{j}^{\prime}$
that preserve the orientation. We have a homeomorphism $\sigma_{Y}\colon
Y_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}Y_{u^{\prime}}$ that covers
$\sigma$, and $\sigma_{Y}$ induces an isometry

$\sigma_{SY}\colon
SY_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{u^{\prime}},$

which maps $h_{u}$ to $h_{u^{\prime}}$. If $l_{i}\subset Y_{u}$ is a $Y$-lift
of a bitangent $\bar{l}_{i}\subset Z_{\zeta}$, there exists a $Y$-lift
$l_{i}^{\prime}\subset Y_{u^{\prime}}$ of the bitangent
$\bar{l}_{i}^{\prime}\subset Z_{\zeta^{\prime}}$ such that $\sigma_{Y}$
induces a homeomorphism
$l_{i}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}l_{i}^{\prime}$ preserving
the orientation. In particular, we have
$\sigma_{SY}([l_{i}])=[l_{i}^{\prime}]$. The same holds for a $Y$-lift
$c_{j}\subset Y_{u}$ of a $4$-tangent conic $\bar{c}_{j}\subset Z_{\zeta}$.
Hence the bijection
$P^{(m,n)}_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}P^{(m,n)}_{u^{\prime}}$
induced by the isometry $\sigma_{SY}$ maps $\theta(\zeta)$ to
$\theta(\zeta^{\prime})$. Thus ${\rm(iii)}$ holds.

We show ${\rm(iii)}\Longrightarrow{\rm(ii)}$. Suppose that ${\rm(iii)}$ holds.
We choose a path $\beta\colon I\to\mathord{\mathcal{U}}$ from $u$ to the base-
point $b$ and a path $\beta^{\prime}\colon I\to\mathord{\mathcal{U}}$ from
$u^{\prime}$ to $b$, and consider the isometries

$\beta_{*}\colon
SY_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{b},\quad\beta^{\prime}_{*}\colon
SY_{u^{\prime}}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{b}$

obtained by the parallel transports along $\beta$ and $\beta^{\prime}$. Note
that $\beta_{*}(h_{u})=h_{b}$ and $\beta^{\prime}_{*}(h_{u^{\prime}})=h_{b}$.
Hence $\beta^{\prime}_{*}\circ g\circ\beta^{-1}_{*}$ is an element of
$\mathord{\mathrm{O}}(SY_{b},h_{b})$. By Theorem 3.1, there exists a loop
$\alpha\colon I\to\mathord{\mathcal{U}}$ with the base point $b$ such that

$\alpha_{*}=\beta^{\prime}_{*}\circ g\circ\beta^{-1}_{*}.$

Therefore the isometry $g\colon
SY_{u}\raise-1.0pt\hbox{\,$\xrightarrow{\sim\,}$\,}SY_{u^{\prime}}$ is equal
to the parallel transport $\gamma_{*}$ along the path
$\gamma:=\beta^{\prime-1}\alpha\beta$ from $u$ to $u^{\prime}$. Let

$\tilde{\gamma}\colon I\to\mathord{\mathcal{P}}^{(m,n)}$

be the lift of $\gamma$ such that $\tilde{\gamma}(0)=\theta(\zeta)$. Since
$g=\gamma_{*}$ maps $\theta(\zeta)$ to $\theta(\zeta^{\prime})$, we see that
$\tilde{\gamma}(1)=\theta(\zeta^{\prime})$. Therefore $\theta(\zeta)$ and
$\theta(\zeta^{\prime})$ are in the same connected component of
$\mathord{\mathcal{P}}^{(m,n)}$. $\Box$

##  6\. Geometry of the $K3$ surface $X_{u}$

We investigate the connected families of $4$-tangent conics more closely for a
_general_ point $u\in\mathord{\mathcal{U}}$. Our main result of this section
is as follows.

######  Theorem 6.1.

Suppose that $u\in\mathord{\mathcal{U}}$ is general. Then each connected
family of $4$-tangent conics $\bar{c}$ of $Q_{u}$ is parameterized by a
rational curve minus $12+6$ points. A member $\bar{c}$ of this family becomes
a $3$-tangent conic at each of $12$ punctured points, and $\bar{c}$
degenerates into a union of two distinct bitangents at each of the remaining
$6$ punctured points.

For the proof, we add the following easy result to Proposition 2.4.

######  Proposition 6.2.

Let $\bar{c}$ be a $3$-tangent conic of $Q_{u}$. Then
$\gamma_{u}^{*}(\bar{c})$ is a union of two one-nodal rational curves. $\Box$

Recall that the double covering $\eta_{u}\colon X_{u}\to Y_{u}$ induces a
primitive embedding of lattices $\eta_{u}^{*}\colon SY_{u}(2)\hookrightarrow
S\hskip-0.9ptX_{u}$.

######  Proposition 6.3.

If $u\in\mathord{\mathcal{U}}$ is general, then $\eta_{u}^{*}$ is an
isomorphism.

###### Proof.

Kondo [10] studied the moduli of genus-$3$ curves by considering the periods
of $K3$ surfaces $X$ that are cyclic covers of $\mathord{\mathbb{P}}^{2}$ of
degree $4$ branched along quartic curves $Q\subset\mathord{\mathbb{P}}^{2}$.
Let $j$ denote the generator of
$\operatorname{\mathrm{Gal}}\nolimits(X/\mathord{\mathbb{P}}^{2})\cong\mu_{4}$
that acts on $H^{2,0}(X)$ as $\sqrt{-1}$. Kondo exhibits an action of the
cyclic group $\mu_{4}$ on the $K3$ lattice

$\mathord{\mathbf{L}}:=E_{8}^{\oplus 2}\oplus U^{\oplus 3}$

that is obtained by a marking
$H^{2}(X,\mathord{\mathbb{Z}})\cong\mathord{\mathbf{L}}$. Let
$\mathord{\mathbf{L}}_{S}$ and $\mathord{\mathbf{L}}_{T}$ be the kernel of
$j^{*2}-1$ and of $j^{*2}+1$ on $\mathord{\mathbf{L}}$, respectively. Then
$\mathord{\mathbf{L}}_{S}$ is of rank $8$, and, via the marking, equal to the
image of the pull-back of $H^{2}(Y,\mathord{\mathbb{Z}})(2)$ by the double
covering $X\to Y:=X/\langle j^{2}\rangle$. The period $H^{2,0}(X)$ is a point
of $\mathord{\mathbb{P}}_{*}(V_{\sqrt{-1}})$, where $V_{\sqrt{-1}}$ is the
kernel of $j^{*}-\sqrt{-1}$ on
$\mathord{\mathbf{L}}_{T}\otimes\mathord{\mathbb{C}}$. We have
$\dim\mathord{\mathbb{P}}_{*}(V_{\sqrt{-1}})=6$. The result of [10] implies
that, when $Q$ varies, the point $H^{2,0}(X)$ of
$\mathord{\mathbb{P}}_{*}(V_{\sqrt{-1}})$ sweeps an open subset of
$\mathord{\mathbb{P}}_{*}(V_{\sqrt{-1}})$.

