Elliptic fibrations on toric $K3$ hypersurfaces and mirror symmetry derived
from Fano polytopes Tomonao Matsumura and Atsuhira Nagano

###### Abstract

We determine the Néron-Severi lattices of $K3$ hypersurfaces with large Picard
number in toric three-folds derived from Fano polytopes. Such $K3$ surfaces
are important objects of research in mirror symmetry. On each $K3$ surface, we
introduce a particular elliptic fibration which induces a Mordell-Weil group
of rank $1$. In the proof of the main theorem, we show that the Néron-Severi
lattice of each $K3$ surface is generated by a general fibre, sections and
appropriately selected components of the singular fibres of our elliptic
fibration. Our argument gives a down-to-earth proof of the Dolgachev
conjecture for Fano polytopes, which is a conjecture on mirror symmetry for
$K3$ surfaces.

000Keywords: $K3$ surfaces ; elliptic fibrations ; mirror symmetry ; Fano
polytopes. 000Mathematics Subject Classification 2020: Primary 14J28 ;
Secondary 14J27, 14J33, 52B10.

## Introduction

In this paper, we determine the Néron-Severi lattices and the transcendental
lattices of $K3$ surfaces which are given as hypersurfaces in toric three-
folds derived from three-dimensional Fano polytopes. These lattices are
important from the viewpoint of mirror symmetry. Our argument gives a down-to-
earth proof of a conjecture in mirror symmetry for $K3$ surfaces. The proof of
our main theorem is based on precise studies of appropriate Jacobian elliptic
fibrations on our $K3$ surfaces.

Mirror symmetry is a phenomenon originally discovered by physicists studying
superstring theory. It suggests non-trivial relationships between geometric
objects and provides a lots of interesting problems in algebraic geometry.
Especially, many mathematicians have intensively studied Calabi-Yau
hypersurfaces in toric varieties coming from reflexive polytopes since the
publication of [B2]. Calabi-Yau hypersurfaces in toric three-folds are $K3$
surfaces. There is a famous conjecture on mirror symmetry for such toric $K3$
hypersurfaces by Dolgachev [D]. This conjecture is formulated in terms of the
following lattices of $K3$ surfaces. The $2$-cohomology group
$H^{2}(S,\mathbb{Z})$ of a $K3$ surface $S$ gives an even unimodular lattice
$L_{K3}$ of signature $(3,19).$ The Néron-Severi lattice ${\rm NS}(S)$ is a
sublattice of $H^{2}(S,\mathbb{Z})$ defined as $H^{2}(S,\mathbb{Z})\cap
H^{1,1}(S,\mathbb{R})$. We call $\rho={\rm rank}({\rm NS}(S))$ the Picard
number of $S$. Then, ${\rm NS}(S)$ is of signature $(1,\rho-1)$. We will
identify $H^{2}(S,\mathbb{Z})$ with the $2$-homology group
$H_{2}(S,\mathbb{Z})$ by the Poincaré duality. Then, ${\rm NS}(S)$ is regarded
as the sublattice of $H_{2}(S,\mathbb{Z})$ generated by divisors on $S$. The
transcendental lattice ${\rm Tr}(S)$ is the orthogonal complement of ${\rm
NS}(S)$ in $H^{2}(S,\mathbb{Z})$. Here, arithmetic techniques of even lattices
are powerful tools to study the above lattices (for example, see [Ni]). Thus,
mirror symmetry for toric $K3$ hypersurfaces is an interesting research theme
involving researchers from various fields.

Fano polytopes are special reflexive polytopes (see Definition 1.1). There
exists a smooth Fano $n$-fold attached to each $n$-dimensional Fano polytope.
Since Fano polytopes have nice combinatorial properties, many researchers have
convergently studied them (for example, see [B1], [B3], [WW], [Sa], and [Øb]).
In particular, three-dimensional Fano polytopes are classified into $18$ types
up to ${\rm GL}_{3}(\mathbb{Z})$-action. The number of vertices of such a
polytope is one of $4,5,6,7$ or $8$. See Table 1. Here, each column of the
matrix in the table stands for the coordinates of each vertex of the
corresponding polytope.

Table 1: Fano polytopes $P_{k}$ and Lattices $L_{k}$ $k$ | $P_{k}$ | $L_{k}$
---|---|---
$1$ | $\left(\begin{array}[]{cccccc}1&0&0&-1\\\ 0&1&0&-1\\\ 0&0&1&-1\\\ \end{array}\right)$ | $(4)$
$2$ | $\left(\begin{array}[]{cccccc}1&0&0&-1&0\\\ 0&1&0&-1&0\\\ 0&0&1&0&-1\\\ \end{array}\right)$ | $\begin{pmatrix}0&3\\\ 3&2\end{pmatrix}$
$3$ | $\left(\begin{array}[]{cccccc}1&0&0&0&-1\\\ 0&1&0&0&-1\\\ 0&0&1&-1&-1\\\ \end{array}\right)$ | $\begin{pmatrix}-2&2\\\ 2&-2\end{pmatrix}$
$4$ | $\left(\begin{array}[]{cccccc}1&0&0&0&-1\\\ 0&1&0&0&-1\\\ 0&0&1&-1&-2\\\ \end{array}\right)$ | $\begin{pmatrix}-2&1\\\ 1&-2\end{pmatrix}$
$5$ | $\left(\begin{array}[]{cccccc}1&0&0&-1&-1\\\ 0&1&0&-1&-1\\\ 0&0&1&0&-1\\\ \end{array}\right)$ | $\begin{pmatrix}0&3\\\ 3&-2\end{pmatrix}$
$6$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&-1&0\\\ 0&0&1&-1&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}0&2&2\\\ 2&0&2\\\ 2&2&0\\\ \end{array}\right)$
$7$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&-1&0\\\ 0&0&1&-1&-1&-1\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}-2&1&1\\\ 1&0&2\\\ 1&2&0\\\ \end{array}\right)$
$8$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&-1&0\\\ 0&0&1&-1&1&-1\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}-2&1&3\\\ 1&0&2\\\ 3&2&0\\\ \end{array}\right)$
$9$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&-1&-1\\\ 0&0&1&-1&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}0&1&2\\\ 1&-2&2\\\ 2&2&0\\\ \end{array}\right)$
$10$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&-1&-1\\\ 0&0&1&-1&-1&-1\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}-2&1&1\\\ 1&-2&2\\\ 1&2&0\\\ \end{array}\right)$
$11$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&1&-1\\\ 0&0&1&-1&-1&-1\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}-2&1&1\\\ 1&-2&1\\\ 1&1&2\\\ \end{array}\right)$
$12$ | $\left(\begin{array}[]{cccccc}1&0&0&0&0&-1\\\ 0&1&0&0&1&-1\\\ 0&0&1&-1&-1&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccc}-2&2&2\\\ 2&-2&1\\\ 2&1&2\\\ \end{array}\right)$
$13$ | $\left(\begin{array}[]{ccccccc}1&0&0&0&0&-1&-1\\\ 0&1&0&0&-1&0&-1\\\ 0&0&1&-1&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}0&1&1&1\\\ 1&-2&0&2\\\ 1&0&-2&2\\\ 1&2&2&-2\\\ \end{array}\right)$
$14$ | $\left(\begin{array}[]{ccccccc}1&0&0&-1&0&-1&-1\\\ 0&1&0&0&-1&0&-1\\\ 0&0&1&-1&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}0&1&1&1\\\ 1&-2&0&2\\\ 1&0&-2&1\\\ 1&2&1&-2\\\ \end{array}\right)$
$15$ | $\left(\begin{array}[]{ccccccc}1&0&0&1&0&-1&-1\\\ 0&1&0&0&-1&0&-1\\\ 0&0&1&-1&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}0&1&1&1\\\ 1&-2&0&2\\\ 1&0&-2&3\\\ 1&2&3&-2\\\ \end{array}\right)$
$16$ | $\left(\begin{array}[]{ccccccc}1&0&0&-1&0&-1&-1\\\ 0&1&0&-1&-1&0&-1\\\ 0&0&1&-1&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{cccc}0&1&1&1\\\ 1&-2&0&1\\\ 1&0&-2&1\\\ 1&1&1&-2\\\ \end{array}\right)$
$17$ | $\left(\begin{array}[]{cccccccc}1&0&0&0&0&-1&1&-1\\\ 0&1&0&0&-1&0&-1&1\\\ 0&0&1&-1&0&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccccc}0&1&1&1&1\\\ 1&-2&2&2&0\\\ 1&2&-2&0&2\\\ 1&2&0&-2&0\\\ 1&0&2&0&-2\\\ \end{array}\right)$
$18$ | $\left(\begin{array}[]{cccccccc}1&0&0&1&0&-1&1&-1\\\ 0&1&0&-1&-1&0&-1&1\\\ 0&0&1&-1&0&0&0&0\\\ \end{array}\right)$ | $\left(\begin{array}[]{ccccc}0&1&1&1&1\\\ 1&-2&2&1&0\\\ 1&2&-2&0&3\\\ 1&1&0&-2&0\\\ 1&0&3&0&-2\\\ \end{array}\right)$

From a three-dimensional Fano polytope $P$, we obtain another polytope
$P^{\circ}$, which is called the polar dual. The duality between $P$ and
$P^{\circ}$ derives two different families of $K3$ surfaces. Namely, we have
the family of $S_{P^{\circ}}$ with small Picard number and the family of
$S_{P}$ with large Picard number. Dolgachev [D] conjectures that
$(S_{P},S_{P^{\circ}})$ gives a mirror pair. This means that there is a non-
trivial relationship between the lattices of $S_{P}$ and those of
$S_{P^{\circ}}$ (for detail, see Section 1). Here, the $K3$ surface
$S_{P^{\circ}}$ is a hypersurface in a smooth toric Fano three-fold. One can
apply algebrogeometric techniques to smooth toric varieties. Then, it is
possible to determine ${\rm NS}(S_{P^{\circ}})$ systematically. Indeed, Koike
[Koi1] and Mase [Mas] determine the Néron-Severi lattices for all
$S_{P^{\circ}}$ (see Proposition 1.1). On the other hand, to the best of the
authors’ knowledge, it is not easy to determine the Néron-Severi lattice of
$S_{P}$ for every $P$, because its Picard number is large and its ambient
toric variety has singularities.

The subjects of the present paper are the $K3$ surfaces $S_{P}$. Here, we note
that Narumiya and Shiga [NS] determine the lattice structure of $S_{P}$ if the
number of vertices of $P$ is $4$. Also, if the number of vertices of $P$ is
$5$, the second author [Na1] determines the lattice structure of $S_{P}$ (see
also [IIT]). In this paper, we will determine the lattice structure of $S_{P}$
for every three-dimensional Fano polytope $P$ with $6,7$ or $8$ vertices. Our
main theorem is given as follows.

######  Theorem 0.1.

Let $P_{k}$ ($k\in\\{1,\ldots,18\\}$) be the Fano polytope in Table 1. Then,
the intersection matrix of the transcendental lattice ${\rm Tr}(S_{P_{k}})$ is
isometric to $U\oplus L_{k}$, where $U$ is the hyperbolic lattice of rank $2$
and $L_{k}$ is the lattice in Table 1.

