Sequence of families of lattice polarized $K3$ surfaces, modular forms and
degrees of complex reflection groups Atsuhira Nagano

###### Abstract

We introduce a sequence of families of lattice polarized $K3$ surfaces. This
sequence is closely related to complex reflection groups of exceptional type.
Namely, we obtain modular forms coming from the inverse correspondences of the
period mappings attached to our sequence. We study a non-trivial relation
between our modular forms and invariants of complex reflection groups.
Especially, we consider a family concerned with the Shepherd-Todd group of
No.34 based on arithmetic properties of lattices and algebro-geometric
properties of the period mappings.

000Keywords: $K3$ surfaces ; Modular forms ; Complex reflection groups ;
Compactifications defined by arrangements. 000Mathematics Subject
Classification 2020: Primary 14J28 ; Secondary 11F11, 20F55, 32S22.

## Introduction

In the 20th century, Brieskorn founded an interesting theory which connects
finite real reflection groups, Klein singularities and families of rational
surfaces (see [Br]). For example, according to this theory, a family of
rational surfaces defined by the equation

$\displaystyle
z^{2}=y^{3}+(\alpha_{2}x^{3}+\alpha_{8}x^{2}+\alpha_{14}x+\alpha_{20})y+(x^{5}+\alpha_{12}x^{3}+\alpha_{18}x^{2}+\alpha_{24}x+\alpha_{30})$
(0.1)

is characterized by the real reflection group $W(E_{8})$. Namely, the theory
enables us to interpret the parameters
$\alpha_{2},\alpha_{8},\alpha_{12},\alpha_{14},\alpha_{18},\alpha_{20},\alpha_{24}$
and $\alpha_{30}$ as the invariants of $W(E_{8})$ (see [Sl] Chapter IV or [H]
Chapter 5). Many researchers have been attracted by this theory and they have
tried to generalize it. Indeed, Arnold suggested that it is an interesting
problem to obtain an analogous theory for finite complex reflection groups
(see [A], p.20). There are many works for that problem based on various ideas,
viewpoints and techniques.

On the other hand, the author has studied modular forms derived from periods
of $K3$ surfaces and realized a potential of those modular forms to be applied
to researches for complex reflection groups. In this paper, we introduce a
sequence of families of $K3$ surfaces whose period mappings are closely
related to complex reflection groups.

Let $U$ be the hyperbolic lattice of rank $2$. Let $A_{m}$ or $E_{m}$ be the
root lattices of rank $m$. Then, the $K3$ lattice $L_{K3}$ is given by
$U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}$. Here, if a lattice $\Lambda$ with
the intersection matrix $(c_{ij})$ is given, the lattice given by $(nc_{ij})$
is denoted by $\Lambda(n)$. The $2$-homology group of a $K3$ surface $S$ is
isometric to $L_{K3}$. The Néron-Severi lattice, which is denoted by ${\rm
NS}(S)$, is a sublattice of $H_{2}(S,\mathbb{Z})$ of signature $(1,\rho-1)$.
The transcendental lattice ${\rm Tr}(S)$ is the orthogonal complement of ${\rm
NS}(S)$ in $L_{K3}$. Then, ${\rm Tr}(S)$ is an even lattice of signature
$(2,20-\rho)$.

In this paper, we study a sequence of even lattices

${\bf A}_{3}\subset{\bf A}_{2}\subset{\bf A}_{1}\subset{\bf A}_{0}={\bf A},$

where

$\displaystyle\begin{cases}&{\bf A}_{0}=U\oplus U\oplus A_{2}(-1)\oplus
A_{1}(-1),\\\ &{\bf A}_{1}=U\oplus U\oplus A_{2}(-1),\\\ &{\bf A}_{2}=U\oplus
U\oplus A_{1}(-1),\\\ &{\bf A}_{3}=U\oplus\begin{pmatrix}2&1\\\
1&-2\end{pmatrix}.\end{cases}$ (0.2)

Set ${\bf M}_{j}={\bf A}_{j}^{\perp}$ in $L_{K3}$. Then, ${\bf A}_{j}$ is of
type $(2,5-j)$ and ${\bf M}_{j}$ is of type $(1,14+j).$ We introduce a
sequence of analytic sets

$\mathfrak{A}_{3}\subset\mathfrak{A}_{2}\subset\mathfrak{A}_{1}\subset\mathfrak{A}_{0}=\mathfrak{A}$

with ${\rm dim}(\mathfrak{A}_{j})=7-j$ (for detail, see Lemma 2.1 and (4.1)).
Then, we obtain an explicit family

$\displaystyle\varpi_{j}:\mathfrak{F}_{j}\rightarrow\mathfrak{A}_{j}\quad(j\in\\{0,1,2,3\\})$
(0.3)

of ${\bf M}_{j}$-polarized $K3$ surfaces. This roughly means that the Néron-
Severi lattice of a generic member of $\mathfrak{F}_{j}$ is ${\bf M}_{j}.$
These families (0.3) constitute a sequence of $K3$ surfaces indicated in the
following diagram:

(0.8)

Here, $i_{j}$ and $\tilde{i}_{j}$ are natural inclusion. The period domain for
$\mathfrak{F}_{j}$ is given by a connected component $\mathcal{D}_{j}$ of

$\displaystyle\mathcal{D}_{{\bf M}_{j}}=\\{[\xi]\in\mathbb{P}({\bf
A}_{j}\otimes\mathbb{C})\hskip 2.84526pt|\hskip 2.84526pt{}^{t}\xi{\bf
A}_{j}\xi=0,{}^{t}\xi{\bf A}_{j}\overline{\xi}>0\\},$ (0.9)

which is $(5-j)$-dimensional.

The main theme of the present paper is to show that there is an interesting
and non-trivial relation between the sequence (0.8) and finite complex
reflection groups of exceptional type.

The first purpose of this paper is to study the period mapping of the family
$\mathfrak{F}_{0}$. The inverse of the period mapping gives a pair of
meromorphic modular forms on $\mathcal{D}_{0}$ (see Definition 5.1). We will
obtain a system of generators of the ring of these modular forms by applying
techniques for periods of $K3$ surfaces (Theorem 5.1 and 5.2). In short, we
study a family of $K3$ surfaces defined by the equation

$\displaystyle
z^{2}=y^{3}+(a_{0}x^{5}+a_{4}x^{4}+a_{8}x^{3})y+(a_{2}x^{7}+a_{6}x^{6}+a_{10}x^{5}+a_{14}x^{4})$
(0.10)

and we show that the parameters $a_{2},a_{4},a_{6},a_{8},a_{10}$ and $a_{14}$
with positive weight induce a system of generators of the ring of modular
forms.

Our modular forms are highly expected to have a closed relation with the
complex reflection group of No.34 in the list of Shepherd-Todd [ST] (see also
[LT] Appendix D), because three times the weights of the modular forms
(namely, $6,12,18,24,30$ and $42$) are equal to the degrees of the group. This
group has the maximal rank among finite complex reflection groups of
exceptional type. This expectation is based on not only the above mentioned
apparent similarity between the weights and the degrees, but also the
following fact. There are exact descriptions of the period mappings for the
subfamilies $\mathfrak{F}_{1},\mathfrak{F}_{2},\mathfrak{F}_{3}$ via
invariants of complex reflection groups. Precisely, the period mappings for
$\mathfrak{F}_{1}$ ($\mathfrak{F}_{2},\mathfrak{F}_{3}$, resp.) derive
Hermitian (Siegel, Hilbert, resp.) modular forms with explicit expressions via
the invariants of the complex reflection group No.33 (No.31, No.23, resp.) of
rank $r_{j}$ and a system of appropriate theta functions (for detail, see
Section 4.3; see also Remark 5.1). Letting $(w_{1}^{(j)},\ldots,w_{r}^{(j)})$
be the weights of the modular forms as in Table 1, then the degrees of the
complex reflection groups are given by
$(\kappa_{j}w_{1}^{(j)},\ldots,\kappa_{j}w_{r}^{(j)}).$ Here, the weights of
the theta functions account for the integer $\kappa_{j}$.

$j$ | Modular forms | Weights |  $K3$ surfaces | Reflection groups | $r_{j}$ | $\kappa_{j}$
---|---|---|---|---|---|---
$0$ | see Definition 5.1 | $2,4,6,8,10,14$ | This paper | No.34 | $6$ | $3$
$1$ | Hermitian | $4,6,10,12,18$ | [NS2] | No.33 | $5$ | $1$
$2$ | Siegel | $4,6,10,12$ | [CD] | No.31 | $4$ | $2$
$3$ | Hilbert | $2,6,10$ | [Na1] | No.23 | $3$ | $1$
Table 1: Modular forms coming from $K3$ surfaces and complex reflection groups

In these works for the families $\mathfrak{F}_{j}$ $(j\in\\{1,2,3\\})$, the
Satake-Baily-Borel compactifications for bounded symmetric domains play big
roles. However, in the case of $\mathfrak{F}_{0}$, the Satake-Baily-Borel
compactifications are inadequate, because we need to consider modular forms
defined on a complement of an arrangement of hyperplanes (see Section 3 and
5). So, instead of the Satake-Baily-Borel compactifications, we will consider
the Looijenga compactifications constructed in [L2]. The Looijenga
compactifications are coming from arithmetic arrangements of hyperplanes. We
can regard them as interpolations between the Satake-Baily-Borel
compactifications and toroidal compactifications. Their properties are
essential for our construction of modular forms.

By the way, there exists a double covering of every member of
$\mathfrak{F}_{0}$, which is a $K3$ surface also. We obtain the family

$\overline{\varpi_{j}}:\mathfrak{G}_{j}\rightarrow\mathfrak{A}_{j}\quad(j\in\\{0,1,2,3\\})$

whose members are such double coverings and we have the following diagram:

(0.15)

Here, $\varphi_{j}$ is a correspondence given by the double covering.

The second purpose of this paper is to determine the transcendental lattice
for $\mathfrak{G}_{0}.$ The family $\mathfrak{G}_{0}$ has interesting
features. For example, it naturally contains the famous family of Kummer
surfaces coming from principally polarized Abelian surfaces. Moreover,
$\mathfrak{G}_{0}$ is a natural extension of the family studied by Matsumoto-
Sasaki-Yoshida [MSY], whose periods are solutions of the hypergeometric
equation of type $(3,6)$. In spite of interesting properties of
$\mathfrak{G}_{j}$ $(j\in\\{0,1,2,3\\})$, it is not straightforward to
determine the lattices for them. For example, if $j\in\\{1,2,3\\}$, the
lattices for $\mathfrak{G}_{j}$ were determined via precise arguments or heavy
calculations (for detail, see Section 6). In the present paper, we will
determine the transcendental lattice for $\mathfrak{G}_{0},$ based on the
result of $\mathfrak{G}_{1}$ and arithmetic properties of even lattices
(Theorem 6.1). As a result, the transcendental lattices ${\bf B}_{j}$ for
$\mathfrak{G}_{j}$ are given as follows.

$\displaystyle\begin{cases}&{\bf B}_{0}=U(2)\oplus
U(2)\oplus\begin{pmatrix}-2&0&1\\\ 0&-2&1\\\ 1&1&-4\end{pmatrix},\\\ &{\bf
B}_{1}={\bf A}_{1}(2)=U(2)\oplus U(2)\oplus A_{2}(-2),\\\ &{\bf B}_{2}={\bf
A}_{2}(2)=U(2)\oplus U(2)\oplus A_{1}(-2),\\\ &{\bf B}_{3}={\bf
A}_{3}(2)=U(2)\oplus\begin{pmatrix}4&2\\\ 2&-4\end{pmatrix}.\end{cases}$
(0.16)

Especially, we note that ${\bf B}_{0}$ is not just ${\bf A}_{0}(2).$

It is an interesting problem to describe our meromorphic modular forms via the
invariants of the group No.34 and explicit special functions (like theta
functions). Furthermore, it may be quite meaningful to understand why the
period mappings for the sequence (0.8) are related with complex reflection
groups. While the methods in this paper and preceding papers [CD], [Na1] and
[NS2] are just based on algebro-gemetric properties of $K3$ surfaces and
arithmetic properties of modular forms, the author does not know the
fundamental reason why the complex reflection groups work effectively as in
Table 1. The author expects that there exists an unrevealed principle
underlying the relation between our sequence of the families and complex
reflection groups. There are indications which support this expectation. For
example, Sekiguchi [Se] studies Arnold’s problem based on methods of Frobenius
potentials and he obtains a family of rational surfaces. Although his
standpoint and methods are widely different from ours, a direct calculation
shows that our $K3$ surface of (0.10) is related to Sekiguchi’s rational
surface (see Remark 2.2). The author is hoping that a new principle will
rationalize our families of $K3$ surfaces from the viewpoint of complex
reflection groups in near future, as Brieskorn’s theory enables us to explain
the essence of the family of the rational surfaces of (0.1).

##  1 Arithmetic arrangement of hyperplanes and Looijenga compactification

Looijenga constructed compactifications for bounded symmetric domains of type
$IV$ derived from arithmetic arrangements of hyperplanes. First, we will
survey his result. For detail, see [L2].

Let $V$ be an $(n+2)$-dimensional vector space over $\mathbb{C}$ with a non-
degenerated symmetric bilinear form $\varphi.$ Now, we suppose that
$(V,\varphi)$ has been defined over $\mathbb{Q}$ and $\varphi$ is of signature
$(2,n)$ for the $\mathbb{Q}$-structure. Then, the set $\\{[v]\in V|\hskip
2.84526pt\varphi(v,v)=0,\varphi(v,\overline{v})>0\\}$ has two connected
components $\mathscr{D}$ and $\mathscr{D}_{-}$. We take $\mathscr{D}$ from
$\\{\mathscr{D},\mathscr{D}_{-}\\}$. This is a bounded symmetric domain of
type $IV$. For a linear subspace $L$ of $V$, we set
$\mathscr{D}_{L}=\mathscr{D}\cap\mathbb{P}(L).$ The orthogonal group
$O(\varphi)$ is an algebraic group over $\mathbb{Q}$. Let $\Gamma$ be an
arithmetic subgroup of $O(\varphi)$. We set $X=\mathscr{D}/\Gamma$.