We fix a marking $H^{2}(X_{u},\mathord{\mathbb{Z}})\cong\mathord{\mathbf{L}}$.
Since $u\in\mathord{\mathcal{U}}$ is general, the period $H^{2,0}(X_{u})$ is
general in $\mathord{\mathbb{P}}_{*}(V_{\sqrt{-1}})$. Since
$\mathord{\mathbf{L}}_{T}\otimes\mathord{\mathbb{C}}=V_{\sqrt{-1}}\oplus\overline{V_{\sqrt{-1}}}$,
the minimal $\mathord{\mathbb{Z}}$-submodule $M$ of $\mathord{\mathbf{L}}$
such that $M\otimes\mathord{\mathbb{C}}$ contains $H^{2,0}(X_{u})$ is equal to
$\mathord{\mathbf{L}}_{T}$, and hence its orthogonal complement
$M^{\perp}=S\hskip-0.9ptX_{u}$ is equal to
$\mathord{\mathbf{L}}_{S}=\eta_{u}^{*}(SY_{u}(2))$. $\Box$

Let $\mathord{\mathit{Rats}}(X_{u})$ denote the set of rational curves on
$X_{u}$, and $\mathord{\mathit{Ells}}(X_{u})$ the set of elliptic fibrations
on $X_{u}$.

######  Proposition 6.4.

Suppose that $u\in\mathord{\mathcal{U}}$ is general. Then
$\mathord{\mathit{Rats}}(X_{u})$ is equal to the set
$\widetilde{L}_{u}:=\\{\,{\eta_{u}^{*}(l)}\,|\,{l\in L_{u}}\,\\}$ of $56$
smooth rational curves on $X_{u}$.

This proposition is proved by Proposition 6.3 and [13, Proposition 98]. See
also [13, Remark 99]. We give a proof, however, because the argument is also
used in the proof of Proposition 6.5 below. Recall from the proof of Theorem
4.1 that, for $v\in F_{u}$, there exists an elliptic fibration $\phi_{v}\colon
X_{u}\to\mathord{\mathbb{P}}^{1}$ such that the class of a fiber of $\phi_{v}$
is $\eta_{u}^{*}(v)$.

######  Proposition 6.5.

Suppose that $u\in\mathord{\mathcal{U}}$ is general. Then $v\mapsto\phi_{v}$
gives a bijection $F_{u}\cong\mathord{\mathit{Ells}}(X_{u})$. Each fibration
$\phi_{v}$ has no section. The singular fibers of $\phi_{v}$ consist of $6$
fibers of type $\mathord{\mathrm{I}}_{2}$ and $12$ fibers of type
$\mathord{\mathrm{I}}_{1}$.

###### Proof of Propositions 6.4 and 6.5.

The space

$\\{\,{v\in S\hskip-0.9ptX_{u}\otimes\mathord{\mathbb{R}}}\,|\,{\langle
v,v\rangle_{X}>0}\,\\}$

has two connected components. Let $\mathord{\mathcal{P}}_{u}$ be the connected
component containing the ample class $\tilde{h}_{u}$. For a vector $v\in
S\hskip-0.9ptX_{u}\otimes\mathord{\mathbb{R}}$ with $\langle
v,v\rangle_{X}<0$, let $[v]^{\perp}$ be the hyperplane of
$S\hskip-0.9ptX_{u}\otimes\mathord{\mathbb{R}}$ defined by $\langle
x,v\rangle_{X}=0$, and we put
$(v)^{\perp}:=[v]^{\perp}\cap\mathord{\mathcal{P}}_{u}$. We then put

$N_{u}:=\\{\,{v\in\mathord{\mathcal{P}}_{u}}\,|\,{\langle
v,\Gamma\rangle_{X}\geq 0\;\;\textrm{for all curves}\;\;\Gamma\subset
X_{u}}\,\\}.$

It is well known that $N_{u}$ is equal to

$\\{\,{v\in\mathord{\mathcal{P}}_{u}}\,|\,{\langle v,\Gamma\rangle_{X}\geq
0\;\;\textrm{for all}\;\;\Gamma\in\mathord{\mathit{Rats}}(X_{u})}\,\\},$

and that each $\Gamma\in\mathord{\mathit{Rats}}(X_{u})$ defines a wall of the
cone $N_{u}$, that is, $(\Gamma)^{\perp}\cap N_{u}$ contains a non-empty open
subset of $(\Gamma)^{\perp}$. Let $\overline{N}_{u}$ be the closure of $N_{u}$
in $S\hskip-0.9ptX_{u}\otimes\mathord{\mathbb{R}}$. For the proof of
Proposition 6.4, it is enough to show that $\overline{N}_{u}$ is equal to

$\overline{N}^{\,\lower 1.0pt\hbox{\scriptsize$\prime$}}_{u}:=\\{\,{v\in
S\hskip-0.9ptX_{u}\otimes\mathord{\mathbb{R}}}\,|\,{\langle
v,\tilde{l}\rangle_{X}\geq 0\;\;\textrm{for
all}\;\;\tilde{l}\in\widetilde{L}_{u}}\,\\}.$

A _face_ of the cone $\overline{N}^{\,\lower
1.0pt\hbox{\scriptsize$\prime$}}_{u}$ is a closed subset $F$ of
$\overline{N}^{\,\lower 1.0pt\hbox{\scriptsize$\prime$}}_{u}$ of the form
$F=V\cap\overline{N}^{\,\lower 1.0pt\hbox{\scriptsize$\prime$}}_{u}$, where
$V$ is an intersection of some of the hyperplanes $[\tilde{l}]^{\perp}$
($\tilde{l}\in\widetilde{L}_{u}$) such that $F$ contains a non-empty open
subset of $V$. We say that $V$ is the _supporting linear subspace_ of the face
$F$, and put $\dim F:=\dim V$. A _ray_ is a $1$-dimensional face. For the
proof of $\overline{N}_{u}=\overline{N}^{\,\lower
1.0pt\hbox{\scriptsize$\prime$}}_{u}$, it is enough to show that all rays of
$\overline{N}^{\,\lower 1.0pt\hbox{\scriptsize$\prime$}}_{u}$ are contained in
$\overline{N}_{u}$. We can calculate all the faces $F$ of
$\overline{N}^{\,\lower 1.0pt\hbox{\scriptsize$\prime$}}_{u}$ by descending
induction on $d:=\dim F$ using linear programming method (see [17, Section
2.2]). The result is as follows. Suppose that $d\geq 2$. Then a linear
subspace

(6.1) $V=[\tilde{l}_{1}]^{\perp}\cap\dots\cap[\tilde{l}_{k}]^{\perp}$

with $\tilde{l}_{1},\dots,\tilde{l}_{k}\in\widetilde{L}_{u}$ is the supporting
linear subspace of a face $F$ with $\dim F=d$ if and only if $k=8-d$ and
$\tilde{l}_{1},\dots,\tilde{l}_{k}$ are disjoint from each other, that is,
their dual graph is the Dynkin diagram of type $(8-d)A_{1}$. Suppose that
$d=1$. Then a linear subspace $V$ as (6.1) is the supporting linear subspace
of a ray $F$ if and only if one of the following holds:

  * ($7A_{1}$)

$k=7$ and the dual graph of $\tilde{l}_{1},\dots,\tilde{l}_{7}$ is the Dynkin
diagram of type $7A_{1}$. In this case, $F$ is generated by a vector $v\in
S\hskip-0.9ptX_{u}$ with $\langle\tilde{h}_{u},v\rangle_{X}=6$ and $\langle
v,v\rangle_{X}=2$. There exist exactly $576$ rays of this type.