Indeed, we will determine the structure of ${\rm NS}(S_{P_{k}})$
$(k\in\\{6,\ldots,18\\})$ in Theorem 4.1. Theorem 0.1 immediately follows from
Theorem 4.1 (see Corollary 4.1). Also, this main theorem establishes that the
Dolgachev conjecture holds for Fano polytopes.

In our proof, it is important to study divisors on $S_{P}$ with large Picard
number. For the cases of $S_{P}$, unlike the cases of $S_{P^{\circ}}$,
algebrogeometric techniques for toric divisors are not so effective.
Therefore, our mathematical argument is much different to those of [Koi1] or
[Mas]. We will explicitly take a divisor $F$ on each $S_{P}$ whose self
intersection number is $0$. Then, we obtain an elliptic fibration
$\pi_{P}:S_{P}\rightarrow\mathbb{P}^{1}(\mathbb{C})$ with a general fibre $F$.
Each fibration $\pi_{P}$ has sections $\mathbb{P}^{1}(\mathbb{C})\rightarrow
S_{P}$. Such a fibration is called a Jacobian elliptic fibration. We will make
full use of arithmetic properties of even lattices and Mordell-Weil groups in
order to determine ${\rm NS}(S_{P})$ in Section 4 and 5.

As a matter of fact, our proof of the main theorem is heavily due to nice
properties of the fibration $\pi_{P}$. Let us summarize our argument. For each
$P$, we will introduce a sublattice $E_{P}$ of ${\rm NS}(S_{P})$ generated by
a general fibre, sections and appropriately selected irreducible components of
the singular fibres of $\pi_{P}$ (see Table 8 and 9). The lattice $E_{P}$ will
be called the evident lattice attached to $\pi_{P}.$ The absolute value
$|\det(E_{P})|$ of the discriminant of our lattice $E_{P}$ is equal to
$|\det({\rm NS})(S_{P^{\circ}})|$, which is already calculated in [Koi1] or
[Mas]. Moreover, our lattice $E_{P}$ defines a period mapping for the family
of $S_{P}$ as in Section 3. By using these properties, we will prove that
$E_{P}$ is equal to ${\rm NS}(S_{P})$ in Theorem 4.1.

We note that each $S_{P}$ has several Jacobian elliptic fibrations. However,
not every Jacobian elliptic fibration is useful to determine the lattice
structure in question. Our fibration $\pi_{P}$ is necessary for our proof of
the main theorem, because it has particular properties as we saw in the above
paragraph. It is highly non-trivial to find such a fibration. Actually, our
$\pi_{P}$ for every $P$ was explicitly constructed by the first author with
the greatest care and it first appeared in his master thesis [Mat1].

The authors expect that the $K3$ surfaces $S_{P}$ with large Picard number
have various applications. For example, the second author shows that the
period mapping for the family of $K3$ surfaces $S_{P_{4}}$, which is derived
from the polytope $P_{4}$ with $5$ vertices, has a good arithmetic properties
(see [Na2]). Hence, it would be natural to expect that $S_{P}$ for other $P$
are also interesting objects of research. In particular, it is desirable to
comprehend the structures of the Mordell-Weil groups attached to our elliptic
fibrations in order to study arithmeticity of our $K3$ surfaces. In Section 5,
we will determine the structure of the Mordell-Weil group ${\rm
MW}(\pi_{P},O)$ attached to $\pi_{P}$ for each Fano polytope $P$. Especially,
the rank of ${\rm MW}(\pi_{P},O)$ is equal to $1$.

##  1 Mirror pairs of toric $K3$ hypersurfaces from Fano polytopes

###  1.1 Preliminaries and notations

We start this paper with surveying the construction of toric varieties from
reflexive polytopes. For detailed arguments or proofs, see [B2], [Od] and
[CLS].

Let $M$ be a free abelian group of rank $n$. We identify $M$ with
$\mathbb{Z}^{n}.$ Let $N$ be the dual of $M$: $N={\rm Hom}(M,\mathbb{Z}).$ We
set $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$ and
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}.$ The non-degenerate pairing
$N_{\mathbb{R}}\times M_{\mathbb{R}}\rightarrow\mathbb{R}$ is denoted by
$\langle n,m\rangle$ for $n\in N_{\mathbb{R}}$ and $m\in M_{\mathbb{R}}$. In
this paper, $m\in M_{\mathbb{R}}$ ($n\in N_{\mathbb{R}}$, resp.) will be
usually regarded as an $n$-component column (row, resp.) vector and $\langle
n,m\rangle$ is regarded as the canonical inner product of $\mathbb{R}^{n}$.
Let $\mathbb{T}$ be a torus: $\mathbb{T}\simeq(\mathbb{C}^{\times})^{n}$. For
$m=(m_{1},\ldots,m_{n})\in M$, let $\chi^{m}$ be a character
$\mathbb{T}\rightarrow\mathbb{C}^{\times}$ defined by
$\chi^{m}(t_{1},\ldots,t_{n})=t_{1}^{m_{1}}\cdots t_{n}^{m_{n}}$.

Let $P\subset M_{\mathbb{R}}$ be an $n$-dimensional convex lattice polytope.
Here, a lattice polytope means a polytope whose vertices are integral. Let
$\mathfrak{F}$ be a face of $P$ and $\mathfrak{F}_{0}$ be a facet which
contains $\mathfrak{F}$. We can take the unique primitive vector
$n_{\mathfrak{F}_{0}}\in N$ which gives the normal vector of
$\mathfrak{F}_{0}$ pointing towards the interior of $P$. Let
$\sigma_{\mathfrak{F}}\subset N_{\mathbb{R}}$ be the cone generated by
$n_{\mathfrak{F}_{0}}$ for all facets $\mathfrak{F}_{0}$ satisfying
$\mathfrak{F}\subset\mathfrak{F}_{0}.$ Then,
$\Sigma(P)=\\{\sigma_{\mathfrak{F}}|\mathfrak{F}\text{ is a face of }P\\}$
gives a complete rational fan in $N_{\mathbb{R}}$. The fan $\Sigma(P)$
determines an $n$-dimensional toric variety $X_{\Sigma(P)}$, which will be
shortly denoted by $X_{P}$. In this paper, we call $X_{P}$ the toric variety
determined by $P$.

If an $n$-dimensional convex polytope $P\subset M_{\mathbb{R}}$ contains the
origin as an interior point, the polar dual $P^{\circ}\subset N_{\mathbb{R}}$
of $P$ is defined by $\\{n\in N_{\mathbb{R}}|\langle n,m\rangle\geq-1\text{
for all }m\in P\\}$. We can see that $P^{\circ}$ is also an $n$-dimensional
convex polytope and contains the origin as the interior point. Moreover, it
holds $(P^{\circ})^{\circ}=P.$ If a convex lattice polytope $P$ contains the
origin $0\in M$ in its interior and $P^{\circ}$ gives a lattice polytope, then
$P$ is called a reflexive polytope. Batyrev [B2] proves that the
$n$-dimensional toric variety $X_{P}$ is a Fano variety with Gorenstein
singularities whose anti-canonical sections are given by

$\displaystyle H^{0}(X_{P},\mathcal{O}_{X_{P}}(-K_{X_{P}}))=\bigoplus_{m\in
P\cap M}\mathbb{C}\chi^{m}.$ (1.1)

Then, a generic member of the linear system $|-K_{X_{P}}|$ defines an
$(n-1)$-dimensional Calabi-Yau hypersurface in $X_{P}$ as in [B2] Section 3.

Let $P\subset M_{\mathbb{R}}$ be a reflexive polytope. For a face
$\mathfrak{F}\subset P^{\circ}\subset N_{\mathbb{R}}$, let $\mathbb{R}_{\geq
0}\mathfrak{F}$ be the cone $\\{\lambda
x|x\in\mathfrak{F},\lambda\in\mathbb{R}_{\geq 0}\\}$. Then, the set
$\Sigma[P^{\circ}]$ of $\mathbb{R}_{\geq 0}\mathfrak{F}$ for all faces
$\mathfrak{F}$ of $P^{\circ}$ gives a complete fan in $N_{\mathbb{R}}$ such
that $X_{\Sigma[P^{\circ}]}=X_{\Sigma(P)}=X_{P}$ ([B2] Proposition 2.1.1,
Corollary 4.1.11).

We introduce Fano polytopes as follows. This terminology is due to [B3], [Sa]
and [Øb].

######  Definition 1.1.

An $n$-dimensional convex integral polytope $P\subset M_{\mathbb{R}}$ is
called a Fano polytope, if $P$ satisfies the following conditions:

  * (i)

$0\in M_{\mathbb{R}}$ is a inner point of $P$,

  * (ii)

the vertices of every facet of $P$ give a $\mathbb{Z}$-basis of $M$.

If $P$ is a Fano polytope, $X_{P^{\circ}}=X_{\Sigma(P^{\circ})}=X_{\Sigma[P]}$
is a smooth toric Fano $n$-fold.

###  1.2 Dolgachev conjecture for toric $K3$ hypersurfaces

In this paper, we will study cases of $n=3$. In these cases, we will consider
$K3$ hypersurfaces corresponding to the anti-canonical sections of three-
dimensional toric varieties. We need to study lattice theoretic properties of
$K3$ surfaces. In this paper, let $U$ be the unimodular hyperbolic lattice of
rank $2$. Also, let $E_{8}(-1)$ be the root lattices whose self intersection
numbers are $-2$. Then, the $K3$ lattice $L_{K3}$, which is isomorphic to the
$2$-homology group of a $K3$ surface, is isometric to the unimodular lattice
$II_{3,19}=U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}.$ For a $K3$ surface $S$,
its Néron-Severi lattice is denoted by ${\rm NS}(S)$ and its transcendental
lattice is denoted by ${\rm Tr}(S).$

For a three-dimensional reflexive polytope $P$, a generic member of
$|-K_{X_{P}}|$ defines a $K3$ surface $S_{P}$. Also, we obtain another $K3$
surface $S_{P^{\circ}}$ from a generic member of $|-K_{X_{P^{\circ}}}|$. Let
$L_{P}$ ($L_{P^{\circ}}$, resp.) be the lattice in $L_{K3}$ which is isometric
to the sublattice generated by the elements of $\iota^{*}({\rm NS}(X_{P}))$
($\iota_{\circ}^{*}({\rm NS}(X_{P^{\circ}}))$, resp.), where
$\iota^{*}:H^{2}(X_{P},\mathbb{Z})\rightarrow H^{2}(S_{P},\mathbb{Z})$
($\iota_{\circ}^{*}:H^{2}(X_{P^{\circ}},\mathbb{Z})\rightarrow
H^{2}(S_{P^{\circ}},\mathbb{Z})$, resp.) is the pull back of
$\iota:S_{P}\hookrightarrow X_{P}$
($\iota_{\circ}:S_{P^{\circ}}\hookrightarrow X_{P^{\circ}}$, resp.). Let
$L_{P}^{\perp}$ ($L_{P^{\circ}}^{\perp}$, resp.) be the orthogonal complement
of $L_{P}$ ($L_{P^{\circ}}$, resp.) in $L_{K3}$.