If a hyperplane $H$ of $V$ is defined over $\mathbb{Q}$ and of signature
$(2,n-1)$, it gives a hypersurface $\mathscr{D}_{H}\not=\phi.$ Suppose
$\mathscr{H}$ is a $\Gamma$-invariant arrangement of hyperplanes satisfying
this property. Such an arrangement is said to be arithmetic if it is given by
a finite union of $\Gamma$-orbits.

Set

$\mathscr{D}^{\circ}=\mathscr{D}-\bigcup_{H\in\mathscr{H}}\mathscr{D}_{H}.$

Looijenga [L2] constructs a natural compactification of
$X^{\circ}=\mathscr{D}^{\circ}/\Gamma$. Namely, the Looijenga compactification
is given by

$\displaystyle\widehat{X^{\circ}}^{\bf L}:=\widehat{\mathscr{D}^{\circ}}^{\bf
L}/\Gamma,$ (1.1)

where

$\displaystyle\widehat{\mathscr{D}^{\circ}}^{\bf
L}=\mathscr{D}^{\circ}\sqcup\coprod_{L\in{\bf
PO}(\mathscr{H}|_{\mathscr{D}})}\pi_{L}(\mathscr{D}^{\circ})\sqcup\coprod_{\sigma\in\Sigma(\mathscr{H})}\pi_{\sigma}(\mathscr{D}^{\circ}).$
(1.2)

The disjoint union of (1.2) admits an appropriate topology and
$\mathscr{D}^{\circ}$ is a open and dense set in
$\widehat{\mathscr{D}^{\circ}}^{\bf L}.$ We remark that the Looijenga
compactification $\widehat{X^{\circ}}^{\bf L}$ coincides with the Satake-
Baily-Borel compactification $\widehat{X}^{\bf SBB}$ when $\mathscr{H}=\phi.$
In the following, we will see the meaning of (1.1) and (1.2).

Letting $L$ be a subspace of $V$, we have a natural projection
$\pi_{L}:\mathbb{P}(V)-\mathbb{P}(L)\rightarrow\mathbb{P}(V/L)$. Let ${\bf
PO}(\mathscr{H}|_{\mathscr{D}})$ be a set of subspaces $L$ of $V$ such that
there exists $z\in L$ with $\varphi(z,\overline{z})>0$. We note that $L\in{\bf
PO}(\mathscr{H}|_{\mathscr{D}})$ if and only if $\mathscr{D}_{L}\not=\phi.$
Also, we remark that ${\bf PO}(\mathscr{H}|_{\mathscr{D}})$ is a partially
ordered set (see [L1] Section 2 and 3).

For a $\mathbb{Q}$-isotropic line $I$ in $V$, $\varphi$ defines a bilinear
form on the $n$-dimensional space $I^{\perp}/I$ of signature $(1,n-1)$. Taking
the choice of $\mathscr{D}$ from $\\{\mathscr{D},\mathscr{D}_{-}\\}$ into
account, we have an $n$-dimensional cone $C_{I}$ in $I^{\perp}/I$. Let
$C_{I,+}(\subset I^{\perp}/I)$ be the convex hull of
$(I^{\perp}/I)\cap\overline{C_{I}}$. If a member $H\in\mathscr{H}$ contains
$I$, then it naturally determines a hyperplane $H_{I^{\perp}/I}$ of
$I^{\perp}/I$ of signature $(1,n-2).$ The hyperplanes $H_{I^{\perp}/I}$ with
$H_{I^{\perp}/I}\cap C_{I,+}\not=\phi$ give a decomposition
$\Sigma(\mathscr{H})_{I}$ of the cone $C_{I,+}$ into locally rational cones.

We set $\Sigma(\mathscr{H})=\bigcup_{I}\Sigma(\mathscr{H})_{I}$. For a cone
$\sigma\in\Sigma(\mathscr{H})_{I}(\subset\Sigma(\mathscr{H})),$ we have the
$\Sigma$-support space $V_{\sigma}(\subset I^{\perp})$. Namely, $V_{\sigma}$
contains $I$ and corresponds to the $\mathbb{C}$-span of the cone
$\sigma(\subset I^{\perp}/I)$. Hence, $V_{\sigma}$ is given by the
intersection of $I^{\perp}$ and the members $H$ of $\mathscr{H}$ such that
$H\supset I$. We put $\pi_{\sigma}=\pi_{V_{\sigma}}$.

Set

$\displaystyle\mathscr{D}^{\Sigma(\mathscr{H})}=\coprod_{\sigma\in\Sigma(\mathscr{H})}\pi_{\sigma}(\mathscr{D}).$
(1.3)

Then, $X^{\Sigma(\mathscr{H})}=\mathscr{D}^{\Sigma(\mathscr{H})}/\Gamma$ is a
normal analytic space.

We have a blowing up $\widetilde{X^{\circ}}\rightarrow
X^{\Sigma(\mathscr{H})}$, which is coming from the connected components of
intersections of members $H\in\mathscr{H}$. The Looijenga compactification
$\widehat{X^{\circ}}^{\bf L}$ of (1.1) is equal to a blowing down
$\widetilde{X^{\circ}}\rightarrow\widehat{X^{\circ}}^{\bf L}.$

######  Theorem 1.1.

([L2] Corollary 7.5) Suppose that every $\pi_{\sigma}(\mathscr{D})$ in (1.3)
is not $(n-1)$-dimensional. Then, the algebra

$\bigoplus_{k\in\mathbb{Z}}H^{0}(\mathscr{D}^{\circ},\mathcal{O}(\mathscr{L}^{k}))^{\Gamma},$

where $\mathscr{L}$ is the natural automorphic bundle over $\mathscr{D}$, is
finitely generated with positive degree generators. Its Proj gives the
Looijenga compactification $\widehat{X^{\circ}}^{\bf L}$ of (1.1). The
boundary $\widehat{X^{\circ}}^{\bf L}-X^{\circ}$ is the strict transform of
the boundary $X^{\Sigma(\mathscr{H})}-X$. Especially,

${\rm codim}\left(\widehat{X^{\circ}}^{\bf L}-X^{\circ}\right)\geq 2$

holds.

Let $\widehat{\mathscr{L}}$ be an ample line bundle on
$\widehat{X^{\circ}}^{\bf L}$ such that
$\widehat{\mathscr{L}}|_{X^{\circ}}=\mathscr{L}|_{X^{\circ}}$. It is shown in
[L2] that every meromorphic $\Gamma$-invariant automorphic form whose poles
are contained in $\mathscr{H}$ is corresponding to a meromorphic section $s$
of $\widehat{\mathscr{L}}$ such that $s|_{X^{\circ}}$ is holomorphic.

###  1.1 Lattice ${\bf A}$ and arrangement $\mathcal{H}$

We set

$\displaystyle{\bf A}={\bf A}_{0}=U\oplus U\oplus A_{2}(-1)\oplus A_{1}(-1).$
(1.4)

For this lattice ${\bf A}$, we set

$\displaystyle\Gamma=\tilde{O}({\bf A})\cap O^{+}({\bf A}).$ (1.5)

Here, $\tilde{O}({\bf A})$ is the stable orthogonal group: $\tilde{O}({\bf
A})={\rm Ker}\left(O({\bf A})\rightarrow{\rm Aut}({\bf A}^{\vee}/{\bf
A})\right)$, where ${\bf A}^{\vee}={\rm Hom}({\bf A},\mathbb{Z}).$ Also,
$O^{+}({\bf A})$ is the subgroup of $O({\bf A})$ which preserves the connected
component $\mathcal{D}$. The lattice ${\bf A}$ satisfies the Kneser conditions
in the sense of Gritsenko-Hulek-Sankaran [GHS]. Therefore, we have the
following result.

######  Proposition 1.1.

Let $\Delta({\bf A})$ be the set of vectors $v\in{\bf A}$ such that $(v\cdot
v)=-2$. The group $\Gamma$ of (1.5) is generated by reflections
$\sigma_{\delta}:z\mapsto z+(z\cdot\delta)\delta$ for $\delta\in\Delta({\bf
A})$ and it holds ${\rm Char}(\Gamma)=\\{{\rm id},{\rm det}\\}$.

Here, we note that the intersection number of elements $v_{1}$ and $v_{2}$ of
lattices are often denoted by $(v_{1}\cdot v_{2})$ in this paper.

######  Lemma 1.1.

The group $\Gamma$ is isomorphic to the projective orthogonal group
$PO^{+}({\bf A})$.

###### Proof.

Letting $\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$ be a system of basis of
$A_{2}(-1)\oplus A_{1}(-1)$ with $(\alpha_{j}\cdot\alpha_{j})=-2$
$(j\in\\{1,2,3\\})$, $(\alpha_{1}\cdot\alpha_{2})=1$ and
$(\alpha_{k}\cdot\alpha_{3})=0$ $(k\in\\{1,2\\})$. Then,
$y_{1}=\frac{1}{3}\alpha_{1}+\frac{2}{3}\alpha_{2},y_{2}=\frac{2}{3}\alpha_{1}+\frac{1}{3}\alpha_{2}$
and $y_{3}=\frac{1}{2}\alpha_{3}$ generate the discriminant group ${\bf
A}^{\vee}/{\bf A}$. It follows that $-id_{O^{+}({\bf A})}\not\in\Gamma$ and
$O^{+}({\bf A})/\Gamma\simeq\mathbb{Z}/2\mathbb{Z}$. Hence, the assertion
follows. ∎

We consider the case $V={\bf A}\otimes\mathbb{C}$. Let $\\{e_{j},f_{j}\\}$
($j\in\\{1,2\\}$) be the system of basis of $U$. Let
$\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$ be the system of basis of
$A_{2}(-1)\oplus A_{1}(-1)$ as in Lemma 1.1. A vector $v\in V$ is given by the
form

$\displaystyle
v=\xi_{1}e_{1}+\xi_{2}f_{1}+\xi_{3}e_{2}+\xi_{4}f_{2}+\xi_{5}\alpha_{1}+\xi_{6}\alpha_{2}+\xi_{7}\alpha_{3}.$
(1.6)

Let us consider a hyperplane $H_{0}=\\{\xi_{7}=0\\}$ in $V$. For $\Gamma$ of
(1.5), the $\Gamma$-orbits of $H_{0}$ give an arithmetic arrangement
$\mathcal{H}$.

######  Lemma 1.2.

The above arrangement $\mathcal{H}$ of hyperplanes satisfies the condition of
Theorem 1.1.

###### Proof.

We will prove that all members of $\mathcal{H}$ contain a non-zero common
subspace of the negative-definite vector space
$\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{\mathbb{C}}$. Then, it is
guaranteed that our arrangement $\mathcal{H}$ satisfies the condition of
Theorem 1.1, as in the argument of [L1] Section 6.

The hyperplane $H_{0}=\\{\xi_{7}=0\\}$ of $V$ is generated by the system
$\\{e_{1},f_{1},e_{2},f_{2},\alpha_{1},\alpha_{2}\\}$ of basis. Using the
notation in the proof of Lemma 1.1, ${\bf A}^{\vee}/{\bf A}$ is generated by
$y_{1},y_{2}$ and $y_{3}$. We note that every $\gamma\in\Gamma$ fixes
$y_{j}\in{\bf A}^{\vee}/{\bf A}$. This implies that we can take a system of
basis of the subspace $\gamma H_{0}$ such that this system is an extension of
$\\{\alpha_{1},\alpha_{2}\\}$. Therefore, every member of $\mathcal{H}$
contains the $2$-dimensional subspace
$\langle\alpha_{1},\alpha_{2}\rangle_{\mathbb{C}}$ of
$\langle\alpha_{1},\alpha_{2},\alpha_{3}\rangle_{\mathbb{C}}$. ∎

######  Remark 1.1.

The condition of Theorem 1.1 is not always satisfied. For example, under the
notation (1.6), let us take a hyperplane
$H_{0}^{\prime}=\\{\xi_{5}-\xi_{6}=0\\}$. We can see that the arrangement
$\mathcal{H}^{\prime}$ of $\gamma H_{0}^{\prime}$ for $\gamma\in\Gamma$ does
not satisfy the condition of Theorem 1.1. This condition is closely related to
whether an arrangement gives the zero of a modular form or not. In fact, we
can see that $\mathcal{H}$ is not the zero set of a modular form, whereas
$\mathcal{H}^{\prime}$ does.

For our symmetric space $\mathcal{D}=\mathcal{D}_{0}$, which is a connected
component of $\mathcal{D}_{{\bf M}_{0}}$ of (0.9), set

$\displaystyle\mathcal{D}^{\circ}=\mathcal{D}-\bigcup_{H\in\mathcal{H}}\mathcal{D}_{H}.$
(1.7)

Due to Theorem 1.1 and Proposition 1.2, we have the following result for our
arrangement $\mathcal{H}$.

######  Proposition 1.2.

${\rm codim}\left(\widehat{\mathcal{D}^{\circ}/\Gamma}^{\bf
L}-\mathcal{D}^{\circ}/\Gamma\right)\geq 2.$

##  2 Family $\mathfrak{F}_{0}$ of $K3$ surfaces with Picard number $15$

In the present paper, we will consider the Looijenga compactification coming
from the arithmetic arrangement $\mathcal{H}$ in Section 1.1. In order to
obtain an explicit model of the compactification, we will introduce a family
of elliptic $K3$ surfaces whose transcendental lattice is ${\bf A}$ in (1.4)
(see Theorem 2.1). Periods for $K3$ surfaces are very important in our
argument. We remark that the period mapping for a family of $K3$ surfaces is
essentially related to the arithmetic property of the transcendental lattice
for a generic member of the family.