  * ($6\widetilde{A}_{1}$)

$k=12$ and the dual graph of $\tilde{l}_{1},\dots,\tilde{l}_{12}$ is the
Dynkin diagram of type $6\widetilde{A}_{1}$, where $\widetilde{A}_{1}$ is . In
this case, $F$ is generated by a primitive vector $\tilde{v}$ with
$\langle\tilde{h}_{u},\tilde{v}\rangle_{X}=4$ and
$\langle\tilde{v},\tilde{v}\rangle_{X}=0$. There exist exactly $126$ rays of
this type, and these generators $\tilde{v}$ are equal to $\eta_{u}^{*}(v)$ for
some $v\in F_{u}$.

In Table 6.1, the numbers of faces of $\overline{N}^{\,\lower
1.0pt\hbox{\scriptsize$\prime$}}_{u}$ are given.

Suppose that there exists a ray $F$ of $\overline{N}^{\,\lower
1.0pt\hbox{\scriptsize$\prime$}}_{u}$ not contained in $\overline{N}_{u}$.
Then the generating class $v\in S\hskip-0.9ptX_{u}$ of $F$ given above is
effective but not nef. Let $D$ be an effective divisor of $X_{u}$ such that
$[D]=v$. Then $D$ contains a smooth rational curve $\Gamma$ with
$\langle\Gamma,v\rangle_{X}<0$ as an irreducible component. Since
$\tilde{h}_{u}$ is ample, the $(-2)$-vector $r=[\Gamma]$ satisfies
$\langle\tilde{h}_{u},r\rangle_{X}<\langle\tilde{h}_{u},v\rangle_{X}\leq 6$.
We make the set of all $(-2)$-vectors $r^{\prime}\in S\hskip-0.9ptX_{u}$ with
$\langle\tilde{h}_{u},r^{\prime}\rangle_{X}=1,\dots,5$, and confirm that this
set is equal to the set of classes of $\widetilde{L}_{u}$. In particular, it
contains no element $r^{\prime}$ satisfying $\langle
r^{\prime},v\rangle_{X}<0$. This contradiction shows $\overline{N}^{\,\lower
1.0pt\hbox{\scriptsize$\prime$}}_{u}=\overline{N}_{u}$, and
$\mathord{\mathit{Rats}}(X_{u})=\widetilde{L}_{u}$ is proved.

It is well known that there exists a bijection between
$\mathord{\mathit{Ells}}(X_{u})$ and the set of rays contained in
$\overline{N}_{u}\cap\partial\,\overline{\mathord{\mathcal{P}}}_{u}$. Hence we
have $|\mathord{\mathit{Ells}}(X_{u})|=126$, and $v\mapsto\phi_{v}$ gives a
bijection from $F_{u}$ to $\mathord{\mathit{Ells}}(X_{u})$. Therefore, as was
shown in the proof of Theorem 4.1, every fiber $f$ of any elliptic fibration
$\phi_{v}$ is a double cover of a splitting conic of $Q_{u}$. The class of $f$
is equal to $\eta_{u}^{*}(v)$. Since no element
$\tilde{l}\in\mathord{\mathit{Rats}}(X_{u})$ satisfies $\langle
f,\tilde{l}\rangle_{X}=1$, the fibration $\phi_{v}$ has no section. Since the
dual graph of the set of $\tilde{l}\in\mathord{\mathit{Rats}}(X_{u})$ with
$\langle f,\tilde{l}\rangle_{X}=0$ is of type $6\widetilde{A}_{1}$, the
fibration $\phi_{v}$ has exactly $6$ reducible fibers, each of which is either
of type $\mathord{\mathrm{I}}_{2}$ or of type $\mathord{\mathrm{III}}$. If
$\tilde{l}_{i},\tilde{l}_{j}\in\mathord{\mathit{Rats}}(X_{u})$ are in the same
fiber of $\phi_{v}$, then they satisfy
$\langle\tilde{l}_{i},\tilde{l}_{j}\rangle_{X}=2$ and hence
$\bar{l}_{i}:=\gamma_{u}(\tilde{l}_{i})$ and
$\bar{l}_{j}:=\gamma_{u}(\tilde{l}_{j})$ are distinct bitangents of $Q_{u}$ by
Table 2.1. Since $u\in\mathord{\mathcal{U}}$ is general, the intersection
point of $\bar{l}_{i}$ and $\bar{l}_{j}$ is not on $Q_{u}$. Hence every
reducible fiber of $\phi_{v}$ is of type $\mathord{\mathrm{I}}_{2}$. The
irreducible singular fibers are either of type $\mathord{\mathrm{I}}_{1}$ or
of type $\mathord{\mathrm{II}}$. By Lemma 2.2 and Proposition 6.2, we see that
all irreducible singular fibers must be of type $\mathord{\mathrm{I}}_{1}$.
Calculating the Euler number, we conclude that the number of singular fibers
of type $\mathord{\mathrm{I}}_{1}$ is $12$. $\Box$

$\begin{array}[]{c | lllllll}\dim F&\;\;7&6&5&4&3&2&1\\\ \hline\cr\\#&\;\;56&756&4032&10080&12096&6048&576+126\end{array}$
Table 6.1. Numbers of faces $F$

######  Remark 6.6.

The set of $576$ rays of type $7A_{1}$ is in bijective correspondence with the
set $L^{\\{7\\}}_{u}$ in the proof of Theorem 3.1. A ray $F$ of type $7A_{1}$
corresponds to a $7$-tuple $\\{l_{1},\dots,l_{7}\\}\in L^{\\{7\\}}_{u}$ as
follows. The generator $v$ of $F$ with $\langle v,v\rangle_{X}=2$ is the class
of the pull-back of a line of a plane $\mathord{\mathbf{P}}$ by the double
covering $X_{u}\to Y_{u}\to\mathord{\mathbf{P}}$, where
$Y_{u}\to\mathord{\mathbf{P}}$ is the blowing down of the $(-1)$-curves
$l_{1},\dots,l_{7}$.

###### Proof of Theorem 6.1.

In fact, the proof was already given in the last paragraph of the proof of
Proposition 6.5. $\Box$

##  7\. Configurations of $Y$-lifts

Throughout this section, let $u$ be a general point of
$\mathord{\mathcal{U}}$.

###  7.1. Lemmas on quartic polynomials

Let $[d_{1},\dots,d_{m}]$ be a list of positive integers satisfying
$d_{1}+\dots+d_{m}=4$. We put

$\Gamma(d_{1},\dots,d_{m}\colon
2):=\Gamma(d_{1})\times\dots\times\Gamma(d_{m})\times\Gamma(2),$

and denote by $\psi_{[d_{1},\dots,d_{m}]}\colon\Gamma(d_{1},\dots,d_{m}\colon
2)\to\Gamma(4)$ the morphism

$(f_{1},\cdots,f_{m},q)\mapsto f_{1}\cdots f_{m}+q^{2}.$

######  Lemma 7.1.

The morphism $\psi_{[d_{1},\dots,d_{m}]}$ is dominant.