The following conjecture is famous and important to study mirror symmetry for
toric $K3$ hypersurfaces.

######  Conjecture 1.1.

(The Dolgachev conjecture, [D] Conjecture (8.6)) The notation being as above,
the lattice $L_{P^{\circ}}^{\perp}$ has an orthogonal decomposition
$L_{P^{\circ}}^{\perp}\simeq U\oplus\widecheck{L}_{P^{\circ}}$ with the
summand $U$ and there is a primitive embedding
$L_{P}\hookrightarrow\widecheck{L}_{P^{\circ}}$. Moreover,
$L_{P}=\widecheck{L}_{P^{\circ}}$ holds if and only if
$L_{P^{\circ}}\simeq{\rm NS}(S_{P^{\circ}})$.

We will study the cases for three-dimensional Fano polytopes. As in Table 1,
three-dimensional Fano polytopes are classified into 18 types up to the action
of ${\rm GL}_{3}(\mathbb{Z})$ by [B1] and [WW]. Suppose $P$ is a three-
dimensional Fano polytope. One can determine the structure of the Néron-Severi
lattice ${\rm NS}(S_{P^{\circ}})$ of the $K3$ surface $S_{P^{\circ}}$ on the
basis of algebrogeometric techniques, like intersection theory for divisors of
toric manifolds. We already have the following fact.

######  Proposition 1.1.

([Koi1] or [Mas]) For $k\in\\{1,\ldots,18\\},$ let $P_{k}$ be the Fano
polytope of Table 1. Then, the Neron-Séveri lattice ${\rm
NS}(S_{P_{k}^{\circ}})$ is isometric to the lattice $L_{k}$ of signature
$(1,\ell_{k}-4)$ of Table 1, where $L_{k}$ is equal to the lattice
$\iota_{\circ}^{*}({\rm NS}(X_{P_{k}^{\circ}}))$.

In practice, [Koi1] proves the proposition by applying the following
algebrogeometric result.

######  Proposition 1.2.

([Moi] Theorem 7.5) Let $X$ be a three-dimensional smooth projective algebraic
variety. Let $\iota:S\hookrightarrow X$ be an embedding of a general
hyperplane section. Then, the pull-back $\iota^{*}:{\rm NS}(X)\rightarrow{\rm
NS}(S)$ is a surjective mapping if and only if $X$ and $S$ satisfy the
following (i) or (ii):

  * (i)

$b_{2}(X)=b_{2}(S)$,

  * (ii)

$h^{2,0}(S)>h^{2,0}(X)$.

It holds $h^{2,0}(S_{P^{\circ}})=1$. Also, a smooth Fano three-fold
$X_{P^{\circ}}$ satisfies $h^{2,0}(X_{P^{\circ}})=0$. So, one can apply
Proposition 1.2 and calculate the intersection form of ${\rm
NS}(S_{P^{\circ}})$ from that of ${\rm NS}(X_{P^{\circ}})$.

######  Remark 1.1.

For each $k$, we can see that the orthogonal complement $L_{k}^{\perp}$ admits
a decomposition in the form $U\oplus\widecheck{L}_{k}$, where
$\widecheck{L}_{k}$ is a lattice of signature $(1,22-\ell_{k})$.

Furthermore, according to [Kob] Corollary 4.3.6, Proposition 1.1 implies the
isomorphism $L_{P_{k}}\simeq{\rm NS}(S_{P_{k}})$. Therefore, in order to see
that the Dolgachev conjecture holds for three-dimensional Fano polytopes, it
is enough to prove

$\displaystyle{\rm Tr}(S_{P})\simeq U\oplus{\rm NS}(S_{P^{\circ}})=U\oplus
L_{P^{\circ}}$ (1.2)

for each three-dimensional Fano polytope $P$.

By the way, a pair $(S,S^{\prime})$ of $K3$ surfaces is usually called a
mirror pair if ${\rm Tr}(S)\simeq U\oplus{\rm NS}(S^{\prime}).$ The relation
(1.2) means that $(S_{P},S_{P^{\circ}})$ is a mirror pair.

###  1.3 $K3$ hypersurfaces with explicit parameters

For each Fano polytope $P_{k}$ in Table 1, let $S_{k}=S_{P_{k}}$ be a generic
member of the linear system $|-K_{X_{P_{k}}}|$ in the toric three-fold
$X_{P_{k}}$. Due to Table 1, Proposition 1.1 and (1.2), it is enough to prove
the relation

$\displaystyle{\rm Tr}(S_{k})\simeq U\oplus L_{k}$ (1.3)

for $k\in\\{1,\ldots,18\\}$ to see that the Dolgachev conjecture holds for the
Fano polytopes. As in Table 1, the numbers of vertices of a three-dimensional
Fano polytopes is one of $4,5,6,7$ or $8$.

The Néron-Severi lattice of $S_{1}$, which is derived from the unique Fano
polytope $P_{1}$ with $4$ vertices, is determined by Narumiya and Shiga [NS].
The motivation and the method of their paper are based on the study of the
monodromy group of the Picard-Fuchs equation for the family of $S_{1}$. The
second author was inspired by [NS]. Also, he was interested in a numerical
approach of T. Ishige, who studies the monodromy group of the Picard-Fuchs
system for $K3$ surfaces derived from the Fano polytope $P_{3}$ in detail.
Affected by these earlier studies, the second author [Na1] determines the the
Néron-Severi lattices coming from the Fano polytopes with $5$ vertices and
studies the monodromy groups of the Picard-Fuchs systems. His proof is based
on techniques of Mordell-Weil lattices introduced in [Sh]. Anyway, in these
works, it is established that the Dolgachev conjecture is true for the Fano
polytopes with $4$ or $5$ vertices.

######  Remark 1.2.

In [IIT], one can find the above mentioned Ishige’s numerical approach to
study the monodromy group. We remark that there are several other computer-
assisted studies of Picard-Fuchs systems coming from Fano polytopes. For
example, Nakayama and Takayama [NT] give an approximation algorithm to compute
Picard-Fuchs systems attached to three-dimensional Fano polytopes via
techniques of $D$-modules.

In the present paper, we will see that the Dolgachev conjecture is true for
all the Fano polytopes with $6,7$ or $8$ vertices. Namely, we will study toric
$K3$ hypersurfaces $S_{k}$ $(k\in\\{6,\ldots,18\\}).$ In this subsection, we
will give explicit forms of them.

Suppose a three-dimensional Fano polytope $P_{k}$ is given by

$\displaystyle P_{k}=\begin{pmatrix}p_{11}&\cdots&p_{1\ell_{k}}\\\
p_{21}&\cdots&p_{2\ell_{k}}\\\ p_{31}&\cdots&p_{3\ell_{k}}\end{pmatrix},$
(1.4)

where $\ell_{k}$ is the number of vertices of $P_{k}$. According to (1.1), the
family of the $K3$ hypersurfaces $S_{k}$ is explicitly given by

$\displaystyle\Big{\\{}\sum_{i=1}^{\ell_{k}}c_{i}t_{1}^{p_{1i}}t_{2}^{p_{2i}}t_{3}^{p_{3i}}=0\Big{|}\hskip
2.84526ptc_{i}\in\mathbb{C}\Big{\\}}.$ (1.5)

Each Fano polytope in Table 1 contains the origin as the unique inner lattice
point and the standard simplex generated by
${}^{t}(1,0,0),{}^{t}(0,1,0),{}^{t}(0,0,1)$. From the polytope $P_{k}$ of
(1.4), we set

$\displaystyle\widetilde{P}_{k}=\begin{pmatrix}1&1&\cdots&1\\\
p_{10}&p_{11}&\cdots&p_{1\ell_{k}}\\\ p_{20}&p_{21}&\cdots&p_{2\ell_{k}}\\\
p_{30}&p_{31}&\cdots&p_{3\ell_{k}}\end{pmatrix}$ (1.6)

where ${\small\begin{pmatrix}p_{10}&p_{11}&p_{12}&p_{13}\\\
p_{20}&p_{21}&p_{22}&p_{23}\\\
p_{30}&p_{31}&p_{32}&p_{33}\end{pmatrix}=\begin{pmatrix}0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}}$. The matrix $\widetilde{P}_{k}$ is regarded as a
surjective linear mapping $\mathbb{R}^{\ell_{k}+1}\rightarrow\mathbb{R}\oplus
M_{\mathbb{R}}$ which induces the exact sequence

$\displaystyle
0\rightarrow\mathbb{Z}^{\ell_{k}-3}\overset{K_{k}}{\longrightarrow}\mathbb{Z}^{\ell_{k}+1}\overset{\widetilde{P}_{k}}{\longrightarrow}\mathbb{Z}\oplus
M\rightarrow 0,$ (1.7)

where $K_{k}=(\kappa_{1},\ldots,\kappa_{\ell_{k}-3})$ is a
$(\ell_{k}+1)\times(\ell_{k}-3)$-matrix with entries in $\mathbb{Z}$ such that
$\kappa_{k}={}^{t}(\kappa_{k0},\ldots,\kappa_{k\ell_{k}})$
$(k=1,\ldots,\ell_{k}-3)$ satisfies

$\displaystyle\kappa_{k,k^{\prime}+3}=\begin{cases}&1\quad(k^{\prime}=k),\\\
&0\quad(k^{\prime}\not=k).\end{cases}$

Then, by putting $\displaystyle
x=\frac{c_{1}t_{1}}{c_{0}},y=\frac{c_{2}t_{2}}{c_{0}}$ and $\displaystyle
z=\frac{c_{3}t_{3}}{c_{0}},$ the equation in (1.5) is transformed to

$\displaystyle S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3}):\quad
1+x+y+z+\sum_{i=1}^{\ell_{k}-3}\lambda_{i}x^{p_{1,i+3}}y^{p_{2,i+3}}z^{p_{3,i+3}}=0.$
(1.8)

Here, we put

$\displaystyle\lambda_{i}=\prod_{h=0}^{\ell_{k}}c_{h}^{\kappa_{i+3,h}}.$ (1.9)

By clearing fractions of (1.8) for the cases of $k\in\\{6,\ldots,18\\}$, we
obtain explicit defining equations of
$S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})$ as in Table 2.