For
$a=(a_{0},a_{2},a_{4},a_{6},a_{8},a_{10},a_{14})\in\mathbb{C}^{7}-\\{0\\}=:\mathbb{C}_{a}$,
we consider the hypersurface $S_{a}$ defined by an equation

$\displaystyle
S_{a}:z^{2}=y^{3}+(a_{0}x^{5}+a_{4}x^{4}w^{4}+a_{8}x^{3}w^{8})y+(a_{2}x^{7}w^{2}+a_{6}x^{6}w^{6}+a_{10}x^{5}w^{10}+a_{14}x^{4}w^{14})$
(2.1)

of weight $30$ in the weighted projective space ${\rm
Proj}(\mathbb{C}[x,y,z,w])=\mathbb{P}(4,10,15,1)$. We have a natural action of
$\mathbb{C}^{*}$ on $\mathbb{P}(4,10,15,1)$ given by
$(x,y,z,w)\mapsto(x,y,z,\lambda^{-1}w)$ and that on $\mathbb{C}_{a}$ given by
$a\mapsto\lambda\cdot
a=(\lambda^{k}a_{k})=(a_{0},\lambda^{2}a_{2},\lambda^{4}a_{4},\lambda^{6}a_{6},\lambda^{8}a_{8},\lambda^{10}a_{10},\lambda^{14}a_{14})$.

By a direct observation as in Section 1 of [Na3], we have the following
result.

######  Lemma 2.1.

Let $\mathfrak{A}_{0}=\mathfrak{A}$ be the set of parameters
$a\in\mathbb{C}_{a}$ such that $S_{a}$ is a $K3$ surface. Then,
$\mathfrak{A}_{0}$ is a subset of $\\{a\in\mathbb{C}_{a}|\hskip
2.84526pta_{0}\not=0\\}\cup\\{a\in\mathbb{C}_{a}|\hskip
2.84526pta_{2}\not=0\\}$ such that

$\mathbb{C}_{a}-\mathfrak{A}_{0}=\mathcal{C}^{\prime}\sqcup\mathcal{C}^{\prime\prime}$

where

$\displaystyle\begin{cases}\mathcal{C}^{\prime}&=\\{a\in\mathbb{C}_{a}|\hskip
2.84526pta_{0}\not=0,a_{10}=a_{12}=a_{18}=0\\}\subset\\{a_{0}\not=0\\},\\\
\mathcal{C}^{\prime\prime}&=\\{a\in\mathbb{C}_{a}|\hskip
2.84526pta_{2}\not=0,a_{0}=a_{10}=a_{12}=a_{18}=0\\}\subset\\{a_{2}\not=0\\}.\end{cases}$

######  Remark 2.1.

The surface $S_{a}$ is degenerated to a rational surface if $a_{0}=a_{2}=0.$

We have a family

$\varpi_{0}:\mathfrak{F}_{0}=\\{S_{a}\text{ of }(\ref{SK3})|\hskip
2.84526pta\in\mathfrak{A}_{0}\\}\rightarrow\mathfrak{A}_{0}$

of elliptic $K3$ surfaces.

###  2.1 Singular fibres

The Weierstrass equation (2.1) defines an elliptic surface
$\pi_{a}:S_{a}\rightarrow\mathbb{P}^{1}(\mathbb{C}).$ For a generic point
$a\in\mathfrak{A}_{0}$, we have singular fibres for $\pi_{a}$ of Kodaira type

$\displaystyle III^{*}+IV^{*}+7I_{1},$ (2.2)

as illustrated in Figure 1. Now, $\pi_{a}^{-1}(\infty)$ ($\pi_{a}^{-1}(0)$,
resp.) is a singular fibre of Kodaira type $III^{*}$ ($IV^{*}$, resp.). Each
gives an $E_{7}$-singularity and an $E_{6}$-singularity, respectively.

Set $x_{0}=\frac{x}{w^{4}}$ and

$\displaystyle
g_{2}^{\vee}(x_{0},a)=a_{0}x_{0}^{5}+a_{4}x_{0}^{4}+a_{8}x_{0}^{3},\quad
g_{3}^{\vee}(x_{0},a)=a_{2}x_{0}^{7}+a_{6}x_{0}^{6}+a_{10}x_{0}^{5}+a_{14}x_{0}^{4}.$

Let $r(a)$ be the resultant of $g_{2}^{\vee}(x_{0},a)$ and
$g_{3}^{\vee}(x_{0},a)$ in $x_{0}$. Also, let $R(x_{0},a)$ be a polynomial in
$x_{0}$ coming from the discriminant of the right hand side of (2.1) in $y$.
So, the discriminant of $\frac{1}{x_{0}^{8}}R(x_{0},a)$ in $x_{0}$ is given by
$r(a)^{3}d_{84}(a)$, where $d_{84}(a)$ can be calculated as a polynomial in
$a$ of weight $84$.

Figure 1: Singular fibres for
$\pi_{a}:S_{a}\rightarrow\mathbb{P}^{1}(\mathbb{C})$

The following lemma can be proved by arguments of elliptic surfaces like [Na3]
Section 1.

######  Lemma 2.2.

Except for the generic case (2.2), the types of the singular fibres for the
elliptic surface $\pi_{a}:S_{a}\rightarrow\mathbb{P}^{1}(\mathbb{C})$
$(a\in\mathfrak{A}_{0})$ are given by the following.

  * •

If $a\in\mathfrak{A}_{0}$ satisfies $r(a)=0$, there is a new singular fibre of
Kodaira type $II$ on the elliptic surface $S_{a}$. Such a singular fibre does
not acquire any new singularities.

  * •

If $a\in\mathfrak{A}_{0}$ satisfies $d_{84}(a)=0,$ two of singular fibres of
Kodaira type $I_{1}$ in (2.2) collapse into a singular fibre of type $I_{2}$:

$III^{*}+IV^{*}+I_{2}+5I_{1}$

In this case, a new $A_{1}$-singularity appears on the $K3$ surface $S_{a}$.

  * •

If $a_{0}=0$, there are the singular fibres of Kodaira type

$II^{*}+IV^{*}+6I_{1}$

on $S_{a}$. In this situation, the $E_{7}$-singularity of (2.2) terns into an
$E_{8}$-singularity.

  * •

If $a_{14}=0$, there are the singular fibres of Kodaira type

$III^{*}+III^{*}+6I_{1}.$

on $S_{a}$. The $E_{6}$-singularity of (2.2) terns into an
$E_{7}$-singularity.

###  2.2 Local period mapping

Set

$\displaystyle\tilde{\mathfrak{A}}=\mathfrak{A}-\\{a\in\mathbb{C}_{a}\hskip
2.84526pt|\hskip 2.84526pta_{0}a_{14}d_{84}(a)=0\\}.$ (2.3)

Let $F$ be a general fibre for the elliptic surface $\pi_{a}$ and $O$ be the
zero section. Let $C_{1},\ldots,C_{7}$ ($D_{1},\ldots,D_{6}$, resp.) be nodal
curves in the singular fibre of type $III^{*}$ ($IV^{*}$, resp.) indicated
Figure 1. For $a\in\tilde{\mathfrak{A}}$, the lattice generated by

$\displaystyle F,O,C_{1},\ldots,C_{7},D_{1},\ldots,D_{6}$ (2.4)

is a sublattice of ${\rm NS}(S_{a})$ whose intersection matrix is

$\displaystyle{\bf M}={\bf M}_{0}=U\oplus E_{7}(-1)\oplus E_{6}(-1).$ (2.5)

Let $L_{K3}$ be the $K3$ lattice : $L_{K3}=II_{3,19}$. The orthogonal
complement of ${\bf M}$ with respect to the unimodular lattice $L_{K3}$ is
given by ${\bf A}$ of (1.4). We have an isometry
$\psi:H_{2}(S_{a},\mathbb{Z})\rightarrow L_{K3}$ such that

$\displaystyle\psi(F)=\gamma_{8},\quad\psi(O)=\gamma_{9},\quad\psi(C_{j})=\gamma_{9+j},\quad\psi(D_{k})=\gamma_{16+k}\quad(j\in\\{1,\ldots,7\\},k\in\\{1,\ldots,6\\}).$

Then, the sublattice
$\langle\gamma_{8},\ldots,\gamma_{22}\rangle_{\mathbb{Z}}$ in $L_{K3}$ is
isometric to the lattice ${\bf M}$ of (2.5). This is a primitive sublattice,
because $|{\rm det}({\bf M})|=6$ is square-free. Hence, we can take
$\gamma_{1},\ldots,\gamma_{7}\in L_{K3}$ such that
$\\{\gamma_{1},\ldots,\gamma_{7},\gamma_{8},\ldots,\gamma_{22}\\}$ gives a
system of basis of $L_{K3}$. Letting $\\{\delta_{1},\ldots,\delta_{22}\\}$ be
the system of dual basis of $\\{\gamma_{1},\ldots,\gamma_{22}\\}$ with respect
to the unimodular lattice $L_{K3}.$ Then, the intersection matrix of the
sublattice $\langle\delta_{1},\ldots,\delta_{7}\rangle_{\mathbb{Z}}$ is equal
to the intersection matrix of ${\bf A}$.

######  Proposition 2.1.

(The canonical form for $a_{0}\not=0$) If
$a\in\mathfrak{A}\cap\\{a_{0}\not=0\\}$, (2.1) is transformed to the
Weierstrass equation

$\displaystyle
S(u):z^{2}=y^{3}+(x^{5}+u_{4}x^{4}w^{4}+u_{8}x^{3}w^{8})y+(u_{2}x^{7}w^{2}+u_{6}x^{6}w^{6}+u_{10}x^{5}w^{10}+u_{14}x^{4}w^{14}).$
(2.6)

###### Proof.

By putting

$x\mapsto\frac{x}{a_{0}},\quad y\mapsto\frac{y}{a_{0}^{2}},\quad
z\mapsto\frac{z}{a_{0}^{3}},$

and

$u_{2}=\frac{a_{2}}{a_{0}},\quad u_{4}=a_{4},\quad u_{6}=a_{6},\quad
u_{8}=a_{0}a_{8},\quad u_{10}=a_{0}a_{10},\quad u_{14}=a_{0}^{2}a_{14},$

we obtain (2.6). ∎

Here, we put
$u=(u_{k})=(u_{2},u_{4},u_{6},u_{8},u_{10},u_{14})\in\mathbb{C}^{6}-\\{0\\}=:\mathbb{C}_{u}.$
Let $\mathcal{U}^{*}$ be a subset of $\mathbb{C}_{u}$ of codimension $3$ such
that

$\displaystyle\mathbb{C}_{u}-\mathcal{U}^{*}=\\{u\in\mathbb{C}_{u}|\hskip
2.84526ptu_{10}=u_{12}=u_{18}=0\\}.$ (2.7)

For $\lambda\in\mathbb{C}^{*}$ and $u=(u_{k})\in\mathcal{U}$, set
$\lambda\cdot u=(\lambda^{k}u_{k})$. This action induces an isomorphism
$\lambda:S(u)\rightarrow S(\lambda\cdot u)$. Letting
$[u]=(u_{2}:u_{4}:u_{6}:u_{8}:u_{10}:u_{14})\in\mathbb{P}(2,4,6,8,10,14)$ be
the point which is corresponding to $u\in\mathbb{C}_{u}$, set
$\mathcal{U}=\\{[u]\in\mathbb{P}(2,4,6,8,10,14)|\hskip
2.84526ptu\in\mathcal{U}^{*}\\}.$ The above action of $\mathbb{C}^{*}$ on
$\mathcal{U}^{*}$, we naturally defines the family

$\displaystyle\\{S([u])\hskip 2.84526pt|\hskip
2.84526pt[u]\in\mathcal{U}\\}\rightarrow\mathcal{U}.$ (2.8)

######  Definition 2.1.

Let $\pi_{1}:S_{1}\rightarrow\mathbb{P}^{1}(\mathbb{C})$ and
$\pi_{2}:S_{2}\rightarrow\mathbb{P}^{1}(\mathbb{C})$ be two elliptic surfaces.
Suppose that there exist a biholomorphic mapping $f:S_{1}\rightarrow S_{2}$
and $\varphi\in{\rm Aut}(\mathbb{P}^{1}(\mathbb{C}))$ with
$\varphi\circ\pi_{1}=\pi_{2}\circ f$. Then, these two elliptic surfaces are
said to be isomorphic as elliptic surfaces.

The canonical form (2.6) naturally gives an elliptic surface
$\pi_{[u]}:S([u])\rightarrow\mathbb{P}^{1}(\mathbb{C})$.

######  Lemma 2.3.

Two elliptic surfaces
$\pi_{[u_{1}]}:S([u_{1}])\rightarrow\mathbb{P}^{1}(\mathbb{C})$ and
$\pi_{[u_{2}]}:S([u_{2}])\rightarrow\mathbb{P}^{1}(\mathbb{C})$ are isomorphic
as elliptic surfaces if and only if
$[u_{1}]=[u_{2}]\in\mathbb{P}(2,4,6,8,10,14)$.