###### Proof.

It is enough to show that $\psi_{[1,1,1,1]}$ is dominant, and then, it
suffices to find a point $P$ of $\Gamma(1,1,1,1\colon 2)$ at which the
differential of $\psi:=\psi_{[1,1,1,1]}$ is of rank $\dim\Gamma(4)=15$. By
choosing points $P$ randomly and calculating the rank of $d_{P}\psi$, we can
easily find such a point. $\Box$

######  Definition 7.2.

For $[d_{1},\dots,d_{m}]$ with $d_{1}+\dots+d_{m}=4$, we have an open dense
subset
$\mathord{\mathcal{V}}_{[d_{1},\dots,d_{m}]}\subset\Gamma(d_{1},\dots,d_{m}\colon
2)$ and a dominant morphism

$\Psi_{[d_{1},\dots,d_{m}]}\colon\mathord{\mathcal{V}}_{[d_{1},\dots,d_{m}]}\to\mathord{\mathcal{U}}$

such that, for
$p=(f_{1},\dots,f_{m},q)\in\mathord{\mathcal{V}}_{[d_{1},\dots,d_{m}]}$, the
quartic curve corresponding
$\Psi_{[d_{1},\dots,d_{m}]}(p)\in\mathord{\mathcal{U}}$ is defined by
$f_{1}\cdots f_{m}+q^{2}=0$.

######  Lemma 7.3.

If $Q_{u}$ is defined by $f+q^{2}=0$ with $f\in\Gamma(4)$ and $q\in\Gamma(2)$,
then $Y_{u}$ has a divisor that is mapped isomorphically to the divisor
$\\{f=0\\}$ of $\mathord{\mathbb{P}}^{2}$.

###### Proof.

The surface $Y_{u}$ is defined by $w^{2}=f+q^{2}$, where $w$ is a new
variable, and hence contains a divisor defined by $f=w-q=0$. It is obvious
that $\pi_{u}$ maps this divisor to the divisor $\\{f=0\\}$ of
$\mathord{\mathbb{P}}^{2}$ isomorphically. $\Box$

###  7.2. Triangles of bitangents

Recall that $L_{u}$ is the set of $Y$-lifts $l$ of bitangents
$\bar{l}\in\overline{L}_{u}$ of $Q_{u}$.

######  Definition 7.4.

A _triangle on $Y_{u}$_ is a subset $\\{l_{1},l_{2},l_{3}\\}$ of $L_{u}$ such
that $\langle l_{1},l_{2}\rangle_{Y}=\langle l_{2},l_{3}\rangle_{Y}=\langle
l_{3},l_{1}\rangle_{Y}=1$. A _liftable triangle_ of bitangents of $Q_{u}$ is a
subset $\\{\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}\\}$ of $\overline{L}_{u}$ that
is the image of a triangle on $Y_{u}$ by $\pi_{u}$.

Let $\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}$ be bitangents of $Q_{u}$. We choose
$Y$-lifts $l_{1},l_{2},l_{3}\in L_{u}$ of
$\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}$ in such a way that $\langle
l_{1},l_{2}\rangle_{Y}=\langle l_{2},l_{3}\rangle_{Y}=1$. Then
$\\{\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}\\}$ is liftable if and only if
$\langle l_{3},l_{1}\rangle_{Y}=1$.

Let $T_{u}$ be the set of triangles on $Y_{u}$. We have calculated
$L_{u}\subset SY_{u}$ explicitly. Using this data, we enumerate $T_{u}$, and
see that $|T_{u}|=2520$. Let

$\overline{T}_{u}:=T_{u}/\langle\iota_{u}\rangle$

be the set of liftable triangles of bitangents of $Q_{u}$.

######  Corollary 7.5.

There exist exactly $|\overline{T}_{u}|=1260$ liftable triangles. $\Box$

By Theorem 3.1, we obtain the following:

######  Proposition 7.6.

By the monodromy, $\pi_{1}(\mathord{\mathcal{U}},b)$ acts transitively on
$T_{b}$ and hence on $\overline{T}_{b}$. $\Box$

######  Proposition 7.7.

Let $\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}$ be bitangents of $Q_{u}$. Suppose
that $\bar{l}_{i}$ is defined by $f_{i}=0$ for $i=1,\dots,3$, where
$f_{i}\in\Gamma(1)$. Then $\\{\bar{l}_{1},\bar{l}_{2},\bar{l}_{3}\\}$ is
liftable if and only if there exist polynomials $f_{4}\in\Gamma(1)$ and
$q\in\Gamma(2)$ such that $Q_{u}$ is defined by
$f_{1}f_{2}f_{3}f_{4}+q^{2}=0$.

###### Proof.

The if-part follows from Lemma 7.3. Let
$\bar{\tau}\colon\overline{\mathord{\mathcal{T}}}\to\mathord{\mathcal{U}}$ be
the finite étale covering obtained from the family
$\\{\,{\overline{T}_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$. Then
$\overline{\mathord{\mathcal{T}}}$ is irreducible by Proposition 7.6. Let
$p:=(f_{1}^{\prime},\dots,f_{4}^{\prime},q^{\prime})$ be a point of
$\mathord{\mathcal{V}}_{[1,1,1,1]}$, and we put

$u^{\prime}:=\Psi_{[1,1,1,1]}(p)\;\in\;\mathord{\mathcal{U}}$

where $\mathord{\mathcal{V}}_{[1,1,1,1]}$ and $\Psi_{[1,1,1,1]}$ are given in
Definition 7.2. Let $\bar{l}^{\prime}_{i}\subset\mathord{\mathbb{P}}^{2}$ be
the line $\\{f_{i}^{\prime}=0\\}$. By the if-part, we have
$\\{\bar{l}^{\prime}_{1},\bar{l}^{\prime}_{2},\bar{l}^{\prime}_{3}\\}\in\overline{T}_{u^{\prime}}$.
By
$p\mapsto\\{\bar{l}^{\prime}_{1},\bar{l}^{\prime}_{2},\bar{l}^{\prime}_{3}\\}$,
we obtain a morphism
$\Psi_{\overline{\mathord{\mathcal{T}}}}\colon\mathord{\mathcal{V}}_{[1,1,1,1]}\to\overline{\mathord{\mathcal{T}}}$.
Since
$\bar{\tau}\circ\Psi_{\overline{\mathord{\mathcal{T}}}}=\Psi_{[1,1,1,1]}$,
$\bar{\tau}$ is étale, $\overline{\mathord{\mathcal{T}}}$ is irreducible, and
$\Psi_{[1,1,1,1]}$ is dominant, we conclude that
$\Psi_{\overline{\mathord{\mathcal{T}}}}$ is dominant. Since
$u\in\mathord{\mathcal{U}}$ is general, we obtain the proof. $\Box$

######  Corollary 7.8.

There exists a set $\overline{R}_{u}$ consisting of $315$ subsets
$\\{\bar{l}_{a},\bar{l}_{b},\bar{l}_{c},\bar{l}_{d}\\}\subset\overline{L}_{u}$
of size $4$ with the following properties: a subset
$\\{\bar{l}_{i},\bar{l}_{j},\bar{l}_{k}\\}\subset\overline{L}_{u}$ of size $3$
is liftable if and only if there exists an element
$\\{\bar{l}_{a},\bar{l}_{b},\bar{l}_{c},\bar{l}_{d}\\}\in\overline{R}_{u}$
containing $\\{\bar{l}_{i},\bar{l}_{j},\bar{l}_{k}\\}$. $\Box$

###  7.3. Pairs of splitting conics

Recall that $F_{u}\subset SY_{u}$ is the set of classes $[c]$ of $Y$-lifts $c$
of $4$-tangent conics $\bar{c}$ of $Q_{u}$, and that
$\overline{F}_{u}=F_{u}/\langle\iota_{u}\rangle$ is regarded as the set of
connected families of $4$-tangent conics of $Q_{u}$, or equivalently as the
set of connected families of splitting conics of $Q_{u}$. For a splitting
conic $\bar{c}$, let $[\bar{c}]\in\overline{F}_{u}$ denote the connected
family containing $\bar{c}$. By Theorem 3.1, we obtain the following:

######  Proposition 7.9.

By the monodromy, $\pi_{1}(\mathord{\mathcal{U}},b)$ acts transitively on
$F_{b}$ and hence on $\overline{F}_{b}$. $\Box$

######  Definition 7.10.

Let $\bar{c}$ be a splitting conic of $Q_{u}$. We say that a decomposition
$\pi_{u}^{*}(\bar{c})=c+c^{\prime}$ is _normal_ if each of $c$ and
$c^{\prime}$ is a $Y$-lift of $\bar{c}$.