Table 2: Defining equations of toric $K3$ hypersurfaces $S_{k}$ $k$ | Defining equations of $S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})$
---|---
$6$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xz+\lambda_{3}yz=0$
$7$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}x+\lambda_{3}y=0$
$8$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xz^{2}+\lambda_{3}y=0$
$9$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xz+\lambda_{3}z=0$
$10$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}x+\lambda_{3}=0$
$11$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xy^{2}+\lambda_{3}=0$
$12$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xy^{2}+\lambda_{3}z=0$
$13$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}xz+\lambda_{3}yz+\lambda_{4}z=0$
$14$ | $xyz(x+y+z+1)+\lambda_{1}y+\lambda_{2}xz+\lambda_{3}yz+\lambda_{4}z=0$
$15$ | $xyz(x+y+z+1)+\lambda_{1}x^{2}y+\lambda_{2}xz+\lambda_{3}yz+\lambda_{4}z=0$
$16$ | $xyz(x+y+z+1)+\lambda_{1}+\lambda_{2}xz+\lambda_{3}yz+\lambda_{4}z=0$
$17$ | $xyz(x+y+z+1)+\lambda_{1}xy+\lambda_{2}yz+\lambda_{3}xz+\lambda_{4}x^{2}z+\lambda_{5}y^{2}z=0$
$18$ | $xyz(x+y+z+1)+\lambda_{1}x^{2}+\lambda_{2}yz+\lambda_{3}xz+\lambda_{4}x^{2}z+\lambda_{5}y^{2}z=0$

Now, let us characterize the above parameters
$\lambda_{1},\ldots,\lambda_{\ell_{k}-3}$ coming from the polytope $P_{k}$.
The matrix $K_{k}$ of (1.7) is called the Gale transform of
$\widetilde{P}_{k}$. Each row of the matrix $K_{k}$ in (1.7) is a
$\mathbb{Z}$-vector of $\mathbb{R}^{\ell_{k}-3}$. Let $\Sigma^{\prime}(P_{k})$
be a fan in $\mathbb{R}^{\ell_{k}-3}$ whose one-dimensional cones are
generated by these vectors. This fan $\Sigma^{\prime}(P_{k})$ is regarded as
the secondary fan coming from the set of lattice points of the Fano polytope
$P_{k}$ (for detail, see [GKZ]; see also [BFS] Section 4). Then, the tuple of
the parameters $\lambda_{1},\ldots,\lambda_{\ell_{k}-3}$ of (1.9) gives a
system of coordinates of the torus $(\mathbb{C}^{\times})^{\ell_{k}-3}$ of the
$(\ell_{k}-3)$-dimensional toric variety determined by the secondary fan
$\Sigma^{\prime}(P_{k})$. Namely, $(\mathbb{C}^{\times})^{\ell_{k}-3}$ is
regarded as ${\rm
Spec}\left(\mathbb{C}\left[\lambda_{1}^{\pm},\ldots,\lambda_{\ell_{k}-3}^{\pm}\right]\right).$

Thus, for each $j\in\\{6,\ldots,18\\}$, we obtain a family

$\displaystyle\mathscr{F}_{k}:\\{S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})\text{
of (\ref{EqStack})}\hskip 2.84526pt|\hskip
2.84526pt(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})\in(\mathbb{C}^{\times})^{\ell_{k}-3}\\}\rightarrow(\mathbb{C}^{\times})^{\ell_{k}-3}$
(1.10)

of toric $K3$ hypersurfaces. We remark that such a family can also be obtained
via a morphism of fans precisely studied in [L].

##  2 Jacobian elliptic fibrations and evident lattices

Let $S$ be a $K3$ surface. We call a triple $(S,\pi,C)$ an elliptic $K3$
surface, if $C$ is a curve, $\pi:S\rightarrow C$ is a proper mapping such that
the fibres $\pi^{-1}(p)$ are elliptic curves for almost all $p\in C$ and $S$
is relatively minimal. We suppose that an elliptic fibration $\pi:S\rightarrow
C$ has a section $O:C\rightarrow S$ such that $O$ is the identity element of
the Mordell-Weil group ${\rm MW}(\pi,O)$ of sections of $\pi$ (see Section 5).
We call such an elliptic fibration $\pi$ a Jacobian elliptic fibration on $S$.

In this section, we introduce an appropriate Jacobian elliptic fibration on
the $K3$ surface $S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})$ in Table 2
for each $k\in\\{6,\ldots,18\\}$. From now on,
$S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})$ will be shortly denoted by
$S_{k}(\lambda)$.

######  Proposition 2.1.

For each $k\in\\{6,\ldots,18\\},$ there is a Jacobian elliptic fibration
$\pi_{k}=\pi_{k}^{(\lambda)}:S_{k}(\lambda)\rightarrow\mathbb{P}^{1}(\mathbb{C})$
defined by the equation

$\displaystyle
z_{1}^{2}=4y_{1}^{3}+a_{1}(x_{1})y_{1}^{2}+a_{2}(x_{1})y_{1}+a_{3}(x_{1}),$
(2.1)

where $a_{k}(x)$ $(k=1,2,3)$ are polynomials given in Table 3. Here, the
fibration $\pi_{k}$ is given by $(x_{1},y_{1},z_{1})\mapsto x_{1}$ and the
section $O$ is given by $x\mapsto(x,\infty,\infty)$.

Table 3: $a_{1}(x_{1}),a_{2}(x_{1})$ and $a_{3}(x_{1})$ of (2.1) $k$ | $a_{1}(x_{1})$ | $a_{2}(x_{1})$ | $a_{3}(x_{1})$
---|---|---|---
$6$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})+x_{1}^{2}(-4\lambda_{1}-4\lambda_{3}+(1+x_{1})^{2})$ | $4\lambda_{1}\lambda_{3}x_{1}^{4}$ | $0$
$7$ | $x_{1}(-4\lambda_{2}+x_{1}(-4\lambda_{1}+(1+x_{1})^{2}))$ | $-2\lambda_{3}x_{1}^{4}(1+x_{1})$ | $\lambda_{3}^{2}x_{1}^{6}$
$8$ | $x_{1}(x_{1}(1+x_{1})^{2}-4\lambda_{1}(\lambda_{2}+x_{1}))$ | $-2\lambda_{3}x_{1}^{3}(1+x_{1})(\lambda_{2}+x_{1})$ | $\lambda_{3}^{2}x_{1}^{4}(\lambda_{2}+x_{1})^{2}$
$9$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $4\lambda_{1}\lambda_{3}x_{1}^{3}$ | $0$
| $+x_{1}(-4\lambda_{3}+x_{1}(-4\lambda_{1}+(1+x_{1})^{2}))$ |  |
$10$ | $\lambda_{1}^{2}+2\lambda_{1}x_{1}(1+x_{1})+x_{1}(-4\lambda_{2}+x_{1}(1+x_{1})^{2})$ | $-2\lambda_{3}x_{1}^{2}(\lambda_{1}+x_{1}+x_{1}^{2})$ | $\lambda_{3}^{2}x_{1}^{4}$
$11$ | $x_{1}^{2}(1-4\lambda_{1}+(2-4\lambda_{2})x_{1}+x_{1}^{2})$ | $-2\lambda_{3}x_{1}^{3}(1+x_{1})$ | $\lambda_{3}^{2}x_{1}^{4}$
$12$ | $(\lambda_{1}+x_{1}+x_{1}^{2})^{2}$ | $-2\lambda_{3}x_{1}^{2}(\lambda_{2}+x_{1})(\lambda_{1}+x_{1}+x_{1}^{2})$ | $\lambda_{3}^{2}x_{1}^{4}(\lambda_{2}+x_{1})^{2}$
$13$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $4\lambda_{1}x_{1}^{3}(\lambda_{4}+\lambda_{3}x_{1})$ | $0$
| $+x_{1}(-4\lambda_{4}+x_{1}(-4\lambda_{1}-4\lambda_{3}+(1+x_{1})^{2}))$ |  |
$14$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $-2\lambda_{1}x_{1}^{3}(\lambda_{2}+x_{1}+x_{1}^{2})$ | $\lambda_{1}^{2}x_{1}^{6}$
| $+x_{1}(-4\lambda_{4}+x_{1}(-4\lambda_{3}+(1+x_{1})^{2}))$ |  |
$15$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $-2\lambda_{1}x_{1}^{2}(\lambda_{4}+\lambda_{3}x_{1})(\lambda_{2}+x_{1}+x_{1}^{2})$ | $\lambda_{1}^{2}x_{1}^{4}(\lambda_{4}+\lambda_{3}x_{1})^{2}$
| $+x_{1}(-4\lambda_{4}+x_{1}(-4\lambda_{3}+(1+x_{1})^{2}))$ |  |
$16$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $-2\lambda_{1}x_{1}^{2}(\lambda_{2}+x_{1}+x_{1}^{2})$ | $\lambda_{1}^{2}x_{1}^{4}$
| $+x_{1}(-4\lambda_{4}+x_{1}(-4\lambda_{3}+(1+x_{1})^{2}))$ |  |
$17$ | $\lambda_{3}^{2}+2\lambda_{3}x_{1}(1+x_{1})$ | $4\lambda_{1}x_{1}^{3}(\lambda_{4}+x_{1})(\lambda_{2}+\lambda_{5}x_{1})$ | $0$
| $+x_{1}(-4\lambda_{2}(\lambda_{4}+x_{1})$ |  |
| $\hskip 19.91692pt+x_{1}(1-4\lambda_{1}-4\lambda_{4}\lambda_{5}+2x_{1}-4\lambda_{5}x_{1}+x_{1}^{2}))$ |  |
$18$ | $\lambda_{2}^{2}+2\lambda_{2}x_{1}(1+x_{1})$ | $-2\lambda_{1}x_{1}^{3}(\lambda_{5}+x_{1})(\lambda_{2}+x_{1}+x_{1}^{2})$ | $\lambda_{1}^{2}x_{1}^{6}(\lambda_{5}+x_{1})^{2}$
| $+x_{1}(-4\lambda_{3}(\lambda_{5}+x_{1})$ |  |
| $\hskip 19.91692pt+x_{1}((1+x_{1})^{2}-4\lambda_{4}(\lambda_{5}+x_{1})))$ |  |

###### Proof.

By performing the birational transformation
$(x,y,z)\mapsto(x_{1},y_{1},z_{1})$ given by

$x=x(x_{1},y_{1},z_{1}),\quad y=y(x_{1},y_{1},z_{1}),\quad
z=z(x_{1},y_{1},z_{1}),$

as in Table 4, 5 and 6, we have the assertion.