###### Proof.

We can prove it by an argument which is similar to the proof of [Na3] Lemma
1.1. ∎

Let us take a generic point $a\in\tilde{\mathfrak{A}}$ of (2.3). Since
$\tilde{\mathfrak{A}}\subset\mathfrak{A}\cap\\{a_{0}\not=0\\}$, we obtain the
corresponding surface $S([u])$ for a parameter
$[u]\in\mathbb{P}(2,4,6,8,10,14)={\rm
Proj}(\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}])$, which is given by
the canonical form (2.6). We set

$\mathcal{P}(\tilde{\mathfrak{A}})=\\{[u]\in\mathbb{P}(2,4,6,8,10,14)|\hskip
2.84526pt\text{there exists }a\in\tilde{\mathfrak{A}}\text{ such that
}S_{a}\text{ is identified with }S([u])\text{ of }(\ref{SK3Can})\\}.$

By an argument which is similar to [Na3] Section 1, using Lemma 2.3 also, we
can obtain a local period mapping defined on sufficiently small neighborhood
around $[u]$ in $\mathcal{P}(\tilde{\mathfrak{A}})$. By gluing the local
period mappings, we obtain the period mapping

$\displaystyle\Phi_{1}:\mathcal{P}(\tilde{\mathfrak{A}})\rightarrow\mathcal{D}$
(2.9)

given by

$[u]\mapsto\Big{(}\int_{\psi^{-1}_{[u]}(\gamma_{1})}\omega_{[u]}:\cdots:\int_{\psi^{-1}_{[u]}(\gamma_{7})}\omega_{[u]}\Big{)},$

where $\omega_{[u]}$ is the unique holomorphic $2$-form on $S([u])$ up to a
constant factor and

$\psi_{[u]}:H_{2}(S([u]),\mathbb{Z})\rightarrow L_{K3}$

is an appropriate isometry, called $S$-marking. We note that the period
mapping (2.9) is a multivalued analytic mapping.

We call the pair $(S([u]),\psi_{[u]})$ an $S$-marked $K3$ surface. By applying
Torelli’s theorem to the above local period mappings, we can show the
following theorem as in the proof of [Na3] Theorem 1.1 and Corollary 1.1.

######  Theorem 2.1.

For a generic point $a\in\tilde{\mathfrak{A}}$ of (2.3), the Picard number of
$S_{a}$ is $15$. The intersection matrix of the Néron-Severi lattice ${\rm
NS}(S_{a})$ (the transcendental lattice ${\rm Tr}(S_{a})$, resp.) is equal to
${\bf M}$ of (2.5) (${\bf A}$ of (1.4), resp.).

###  2.3 Double covering $K_{a}$ of $S_{a}$

The $K3$ surface $S_{a}$ of (2.1) is transformed to

$\displaystyle
Z^{2}=Y^{3}+\Big{(}a_{4}+a_{0}X+\frac{a_{8}}{X}\Big{)}Y+\Big{(}a_{6}+a_{2}X+\frac{a_{10}}{X}+\frac{a_{14}}{X^{2}}\Big{)}$
(2.10)

by the birational transformation

$x\mapsto X,\quad y\mapsto X^{2}Y,\quad z\mapsto X^{3}Z.$

We have a double covering

$\displaystyle
K_{a}:Z^{2}=Y^{3}+\Big{(}a_{4}+a_{0}U^{2}+\frac{a_{8}}{U^{2}}\Big{)}Y+\Big{(}a_{6}+a_{2}U^{2}+\frac{a_{10}}{U^{2}}+\frac{a_{14}}{U^{4}}\Big{)}$
(2.11)

of (2.10). There is a Nikulin involution on $K$ given by

$\iota_{K_{a}}:(U,Y,Z)\mapsto(-U,Y,-Z).$

This means that it satisfies $\iota_{K_{a}}^{*}\omega_{K}=\omega_{K}$, where
$\omega_{K}$ is the unique holomorphic $2$-form up to a constant factor.

We have a family
$\overline{\varpi_{0}}:\mathfrak{G}_{0}\rightarrow\mathfrak{A}_{0}$ of $K3$
surfaces, where $\mathfrak{G}_{0}=\\{K_{a}\text{ of }(\ref{KL})|\hskip
2.84526pta\in\mathfrak{A}_{0}\\}$. In fact, this surface $K_{a}$ can be
regarded as a natural generalization the Kummer surface for a principally
polarized Abelian surface. Therefore, we call a member of $\mathfrak{G}_{0}$ a
Kummer-like surface in this paper. More precisely, see Section 4. We remark
that the double covering $\varphi_{0}$ in (0.15) is given by this
$\iota_{K_{a}}$.

######  Remark 2.2.

The surface $K_{a}$ of (2.11) has another involution

$\jmath_{K_{a}}:(U,Y,Z)\mapsto(-U,Y,Z).$

This is not a Nikulin involution. The minimal resolution of the quotient
surface $K/\langle\jmath_{K_{a}}\rangle$ is given by the equation

$\Sigma:z^{\prime 2}=y^{\prime 3}+(a_{0}x^{\prime 3}+a_{4}x^{\prime
2}+a_{8}x^{\prime})y^{\prime}+(a_{2}x^{\prime 4}+a_{6}x^{\prime
3}+a_{10}x^{\prime 2}+a_{14}x^{\prime}).$

This is a rational surface. The surface $\Sigma$ is very similar to a surface
appearing in Sekiguchi’s recent work [Se], in which he studies algebraic
Frobenius potentials, deformation of singularities and Arnold’s problem.
Namely, he obtains the equation in the form

$z^{2}=f_{E_{7}(1)}:=y^{3}+(x^{3}+t_{2}x^{2}+t_{4}x)y+(t_{1}x^{4}+t_{3}x^{3}+t_{5}x^{2}+t_{7}x)+s_{3}y^{2}$

of a rational surfaces. Putting $s_{3}=0$, we can see an apparent
correspondence to our $\Sigma.$

The author is expecting that there is a non-trivial and unrevealed theory
connecting our work of the moduli of $K3$ surfaces and Sekiguchi’s result of
Frobenius potentials.

##  3 Moduli space of ${\bf M}$-polarized $K3$ surfaces

In this section, letting ${\bf M}$ be the lattice of (2.5) of signature
$(1,14)$, we consider the moduli space of ${\bf M}$-polarized $K3$ surfaces.
An ${\bf M}$-polarized $K3$ surface is a pair $(S,j)$ of a $K3$ surface $S$
and a primitive embedding $j:{\bf M}\hookrightarrow{\rm NS}(S)$. Two
M-polarized $K3$ surfaces $(S_{1},j_{1})$ and $(S_{2},j_{2})$ are said to be
isomorphic if there exists an isomorphism $f:S_{1}\rightarrow S_{2}$ of $K3$
surfaces such that $j_{2}=f_{*}\circ j_{1}.$ In this paper, ${\rm NS}(S)$ is
often regarded as a sublattice of the homology group $H_{2}(S,\mathbb{Z})$. We
note that ${\rm NS}(S)$ is identified with the sublattice
$H^{2}(S,\mathbb{Z})\cap H^{1,1}(S,\mathbb{R})$ of the cohomology group
$H^{2}(S,\mathbb{Z})$ by the Poincaré duality. This is denoted by the same
notation in the discussion below.

Let $V(S)^{+}$ be the connected component of $V(S)=\\{x\in
H^{1,1}_{\mathbb{R}}(S)|\hskip 2.84526pt(x\cdot x)>0\\}$ which contains the
class of a Kähler form on $S$. Let $\Delta(S)^{+}$ be the subset of effective
classes of $\Delta(S)=\\{\delta\in{\rm NS}(S)|\hskip
2.84526pt(\delta\cdot\delta)=-2\\}.$ Set $C(S)=\\{x\in V(S)^{+}|\hskip
2.84526pt(x,\delta)\geq 0,\text{ for all }\delta\in\Delta(S)^{+}\\}$. The
subset $C(S)^{+}$ of $C(S)$, which is defined by the condition
$(x\cdot\delta)>0$, is called the Kähler cone. We set ${\rm
NS}(S)^{+}=C(S)\cap H^{2}(S,\mathbb{Z})$ and ${\rm NS}(S)^{++}=C(S)^{+}\cap
H^{2}(S,\mathbb{Z}).$

Due to Theorem 2.1, we can take a point $\bar{a}\in\tilde{\mathfrak{A}}$ and
an $S$-marking $\psi_{0}:H_{2}(S_{0},\mathbb{Z})\rightarrow L_{K3}$ such that
$\psi_{0}^{-1}({\bf M})={\rm NS}(S_{0})$, where $S_{0}=S_{\bar{a}}$ is called
a reference surface. Letting $\Delta({\bf M})=\\{\delta\in{\bf M}|\hskip
2.84526pt(\delta\cdot\delta)=-2\\}$, we set $\Delta({\bf
M})^{+}=\\{\delta\in\Delta({\bf M})|\hskip
2.84526pt\psi_{0}^{-1}(\delta)\in{\rm NS}(S_{0})\text{ gives an effective
class}\\}$. The set $V({\bf M})=\\{y\in{\bf M}_{\mathbb{R}}|\hskip
2.84526pt(y\cdot y)>0\\}$ has two connected components. We suppose the
component $V({\bf M})^{+}$ contains $\psi_{0}(x)$ for $x\in V(S_{0})^{+}$. Set
$C({\bf M})^{+}=\\{y\in V({\bf M})^{+}|\hskip 2.84526pt(y\cdot\delta)>0,\text{
for all }\delta\in\Delta({\bf M})^{+}\\}$. An ${\bf M}$-polarized $K3$ surface
$(S,j)$ is called a pseudo-ample ${\bf M}$-polarized $K3$ surface if $j(C({\bf
M})^{+})\cap{\rm NS}(S)^{+}\not=\phi.$

For a $K3$ surface $S$, let $\psi:H_{2}(S,\mathbb{Z})\rightarrow L_{K3}$ be an
isometry of lattices with $\psi^{-1}({\bf M})\subset{\rm NS}(S)$. We call the
pair $(S,\psi)$ of such $S$ and $\psi$ is called a marked $K3$ surface. If
$(S,\psi^{-1}|_{\bf M})$ is a pseudo-ample ${\bf M}$-polarized $K3$ surface,
then $(S,\psi)$ is called a pseudo-ample marked ${\bf M}$-polarized $K3$
surface. For two pseudo-ample marked ${\bf M}$-polarized $K3$ surfaces
$(S_{1},\psi_{1})$ and $(S_{2},\psi_{2})$, we suppose
$(S_{1},\psi_{1}^{-1}|_{\bf M})$ and $(S_{2},\psi_{2}^{-1}|_{\bf M})$ are
isomorphic as ${\bf M}$-polarized $K3$ surfaces. Then, $(S_{1},\psi_{1})$ and
$(S_{2},\psi_{2})$ are said to be isomorphic as pseudo-ample ${\bf
M}$-polarized $K3$ surfaces. Also, if there is an isomorphism
$f:S_{1}\rightarrow S_{2}$ such that $\psi_{1}=\psi_{2}\circ f_{*}$, we say
$(S_{1},\psi_{1})$ and $(S_{2},\psi_{2})$ are isomorphic as pseudo-ample
marked ${\bf M}$-polarized $K3$ surfaces.

By gluing local moduli spaces of marked ${\bf M}$-polarized $K3$ surfaces, we
have the fine moduli space $\mathcal{M}_{\bf M}$ of marked ${\bf M}$-polarized
$K3$ surfaces. Then, we have the period mapping

$\displaystyle{\rm per}:\mathcal{M}_{\bf M}\rightarrow\mathcal{D}_{\bf M},$
(3.1)

where $\mathcal{D}_{\bf M}$ is given in (0.9).

Let $\mathcal{M}_{\bf M}^{\rm pa}(\subset\mathcal{M}_{\bf M})$ be the set of
isomorphism classes of pseudo-ample marked M-polarized $K3$ surfaces. By
restricting (3.1) to $\mathcal{M}_{\bf M}^{\rm pa}$, we have a surjective
mapping

$\displaystyle{\rm per}^{\prime}:\mathcal{M}_{\bf M}^{\rm
pa}\rightarrow\mathcal{D}_{\bf M}.$ (3.2)

The group $\Gamma({\bf M})=\\{\sigma\in O(L_{K3})|\hskip
2.84526pt\sigma(m)=m\text{ for all }m\in{\bf M}\\}$ acts on $\mathcal{M}_{\bf
M}$ by $(S,\psi)\mapsto(S,\psi\circ\sigma).$ Then, $\mathcal{M}_{\bf M}^{\rm
pa}/\Gamma({\bf M})$ gives the set of isomorphism classes of pseudo-ample
${\bf M}$-polarized $K3$ surfaces.

######  Theorem 3.1.

(Dolgachev [D], Section 3) The period mapping (3.2) induces the bijection

$\mathcal{M}_{\bf M}^{\rm pa}/\Gamma({\bf M})\simeq\mathcal{D}_{\bf
M}/\tilde{O}({\bf A})=\mathcal{D}/\Gamma,$

where $\Gamma$ is given in (1.5). Especially, $\mathcal{D}/\Gamma$ gives the
set of isomorphism classes of pseudo-ample ${\bf M}$-polarized $K3$ surfaces.

Le us take a reference surface $S_{0}=S_{\bar{a}}$ for
$\bar{a}\in\mathfrak{A}$ with the divisors (2.4) and the $S$-marking
$\psi_{0}:H_{2}(S_{0},\mathbb{Z})\rightarrow L_{K3}$ such that ${\rm
NS}(S_{0})=\psi_{0}^{-1}({\bf M}).$ For a pseudo-ample marked ${\bf
M}$-polarized $K3$ surface $(S,\psi)$, as in the proof of [Na3] Theorem 2.3,
we can show that there is an isometry $\psi:H_{2}(S,\mathbb{Z})\rightarrow
L_{K3}$ satisfying the following conditions:

  * (i)

$\psi^{-1}({\bf M})\subset{\rm NS}(S)$,

  * (ii)

$\psi^{-1}\circ\psi_{0}(F)$,$\psi^{-1}\circ\psi_{0}(O)$,$\psi^{-1}\circ\psi_{0}(C_{j})$
$(j\in\\{1,\ldots,7\\})$, $\psi^{-1}\circ\psi_{0}(D_{k})$
$(k\in\\{1,\ldots,6\\})$ are effective divisors,

  * (iii)

$\psi^{-1}\circ\psi_{0}(F)$ is a nef divisor.

By an argument which is similar to the proof of [Na3] Lemma 2.1, we can prove
the following lemma.