Note that, if $\bar{c}$ is smooth, then the decomposition
$\pi_{u}^{*}(\bar{c})=c+c^{\prime}$ is normal, whereas if $\bar{c}$ is a sum
of two bitangents $\bar{l}+\bar{l}^{\prime}$, then
$\pi_{u}^{*}(\bar{c})=c+c^{\prime}$ being normal means that $c=l+l^{\prime}$
with $\langle l,l^{\prime}\rangle_{Y}=1$.

######  Definition 7.11.

Let $\bar{c}_{1}$ and $\bar{c}_{2}$ be splitting conics of $Q_{u}$, and let
$\pi_{u}^{*}(\bar{c}_{1})=c_{1}+c_{1}^{\prime}$ and
$\pi_{u}^{*}(\bar{c}_{2})=c_{2}+c_{2}^{\prime}$ be the normal decompositions.
We put

$I([\bar{c}_{1}],[\bar{c}_{2}]):=\left[\begin{array}[]{cc}\langle
c_{1},c_{2}\rangle_{Y}&\langle c_{1},c_{2}^{\prime}\rangle_{Y}\\\ \langle
c_{1}^{\prime},c_{2}\rangle_{Y}&\langle
c_{1}^{\prime},c_{2}^{\prime}\rangle_{Y}\end{array}\right].$

Since we can make switchings $c_{1}\leftrightarrow c_{1}^{\prime}$ and
$c_{2}\leftrightarrow c_{2}^{\prime}$, the matrix
$I([\bar{c}_{1}],[\bar{c}_{2}])$ is well-defined only up to the transpositions
of the two rows and of the two columns.

We have calculated $F_{u}\subset S_{u}$ explicitly. Using this data, we see
that the matrix $I([\bar{c}_{1}],[\bar{c}_{2}])$ is one of the following:

$\displaystyle I_{A}$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}0&4\\\
4&0\end{array}\right]\textrm{or}\left[\begin{array}[]{cc}4&0\\\
0&4\end{array}\right],$ $\displaystyle I_{B}$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}2&2\\\ 2&2\end{array}\right],$
$\displaystyle I_{C}$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}1&3\\\
3&1\end{array}\right]\textrm{or}\left[\begin{array}[]{cc}3&1\\\
1&3\end{array}\right].$

######  Proposition 7.12.

Let $\bar{c}_{1}=\\{g_{1}=0\\}$ and $\bar{c}_{2}=\\{g_{2}=0\\}$ be splitting
conics of $Q_{u}$. Consider the following conditions:

  1. (i)

$[\bar{c}_{1}]=[\bar{c}_{2}]$, that is, $\bar{c}_{1}$ and $\bar{c}_{2}$ belong
to the same connected family.

  2. (ii)

The matrix $I([\bar{c}_{1}],[\bar{c}_{2}])$ is equal to $I_{A}$.

  3. (iii)

There exists a polynomial $q\in\Gamma(2)$ such that $Q_{u}$ is defined by
$g_{1}g_{2}+q^{2}=0$.

Then we have ${\rm(iii)}\Longrightarrow{\rm(ii)}\Longleftrightarrow{\rm(i)}$.
If ${\rm(i)}$ holds and $\bar{c}_{1}$ and $\bar{c}_{2}$ are general in the
connected family $[\bar{c}_{1}]=[\bar{c}_{2}]\in\overline{F}_{u}$ of splitting
conics, then ${\rm(iii)}$ holds.

###### Proof.

The implication ${\rm(i)}\Longrightarrow{\rm(ii)}$ follows immediately from
Table 2.1, and the implication ${\rm(iii)}\Longrightarrow{\rm(ii)}$ follows
from Lemma 7.3. Suppose that ${\rm(ii)}$ holds. Let
$\pi_{u}^{*}(\bar{c}_{1})=c_{1}+c_{1}^{\prime}$ and
$\pi_{u}^{*}(\bar{c}_{2})=c_{2}+c_{2}^{\prime}$ be the normal decompositions.
Interchanging $c_{2}$ and $c_{2}^{\prime}$ if necessary, we can assume that
$\langle c_{1},c_{2}\rangle_{Y}=0$. We put $f_{1}:=\eta_{u}^{*}(c_{1})$ and
$f_{2}:=\eta_{u}^{*}(c_{2})$. Note that $f_{1}$ is a fiber of the elliptic
fibration $\phi_{1}\in\mathord{\mathit{Ells}}(X_{u})$ corresponding to the
class $[c_{1}]\in F_{u}$ of $c_{1}$ by
$F_{u}\cong\mathord{\mathit{Ells}}(X_{u})$. Since $\langle
f_{1},f_{2}\rangle_{X}=2\langle c_{1},c_{2}\rangle_{Y}=0$, we conclude that
$f_{2}$ is a fiber of $\phi_{1}$, that is, the elliptic fibration
corresponding to $[c_{2}]\in F_{u}\cong\mathord{\mathit{Ells}}(X_{u})$ is
equal to $\phi_{1}$. Therefore $\bar{c}_{1}$ and $\bar{c}_{2}$ belong to the
same connected family of splitting conics, and ${\rm(i)}$ holds. Thus
${\rm(iii)}\Longrightarrow{\rm(ii)}\Longleftrightarrow{\rm(i)}$ is proved.

Suppose that $[\bar{c}_{1}]=[\bar{c}_{2}]$. Let
$\sigma\colon\overline{\mathord{\mathcal{F}}}\to\mathord{\mathcal{U}}$ be the
finite étale covering defined by the family
$\\{\,{\overline{F}_{u}}\,|\,{u\in\mathord{\mathcal{U}}}\,\\}$. By Proposition
7.9, we see that $\overline{\mathord{\mathcal{F}}}$ is irreducible. Let
$p:=(g_{1}^{\prime},g_{2}^{\prime},q^{\prime})$ be a point of
$\mathord{\mathcal{V}}_{[2,2]}$, and we put
$u^{\prime}:=\Psi_{[2,2]}(p)\in\mathord{\mathcal{U}}$;