Table 4: Rational functions $x(x_{1},y_{1},z_{1})$ $k$ | $x(x_{1},y_{1},z_{1})$
---|---
$6$ | $\frac{2y_{1}(-\lambda_{3}x_{1}^{2}+y_{1})}{x_{1}(\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$7$ | $\frac{2y_{1}^{2}}{x_{1}(-\lambda_{3}x_{1}^{3}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1})}$
$8$ | $\frac{2y_{1}^{2}}{x_{1}(-\lambda_{2}\lambda_{3}x_{1}^{2}-\lambda_{3}x_{1}^{3}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$9$ | $\frac{2y_{1}(-\lambda_{3}x_{1}+y_{1})}{x_{1}(\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1})}$
$10$ | $\frac{2y_{1}^{2}}{x_{1}(-\lambda_{3}x_{1}^{2}+\lambda_{1}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$11$ | $\frac{2y_{1}^{2}}{x_{1}(-\lambda_{3}x_{1}^{2}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$12$ | $\frac{2y_{1}^{2}}{x_{1}(-\lambda_{2}\lambda_{3}x_{1}^{2}-\lambda_{3}x_{1}^{3}+\lambda_{1}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$13$ | $\frac{2y_{1}(-\lambda_{4}x_{1}-\lambda_{3}x_{1}^{2}+y_{1})}{x_{1}(\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1})}$
$14$ | $\frac{2(\lambda_{4}x_{1}+\lambda_{3}x_{1}^{2}-y_{1})y_{1}}{x_{1}(\lambda_{1}x_{1}^{3}-\lambda_{2}y_{1}-x_{1}y_{1}-x_{1}^{2}y_{1}-z_{1})}$
$15$ | $\frac{(\lambda_{4}+\lambda_{3}x_{1})(-\lambda_{1}\lambda_{4}x_{1}^{2}-\lambda_{1}\lambda_{3}x_{1}^{3}+\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}{2y_{1}(-\lambda_{4}x_{1}-\lambda_{3}x_{1}^{2}+y_{1})}$
$16$ | $\frac{2y_{1}(-\lambda_{4}x_{1}-\lambda_{3}x_{1}^{2}+y_{1})}{x_{1}(-\lambda_{1}x_{1}^{2}+\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$17$ | $\frac{2y_{1}(-\lambda_{2}\lambda_{4}x_{1}-\lambda_{2}x_{1}^{2}-\lambda_{4}\lambda_{5}x_{1}^{2}-\lambda_{5}x_{1}^{3}+y_{1})}{(\lambda_{4}+x_{1})(\lambda_{3}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1})}$
$18$ | $x_{1}$
Table 5: Rational functions $y(x_{1},y_{1},z_{1})$ $k$ | $y(x_{1},y_{1},z_{1})$
---|---
$6$ | $x_{1}$
$7$ | $x_{1}$
$8$ | $x_{1}$
$9$ | $x_{1}$
$10$ | $-\frac{-\lambda_{3}x_{1}^{2}+\lambda_{1}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}y_{1}}$
$11$ | $x_{1}$
$12$ | $\frac{\lambda_{2}\lambda_{3}x_{1}^{2}+\lambda_{3}x_{1}^{3}-\lambda_{1}y_{1}-x_{1}y_{1}-x_{1}^{2}y_{1}-z_{1}}{2(\lambda_{2}+x_{1})y_{1}}$
$13$ | $x_{1}$
$14$ | $x_{1}$
$15$ | $x_{1}$
$16$ | $x_{1}$
$17$ | $x_{1}$
$18$ | $\frac{\lambda_{1}\lambda_{5}x_{1}^{3}+\lambda_{1}x_{1}^{4}-\lambda_{2}y_{1}-x_{1}y_{1}-x_{1}^{2}y_{1}+z_{1}}{2(\lambda_{5}+x_{1})y_{1}}$
Table 6: Rational functions $z(x_{1},y_{1},z_{1})$ $k$ | $z(x_{1},y_{1},z_{1})$
---|---
$6$ | $-\frac{\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}(-\lambda_{3}x_{1}^{2}+y_{1})}$
$7$ | $-\frac{-\lambda_{3}x_{1}^{3}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1}}{2x_{1}y_{1}}$
$8$ | $\frac{\lambda_{2}\lambda_{3}x_{1}^{2}+\lambda_{3}x_{1}^{3}-x_{1}y_{1}-x_{1}^{2}y_{1}-z_{1}}{2(\lambda_{2}+x_{1})y_{1}}$
$9$ | $\frac{\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1}}{2x_{1}(\lambda_{3}x_{1}-y_{1})}$
$10$ | $x_{1}$
$11$ | $-\frac{-\lambda_{3}x_{1}^{2}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}y_{1}}$
$12$ | $x_{1}$
$13$ | $\frac{\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}-z_{1}}{2x_{1}(\lambda_{4}x_{1}+\lambda_{3}x_{1}^{2}-y_{1})}$
$14$ | $-\frac{\lambda_{1}x_{1}^{3}-\lambda_{2}y_{1}-x_{1}y_{1}-x_{1}^{2}y_{1}-z_{1}}{2x_{1}(\lambda_{4}x_{1}+\lambda_{3}x_{1}^{2}-y_{1})}$
$15$ | $-\frac{-\lambda_{1}\lambda_{4}x_{1}^{2}-\lambda_{1}\lambda_{3}x_{1}^{3}+\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}(-\lambda_{4}x_{1}-\lambda_{3}x_{1}^{2}+y_{1})}$
$16$ | $\frac{-\lambda_{1}x_{1}^{2}+\lambda_{2}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}(\lambda_{4}x_{1}+\lambda_{3}x_{1}^{2}-y_{1})}$
$17$ | $\frac{\lambda_{3}y_{1}+x_{1}y_{1}+x_{1}^{2}y_{1}+z_{1}}{2x_{1}(\lambda_{2}\lambda_{4}x_{1}+\lambda_{2}x_{1}^{2}+\lambda_{4}\lambda_{5}x_{1}^{2}+\lambda_{5}x_{1}^{3}-y_{1})}$
$18$ | $-\frac{\lambda_{1}x_{1}^{2}(\lambda_{5}+x_{1})}{y_{1}}$

∎

######  Remark 2.1.

We can naturally extend the family $\mathscr{F}_{k}$ to
$\tilde{\mathscr{F}}_{k}:\\{S_{k}(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})\text{
of
}(\ref{EqStack})|(\lambda_{1},\ldots,\lambda_{\ell_{k}-3})\in\mathbb{C}^{\ell_{k}-3}\\}\rightarrow\mathbb{C}^{\ell_{k}-3}$.
From Table 3, one can find inclusion relationships among our families:
$\tilde{\mathscr{F}}_{7}$ ($\tilde{\mathscr{F}}_{9}$,
$\tilde{\mathscr{F}}_{10}$, resp.) is the subfamily of
$\tilde{\mathscr{F}}_{14}$ ($\tilde{\mathscr{F}}_{13}$,
$\tilde{\mathscr{F}}_{16}$, resp.) over the locus $\\{\lambda_{2}=0\\}$
($\\{\lambda_{3}=0\\}$, $\\{\lambda_{3}=0\\}$, resp.).

######  Remark 2.2.

There are several Jacobian elliptic fibrations of $S_{k}$ for each
$k\in\\{6,\ldots,18\\}$. Our fibration $\pi_{k}$ of $S_{k}$ is carefully
chosen in the first author’s master thesis [Mat1]. In fact, $\pi_{k}$ enable
us to determine the lattice structure of $S_{k}$. Namely, our mathematical
arguments in Section 3, 4 and 5 heavily depend on the properties of $\pi_{k}$.

We have the following proposition by direct observations.

######  Proposition 2.2.

For each $k\in\\{6,\ldots,18\\},$ there are singular fibres of the elliptic
fibration $\pi_{k}^{(\lambda)}$ on a generic member $S_{k}(\lambda)$ of the
family $\mathscr{F}_{k}$ induced from the equation (2.1) as listed in Table 7.
Also, there are non-trivial sections $\mathbb{P}^{1}(\mathbb{C})\rightarrow
S_{k}$ of $\pi_{k}^{(\lambda)}$ given by $x_{1}\mapsto(x_{1},y_{1},z_{1})$ as
in Table 7.

Table 7: Singular fibres and non-trivial sections of $\pi_{k}$ $k$ | Kodaira type of singular fibres |  Non-trivial sections $x_{1}\mapsto(x_{1},y_{1},z_{1})$
---|---|---
$6$ | $I_{8}+I_{8}+8I_{1}$ | $O^{\prime}:x_{1}\mapsto(x_{1},0,0)$
|  | $Q:x\mapsto(x_{1},\lambda_{1}x_{1}^{2},\lambda_{1}x_{1}^{2}(\lambda_{2}+x_{1}+x_{1}^{2}))$
$7$ | $I_{3}^{*}+I_{8}+7I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{3}x_{1}^{3})$
$8$ | $I_{1}^{*}+I_{3}+I_{8}+6I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{3}x_{1}^{2}(\lambda_{2}+x_{1}))$
$9$ | $I_{6}+I_{10}+8I_{1}$ | $O^{\prime}:x_{1}\mapsto(x_{1},0,0)$
|  | $Q:x_{1}\mapsto(x_{1},\lambda_{1}x_{1}^{2},\lambda_{1}x_{1}^{2}(\lambda_{2}+x_{1}+x_{1}^{2})$
$10$ | $I_{5}+I_{11}+8I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{3}x_{1}^{2})$
$11$ | $IV^{*}+I_{9}+7I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{3}x_{1}^{2})$
$12$ | $I_{6}+I_{3}+I_{9}+6I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{3}x_{1}^{2}(\lambda_{2}+x_{1}))$
$13$ | $I_{6}+I_{2}+I_{8}+8I_{1}$ | $O^{\prime}:x_{1}\mapsto(x_{1},0,0)$
|  | $Q:x_{1}\mapsto(x_{1},x_{1}(\lambda_{4}+\lambda_{3}x_{1}),x_{1}(\lambda_{4}+\lambda_{3}x_{1})(\lambda_{2}+x_{1}+x_{1}^{2})$
$14$ | $I_{7}+I_{8}+9I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{1}x_{1}^{3})$
$15$ | $I_{5}+I_{3}+I_{8}+8I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{1}x_{1}^{2}(\lambda_{4}+\lambda_{3}x))$
$16$ | $I_{5}+I_{10}+9I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{1}x_{1}^{2})$
$17$ | $I_{6}+I_{2}+I_{2}+I_{6}+8I_{1}$ | $O^{\prime}:x_{1}\mapsto(x_{1},0,0)$
|  | $Q:x_{1}\mapsto(x_{1},x_{1}(\lambda_{2}+\lambda_{5}x_{1})(\lambda_{4}+x_{1}),x_{1}(\lambda_{2}+\lambda_{5}x_{1})(\lambda_{4}+x_{1})(\lambda_{3}+x_{1}+x_{1}^{2}))$
$18$ | $I_{7}+I_{3}+I_{5}+9I_{1}$ | $Q:x_{1}\mapsto(x_{1},0,\lambda_{1}x_{1}^{3}(\lambda_{5}+x_{1}))$

From now on, for a $K3$ surface $S$, let us regard ${\rm NS}(S)$ as a
sublattice of the $2$-homology group $H_{2}(S,\mathbb{Z}).$ For $P\in{\rm
MW}(\pi,O)$, let $(P)\in{\rm NS}(S)$ denote the corresponding divisor.

For our elliptic fibrations $\pi_{k}$, a general fibre is denoted by $F$. For
each $k\in\\{6,\ldots,18\\}$, the dual graph of the sections and the fibres of
$\pi_{k}$ is illustrated in Table 8. Here, each dot $\bullet$ ($\odot$, resp.)
stands for a rational nodal curve (a general fibre of the elliptic fibration,
resp.).