######  Lemma 3.1.

For any pseudo-ample marked ${\bf M}$-polarized $K3$ surface $(S,\psi)$, there
exists $a\in\mathfrak{A}$ such that $(S,\psi)$ is given by the elliptic $K3$
surface $\pi_{a}:S_{a}\rightarrow\mathbb{P}^{1}(\mathbb{C})$ given by the
Weierstrass equation (2.1). Especially, the divisor
$\psi^{-1}\circ\psi_{0}(F)$, which is effective and nef, gives a general fibre
for $\pi_{a}$.

Let us take two pseudo-ample marked ${\bf M}$-polarized $K3$ surfaces given by
elliptic $K3$ surfaces $\pi_{a}$ and $\pi_{a^{\prime}}$ in the sense of Lemma
3.1. If they are isomorphic, then the types of the singular fibres for
$\pi_{a}$ coincide with those for $\pi_{a^{\prime}}$. According to Lemma 2.2,
we have the following facts:

  * •

for $a\in\\{a_{0}\not=0\\}$, then $\pi_{a}^{-1}(\infty)$ is of Kodaira type
$III^{*}$,

  * •

for $a\in\\{a_{0}=0\\}$, then $\pi_{a}^{-1}(\infty)$ is of Kodaira type
$II^{*}$.

Hence, $S_{a}$ for $a\in\\{a_{0}\not=0\\}$ is not isomorphic to
$S_{a^{\prime}}$ for $a^{\prime}\in\\{a_{0}\not=0\\}$ as pseudo-ample ${\bf
M}$-polarized $K3$ surfaces. If $a_{0}\not=0$, we have a canonical form given
by (2.6). On the other hand, if $a_{0}=0$, we have the following result.

######  Proposition 3.1.

(The canonical form for $a_{0}=0$) If $a\in\mathfrak{A}\cap\\{a_{0}=0\\}$,
(2.1) is transformed to the Weierstrass equation

$\displaystyle
S_{1}(t):z^{2}=y^{3}+(t_{4}x^{4}w^{4}+t_{10}x^{3}w^{10})y+(x^{7}+t_{6}x^{6}w^{6}+t_{12}x^{5}w^{12}+t_{18}x^{4}w^{18}).$
(3.3)

###### Proof.

By putting

$x\mapsto\frac{x}{a_{2}},\quad y\mapsto\frac{y}{a_{2}^{2}},\quad
z\mapsto\frac{z}{a_{2}^{3}},$

and

$t_{4}=a_{4},\quad t_{6}=a_{6},\quad t_{10}=a_{2}a_{8},\quad
t_{12}=a_{2}a_{10},\quad t_{18}=a_{2}^{2}a_{14},$

we obtain (3.3). ∎

Now, we naturally obtain $S_{1}([t])$ for $[t]\in\mathbb{P}(4,6,10,12,18)={\rm
Proj}(\mathbb{C}[t_{4},t_{6},t_{10},t_{12},t_{18}])$. Let $\mathcal{T}$ be a
subset of $\mathbb{P}(4,6,10,12,18)$ such that
$\mathbb{P}(4,6,10,12,18)-\mathcal{T}=\\{t_{10}=t_{12}=t_{18}=0\\}$.

The above argument guarantees that the set of isomorphism classes of pseudo-
ample ${\bf M}$-polarized $K3$ surfaces is given by the disjoint union
$\mathcal{U}\sqcup\mathcal{T}.$ Therefore, we have the following result.

######  Theorem 3.2.

The period mapping in Theorem 3.1 has an explicit form

$\displaystyle\Phi:\mathcal{U}\sqcup\mathcal{T}\simeq\mathcal{D}/\Gamma,$
(3.4)

where $\Gamma$ is given in (1.5). Especially, the injection
$\mathcal{P}(\tilde{\mathfrak{A}})\hookrightarrow\mathcal{D}/\Gamma$, which is
induced by the visualized period mapping $\Phi_{1}$ of (2.9), is extended to
(3.4).

Let $\mathcal{H}$ be the arithmetic arrangement of hyperplanes defined in
Section 1.1. By virtue of Lemma 2.2, $\mathcal{H}$ corresponds to pseudo-ample
${\bf M}$-polarized $K3$ surfaces given by the canonical form (3.3). So, from
Theorem 3.2, the restriction of the period mapping $\Phi$ of (3.4) gives an
isomorphism

$\displaystyle\Phi|_{\mathcal{T}}:\mathcal{T}\simeq\left(\bigcup_{H\in\mathcal{H}}\mathcal{D}_{H}\right)/\Gamma.$
(3.5)

We remark that (3.5) coincides with the period mapping of [Na3] Corollary 2.1.
The detailed results of (3.5) will be summarized in Section 4.

For $\mathcal{D}^{\circ}$ of (1.7), we have an isomorphism

$\displaystyle\Phi|_{\mathcal{U}}:\mathcal{U}\simeq\mathcal{D}^{\circ}/\Gamma.$
(3.6)

Since $\mathcal{P}(\tilde{\mathfrak{A}})\subset\mathcal{U}$, (3.6) is an
extension of
$\mathcal{P}(\tilde{\mathfrak{A}})\hookrightarrow\mathcal{D}/\Gamma$, which is
derived from $\Phi_{1}$ of (2.9). By abuse of notation, this
$\Phi|_{\mathcal{U}}$ will be denoted by $\Phi$ in Section 5.

##  4 Sequence of families of $K3$ surfaces and complex reflection groups

Let $\mathfrak{A}_{1},\mathfrak{A}_{2},\mathfrak{A}_{3}$ be subvarieties of
$\mathfrak{A}_{0}$ explicitly given by

$\displaystyle\begin{cases}&\mathfrak{A}_{1}=\\{a\in\mathfrak{A}_{0}|\hskip
2.84526pta_{0}=0\\},\\\ &\mathfrak{A}_{2}=\\{a\in\mathfrak{A}_{0}|\hskip
2.84526pta_{0}=a_{14}=0\\},\\\
&\mathfrak{A}_{3}=\\{a\in\mathfrak{A}_{0}|\hskip
2.84526pta_{0}=a_{14}=\mathfrak{M}(a)=0\\}.\end{cases}$ (4.1)

Here,

$\mathfrak{M}(a):=\left(a_{10}a_{2}+\frac{a_{4}^{3}}{27}-\frac{a_{6}^{2}}{4}\right)^{2}+\frac{1}{27}a_{4}(a_{4}a_{6}+6a_{2}a_{8})^{2}=0$

coincides with the modular equation of the Humbert surface for the minimal
discriminant which is studied in [NS1] Theorem 5.4 via an appropriate
transformation.

We have the subfamilies

$\varpi_{j}:\mathfrak{F}_{j}\rightarrow\mathfrak{A}_{j}\quad\quad(j\in\\{1,2,3\\})$

of $\varpi_{0}:\mathfrak{F}_{0}=\\{S_{a}|\hskip
2.84526pta\in\mathfrak{A}_{0}\\}\rightarrow\mathfrak{A}_{0}$. They are
indicated in the diagram (0.8). Also, we need another subvariety
$\mathfrak{A}_{1}^{\prime}=\\{a\in\mathfrak{A}_{0}|\hskip
2.84526pta_{14}=0\\}$ and another subfamily
$\varpi^{\prime}_{1}:\mathfrak{F}_{1}^{\prime}\rightarrow\mathfrak{A}_{1}^{\prime}$
such that $\mathfrak{A}_{1}\cap\mathfrak{A}_{1}^{\prime}=\mathfrak{A}_{2}$.

###  4.1 Transcendental lattices for subfamilies of $\mathfrak{F}_{0}$

The above mentioned subfamilies of $\mathfrak{F}_{0}$ are precisely studied in
[CD], [Na1], [CMS] and [Na3]. For each case, it is important to determine the
transcendental lattice in order to study the moduli space of the corresponding
lattice polarized $K3$ surfaces. By surveying the results of those papers, we
have the following result.

######  Proposition 4.1.

The intersection matrices of a generic member of the subfamilies
$\mathfrak{F}_{1},\mathfrak{F}_{1}^{\prime},\mathfrak{F}_{2},\mathfrak{F}_{3}$
of $\mathfrak{F}_{0}$ are given in Table 2.

Parameters | Transcendental lattices |  $K3$ surfaces
---|---|---
$\mathfrak{A}_{1}$ | ${\bf A}_{1}=U\oplus U\oplus A_{2}(-1)$ | [Na3]
$\mathfrak{A}_{1}^{\prime}$ | ${\bf A}_{1}^{\prime}=U\oplus U\oplus A_{1}(-1)^{\oplus 2}$ | [CMS]
$\mathfrak{A}_{2}$ | ${\bf A}_{2}=U\oplus U\oplus A_{1}(-1)$ | [CD]
$\mathfrak{A}_{3}$ | ${\bf A}_{3}=U\oplus\begin{pmatrix}2&1\\\ 1&-2\end{pmatrix}$ | [Na1]
Table 2: Subfamilies of $\mathfrak{F}_{0}$

The lattices in (0.2) are based on Proposition 4.1 and Theorem 2.1.

###  4.2 Transcendental lattices for subfamilies of $\mathfrak{G}_{0}$ of
Kummer-like surfaces

In Section 2.3, we have the family
$\overline{\varpi_{0}}:\mathfrak{G}_{0}=\\{K_{a}|\hskip
2.84526pta\in\mathfrak{A}_{0}\\}\rightarrow\mathfrak{A}_{0}$ of $K3$ surfaces.
This family contains interesting and important families of algebraic $K3$
surfaces. For example, the subfamily $\mathfrak{G}_{2}$ over
$\mathfrak{A}_{2}$ coincides with the well-known family of Kummer surfaces
derived from principally polarized Abelian surfaces. Also, the subfamily
$\mathfrak{G}_{1}^{\prime}$ over $\mathfrak{A}_{1}^{\prime}$ is the family of
$K3$ surfaces given by the double covering of $\mathbb{P}^{2}(\mathbb{C})$
branched along six lines, studied by Matsumoto-Sasaki-Yoshida [MSY].

######  Proposition 4.2.

The intersection matrices of a generic member of the subfamilies
$\mathfrak{G}_{1},\mathfrak{G}_{1}^{\prime},\mathfrak{G}_{2},\mathfrak{G}_{3}$
of $\mathfrak{G}_{0}$ are given in Table 3.

Parameters | Transcendental lattices |  $K3$ surfaces
---|---|---
$\mathfrak{A}_{1}$ | ${\bf B}_{1}=U(2)\oplus U(2)\oplus A_{2}(-2)$ | [NS3]
$\mathfrak{A}_{1}^{\prime}$ | ${\bf B}_{1}^{\prime}=U(2)\oplus U(2)\oplus A_{1}(-1)^{\oplus 2}$ | [MSY]
$\mathfrak{A}_{2}$ | ${\bf B}_{2}=U(2)\oplus U(2)\oplus A_{1}(-2)$ | Kummer surface
$\mathfrak{A}_{3}$ | ${\bf B}_{3}=U(2)\oplus\begin{pmatrix}4&2\\\ 2&-4\end{pmatrix}$ | [Na2]
Table 3: Subfamilies of $\mathfrak{G}_{0}$

By the way, we are able to determine the transcendental lattices in Table 2
for the subfamilies $\mathfrak{F}_{0}$ by a relatively simple way. However,
the proof of Proposition 4.2 is much more complicated. We need a delicate
argument for each case. For detail, see the beginning of Section 6.

Proposition 4.2 is necessary for the proof of Theorem 6.1, which is the main
theorem of Section 6. The latices in (0.16) are based on Proposition 4.2 and
Theorem 6.1.

###  4.3 Relation between modular forms and invariants of complex reflection
groups via theta functions

For the subfamilies $\mathfrak{F}_{1},\mathfrak{F}_{2}$ and
$\mathfrak{F}_{3}$, there is a non-trivial relationship between the period
mappings for them and a complex reflection group of rank $r_{j}=6-j$.

In each family of $K3$ surfaces in Table 1, we have the period mapping

$\displaystyle\Phi_{j}:\mathcal{P}_{j}\simeq\mathcal{D}_{j}/\Gamma_{j}$ (4.2)

where $\mathcal{P}_{j}$ is a Zariski open set in the weighted projective space
whose weights are given in Table 1, $\mathcal{D}_{j}$ is a $(5-j)$-dimensional
symmetric domain and $\Gamma_{j}$ is a subgroup of the orthogonal group of the
lattice ${\bf A}_{j}$ in Table 2. We note that $\Phi_{j}$ of (4.2) can be
obtained as a restriction of (3.4).

$j$ | Reflection groups | Theta functions | Theta expressions via $K3$ surfaces
---|---|---|---
$1$ | No.33 | [DK] | [NS2]
$2$ | No.31 | [R] |  [CD] and [R]
$3$ | No.23 | [Mu] | [Na1]
Table 4: $K3$ surfaces, reflection groups and theta functions

Finite complex reflection groups are listed by Shepherd-Todd [ST] (see also
[LT]). Note that real reflection groups are contained in this list. The group
No.33 (No.31, No.23, resp.) has the simplest structure among the complex
reflection groups which are of exceptional type, not real reflection groups
and of rank $6$ ($5$, $4$, resp.).

A complex reflection group of rank $r_{j}$ acts on the polynomial ring
$\mathbb{C}[X_{1},\cdots,X_{r_{j}}]$. We can find generators of the ring of
invariants for this action. Letting $(w_{1}^{(j)},\ldots,w_{r_{j}}^{(j)})$
$(j\in\\{1,2,3\\})$ be the set of weights given in Table 1, there is a system
$\\{g_{\kappa_{j}w_{1}}^{(j)}(X_{1},\cdots,X_{r_{j}}),\ldots,g_{\kappa_{j}w_{r_{j}}}^{(j)}(X_{1},\cdots,X_{r_{j}})\\}$
of generators of the ring of invariants. Here, $\kappa_{j}\in\mathbb{Z}$ is
the integer in Table 1 and $g^{(j)}_{\kappa_{j}w_{l}}$ is a polynomial of
degree $\kappa_{j}w_{l}^{(j)}$. For example, if $j=3$, the invariants for the
group No.23 are famous Klein’s icosahedral invariants introduced in [K]. In
the preceding papers in Table 4, we have the explicit theta expressions of the
inverse of $\Phi_{j}$ of (4.2) via appropriate systems of theta functions.