$Q_{u^{\prime}}=\\{g_{1}^{\prime}g_{2}^{\prime}+q^{\prime\,2}=0\\}.$

Let $\bar{c}^{\prime}_{i}$ be the splitting conic $\\{g_{i}^{\prime}=0\\}$ of
$Q_{u^{\prime}}$ for $i=1,2$. By the implication
${\rm(iii)}\Longrightarrow{\rm(i)}$, we have
$[\bar{c}_{1}^{\prime}]=[\bar{c}_{2}^{\prime}]$ in
$\overline{F}_{u^{\prime}}$. By $p\mapsto[\bar{c}_{1}^{\prime}]$, we obtain a
morphism
$\Psi_{\overline{\mathord{\mathcal{F}}}}\colon\mathord{\mathcal{V}}_{[2,2]}\to\overline{\mathord{\mathcal{F}}}$.
By the same argument as in the proof of Proposition 7.7, we see that
$\Psi_{\overline{\mathord{\mathcal{F}}}}$ is dominant. Since $u$ is general in
$\mathord{\mathcal{U}}$, the point $(u,[\bar{c}_{1}])=(u,[\bar{c}_{2}])$ is
general in $\overline{\mathord{\mathcal{F}}}$ and the fiber $W$ of
$\Psi_{\overline{\mathord{\mathcal{F}}}}$ over $(u,[\bar{c}_{1}])$ is of
dimension

$\dim\Gamma(2,2\colon 2)-\dim\mathord{\mathcal{U}}=18-14=4.$

Let $S:=\\{\,{\bar{c}(t)}\,|\,{t\in\mathord{\mathbb{P}}^{1}}\,\\}$ be the
connected family of splitting conics containing $\bar{c}_{1}$ and
$\bar{c}_{2}$. If $(g_{1}^{\prime},g_{2}^{\prime},q^{\prime})$ is a point of
the fiber $W$, then we have two members
$\bar{c}^{\prime}_{1}=\\{g_{1}^{\prime}=0\\}$ and
$\bar{c}^{\prime}_{2}=\\{g_{2}^{\prime}=0\\}$ of $S$, and thus we have a
morphism $W\to\mathord{\mathbb{P}}^{1}\times\mathord{\mathbb{P}}^{1}$, where
$\mathord{\mathbb{P}}^{1}$ is the base curve of the family $S$. If two points
$(g_{1}^{\prime},g_{2}^{\prime},q^{\prime})$ and
$(g_{1}^{\prime\prime},g_{2}^{\prime\prime},q^{\prime\prime})$ of $W$ are
mapped to the same point of
$\mathord{\mathbb{P}}^{1}\times\mathord{\mathbb{P}}^{1}$, then there exist
scalars $\lambda_{1},\lambda_{2}\in\mathord{\mathbb{C}}^{\times}$ such that
$g_{1}^{\prime\prime}=\lambda_{1}g_{1}^{\prime}$ and
$g_{2}^{\prime\prime}=\lambda_{2}g_{2}^{\prime}$. By the dimension reason, we
see that $W\to\mathord{\mathbb{P}}^{1}\times\mathord{\mathbb{P}}^{1}$ is
dominant. Hence, if $\bar{c}_{1}=\\{g_{1}=0\\}$ and
$\bar{c}_{2}=\\{g_{2}=0\\}$ are general members of the family $S$, there
exists a polynomial $q\in\Gamma(2)$ such that $(g_{1},g_{2},q)\in W$, that is,
$Q_{u}$ is defined by $g_{1}g_{2}+q^{2}=0$. $\Box$

The following two propositions are confirmed by direct computation.

######  Proposition 7.13.

Among the $1953$ non-ordered pairs $\\{[\bar{c}_{1}],[\bar{c}_{2}]\\}$ of
distinct elements $[\bar{c}_{1}],[\bar{c}_{2}]$ of $\overline{F}_{u}$, exactly
$945$ pairs satisfy $I([\bar{c}_{1}],[\bar{c}_{2}])=I_{B}$, and the remaining
$1008$ pairs satisfy $I([\bar{c}_{1}],[\bar{c}_{2}])=I_{C}$. When $u=b$, these
two sets of pairs are the orbits of the monodromy action of
$\pi_{1}(\mathord{\mathcal{U}},b)$ on the set of non-ordered pairs of elements
of $\overline{F}_{b}$. $\Box$

Recall that each connected family $[c]\in F_{u}$ of $Y$-lifts of splitting
conics contains exactly $6$ reducible members, and the irreducible components
$l,l^{\prime}$ of a reducible member satisfy $\langle
l,l^{\prime}\rangle_{Y}=1$. We have a surjective map

$\\{\,{\\{l,l^{\prime}\\}\;}\,|\,{\;l,l^{\prime}\in L_{u},\;\langle
l,l^{\prime}\rangle_{Y}=1\;}\,\\}\to F_{u}$

defined by $\\{l,l^{\prime}\\}\mapsto[l]+[l^{\prime}]$. Each fiber of size
$6$. The following gives how the cases $I([\bar{c}_{1}],[\bar{c}_{2}])=I_{B}$
and $I([\bar{c}_{1}],[\bar{c}_{2}])=I_{C}$ are distinguished.

######  Proposition 7.14.

Let $[c_{1}]$ and $[c_{2}]$ be elements of $F_{u}$, and let $[\bar{c}_{1}]$
and $[\bar{c}_{2}]$ be their images by $F_{u}\to\overline{F}_{u}$. Then
$I([\bar{c}_{1}],[\bar{c}_{2}])=I_{B}$ holds if and only if there exists a
triangle $\\{l_{1},l_{2},l_{3}\\}$ on $Y_{u}$ such that
$[c_{1}]=[l_{1}]+[l_{3}]$ and $[c_{2}]=[l_{2}]+[l_{3}]$. $\Box$

###  7.4. Pairs of a bitangent and a splitting conic

Let $\bar{l}$ be a bitangent of $Q_{u}$ with
$\pi_{u}^{*}(\bar{l})=l+l^{\prime}$, and let $\bar{c}$ be a splitting conic of
$Q_{u}$ with the normal decomposition $\pi_{u}^{*}(\bar{c})=c+c^{\prime}$. We
put

$J(\bar{l},[\bar{c}]):=\left[\begin{array}[]{cc}\langle l,c\rangle_{Y}&\langle
l,c^{\prime}\rangle_{Y}\\\ \langle l^{\prime},c\rangle_{Y}&\langle
l^{\prime},c^{\prime}\rangle_{Y}\end{array}\right].$

The matrix $J(\bar{l},[\bar{c}])$ is one of the following:

$\displaystyle J_{\alpha}$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}0&2\\\
2&0\end{array}\right]\textrm{or}\left[\begin{array}[]{cc}2&0\\\
0&2\end{array}\right],$ $\displaystyle J_{\beta}$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}1&1\\\ 1&1\end{array}\right].$

By direct computation, we confirm the following:

######  Proposition 7.15.

Let $\bar{l}$ be a bitangent of $Q_{u}$, and $\bar{c}$ a splitting conic of
$Q_{u}$. Then $J(\bar{l},[\bar{c}])$ is equal to $J_{\alpha}$ if and only if
the connected family $[\bar{c}]\in\overline{F}_{u}$ of splitting conics has a
singular member containing $\bar{l}$ as an irreducible component.