Table 8: Dual graphs of the elliptic fibration $\pi_{k}$ coming from (2.1) $k$ | Dual graphs | Divisors in singular fibres
---|---|---
$6$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{0}$$(Q)$$(O)$$(O^{\prime})$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{5}$$b_{6}$$b_{7}$$F$ | $\begin{matrix}\pi_{6}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{7},\\\ \pi_{6}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{7}.\end{matrix}$
$7$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{7}$$a_{0}$$(O)$$(Q)$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{7}$$b_{6}$$b_{5}$$F$ | $\begin{matrix}\pi_{7}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{7},\\\ \pi_{7}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{7}.\end{matrix}$
$8$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{5}$$a_{4}$$a_{0}$$b_{0}$$b_{1}$$b_{2}$$(O)$$(Q)$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{7}$$c_{6}$$c_{5}$$F$ | $\begin{matrix}\pi_{8}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{5},\\\ \pi_{8}^{-1}(-\lambda_{2})=b_{0}+b_{1}+b_{2},\\\ \pi_{8}^{-1}(\infty)=c_{0}+c_{1}+\cdots+c_{7}.\end{matrix}$
$9$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{0}$$(Q)$$(O)$$(O^{\prime})$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{5}$$b_{6}$$b_{7}$$b_{8}$$b_{9}$$F$ | $\begin{matrix}\pi_{9}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{5},\\\ \pi_{9}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{9}.\end{matrix}$
$10$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{0}$$(O)$$(Q)$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{5}$$b_{10}$$b_{9}$$b_{8}$$b_{7}$$b_{6}$$F$ | $\begin{matrix}\pi_{10}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{4},\\\ \hskip 5.69054pt\pi_{10}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{10}.\end{matrix}$
$11$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{2}$$a_{1}$$a_{3}$$a_{4}$$a_{0}$$a_{5}$$a_{6}$$(O)$$(Q)$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{8}$$b_{7}$$b_{6}$$b_{5}$$F$ | $\begin{matrix}\pi_{11}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{6},\\\ \pi_{11}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{8}.\end{matrix}$
$12$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{0}$$b_{0}$$b_{1}$$b_{2}$$(O)$$(Q)$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{5}$$c_{6}$$c_{7}$$c_{8}$$F$ | $\begin{matrix}\pi_{12}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{5},\\\ \pi_{12}^{-1}(-\lambda_{2})=b_{0}+b_{1}+b_{2},\\\ \pi_{12}^{-1}(\infty)=c_{0}+c_{1}+\cdots+c_{8}.\end{matrix}$
$13$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{0}$$(Q)$$(O)$$(O^{\prime})$$b_{0}$$b_{1}$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{5}$$c_{6}$$c_{7}$$F$ | $\begin{matrix}\pi_{13}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{5},\\\ \pi_{13}^{-1}(-\frac{\lambda_{4}}{\lambda_{3}})=b_{0}+b_{1},\\\ \pi_{13}^{-1}(\infty)=c_{0}+c_{1}+\cdots+c_{7}.\end{matrix}$
$14$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{0}$$(O)$$(Q)$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{7}$$b_{6}$$b_{5}$$F$ | $\begin{matrix}\pi_{14}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{6},\\\ \pi_{14}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{7}.\end{matrix}$
$15$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{0}$$b_{0}$$b_{1}$$b_{2}$$(O)$$(Q)$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{7}$$c_{6}$$c_{5}$$F$ | $\begin{matrix}\pi_{15}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{4},\\\ \pi_{15}^{-1}(-\lambda_{2})=b_{0}+b_{1}+b_{2},\\\ \pi_{15}^{-1}(\infty)=c_{0}+c_{1}+\cdots+c_{7}.\end{matrix}$
$16$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{0}$$(O)$$(Q)$$b_{0}$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$b_{5}$$b_{9}$$b_{8}$$b_{7}$$b_{6}$$F$ | $\begin{matrix}\pi_{16}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{4},\\\ \pi_{16}^{-1}(\infty)=b_{0}+b_{1}+\cdots+b_{9}.\end{matrix}$
$17$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{0}$$(Q)$$(O)$$(O^{\prime})$$b_{0}$$b_{1}$$c_{0}$$c_{1}$$d_{0}$$d_{1}$$d_{2}$$d_{3}$$d_{4}$$d_{5}$$F$ | $\begin{matrix}\pi_{17}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{5},\\\ \pi_{17}^{-1}(-\lambda_{4})=b_{0}+b_{1},\\\ \pi_{17}^{-1}(-\frac{\lambda_{2}}{\lambda_{5}})=c_{0}+c_{1},,\\\ \pi_{17}^{-1}(\infty)=d_{0}+d_{1}+\cdots+c_{5}.\end{matrix}$
$18$ | $\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\odot$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$$a_{6}$$a_{0}$$b_{0}$$b_{1}$$b_{2}$$(O)$$(Q)$$c_{0}$$c_{1}$$c_{2}$$c_{3}$$c_{4}$$F$ | $\begin{matrix}\pi_{18}^{-1}(0)=a_{0}+a_{1}+\cdots+a_{6},\\\ \pi_{18}^{-1}(-\lambda_{5})=b_{0}+b_{1}+b_{2},\\\ \pi_{18}^{-1}(\infty)=c_{0}+c_{1}+\cdots+c_{4}.\end{matrix}$

######  Proposition 2.3.

For each $k\in\\{6,\ldots,18\\},$ let $\ell_{k}$ be the number of vertices of
the Fano polytope $P_{k}$, $L_{k}$ be the lattice given in Table 1 and $S_{k}$
be a generic member of the family $\mathscr{F}_{k}$. Then, there is a
sublattice $E_{k}$ of ${\rm NS}(S_{k})$, which we call the evident lattice of
$S_{k}$, satisfying ${\rm rank}(E_{k})=23-\ell_{k}$ and
$|\det(E_{k})|=|\det(L_{k})|.$ The generators of $E_{k}$ are explicitly given
in Table 9.

Table 9: Evident lattices $E_{k}$ $k$ | $E_{k}$ | ${\rm rank}(E_{k})$ | $|\det(E_{k})|=|\det(L_{k})|$
---|---|---|---
$6$ | $\langle F,(O),(Q),(O^{\prime}),a_{2},\ldots,a_{7},b_{1},\ldots,b_{7}\rangle_{\mathbb{Z}}$ |  $17$ |  $16$
$7$ | $\langle F,(O),(Q),a_{1},\ldots,a_{7},b_{1},\ldots,b_{7}\rangle_{\mathbb{Z}}$ |  $17$ |  $12$
$8$ | $\langle F,(O),(Q),a_{1},\ldots,a_{5},b_{1},b_{2},c_{1}\ldots,c_{7}\rangle_{\mathbb{Z}}$ |  $17$ |  $20$
$9$ | $\langle F,(O),(Q),(O^{\prime}),a_{1},\ldots,a_{5},b_{1},\ldots,b_{8}\rangle_{\mathbb{Z}}$ |  $17$ |  $16$
$10$ | $\langle F,(O),(Q),a_{1},\ldots,a_{4},b_{1},\ldots,b_{10}\rangle_{\mathbb{Z}}$ |  $17$ |  $14$
$11$ | $\langle F,(O),(Q),a_{1},\ldots,a_{6},b_{1},\ldots,b_{8}\rangle_{\mathbb{Z}}$ |  $17$ |  $12$
$12$ | $\langle F,(O),(Q),a_{1},\ldots,a_{5},b_{1},c_{1}\ldots,c_{7}\rangle_{\mathbb{Z}}$ |  $17$ |  $18$
$13$ | $\langle F,(O),(Q),(O^{\prime}),a_{1},\ldots,a_{4},b_{1},c_{1},\ldots,c_{7}\rangle_{\mathbb{Z}}$ |  $16$ |  $28$
$14$ | $\langle F,(O),(Q),a_{1},\ldots,a_{6},b_{1},\ldots,b_{7}\rangle_{\mathbb{Z}}$ |  $16$ |  $23$
$15$ | $\langle F,(O),(Q),a_{1},\ldots,a_{4},b_{1},b_{2},c_{1},\ldots,c_{7}\rangle_{\mathbb{Z}}$ |  $16$ |  $31$
$16$ | $\langle F,(O),(Q),a_{1},\ldots,a_{4},b_{1},\ldots,b_{9}\rangle_{\mathbb{Z}}$ |  $16$ |  $20$
$17$ | $\langle F,(O),(Q),(O^{\prime}),a_{1},\ldots,a_{4},b_{1},c_{1},d_{1},\ldots,d_{5}\rangle_{\mathbb{Z}}$ |  $15$ |  $48$
$18$ | $\langle F,(O),(Q),a_{1},\ldots,a_{6},b_{1},b_{2},c_{1},\ldots,c_{4}\rangle_{\mathbb{Z}}$ |  $15$ |  $44$

###### Proof.

For an elliptic $K3$ surface $(S,\pi,\mathbb{P}^{1}(\mathbb{C}))$, a general
fibre $F$ and a section $R$ of $\pi$ satisfies $(F\cdot F)=0$, $(R\cdot R)=-2$
and $(F\cdot R)=1$ (see [Sh]). Then, we can calculate the intersection matrix
of the evident lattices $E_{k}$ for each $k$. (One can find explicit forms of
the intersection matrices in [Mat1].) ∎

##  3 Period mappings and Picard numbers

For $k\in\\{6,\ldots,18\\},$ take a generic point
$\lambda^{0}=(\lambda_{1}^{0},\ldots,\lambda_{l_{k}-3}^{0})\in(\mathbb{C}^{\times})^{\ell_{k}-3}.$
We call $S_{k}^{0}=S_{k}(\lambda^{0})$ a reference surface of the family
$\mathscr{F}_{k}$ of (1.10). Let us introduce a marking for
$S_{k}(\lambda^{0})$ using the evident lattice $E_{k}$ in Proposition 2.3.
According to the procedure, $E_{k}$ is a sublattice of the Néron-Severi
lattice ${\rm NS}(S_{k}^{0})$. Let $\widehat{E}_{k}$ be the primitive closure
of $E_{k}$ in ${\rm NS}(S_{k}^{0})$:
$\widehat{E}_{k}=(E_{k}\otimes_{\mathbb{Z}}\mathbb{Q})\cap{\rm
NS}(S_{k}^{0})$. Take a basis
$\\{\Gamma_{k,1},\ldots,\Gamma_{k,23-\ell_{k}}\\}$ of $\widehat{E}_{k}$. Let
us identify $H_{2}(S_{k}^{0},\mathbb{Z})$ with $L_{K3}$. This identification
gives an isometry $\psi^{0}:H_{2}(S_{k}^{0},\mathbb{Z})\rightarrow L_{K3}$.
Put $\mathcal{E}_{k}=\psi^{0}\left(\widehat{E}_{k}\right)$ and
$\gamma_{k,j}=\psi\left(\Gamma_{k,j}\right)$
($j\in\\{1,\ldots,23-\ell_{k}\\}$). Then, $\mathcal{E}_{k}$ satisfies
$\left(\psi^{0}\right)^{-1}(\mathcal{E}_{k})\subset{\rm NS}(S_{k}^{0})$. We
can extend $\left\\{\gamma_{k,1},\ldots,\gamma_{k,23-\ell_{k}}\right\\}$ to a
basis $\left\\{\gamma_{k,1},\ldots,\gamma_{k,22}\right\\}$ of $L_{K3}$. Let
$\left\\{\delta_{k,1},\ldots,\delta_{k,22}\right\\}$ be the dual basis of
$\left\\{\gamma_{k,1},\ldots,\gamma_{k,22}\right\\}$ with respect to the
unimodular lattice $L_{K3}$.