For $j=1,2$, we have simple expressions of the above mentioned results. There
exists a system
$\\{\vartheta_{1}^{(j)}(Z_{j}),\cdots,\vartheta_{r_{j}}^{(j)}(Z_{j})\\}$ of
theta functions $\mathcal{D}_{j}\ni
Z_{j}\mapsto\vartheta_{\ell}^{(k)}(Z_{j})\in\mathbb{C}$ of weight
$1/\kappa_{j}$ such that

$\displaystyle\mathcal{D}_{j}\ni
Z_{j}\mapsto\left(g_{\kappa_{j}w_{1}}^{(j)}\left(\vartheta_{1}^{(j)}(Z_{j}),\cdots,\vartheta_{r_{j}}^{(j)}(Z_{j})\right):\cdots:g_{\kappa_{j}w_{r_{j}}}^{(j)}\left(\vartheta_{1}^{(j)}(Z_{j}),\cdots,\vartheta_{r_{j}}^{(j)}(Z_{j})\right)\right)\in\mathcal{P}_{j}$
(4.3)

gives a ratio of modular forms on $\mathcal{D}_{j}$ with respect to
$\Gamma_{j}$ and it coincides with the inverse of the period mapping
$\Phi_{j}$ of (4.2).

  * •

If $j=1$, the invariants for the group No.33 are given in [Bu] and the
explicit form (4.3) is established in [NS2] Theorem 4.1, using the theta
functions of [DK]. In this case, (4.3) is given by a ratio of Hermitian
modular forms for the unitary group $U(2,2)$ concerned with the imaginary
quadratic field of the simplest discriminant.

  * •

If $j=2$, using the invariants for the group No.31 and the theta functions
given in [R] Section 4, one can obtain the expression (4.3) by combining the
results [CD] Theorem 3.5 and [R] Section 4 (see also, [NS2] Section 4.1). In
this case, (4.3) is given by a ratio of well-known Siegel modular forms of
degree $2$.

Also, refer to Remark 5.1.

##  5 Meromorphic modular forms

Let $\mathcal{D}^{*}$ be the connected component of $\\{\xi\in{\bf
A}\otimes\mathbb{C}\hskip 2.84526pt|\hskip
2.84526pt(\xi\cdot\xi)=0,(\xi\cdot\overline{\xi})>0\\}$ which projects to
$\mathcal{D}$. For $\mathcal{D}^{\circ}$ of (1.7), let
$(\mathcal{D}^{\circ})^{*}$ be a subset of $\mathcal{D}^{*}$ which projects to
$\mathcal{D}^{\circ}$.

Based on the fact stated in Theorem 1.1 below, we will use the following
terminology.

######  Definition 5.1.

A holomorphic function $f:(\mathcal{D}^{\circ})^{*}\rightarrow\mathbb{C}$
given by $Z\mapsto f(Z)$ is called a meromorphic modular form of weight
$k\in\mathbb{Z}$ and character $\chi\in{\rm Char}(\Gamma)$ with poles in
$\mathcal{H}$, if $f$ satisfies

  * (i)

$f(\lambda Z)=\lambda^{-k}f(Z)\quad(\text{for all }\lambda\in\mathbb{C}^{*}),$

  * (ii)

$f(\gamma Z)=\chi(\gamma)f(Z)\quad(\text{for all }\gamma\in\Gamma).$

The vector space of the meromorphic modular forms of weight $k\in\mathbb{Z}$
and $\chi\in{\rm Char}(\Gamma)$ with poles in $\mathcal{H}$ is denoted by
$\mathcal{A}^{\circ}_{k}(\Gamma,\chi)$. Then, the ring of the meromorphic
modular forms is given by

$\mathcal{A}^{\circ}(\Gamma)=\bigoplus_{k\in\mathbb{Z}}\bigoplus_{\chi\in{\rm
Char}(\Gamma)}\mathcal{A}^{\circ}_{k}(\Gamma,\chi).$

In this section, we will construct the generator of this ring. Recalling
Proposition 1.1, we will consider the cases of $\chi={\rm id}$ and $\chi={\rm
det}$. The structure of the ring $\mathcal{A}^{\circ}(\Gamma)$ is determined
by Theorem 5.1 and 5.2.

By the way, in [HU] (see also [Na3]), period mappings for lattice polarized
$K3$ surfaces and the canonical orbibundles on the Satake-Baily-Borel
compactifications of symmetric spaces are effective to construct holomorphic
modular forms. However, the Satake-Baily-Borel compactifications are useless
for our purpose, because we want to obtain not holomorphic modular forms but
meromorphic modular forms. Accordingly, we will use the Looijenga
compactification for $\mathcal{D}^{\circ}$ of (1.7) and $\Gamma$ of (1.5),
instead of the Satake-Baily-Borel compactification.

Since Proposition 1.2 and (2.7) hold, by Hartogus’s extension theorem, the
period mapping $\Phi$ of (3.6) is extended to the isomorphism

$\displaystyle\widehat{\Phi}:\widehat{\mathcal{U}}\simeq\widehat{\mathcal{D}^{\circ}/\Gamma}^{\bf
L}$ (5.1)

between the weighted projective space
$\widehat{\mathcal{U}}=\mathbb{P}(2,4,6,8,10,14)$ and the Looijenga
compactification. Our construction of meromorphic modular forms is based on
this period mapping.

###  5.1 Meromorphic modular forms of character ${\rm id}$

There exists the unique holomorphic $2$-form $\omega_{u}$ on $S(u)$ of (2.6)
up to a constant factor. This is explicitly given by $\frac{dx_{0}\wedge
dy_{0}}{z_{0}},$ where
$x_{0}=\frac{x}{w^{4}},y_{0}=\frac{y}{w^{10}},z_{0}=\frac{z}{w^{15}}.$ The
action of $\lambda\in\mathbb{C}^{*}$ given by $S(u)\rightarrow S(\lambda\cdot
u)$, which defines the family (2.8), induces the relation

$\displaystyle\lambda^{*}\omega_{\lambda\cdot u}=\lambda^{-1}\omega_{u}.$
(5.2)

######  Theorem 5.1.

The ring $\mathcal{A}^{\circ}(\Gamma,{\rm id})$ of meromorphic modular forms
of character ${\rm id}$ is isomorphic to the polynomial ring
$\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}].$ Here, a polynomial of
weight $k$ defines a modular form of weight $k$.

###### Proof.

We have a principal $\mathbb{C}^{*}$-bundle ${\rm
pr}:(\mathcal{D}^{\circ})^{*}\rightarrow\mathcal{D}^{\circ}$. The quotient
space $Q=\mathcal{D}^{\circ}/\Gamma$ is identified with a Zariski open set
$\mathcal{U}$ of the weighted projective space
$\widehat{\mathcal{U}}=\mathbb{P}(2,4,6,8,10,14)$ via the period mapping.
Since ${\rm pr}$ is equivalent under the action of $\Gamma=\tilde{O}^{+}({\bf
A}),$ we have a principal $\mathbb{C}^{*}$-bundle $\overline{{\rm
pr}}:(\mathcal{D}^{\circ})^{*}/\Gamma\rightarrow Q.$ Let $\mathcal{O}_{Q}(1)$
be the line bundle over $Q$ associated with $\overline{{\rm pr}}$ and set
$\mathcal{O}_{Q}(k)=\mathcal{O}_{Q}(1)^{\otimes k}.$ Recalling the definition
of the associated bundle, we can regard a section of $\mathcal{O}_{Q}(k)$ as a
holomorphic function $(\mathcal{D}^{\circ})^{*}\ni Z\mapsto s(Z)\in\mathbb{C}$
satisfying

$\displaystyle s(\lambda Z)=\lambda^{-k}s(Z),\quad s(\gamma Z)=s(Z),$ (5.3)

where $\lambda\in\mathbb{C}^{*}$ and $\gamma\in\Gamma.$ From (2.7),
$\widehat{\mathcal{U}}-\mathcal{U}$ is an analytic subset such that ${\rm
codim}(\widehat{\mathcal{U}}-\mathcal{U})\geq 2.$ So, via Hartogus’s
phenomenon, the inclusion
$\iota_{\mathcal{U}}:\mathcal{U}\hookrightarrow\widehat{\mathcal{U}}$ induces
the isomorphism

$\displaystyle\iota_{\mathcal{U}}^{*}:{\rm
Pic}(\widehat{\mathcal{U}})\simeq{\rm Pic}(\mathcal{U}).$ (5.4)

Now, we have ${\rm Pic}(\widehat{\mathcal{U}})\simeq\mathbb{Z}$ and

$\displaystyle\bigoplus_{k\in\mathbb{Z}}H^{0}(\widehat{\mathcal{U}},\mathcal{O}_{\widehat{\mathcal{U}}}(k))=\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}],$
(5.5)

because $\widehat{\mathcal{U}}={\rm
Proj}(\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}])$ is a weighted
projective space. From (5.4) and (5.5), we have

$\displaystyle\bigoplus_{\mathcal{L}\in{\rm
Pic}(Q)}H^{0}(Q,\mathcal{O}_{Q}(\mathcal{L}))\simeq\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}].$
(5.6)

Due to (5.2), the period mapping gives the following diagram:

(5.11)

From Definition 5.1, together with (5.3) and (5.6), we have the assertion. ∎

The integer $\kappa_{0}=3$ in Table 1 is coming from this theorem. Namely,
$(2\kappa_{0},4\kappa_{0},6\kappa_{0},8\kappa_{0},10\kappa_{0},14\kappa_{0})=(6,12,18,24,30,42)$
is equal to the degrees of the group No.34 (see [LT] Appendix D).

###  5.2 Meromorphic modular forms of character ${\rm det}$

We will study the orbifold

$\mathbb{O}=[(\mathcal{D}^{\circ})^{*}/(\Gamma\times\mathbb{C})]$

in order to construct modular forms for ${\rm det}$.

First, let us observe the action of $\Gamma$ on $\mathcal{D}^{\circ}$
precisely. The action of $\Gamma$ on $\mathcal{D}^{\circ}$ is effective. We
set

$\mathfrak{H}_{\mathcal{D}^{\circ}}=\bigcup_{g\in\Gamma}\\{[Z]\in\mathcal{D}^{\circ}\hskip
2.84526pt|\hskip 2.84526ptg([Z])=[Z]\\}.$

Also, letting $\Gamma_{[Z]}$ be the stabilize subgroup with respect to
$[Z]\in\mathcal{D}^{\circ}$, we set

$\mathfrak{S}_{\mathcal{D}^{\circ}}=\\{[Z]\in\mathcal{D}^{\circ}\hskip
2.84526pt|\hskip 2.84526pt\Gamma_{[Z]}\text{ is neither }\\{{\rm
id}_{\Gamma}\\}\text{ nor }\\{{\rm id}_{\Gamma},\sigma_{\delta}\\}\text{ for
}\delta\in\Delta({\bf A})\\}.$

According to Proposition 1.1, $\mathfrak{H}_{\mathcal{D}^{\circ}}$ is a
countable union of reflection hypersurfaces and
$\mathfrak{S}_{\mathcal{D}^{\circ}}$ is a countable union of analytic subsets
of codimension at least $2$.

Let $\mathfrak{H}_{Q}$ and $\mathfrak{S}_{Q}$ be the images of
$\mathfrak{H}_{\mathcal{D}^{\circ}}$ and $\mathfrak{S}_{\mathcal{D}^{\circ}}$
by the projection $\mathcal{D}^{\circ}\rightarrow
Q=\mathcal{D}^{\circ}/\Gamma$, respectively. From Proposition 1.1, it follows
that

$\displaystyle\Gamma^{\prime}:=\\{\gamma\in\Gamma\hskip 2.84526pt|\hskip
2.84526pt\gamma\text{ is given by a product of reflections of even numbers
}\\}=\\{\gamma\in\Gamma\hskip 2.84526pt|\hskip 2.84526pt{\rm
det}(\gamma)=1\\}.$ (5.12)

We set $Q_{1}=\mathcal{D}^{\circ}/\Gamma^{\prime}$. The action of
$\Gamma^{\prime}$ on $\mathcal{D}^{\circ}-\mathfrak{S}_{\mathcal{D}^{\circ}}$
is free. Let us naturally define $\mathfrak{H}_{Q_{1}}$ and
$\mathfrak{S}_{Q_{1}}$, respectively. Recall that the period mapping $\Phi$
gives an identification $\mathcal{U}\simeq Q$ (see (3.6)). Set
$\mathfrak{H}_{\mathcal{U}}=\Phi^{-1}(\mathfrak{H}_{Q})$ and
$\mathfrak{S}_{\mathcal{U}}=\Phi^{-1}(\mathfrak{S}_{Q})$. Then,
$\mathfrak{H}_{\mathcal{U}}$ gives a divisor on the weighted projective space
$\widehat{\mathcal{U}}$ and there exists a weighted homogeneous polynomial
$\Delta_{\mathcal{U}}(u)\in\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}]$
such that

$\displaystyle\mathfrak{H}_{\mathcal{U}}=\\{[u]\in\widehat{\mathcal{U}}=\mathbb{P}(2,4,6,8,10,14)|\hskip
2.84526pt\Delta_{\mathcal{U}}(u)=0\\}.$ (5.13)