When $u=b$, the monodromy action of $\pi_{1}(\mathord{\mathcal{U}},b)$ acts on
the set of pairs
$(\bar{l},[\bar{c}])\in\overline{L}_{b}\times\overline{F}_{b}$ with
$J(\bar{l},[\bar{c}])=J_{\alpha}$ transitively, and the set of pairs
$(\bar{l},[\bar{c}])$ with $J(\bar{l},[\bar{c}])=J_{\beta}$ also transitively.
$\Box$

##  8\. Intersection graph

######  Definition 8.1.

An _intersection graph_ is a pentad
$(V_{\bar{l}},V_{\bar{c}},T,E_{\bar{c}\bar{c}},E_{\bar{l}\bar{c}})$ such that

  * •

$V_{\bar{l}}$ and $V_{\bar{c}}$ are finite sets,

  * •

$T$ is a subset of $S^{3}_{0}(V_{\bar{l}})$,

  * •

$E_{\bar{c}\bar{c}}$ is a map $S^{2}(V_{\bar{c}})\to\\{A,B,C\\}$, and

  * •

$E_{\bar{l}\bar{c}}$ is a map $V_{\bar{l}}\times
V_{\bar{c}}\to\\{\alpha,\beta\\}$.

Two intersection graphs
$(V_{\bar{l}},V_{\bar{c}},T,E_{\bar{c}\bar{c}},E_{\bar{l}\bar{c}})$ and
$(V^{\prime}_{\bar{l}},V^{\prime}_{\bar{c}},T^{\prime},E^{\prime}_{\bar{c}\bar{c}},E^{\prime}_{\bar{l}\bar{c}})$
are _isomorphic_ if there exists a pair of bijections $V_{\bar{l}}\cong
V^{\prime}_{\bar{l}}$ and $V_{\bar{c}}\cong V^{\prime}_{\bar{c}}$ that induces
$T\cong T^{\prime}$, $E_{\bar{c}\bar{c}}\cong E^{\prime}_{\bar{c}\bar{c}}$,
and $E_{\bar{l}\bar{c}}\cong E^{\prime}_{\bar{l}\bar{c}}$.

######  Definition 8.2.

For a $\mathord{\mathcal{Q}}^{(m,n)}$-curve $Z$ as in (5.4), we define an
intersection graph

$g(Z):=(V_{\bar{l}},V_{\bar{c}},T,E_{\bar{c}\bar{c}},E_{\bar{l}\bar{c}})$

by the following:

  * •

$V_{\bar{l}}$ is $\\{\bar{l}_{1},\dots,\bar{l}_{m}\\}$ and $V_{\bar{c}}$ is
$\\{\bar{c}_{1},\dots,\bar{c}_{n}\\}$,

  * •

$T$ is the set of liftable triangles
$\\{\bar{l}_{i},\bar{l}_{j},\bar{l}_{k}\\}\subset\\{\bar{l}_{1},\dots,\bar{l}_{m}\\}$,

  * •

$E_{\bar{c}\bar{c}}(\bar{c}_{i},\bar{c}_{j})$ is the type of the matrix
$I([\bar{c}_{i}],[\bar{c}_{j}])$ defined in Section 7.3, and

  * •

$E_{\bar{l}\bar{c}}(\bar{l}_{i},\bar{c}_{j})$ is the type of the matrix
$J(\bar{l}_{i},[\bar{c}_{j}])$ defined in Section 7.4.

######  Remark 8.3.

By Proposition 7.12, the relation

$\bar{c}_{i}\sim\bar{c}_{j}\Longleftrightarrow
E_{\bar{c}\bar{c}}(\bar{c}_{i},\bar{c}_{j})=A$

is an equivalence relation on $V_{\bar{c}}$, and the functions
$E_{\bar{c}\bar{c}}$ and $E_{\bar{l}\bar{c}}$ are compatible with this
equivalence relation.

######  Remark 8.4.

When $n=0$, the intersection graph equal to the two-graph in [5].

It is obvious that, if $\zeta$ and $\zeta^{\prime}$ are in the same connected
component of $\mathord{\mathcal{Z}}^{(m,n)}$, the intersection graphs
$g(Z_{\zeta})$ and $g(Z_{\zeta^{\prime}})$ are isomorphic. The converse is not
true in general, as examples in the next section show.

##  9\. Examples

###  9.1. The case $(m,n)=(6,0)$

We have $|P_{b}^{(6,0)}|=376740$. The action of $W(E_{7})$ decomposes
$P_{b}^{(6,0)}$ into orbits as in Table 9.1. For each orbit $o_{i}\subset
P_{b}^{(6,0)}$, we choose a point $\zeta\in o_{i}$ and indicate the following
data of the intersection graph $g(Z_{\zeta})$ of
$Z_{\zeta}=Q_{u}+\bar{l}_{1}+\dots+\bar{l}_{6}$: $|T|=k$ is the number of the
liftable triangles $t_{1},\dots,t_{k}$ in
$\\{\bar{l}_{1},\dots,\bar{l}_{6}\\}$, and $a_{\nu}$ is the number of pairs
$\\{t_{i},t_{j}\\}$ of liftable triangles such that $|t_{i}\cap t_{j}|=\nu$.
The orbit $o_{1}$ and $o_{2}$ cannot be distinguished by the two-graph
$(V_{\bar{l}},T)$, but they belong to different $W(E_{7})$-orbits, and hence
the corresponding $\mathord{\mathcal{Q}}^{(6,0)}$-curves are of different
homeomorphism types.

$\begin{array}[]{c|ccccc}i&|o_{i}|&|T|&a_{0}&a_{1}&a_{2}\\\ \hline\cr
1&2016&0&0&0&0\\\ 2&1008&0&0&0&0\\\ 3&30240&4&0&0&6\\\ 4&60480&6&0&6&9\\\
5&22680&8&2&10&16\\\ 6&181440&8&2&14&12\\\ 7&5040&8&4&12&12\\\
8&12096&10&0&30&15\\\ 9&60480&10&2&24&19\\\ 10&1260&12&6&30&30\\\ \end{array}$
Table 9.1. The orbit decomposition for $(m,n)=(6,0)$

###  9.2. The case $n=0$

We continue to consider the case where $n=0$. From the two-graph
$g=(V_{\bar{l}},T)$, we can construct a graph $\tilde{g}$ whose set of
vertices is $T$ and whose edge connecting $t_{\mu},t_{\nu}\in T$ has weight
$|t_{\mu}\cap t_{\nu}|$. If the graphs $\tilde{g}$ and $\tilde{g}^{\prime}$
are not isomorphic as graphs with weighted edges, then the two-graphs $g$ and
$g^{\prime}$ are not isomorphic. Using this method, we prove the following:

######  Proposition 9.1.

Except for the two orbits $o_{1}$ and $o_{2}$ in the case $m=6$ described in
Section 9.1, all $W(E_{7})$-orbits of $P_{b}^{(m,0)}$ are distinguished by
their two-graphs. $\Box$

######  Example 9.2.

Let $o_{1}^{\prime}$ and $o_{2}^{\prime}$ be the orbits in $P_{b}^{(22,0)}$
containing $22$-tuples obtained by taking the complement in $\overline{L}_{b}$
of $6$-tuples in the orbits $o_{1}\subset P_{b}^{(6,0)}$ and $o_{2}\subset
P^{(6,0)}$ above, respectively. Let $g_{1}^{\prime}$ and $g_{2}^{\prime}$ be
the two-graphs of $o_{1}^{\prime}$ and $o_{2}^{\prime}$. We have $|T|=600$ for
both $g_{1}^{\prime}$ and $g_{2}^{\prime}$. The associated graphs
$\tilde{g}^{\prime}_{1}$ and $\tilde{g}^{\prime}_{2}$ with weighted edges are
not isomorphic. The graph $\tilde{g}^{\prime}_{1}$ has exactly $8203640$
triples $\\{t_{\lambda},t_{\mu},t_{\nu}\\}$ of liftable triangles with weight
$|t_{\lambda}\cap t_{\mu}|=|t_{\mu}\cap t_{\nu}|=|t_{\nu}\cap t_{\lambda}|=0$,
whereas the number of such triples in $\tilde{g}^{\prime}_{2}$ is $8203760$.