There is a sufficiently small neighborhood $\mathcal{U}$ of $\lambda^{0}$ in
$(\mathbb{C}^{\times})^{\ell_{k}-3}$ with the trivialization
$\tau:\\{S_{k}(\lambda)|\lambda\in\mathcal{U}\\}\rightarrow
S_{k}^{0}\times\mathcal{U}$. Letting
$\beta:S_{k}^{0}\times\mathcal{U}\rightarrow S_{k}^{0}$ be the projection, set
$r=\beta\circ\tau.$ Then, for $\lambda\in\mathcal{U}$,
$\left(r^{(\lambda)}_{k}\right)^{\prime}:=r|_{S_{k}(\lambda)}:S_{k}(\lambda)\rightarrow
S_{k}^{0}$ gives a $\mathcal{C}^{\infty}$-isomorphisms of compact complex
surfaces. We obtain an isometry
$\psi^{(\lambda)}=\psi^{0}\circ\left(r^{(\lambda)}_{k}\right)^{\prime}_{*}:H_{2}(S_{k}(\lambda),\mathbb{Z})\rightarrow
L_{K3}$ for $\lambda\in\mathcal{U}$. We call the pair
$(S_{k}(\lambda),\psi^{(\lambda)})$ a $S$-marked $K3$ surfaces on
$\mathcal{U}$. For $\lambda^{1}$ and $\lambda^{2}\in\mathcal{U}$, if there
exists a biholomorphic mapping $f:S_{k}(\lambda^{1})\rightarrow
S_{k}(\lambda^{2})$ satisfying $\left(\psi^{\left(\lambda^{2}\right)}\circ
f_{*}\circ\left(\psi^{\left(\lambda^{1}\right)}\right)^{-1}\right)\Big{|}_{\mathcal{E}_{k}}$,
we say that $\left(S_{k}(\lambda^{1}),\psi^{\left(\lambda^{1}\right)}\right)$
and $\left(S_{k}(\lambda^{2}),\psi^{\left(\lambda^{2}\right)}\right)$ are
equivalent as $S$-marked $K3$ surfaces.

######  Remark 3.1.

We remark the precise meaning of “generic”. In fact, there are certain loci in
$(\mathbb{C}^{\times})^{\ell_{k}-3}$ on which we have degenerated elliptic
$K3$ surfaces. For instance, one can obtain a locus on which two singular
fibres of type $I_{1}$ (see Table 7) collapse into a fibre of type $I_{2}$ for
each $k$. We take a generic point
$\lambda^{0}\in(\mathbb{C}^{\times})^{\ell_{k}-3}$ while avoiding such loci.
Similarly, we should take a neighborhood $\mathcal{U}$ of $\lambda^{0}$ not to
touch them.

Letting the notation be as above, we denote the intersection matrix of the
lattice $\left\\{\delta_{k,23-\ell_{k}+1},\ldots,\delta_{k,22}\right\\}$ by
$\textbf{A}_{k}$. Set
$\mathcal{D}_{\mathcal{E}_{k}}=\\{\xi\in\mathbb{P}^{\ell_{k}-2}(\mathbb{C})|\xi\textbf{A}_{k}{}^{t}\xi=0,\xi\textbf{A}_{k}{}^{t}\overline{\xi}>0\\}$.
We remark that $\mathcal{D}_{\mathcal{E}_{k}}$ has two connected components.
Let $\mathcal{D}_{k}$ be a connected component of
$\mathcal{D}_{\mathcal{E}_{k}}$. Then, we have the local period mapping
$\Phi_{k}:\mathcal{U}\rightarrow\mathcal{D}_{k}$ given by

$\displaystyle\lambda\mapsto\left(\int_{(\psi^{(\lambda)})^{-1}\left(\gamma_{k,23-\ell_{k}+1}\right)}\omega_{k}:\cdots:\int_{(\psi^{(\lambda)})^{-1}\left(\gamma_{k,22}\right)}\omega_{k}\right).$
(3.1)

Here, $\omega_{k}$ is the unique holomorphic $2$-form on $S_{k}$ up to a
constant factor.

Let $\pi:S\rightarrow\mathbb{P}^{1}(\mathbb{C})$ be an elliptic fibration of a
$K3$ surface $S$ with a general fibre $F$. If $\pi$ is defined by the
Weierstrass equation $z^{2}=4y^{3}-g_{2}(x)y-g_{3}(x)$, where $\pi$ is given
by $(x,y,z)\mapsto x$, then the $j$-invariant is given by
$j(x)=\frac{g_{2}^{3}(x)}{g_{2}^{3}(x)-27g_{3}^{2}(x)}$. Suppose
$(S,\pi,\mathbb{P}^{1}(\mathbb{C}))$ and
$(S^{\prime},\pi^{\prime},\mathbb{P}^{1}(\mathbb{C}))$ are two elliptic
surfaces such that there exists a biholomorphic mapping $f:S\rightarrow
S^{\prime}$ and $\varphi\in{\rm Aut}(\mathbb{P}^{1}(\mathbb{C}))$ satisfying
$\varphi\circ\pi=\pi^{\prime}\circ f$. Then, they are said to be isomorphic as
elliptic fibrations. In this case, if $(S,\pi,\mathbb{P}^{1}(\mathbb{C}))$
($(S^{\prime},\pi^{\prime},\mathbb{P}^{1}(\mathbb{C}))$, resp.) is defined by
the Weierstrass equation with the $j$-invariant $j(x)$ ($j^{\prime}(x)$,
resp.), then $j^{\prime}\circ\varphi=j$ holds (see [Kod] or [Ne]). Moreover,
if $\pi^{-1}(x)$ is a singular fibre, then $\pi^{\prime-1}(\varphi(x))$ is a
singular fibre of the same type.

######  Lemma 3.1.

For $k\in\\{6,\ldots,18\\}$, let $\lambda^{0}$ be a general point of
$(\mathbb{C}^{\times})^{\ell_{k}-3}$ and $\mathcal{U}$ be a neighborhood of
$\lambda^{0}$ as above. For each $k$, let $\pi_{k}^{(\lambda)}$ be the
Jacobian elliptic fibration given by the equation (2.1). Take $\lambda^{1}$
and $\lambda^{2}\in\mathcal{U}$. If
$\left(S_{k}(\lambda^{1}),\pi_{k}^{\left(\lambda^{1}\right)},\mathbb{P}^{1}(\mathbb{C})\right)$
and
$\left(S_{k}(\lambda^{2}),\pi_{k}^{\left(\lambda^{2}\right)},\mathbb{P}^{1}(\mathbb{C})\right)$
are isomorphic as elliptic surfaces, then $\lambda^{1}=\lambda^{2}$ holds.

###### Proof.

For each $k$, the elliptic fibration $\pi_{k}^{(\lambda)}$ has singular fibres
for $\lambda\in\mathcal{U}$ as in Table 7 and 8. By a direct calculation, we
can transform the equation (2.1) to a Weierstrass form and obtain the
$j$-invariant $j_{k}^{(\lambda)}(x_{1}).$ There is a polynomial
$\mathcal{P}_{k}^{(\lambda)}(x_{1})\in\mathbb{C}(\lambda)[x_{1}]$ such that
$x_{1}=\xi\in\mathbb{P}^{1}(\mathbb{C})-\\{0,\infty\\}$ is a root of
$\mathcal{P}_{k}^{(\lambda)}(x_{1})$ if and only if
$\left(\pi_{k}^{(\lambda)}\right)^{-1}(\xi)$ gives a singular fibre. We can
explicitly obtain this polynomial. In practice,
$\mathcal{P}_{k}^{(\lambda)}(x_{1})$ appears as an appropriate factor of the
discriminant of the right hand side of (2.1) in the variable $y_{1}$. Note
that $\mathcal{P}_{k}^{(\lambda)}(x_{1})$ factorizes the denominator of
$j_{k}^{(\lambda)}(x_{1}).$

Suppose that $f:S_{k}(\lambda^{1})\rightarrow S_{k}(\lambda^{2})$ is a
biholomorphic mapping which derives an isomorphism of elliptic surfaces.
Namely, there exists $\varphi\in{\rm Aut}(\mathbb{P}^{1}(\mathbb{C}))$
satisfying
$\varphi\circ\pi_{k}^{\left(\lambda^{1}\right)}=\pi_{k}^{\left(\lambda^{2}\right)}\circ
f$. Since the Kodaira type of the singular fibre
$\left(\pi_{k}^{\left(\lambda^{1}\right)}\right)^{-1}(0)$
($\left(\pi_{k}^{\left(\lambda^{1}\right)}\right)^{-1}(\infty)$, resp.) is
same as that of $\left(\pi_{k}^{\left(\lambda^{2}\right)}\right)^{-1}(0)$
($\left(\pi_{k}^{\left(\lambda^{2}\right)}\right)^{-1}(\infty)$, resp.).
Hence, it follows that $\varphi$ must have the form
$x_{1}\mapsto\varphi(x_{1})=\mu x_{1}$ for some $\mu\in\mathbb{C}^{\times}$.
In particular, it holds
$j_{k}^{\left(\lambda^{2}\right)}(x_{1})=j_{k}^{\left(\lambda^{1}\right)}(\mu
x_{1}).$ Also, each root $\xi$ of
$\mathcal{P}_{k}^{\left(\lambda^{1}\right)}(x_{1})$ is sent to a root of
$\mathcal{P}_{k}^{\left(\lambda^{2}\right)}(x_{1})$ by $\varphi$ such that the
Kodaira type of $\left(\pi_{k}^{\left(\lambda^{1}\right)}\right)^{-1}(\xi)$ is
same as that of
$\left(\pi_{k}^{\left(\lambda^{2}\right)}\right)^{-1}(\mu\xi)$. Considering
the above situation and observing the explicit forms of
$\mathcal{P}_{k}^{(\lambda)}(x_{1})$ and $j_{k}^{(\lambda)}(x_{1})$ for each
$k$, we can see that $\mu=1$ and $\lambda^{1}=\lambda^{2}$. ∎

######  Lemma 3.2.

For $k\in\\{6,\ldots,18\\}$, let $\lambda^{0}$ be a general point of
$(\mathbb{C}^{\times})^{\ell_{k}-3}$ and $\mathcal{U}$ be a neighborhood of
$\lambda^{0}$ as above. Take the $S$-marked $K3$ surface
$\left(S_{k}(\lambda),\psi^{(\lambda)}\right)$ for $\lambda\in\mathcal{U}$ as
above. For $\lambda^{1}$ and $\lambda^{2}\in\mathcal{U}$,
$\left(S_{k}(\lambda^{1}),\psi^{\left(\lambda^{1}\right)}\right)$ and
$\left(S_{k}(\lambda^{2}),\psi^{\left(\lambda^{2}\right)}\right)$ are
equivalent as $S$-marked $K3$ surfaces if and only if
$\lambda^{1}=\lambda^{2}$ holds.