We have the double covering $\mathcal{U}_{1}$ of
$\mathcal{U}-\mathfrak{S}_{\mathcal{U}}$ branched along
$\mathfrak{H}_{\mathcal{U}}-\mathfrak{S}_{\mathcal{U}}$:

$\displaystyle\mathcal{U}_{1}=\\{([u],s)\in(\mathcal{U}-\mathfrak{S}_{\mathcal{U}})\times\mathbb{C}|\hskip
2.84526pts^{2}=\Delta_{\mathcal{U}}(u)\\}.$ (5.14)

We can obtain the lift $\Phi_{Q_{1}}:\mathcal{U}_{1}\rightarrow
Q_{1}-\mathfrak{S}_{Q_{1}}$ of
$\Phi|_{\mathcal{U}-\mathfrak{S}_{\mathcal{U}}}$ so that $\Phi_{Q_{1}}$ is
equivalent under the action of
$\Gamma/\Gamma^{\prime}\simeq\mathbb{Z}/2\mathbb{Z}.$ Also, $\Phi_{Q_{1}}$ is
lifted to
$\Phi_{\mathcal{D^{\circ}}}:\mathcal{U}_{\mathcal{D^{\circ}}}\rightarrow\mathcal{D^{\circ}}-\mathfrak{S}_{\mathcal{D^{\circ}}}$,
which is equivalent under the action of $\Gamma$. We can consider the pull-
back
$\mathcal{U}_{(\mathcal{D}^{\circ})^{*}}\rightarrow\mathcal{U}_{\mathcal{D}^{\circ}}$
of the principal bundle $\mathcal{U}^{*}\rightarrow\mathcal{U}$ by the
composition
$\mathcal{U}_{\mathcal{D}^{\circ}}\rightarrow\mathcal{U}_{1}\rightarrow\mathcal{U}-\mathfrak{S}_{\mathcal{U}}\hookrightarrow\mathcal{U}.$
Then, we have the lifted period mapping

$\Phi_{(\mathcal{D}^{\circ})^{*}}:\mathcal{U}_{(\mathcal{D}^{\circ})^{*}}\simeq(\mathcal{D}^{\circ})^{*}-\mathfrak{S}_{(\mathcal{D}^{\circ})^{*}},$

where $\mathfrak{S}_{(\mathcal{D}^{\circ})^{*}}$ is the preimage of
$\mathfrak{S}_{\mathcal{D}^{\circ}}$ under the projection.

######  Lemma 5.1.

The lifted period mapping $\Phi_{(\mathcal{D}^{\circ})^{*}}$ induces an
isomorphism
$[\Phi_{(\mathcal{D}^{\circ})^{*}}]:[\mathcal{U}_{(\mathcal{D}^{\circ})^{*}}/(\mathbb{C}^{*}\times\Gamma)]\simeq[((\mathcal{D}^{\circ})^{*}-\mathfrak{S}_{\mathcal{D}^{\circ}})/(\mathbb{C}^{*}\times\Gamma)].$

###### Proof.

The proof is similar to [Na3] Section 3.3. See the diagram (5.23) also.

(5.23)

∎

######  Proposition 5.1.

The Picard group ${\rm Pic}(\mathbb{O})$ of the orbifold $\mathbb{O}$ is
isomorphic to $\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z}).$

###### Proof.

Set $\mathcal{V}_{1}=\\{([u],s)\in\widehat{\mathcal{U}}\times\mathbb{C}|\hskip
2.84526pts^{2}=\Delta_{\mathcal{U}}(u)\\}$. Then, $\mathcal{U}_{1}$ of (5.14)
satisfies $\mathcal{U}_{1}\subset\mathcal{V}_{1}$ and ${\rm
codim}(\mathcal{V}_{1}-\mathcal{U}_{1})\geq 2$, from (2.7). Recall that $\Phi$
in (5.23) is extended to the identification (5.1). This $\widehat{\Phi}$ is
lifted to $\widehat{\Phi}_{Q_{1}}:\mathcal{V}_{1}\simeq\widehat{Q}_{1}$, which
is equivalent under the $(\mathbb{Z}/2\mathbb{Z})$-action. Here,
$\widehat{Q}_{1}$ is a double covering of
$\widehat{\mathcal{D}^{\circ}/\Gamma}^{\bf L}.$

We consider the orbifold
$\mathbb{V}_{1}=[\mathcal{V}_{1}/(\mathbb{Z}/2\mathbb{Z})]$ with the structure
morphism $p_{\mathbb{V}_{1}}:\mathbb{V}_{1}\rightarrow\widehat{\mathcal{U}}.$
Now, the Picard group ${\rm Pic}(\mathbb{V}_{1})$ is generated by
$\mathcal{O}_{\mathbb{V}_{1}}(1):=p_{\mathbb{V}_{1}}^{*}\mathcal{O}_{\widehat{\mathcal{U}}}(1)$
and the generator $g$ of $\mathbb{Z}/2\mathbb{Z}$. We remark that this
generator $g$ is corresponding to ${\rm det}\in{\rm Char}(\Gamma)$ (see
Proposition 1.1 and (5.12)). The divisor $\\{\Delta_{\mathcal{U}}(u)=0\\}$ is
corresponding to the reflection hypersurfaces for our lattice ${\bf A}$ via
$\Phi$ in (5.23). Therefore, any element of ${\rm Pic}(\mathbb{V}_{1})$ is
given by $\mathcal{O}_{\mathbb{V}_{1}}(1)^{\otimes k}\otimes g^{l}$ for
$k\in\mathbb{Z}$ and $l\in\mathbb{Z}/2\mathbb{Z}$. By considering
$\left[\widehat{\Phi}_{Q_{1}}\right]:\mathbb{V}_{1}\simeq\left[\widehat{Q}_{1}/(\mathbb{Z}/2\mathbb{Z})\right]$,
we obtain ${\rm
Pic}\left(\left[\widehat{Q}_{1}/(\mathbb{Z}/2\mathbb{Z})\right]\right)\simeq\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z}).$

Recall that the action of $\Gamma^{\prime}$ on
$\mathcal{D}^{\circ}-\mathfrak{S}_{\mathcal{D}^{\circ}}$ is free. From the
fact that ${\rm codim}(\widehat{Q}_{1}-Q_{1})\geq 2$, we have

$\displaystyle{\rm Pic}(\mathbb{O})={\rm
Pic}([(\mathcal{D}^{\circ})^{*}/(\mathbb{C}\times\Gamma)])={\rm
Pic}([Q_{1}/(\mathbb{Z}/2\mathbb{Z})])\simeq{\rm
Pic}\left(\left[\widehat{Q}_{1}/(\mathbb{Z}/2\mathbb{Z})\right]\right)\simeq\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z}).$

∎

When we consider holomorphic sections of line bundles, analytic subsets of
codimension at least $2$ do not affect the results due to Hartogus’s
phenomenon. So, in this section, we shall omit such analytic sets. Namely,
from now on, we will often omit such sets (like $\mathfrak{S}_{\mathcal{U}}$
or $\mathfrak{S}_{Q}).$

######  Proposition 5.2.

The weight of $\Delta_{\mathcal{U}}(u)$ is equal to $98.$

###### Proof.

Since $\mathcal{U}$ is a Zariski open set in
$\widehat{\mathcal{U}}=\mathbb{P}(2,4,6,8,10,14)$, the canonical bundle
$\Omega_{\widehat{\mathcal{U}}}$ is calculated as

$\Omega_{\mathcal{U}}\simeq\mathcal{O}_{\mathcal{U}}(-2-4-6-8-10-14)=\mathcal{O}_{\mathcal{U}}(-44).$

Letting $p_{1}$ be the double covering $\mathcal{U}_{1}\rightarrow\mathcal{U}$
branched along $\mathfrak{H}_{\mathcal{U}}$, we obtain the isomorphism
$\Omega_{\mathcal{U}_{1}}\simeq
p_{1}^{*}\Omega_{\mathcal{U}}\otimes\mathcal{O}_{\mathcal{U}_{1}}(\mathfrak{H}_{\mathcal{U}_{1}})$,
by considering the holomorphic differential forms. So, we have

$\displaystyle\Omega_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}\simeq[p_{1}]^{*}\Omega_{\mathcal{U}}\otimes\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(\mathfrak{H}_{\mathcal{U}_{1}})\simeq[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}(-44)\otimes\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(\mathfrak{H}_{\mathcal{U}_{1}}).$

From the proof of Proposition 5.1, the orbifold $\mathbb{O}$ is equivalent to
$[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]$. So, ${\rm
Pic}([\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})])$ is isomorphic to
$\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})$. Let $d$ be the weight of
$\Delta_{\mathcal{U}}(u)$. Then, we have
$[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}(d)\simeq\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(2\mathfrak{H}_{\mathcal{U}_{1}})$.
This implies that the direct summand $\mathbb{Z}/2\mathbb{Z}$ of ${\rm
Pic}([\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})])$ is generated by
$[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}(-d/2)\otimes\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(\mathfrak{H}_{\mathcal{U}_{1}})$.
Here, we do not need to worry about whether $-d/2$ is an integer, for all
weights of $\widehat{\mathcal{U}}$ are even numbers.

Since $\mathcal{D}^{\circ}$ is a Zariski open set in a quadratic hypersurface
in the projective space $\mathbb{P}^{6}(\mathbb{C})$, we apply the adjunction
formula to $\mathcal{D}^{\circ}$ and obtain
$\Omega_{\mathcal{D}^{\circ}}\simeq\mathcal{O}_{\mathcal{D}^{\circ}}(7-2)=\mathcal{O}_{\mathcal{D}^{\circ}}(5)$,
where the weight is concordant with the $\mathbb{C}^{*}$-action indicated in
(5.11). This implies that the canonical orbibundle $\Omega_{\mathbb{O}}$ is
isomorphic to $\mathcal{O}_{\mathbb{O}}(5)\otimes{\rm det}$.

By summarizing the above properties, we have

$\displaystyle\mathcal{O}_{\mathbb{O}}(5)\otimes{\rm
det}\simeq\Omega_{\mathbb{O}}\simeq\Omega_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}$
$\displaystyle\simeq[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}(-44)\otimes\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(\mathfrak{H}_{\mathcal{U}_{1}})$
$\displaystyle\simeq[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}\Big{(}-44+\frac{d}{2}\Big{)}\otimes\Big{(}[p_{1}]^{*}\mathcal{O}_{\mathcal{U}}\Big{(}-\frac{d}{2}\Big{)}\otimes\mathcal{O}_{[\mathcal{U}_{1}/(\mathbb{Z}/2\mathbb{Z})]}(\mathfrak{H}_{\mathcal{U}_{1}})\Big{)}$

and we obtain $d=98$. ∎

######  Theorem 5.2.

(1) There exist holomorphic functions $s_{7}$ of weight $7$ and $s_{42}$ of
weight $42$ on $(\mathcal{D}^{\circ})^{*}$ such that

$s_{7}^{2}=u_{14},\quad\quad s_{42}^{2}=d_{84}(u),$

where $d_{84}(u)$ is the polynomial studied in Lemma 2.2.

(2) The holomorphic function $s_{49}=s_{7}s_{42}$ on
$(\mathcal{D}^{\circ})^{*}$ gives a modular form of weight $49$ and character
${\rm det}$. It holds $\mathcal{A}^{\circ}(\Gamma,{\rm
det})=s_{49}\mathcal{A}^{\circ}(\Gamma,{\rm id})$.

###### Proof.

(1) By the argument in this section, the divisor $\\{[u]\in\mathcal{U}|\hskip
2.84526pt\Delta_{\mathcal{U}}(u)=0\\}$ corresponds to the union of reflection
hyperplanes of $\\{H_{\delta}|\hskip 2.84526pt\delta\in\Delta({\bf
A})\\}-\mathcal{H}$. Here, $H_{\delta}$ is a reflection hyperplane defined by
$\delta\in\Delta({\bf A})$ and $\mathcal{H}$ is the arrangement of $\gamma
H_{0}$ ($\gamma\in\Gamma$, $H_{0}=\\{\xi_{7}=0\\}$) as in Section 1.1.
Recalling the observation of degenerations of our lattice polarized $K3$
surfaces in Lemma 2.2 and the meaning of the canonical form (2.6), we have

$\displaystyle\\{[u]\in\mathcal{U}|\hskip
2.84526ptd_{84}(u)=0\\}\cup\\{[u]\in\mathcal{U}|\hskip
2.84526ptu_{14}=0\\}\subset\\{[u]\in\mathcal{U}|\hskip
2.84526pt\Delta_{\mathcal{U}}(u)=0\\}.$ (5.24)

The inclusion (5.24) means that $u_{14}d_{84}(u)$ divides
$\Delta_{\mathcal{U}}(u)$ in
$\mathbb{C}[u_{2},u_{4},u_{6},u_{8},u_{10},u_{14}].$ On the other hand,
Proposition 5.2 says that $\Delta_{\mathcal{U}}(u)$ is of weight $84$. Thus,
we have the irreducible decomposition

$\displaystyle\Delta_{\mathcal{U}}(u)={\rm const}\cdot u_{14}d_{84}(u).$
(5.25)

Since the double covering $p_{1}:\mathcal{U}_{1}\rightarrow\mathcal{U}$ in the
diagram (5.23) is branched along the divisor $\\{[u]\in\mathcal{U}|\hskip
2.84526pt\Delta_{\mathcal{U}}(u)=0\\}$, (5.25) implies that there is a
holomorphic function $s_{7}$ ($s_{42}$, resp.) on $(\mathcal{D}^{\circ})^{*}$
satisfying $s_{7}^{2}=u_{14}$ ($s_{42}^{2}=d_{84}(u)$, resp.).