###  9.3. The case $(m,n)=(0,3)$

By Remark 8.3, the three edges of the graph $(V_{\bar{c}},E_{\bar{c}\bar{c}})$
are labelled as in the second column of Table 9.2. The set $P^{(0,3)}_{b}$ of
size $43680$ is decomposed into nine $W(E_{7})$-orbits with sizes given in the
third column of Table 9.2.

$\begin{array}[]{c|c|l}i&\textrm{edge labels}&\textrm{orbit sizes}\\\
\hline\cr 1&AAA&63\\\ 2&ABB&1890\\\ 3&ACC&2016\\\ 4&BBB&3780+315\\\
5&BBC&15120\\\ 6&BCC&15120\\\ 7&CCC&5040+336\end{array}$
Table 9.2. The orbit decomposition for $(m,n)=(0,3)$
$\begin{array}[]{c|c|c|l}i&E_{\bar{l}\bar{c}}&E_{\bar{c}\bar{c}}&\textrm{orbit
sizes}\\\ \hline\cr 1&[[\alpha,\alpha],[\alpha,\alpha]]&A&3780+378\\\
2&[[\alpha,\alpha],[\alpha,\alpha]]&B&3780+1890\\\
3&[[\alpha,\alpha],[\alpha,\alpha]]&C&15120\\\
4&[[\alpha,\alpha],[\alpha,\beta]]&B&60480\\\
5&[[\alpha,\alpha],[\alpha,\beta]]&C&60480+12096\\\
6&[[\alpha,\alpha],[\beta,\beta]]&A&12096\\\
7&[[\alpha,\alpha],[\beta,\beta]]&B&30240\\\
8&[[\alpha,\alpha],[\beta,\beta]]&C&60480\\\
9&[[\alpha,\beta],[\alpha,\beta]]&B&45360+7560\\\
10&[[\alpha,\beta],[\alpha,\beta]]&C&30240\\\
11&[[\alpha,\beta],[\beta,\alpha]]&B&60480\\\
12&[[\alpha,\beta],[\beta,\alpha]]&C&30240+6048\\\
13&[[\alpha,\beta],[\beta,\beta]]&B&120960\\\
14&[[\alpha,\beta],[\beta,\beta]]&C&120960\\\
15&[[\beta,\beta],[\beta,\beta]]&A&7560\\\
16&[[\beta,\beta],[\beta,\beta]]&B&22680+3780\\\
17&[[\beta,\beta],[\beta,\beta]]&C&45360\end{array}$
Table 9.3. The orbit decomposition for $(m,n)=(2,2)$

###  9.4. The case $(m,n)=(2,2)$

There exist $17$ intersection graphs indicated in Table 9.3, where
$E_{\bar{c}\bar{c}}$ is shown by the type of $I([\bar{c}_{1}],[\bar{c}_{2}])$,
and

$E_{\bar{l}\bar{c}}:=[[J(\bar{l}_{1},[\bar{c}_{1}]),J(\bar{l}_{1},[\bar{c}_{2}])],[J(\bar{l}_{2},[\bar{c}_{1}]),J(\bar{l}_{2},[\bar{c}_{2}])]].$

The set $P^{(2,2)}_{b}$ of size $762048$ is decomposed into $23$ orbits by the
action of $W(E_{7})$, and their sizes are given in the $4$th column of Table
9.3.

## References

  * [1] Enrique Artal-Bartolo.  Sur les couples de Zariski.  J. Algebraic Geom., 3(2):223–247, 1994.
  * [2] Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga.  Pencils and infinite dihedral covers of $\mathbb{P}^{2}$.  Proc. Amer. Math. Soc., 136(1):21–29, 2008.
  * [3] Enrique Artal Bartolo, José Ignacio Cogolludo, and Hiro-o Tokunaga.  A survey on Zariski pairs.  In Algebraic geometry in East Asia—Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pages 1–100. Math. Soc. Japan, Tokyo, 2008.
  * [4] Shinzo Bannai.  A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces.  Topology Appl., 202:428–439, 2016.
  * [5] Shinzo Bannai and Momoko Ohno.  Two-graphs and the embedded topology of smooth quartics and its bitangent lines.  Canad. Math. Bull., 63(4):802–812, 2020.
  * [6] Shinzo Bannai, Hiro-o Tokunaga, and Momoko Yamamoto.  A note on the topology of arrangements for a smooth plane quartic and its bitangent lines.  Hiroshima Math. J., 49(2):289–302, 2019.
  * [7] Igor V. Dolgachev.  Classical algebraic geometry. A modern view.  Cambridge University Press, Cambridge, 2012.
  * [8] Benoît Guerville-Ballé and Taketo Shirane.  Non-homotopicity of the linking set of algebraic plane curves.  J. Knot Theory Ramifications, 26(13):1750089, 13, 2017.
  * [9] Joe Harris.  Galois groups of enumerative problems.  Duke Math. J., 46(4):685–724, 1979.
  * [10] Shigeyuki Kondo.  A complex hyperbolic structure for the moduli space of curves of genus three.  J. Reine Angew. Math., 525:219–232, 2000.
  * [11] Masato Kuwata.  Twenty-eight double tangent lines of a plane quartic curve with an involution and the Mordell-Weil lattices.  Comment. Math. Univ. St. Pauli, 54(1):17–32, 2005.
  * [12] Daniel Plaumann, Bernd Sturmfels, and Cynthia Vinzant.  Quartic curves and their bitangents.  J. Symbolic Comput., 46(6):712–733, 2011.
  * [13] Xavier Roulleau.  An atlas of K3 surfaces with finite automorphism group, 2021.  arXiv:2003.08985v2.
  * [14] Matthias Schütt and Tetsuji Shioda.  Mordell-Weil lattices, volume 70 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Singapore, 2019.
  * [15] Ichiro Shimada.  Equisingular families of plane curves with many connected components.  Vietnam J. Math., 31(2):193–205, 2003.
  * [16] Ichiro Shimada.  Lattice Zariski $k$-ples of plane sextic curves and $Z$-splitting curves for double plane sextics.  Michigan Math. J., 59(3):621–665, 2010.
  * [17] Ichiro Shimada.  On an Enriques surface associated with a quartic Hessian surface.  Canad. J. Math., 71(1):213–246, 2019.  Corrigendum: _Canad. J. Math._ , 74(2): 603–605, 2022.
  * [18] Taketo Shirane.  A note on splitting numbers for Galois covers and $\pi_{1}$-equivalent Zariski $k$-plets.  Proc. Amer. Math. Soc., 145(3):1009–1017, 2017.
  * [19] Taketo Shirane.  Galois covers of graphs and embedded topology of plane curves.  Topology Appl., 257:122–143, 2019.
  * [20] The GAP Group.  GAP - Groups, Algorithms, and Programming.  Version 4.11.0 (http://www.gap-system.org).