(2) Recall that ${\rm det}\in{\rm Char}(\Gamma)$ is coming from the action of
$\Gamma/\Gamma^{\prime}\simeq\mathbb{Z}/2\mathbb{Z}$ which defines the double
covering $p_{1}:\mathcal{U}_{1}\rightarrow\mathcal{U}$. By the Definition 5.1
and the meaning of (5.14), every modular form of character ${\rm det}$
vanishes on the preimage of $\mathfrak{H}_{\mathcal{D}^{\circ}}$ by the
canonical projection
$(\mathcal{D}^{\circ})^{*}\rightarrow\mathcal{D}^{\circ}$. Since this $p_{1}$
is branched along the divisor $\\{\Delta_{\mathcal{U}}(u)=0\\},$ (5.24) and
(5.25) show that every modular form of character ${\rm det}$ is given by a
product of $s_{49}:=s_{7}s_{42}$ and a modular form of character ${\rm id}.$ ∎

######  Remark 5.1.

This theorem supports the relation between our sequence of the families and
complex reflection groups. The weight $42$ of $s_{42}$ is coming from the
discriminant of the right hand side of (2.6). We note that $126=42\kappa_{0}$,
where $\kappa_{0}=3$ as in Table 1, is equal to the number of reflections of
order $2$ for the group No.34 (see [LT] Appendix D).

Such a phenomenon occurs in each case for $\mathfrak{F}_{j}$
($j\in\\{1,2,3\\}$). In the case for $j=1$ ($2,3$, resp.), there is a
holomorphic function of weight $45$ ($30$, $15$, resp.) coming from the
discriminant of the elliptic $K3$ surfaces of [Na3] ([CD], [Na1], resp.).
Then, $45=45\kappa_{1}$ ($60=30\kappa_{2}$, $15=15\kappa_{3}$, resp.) is equal
to the number of reflections of order $2$ for the group No.33 (No.31, No.23,
resp.). Each of these holomorphic functions gives a factor of a modular form
of a non-trivial character. Also, there are explicit expressions of them by
the theta functions in Table 4.

######  Remark 5.2.

The Weierstrass equation of (2.1) is essential for our purpose.

We have another expression of elliptic $K3$ surfaces with singular fibres of
type (2.2) given by the equation

$\displaystyle
z_{1}^{2}=y_{1}^{3}+(b_{1}x_{1}^{5}w_{1}+b_{4}x_{1}^{4}w_{1}^{4}+b_{7}x_{1}^{3}w_{1}^{7})y_{1}+(b_{0}x_{1}^{8}+b_{3}x_{1}^{7}w_{1}^{3}+b_{6}x_{1}^{6}w_{1}^{6}+b_{9}x_{1}^{5}w_{1}^{9})$
(5.26)

of weight $24$, where $(x:y:z:w)\in\mathbb{P}(3,8,12,1)$. However, the
expression (5.26) is not appropriate to construct modular forms. Although
(5.26) can be birationally transformed to the Weierstrass form (2.1), it is
impossible to construct correct modular forms from the parameters in (5.26).
It seems that we can obtain modular forms of weight $1,3,4,6,7,9$ on
$(\mathcal{D}^{\circ})^{\prime}$ from (5.26), which is a complement of an
arrangement corresponding to the condition $b_{0}=0.$ However, we can see that
this arrangement does not satisfy the condition of Theorem 1.1. So, the
expression (5.26) induces an erroneous use of the theory of the Looijenga
compactifications.

For example, it seems that we can obtain a modular form of weight $7$ coming
from the parameter $b_{7}$. We can see that the zero set of this modular form
coincides with the arrangement $\mathcal{H}$ in Lemma 1.2. However, as stated
in [L2] (see also [L1] Section 6), if an arrangement gives the zero set of a
modular form, then this does not satisfy the condition of Theorem 1.1. This
contradicts to Lemma 1.2.

Thus, our expression of (2.1) is suitable for the theory of the Looijenga
compactifications and effective to construct modular forms.

##  6 Transcendental lattice for family $\mathfrak{G}_{0}$ of Kummer-like
surfaces with Picard number $15$

In Section 2, we determined the lattice structure for the family
$\mathfrak{F}_{0}$ via a natural consideration of singular fibres for elliptic
surfaces. Also, in fact, the lattices for the subfamilies
$\mathfrak{F}_{1},\mathfrak{F}_{2},\mathfrak{F}_{3}$ in Proposition 4.1 can be
determined in a similar way.

However, as for the family $\mathfrak{G}_{j}$ $(j\in\\{1,2,3\\})$ and
$\mathfrak{G}_{1}^{\prime}$ of the Kummer-like surfaces, it is much harder to
determine their lattices correctly. For example,

  * •

The family $\mathfrak{G}_{2}$ is the family of the Kummer surfaces for
principally polarized Abelian surfaces. For a precise study for the lattice
structure of the Kummer surfaces, Nikulin [Ni1] introduces a particular
lattice which is called the Kummer lattice. Also, Morrison [Mo] studies an
interesting viewpoint called the Shioda-Inose structure for $K3$ surfaces
whose Picard numbers are greater than $17$.

  * •

In order to determine the transcendental lattice for the family
$\mathfrak{G}_{1}^{\prime}$ of Kummer-like surfaces with Picard number $16$,
Matsumoto-Sasaki-Yoshida [MSY] (see also [Y]) study hypergeometric integrals
of type $(3,6)$ and calculate intersection numbers of the chambers coming from
the configuration of six lines on $\mathbb{P}^{2}(\mathbb{C})$ by applying a
delicate technique of twisted homologies.

  * •

For the transcendental lattice for the family $\mathfrak{G}_{1}$ of Kummer-
like surfaces with Picard number $16$, Shiga and the author [NS3] have a
geometric construction of $2$-cycles on a generic member of $\mathfrak{G}_{1}$
taking into account the fact that a generic member of $\mathfrak{G}_{1}$ is a
double covering of that of $\mathfrak{F}_{1}$. This construction is also based
on hard calculations of local monodromies for elliptic surfaces.

In this section, we will determine the transcendental lattice for the family
$\mathfrak{G}_{0}$. It is a non-trivial problem to determine it. If there were
a double covering $S_{a}\rightarrow K_{a}$ for generic members of
$\mathfrak{F}_{0}$ and $\mathfrak{G}_{0}$, we could calculate the lattice for
$\mathfrak{G}_{0}$ from the lattice of $\mathfrak{F}_{0}$ in Theorem 2.1 via a
technique of Nikulin [Ni2] Section 2. However, in practice, we can prove that
there is no such a double covering for generic $a\in\mathfrak{A}_{0}$. This
proof can be given in a similar way to the proof of [NS3] Theorem 5.2.

Now, let us remark the fact that our family $\mathfrak{G}_{0}$ naturally
contains the families $\mathfrak{G}_{1}$ and $\mathfrak{G}_{1}^{\prime}.$ We
will determine the transcendental lattice for $\mathfrak{G}_{0}$ based on this
fact. The main result in this section is indebted to heavy calculations for
$\mathfrak{G}_{1}^{\prime}$ in [MSY] and those for $\mathfrak{G}_{1}$ in
[NS3].

The following result for even lattices is necessary for our proof.

######  Lemma 6.1.

([Mo], Corollary 2.10) Suppose $12\leq\rho\leq 20$. Let ${\bf T}$ be an even
lattice of signature $(2,20-\rho).$ Then, the primitive embedding ${\bf
T}\hookrightarrow L_{K3}$ is unique up to isometry.

The following theorem is the main result of this section.

######  Theorem 6.1.

The transcendental lattice of a generic member of $\mathfrak{G}_{0}$ is given
by the intersection matrix $U(2)\oplus U(2)\oplus\begin{pmatrix}-2&0&1\\\
0&-2&1\\\ 1&1&-4\end{pmatrix}$.

###### Proof.

We identify the $K3$ lattice $L_{K3}=U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}$
with the $2$-homology group of $K3$ surfaces.

Let $e_{j},f_{j}$ $(j\in\\{1,2,3\\})$ be elements of $U^{\oplus 3}$ satisfying
$(e_{j}\cdot e_{k})=(f_{j}\cdot f_{k})=0$ and $(e_{j}\cdot
f_{k})=\delta_{j,k}$. Let $p_{j},q_{j}$ $(j\in\\{1,\ldots,8\\})$ be elements
of $E_{8}(-1)^{\oplus 2}$ with the intersection numbers defined by the Dynkin
diagram in Figure 2. Also, put $\nu_{j}=p_{j}+q_{j}$.

Figure 2: Basis of $E_{8}(-1)^{\oplus 2}$

By reference to [Ma], we put

$\displaystyle\begin{cases}&\lambda_{1}=-\nu_{5}+\nu_{7}+2(e_{1}+f_{1}),\quad\mu_{1}=-\nu_{4},\\\
&\lambda_{2}=\nu_{7}+\nu_{8}+2(e_{1}+e_{2}+e_{3}+f_{3}),\quad\mu_{2}=\nu_{6}.\end{cases}$
(6.1)

Then, $\\{\lambda_{1},\mu_{1},\lambda_{2},\mu_{2}\\}$ gives a system of basis
of the lattice $U(2)^{\oplus 2}$. This system defines a primitive embedding
$U(2)^{\oplus 2}\hookrightarrow L_{K3}$.

Extending the system (6.1), we have a primitive embedding of the
transcendental lattice ${\bf B}_{1}=U(2)^{\oplus 2}\oplus A_{2}(-2)$ for
$\mathfrak{G}_{1}$ to $L_{K3}$ given by the system

$\displaystyle{\bf
B}_{1}=\langle\lambda_{1},\mu_{1},\lambda_{2},\mu_{2},\nu_{1},\nu_{2}\rangle_{\mathbb{Z}},$
(6.2)

where the transcendental lattice ${\bf B}_{2}=U(2)^{\oplus 2}\oplus A_{1}(-2)$
for $\mathfrak{G}_{2}$ is a primitive sublattice of $L_{K3}$ defined by the
explicit system

$\displaystyle{\bf
B}_{2}=\langle\lambda_{1},\mu_{1},\lambda_{2},\mu_{2},\nu_{1}\rangle_{\mathbb{Z}}.$
(6.3)

According to Lemma 6.1, this embedding is unique up to isometry. So, we can
fix the embedding given by (6.2) and (6.3) without loss of generality.

We remark that $\mathfrak{G}_{1}^{\prime}$ also contains $\mathfrak{G}_{2}$ as
a subfamily. Hence, its transcendental lattice ${\bf
B}_{1}^{\prime}=U(2)^{\oplus 2}\oplus A_{1}(-1)^{\oplus 2}$ should be an
extension of the lattice ${\bf B}_{2}$ of (6.3). So, we have the following
explicit system of basis:

$\displaystyle{\bf
B}_{1}^{\prime}=\langle\lambda_{1},\mu_{1},\lambda_{2},\mu_{2},p_{1},q_{1}\rangle_{\mathbb{Z}}.$
(6.4)

We note that the expression (6.4) is guaranteed by the fact that the lattice
${\bf B}_{2}$ of (6.3) is invariant under the involution given by
interchanging two $A_{1}(-1)$ summands of the lattice ${\bf B_{1}^{\prime}}$
(see [MSY]; see also [Y] Chapter IX).

Our family $\mathfrak{G}_{0}$ contains $\mathfrak{G}_{1}$ and
$\mathfrak{G}_{1}^{\prime}$. So, the transcendental lattice for a generic
member of $\mathfrak{G}_{0}$ is a primitive lattice of $L_{K3}$, of rank $7$
and given by the system which is an extension of (6.2) and (6.4). Such a
lattice is given by the explicit system
$\langle\lambda_{1},\mu_{1},\lambda_{2},\mu_{2},p_{1},q_{1},\nu_{2}\rangle_{\mathbb{Z}}$
whose intersection matrix $U(2)^{\oplus 2}\oplus\begin{pmatrix}-2&0&1\\\
0&-2&1\\\ 1&1&-4\end{pmatrix}.$ ∎

Since we have a double covering $K_{a}\rightarrow S_{a}$ for generic members
of $\mathfrak{G}_{0}$ and $\mathfrak{F}_{0}$, we can testify the correctness
of Theorem 6.1. Namely, when ${\rm Tr}(K_{a})$ is given, we can calculate the
intersection matrix of ${\rm Tr}(S_{a})$. Letting $\Lambda=({\rm
Tr}(K_{a})\otimes\mathbb{Q})\cap(U^{\oplus 3}\otimes\frac{1}{2}E_{8}(-2))$,
where $E_{8}(-2)$ is the lattice generated by $\nu_{1},\ldots,\nu_{8}$, ${\rm
Tr}(S_{a})$ is isometric to the lattice $\Lambda(2)$ (see [Ni2] Section 2). In
our case, $\Lambda(2)$ is given by the direct sum of $U^{\oplus 2}$ and
$\langle\nu_{1}/2,q_{1},\nu_{2}/2\rangle_{\mathbb{Z}}(2)$. This is isometric
to

$U^{\oplus 2}\oplus\begin{pmatrix}-1&-1&1/2\\\ -1&-2&1/2\\\
1/2&1/2&-1\end{pmatrix}(2)\simeq U^{\oplus 2}\oplus\begin{pmatrix}-2&-2&1\\\
-2&-4&1\\\ 1&1&-2\end{pmatrix}\simeq U^{\oplus
2}\oplus\begin{pmatrix}-2&1&0\\\ 1&-2&0\\\ 0&0&-2\end{pmatrix}={\bf A}.$

This is concordant with Theorem 2.1.

## Acknowledgment

The author would like to thank Professor Manabu Oura and Professor Jiro
Sekiguchi for valuable suggestions from the viewpoint of complex reflection
groups. This work is supported by JSPS Grant-in-Aid for Scientific Research
(18K13383) and MEXT Leading Initiative for Excellent Young Researchers.

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Atsuhira Nagano
Faculty of Mathematics and Physics
Institute of Science and Engineering
Kanazawa University
Kakuma, Kanazawa, Ishikawa
920-1192, Japan
(E-mail<EMAIL_ADDRESS>