###### Proof.

For each $k$, let $F_{k}^{\left(\lambda^{j}\right)}$ $(j\in\\{1,2\\})$ be a
general fibre for the elliptic fibration $\pi_{k}^{\left(\lambda^{j}\right)}$.
We assume that a biholomorphic mapping $f:S_{k}(\lambda^{1})\rightarrow
S_{k}(\lambda^{2})$ induces an equivalence of $S$-marked $K3$ surfaces. Then,
we have $\left(\psi_{k}^{\left(\lambda^{2}\right)}\circ
f_{*}\circ\left(\psi_{k}^{\left(\lambda^{1}\right)}\right)^{-1}\right)\Big{|}_{\mathcal{E}_{k}}={\rm
id}_{\mathcal{E}_{k}}$. This implies
$f_{*}\left(F_{k}^{\left(\lambda^{1}\right)}\right)=F_{k}^{\left(\lambda^{2}\right)}$.
Hence, $F_{k}^{\left(\lambda^{2}\right)}$ is a general fibre of the elliptic
fibration $\pi_{k}^{\left(\lambda^{1}\right)}\circ f^{-1}$ of
$S_{k}(\lambda^{2})$ also. By the way, it hold
$K_{S_{k}(\lambda^{2})}=0,H^{1}(S_{k}(\lambda^{2}),\mathcal{O}_{S_{k}(\lambda^{2})})=0$
and $\left(F_{k}^{\left(\lambda^{2}\right)}\cdot
F_{k}^{\left(\lambda^{2}\right)}\right)=0.$ Due to the Riemann-Roch theorem,
we have
$H^{1}\left(S_{k}(\lambda^{2}),\mathcal{O}_{S_{k}(\lambda^{2})}\left(F_{k}^{\left(\lambda^{2}\right)}\right)\right)=0$
and ${\rm
dim}H^{0}\left(S_{k}(\lambda^{2}),\mathcal{O}_{S_{k}(\lambda^{2})}\left(F_{k}^{\left(\lambda^{2}\right)}\right)\right)=2$.
Therefore, $1$ and $\pi_{k}^{\left(\lambda^{2}\right)}$ generate
$H^{0}\left(S_{k}(\lambda^{2}),\mathcal{O}_{S_{k}(\lambda^{2})}\left(F_{k}^{\left(\lambda^{2}\right)}\right)\right)$
over $\mathbb{C}$. Thereby, $\pi_{k}^{\left(\lambda^{2}\right)}$ coincides
with $\pi_{k}^{\left(\lambda^{1}\right)}\circ f^{-1}$ up to an action of ${\rm
Aut}(\mathbb{P}^{1}(\mathbb{C}))$. Thus,
$\left(S_{k}(\lambda^{1}),\psi^{\left(\lambda^{1}\right)}\right)$ is
equivalent to
$\left(S_{k}(\lambda^{2}),\psi^{\left(\lambda^{2}\right)}\right)$ as
$S$-marked $K3$ surfaces if and only if
$\left(S_{k}(\lambda^{1}),\pi_{k}^{\left(\lambda^{1}\right)},\mathbb{P}^{1}(\mathbb{C})\right)$
is isomorphic to
$\left(S_{k}(\lambda^{2}),\pi_{k}^{\left(\lambda^{2}\right)},\mathbb{P}^{1}(\mathbb{C})\right)$
as elliptic surfaces. According to Lemma 3.1, we have the assertion. ∎

######  Theorem 3.1.

For each $k\in\\{6,\ldots,18\\},$ the Picard number of a generic member of
$\mathscr{F}_{k}$ of (1.10) is equal to ${\rm rank}(E_{k})=23-\ell_{j}$, where
$E_{k}$ is the evident lattice in Proposition 2.3.

###### Proof.

For each $k\in\\{6,\ldots,18\\}$, the lattice $E_{k}$ is a sublattice of ${\rm
NS}(S_{k}(\lambda))$ for generic $\lambda$. Therefore, it holds ${\rm
rank}({\rm NS}(S_{k}(\lambda)))\geq 23-\ell_{k}$. Let us assume that the rank
of ${\rm NS}(S_{k}(\lambda))$ is not equal to $23-\ell_{k}$. Then, the
dimension of the image of our period mapping $\Phi_{k}$ of (3.1) must be
smaller than $\ell_{k}-3$. By the way, letting $\mathcal{U}$ be a small open
set as in the above argument, by virtue of the local Torelli theorem ([D]
Section 3) for $K3$ surfaces polarized by the lattice $\mathcal{E}_{k}$,
$\Phi_{k}(\lambda^{1})=\Phi_{k}(\lambda^{2})$
$(\lambda^{1},\lambda^{2}\in\mathcal{U})$ holds if and only if
$\left(S_{k}(\lambda^{1}),\psi^{\left(\lambda^{1}\right)}\right)$ is
equivalent to
$\left(S_{k}(\lambda^{2}),\psi^{\left(\lambda^{2}\right)}\right)$ as
$S$-marked $K3$ surfaces. Together with Lemma 3.2,
$\mathcal{U}\ni\lambda\mapsto\Phi_{k}(\lambda)\in\mathcal{D}_{k}$ is an
injection. This contradicts to the assumption. ∎

##  4 Lattice structures

Let $L$ be a non-degenerate even lattice. Setting $L^{\vee}={\rm
Hom}(L,\mathbb{Z}),$ the group $\mathscr{A}_{L}=L^{\vee}/L$ becomes to be a
finite abelian group. The length $l(\mathscr{A}_{L})$ of $\mathscr{A}_{L}$ is
defined as the minimum number of generators of $\mathscr{A}_{L}.$ The
intersection form of $L$ induces a quadratic form
$q_{L}:\mathscr{A}_{L}\rightarrow\mathbb{Q}/2\mathbb{Z}.$ This $q_{L}$ is
called the discriminant form of $L$. We note that $q_{L_{1}\oplus L_{2}}\simeq
q_{L_{1}}\oplus q_{L_{2}}$ holds. If the signature of $L$ is of $(s,t)$, the
triple $(s,t,q_{L})$ is called the invariant of $L$.

######  Proposition 4.1.

([Ni] Proposition 1.6.1) Let $L$ be a non-degenerate even lattice. Suppose $L$
is a primitive lattice in a unimodular lattice $\widehat{L}$. Letting
$L^{\perp}$ be the orthogonal complement of $L$ in $\widehat{L}$,
$q_{L^{\perp}}\simeq-q_{L}$ holds.

######  Proposition 4.2.

([Ni] Corollary 1.13.1) Let $L$ be a non-degenerate even lattice with the
invariant $(s,t,q_{L}).$ Suppose $s>0$, $t>0$ and $l(\mathscr{A}_{L})\leq{\rm
rank}(L)-2$ hold. Then, $L$ is the unique lattice with the invariant
$(s,t,q_{L})$ up to isometry.

######  Proposition 4.3.

([Mor] Lemma 2.3) Let $L_{1}$ and $L_{2}$ be even lattices with the same
invariant. Let $\widehat{L}$ be an even lattice which is uniquely determined
by its invariant in the sense of Proposition 4.2. Suppose that there is a
primitive embedding $L_{1}\hookrightarrow\widehat{L}.$ Then, there exists a
primitive embedding $L_{2}\hookrightarrow\widehat{L}.$

We will apply these arithmetic results for lattices to our Néron-Severi
lattices of toric $K3$ hypersurfaces.

######  Lemma 4.1.

For $k\in\\{6,\ldots,18\\},$ let $L_{k}$ be the lattice of Table 1 and $E_{k}$
be the evident lattice of Proposition 2.3. The finite abelian group
$\mathscr{A}_{E_{k}}$ is isomorphic to the group $\mathscr{A}_{L_{k}}.$
Precisely, letting $\mathcal{G}_{k}$ be a finite abelian group as in Table 10,
one has $\mathscr{A}_{E_{k}}\simeq\mathscr{A}_{L_{k}}\simeq\mathcal{G}_{k}$.
Also, it holds $q_{E_{k}}\simeq-q_{L_{k}}.$

Table 10: Finite abelian groups $\mathcal{G}_{k}$ $k$ | $\mathcal{G}_{k}$
---|---
$6$ | $(\mathbb{Z}/4\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})$
$7$ | $\mathbb{Z}/12\mathbb{Z}$
$8$ | $\mathbb{Z}/10\mathbb{Z}$
$9$ | $\mathbb{Z}/16\mathbb{Z}$
$10$ | $\mathbb{Z}/14\mathbb{Z}$
$11$ | $\mathbb{Z}/12\mathbb{Z}$
$12$ | $\mathbb{Z}/18\mathbb{Z}$
$13$ | $(\mathbb{Z}/14\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})$
$14$ | $\mathbb{Z}/23\mathbb{Z}$
$15$ | $\mathbb{Z}/31\mathbb{Z}$
$16$ | $(\mathbb{Z}/10\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})$
$17$ | $(\mathbb{Z}/12\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})\oplus(\mathbb{Z}/2\mathbb{Z})$
$18$ | $\mathbb{Z}/44\mathbb{Z}$

###### Proof.

For each $k\in\\{6,\ldots,18\\},$ let $\\{v_{1},\ldots,v_{\ell_{k}-3}\\}$ be
the basis of $L_{k}$ corresponding to the intersection matrix of Table 1,
where $\ell_{k}$ is the number of vertices of $P_{k}$. Then, we can find an
element $\alpha\in L_{k}^{\vee}/L_{k}$ in the form
$\alpha=\sum_{k=1}^{\ell_{k}-3}c_{k}v_{k}$ $(c_{k}\in\mathbb{Q}/\mathbb{Z})$,
such that $\alpha$ generates a cyclic group which is a direct summand of
$\mathcal{G}_{k}$. In fact, the coefficients $c_{1},\ldots,c_{\ell_{j}-3}$ are
given in Table 11.

Table 11: Coefficients $c_{1},\ldots,c_{\ell_{k}-3}$ and discriminants $q_{L_{k}}(\alpha)$ $k$ | Cyclic groups | Coefficients $c_{1},\ldots,c_{\ell_{k}-3}$ | $q_{L_{k}}(\alpha)$
---|---|---|---
$6$ | $\mathbb{Z}/4\mathbb{Z}$ | $\frac{3}{4},\frac{1}{4},\frac{1}{4}$ | $\frac{7}{4}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,\frac{1}{2},0$ | $0$
| $\mathbb{Z}/2\mathbb{Z}$ | $\frac{1}{2},\frac{1}{2},0$ | $1$
$7$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{1}{6},\frac{11}{12},\frac{5}{12}$ | $\frac{23}{12}$
$8$ | $\mathbb{Z}/20\mathbb{Z}$ | $\frac{1}{10},\frac{7}{20},\frac{19}{20}$ | $\frac{39}{20}$
$9$ | $\mathbb{Z}/16\mathbb{Z}$ | $\frac{3}{8},\frac{1}{8},\frac{15}{16}$ | $\frac{31}{16}$
$10$ | $\mathbb{Z}/14\mathbb{Z}$ | $\frac{2}{7},\frac{5}{14},\frac{3}{14}$ | $\frac{3}{14}$
$11$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{7}{12},\frac{11}{12},\frac{1}{4}$ | $\frac{19}{12}$
$12$ | $\mathbb{Z}/18\mathbb{Z}$ | $\frac{13}{18},\frac{8}{9},\frac{1}{3}$ | $\frac{31}{18}$
$13$ | $\mathbb{Z}/14\mathbb{Z}$ | $\frac{2}{7},\frac{5}{7},\frac{3}{14},\frac{1}{14}$ | $\frac{12}{7}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,\frac{1}{2},0,\frac{1}{2}$ | $0$
$14$ | $\mathbb{Z}/23\mathbb{Z}$ | $\frac{9}{23},\frac{17}{23},\frac{5}{23},\frac{1}{23}$ | $\frac{40}{23}$
$15$ | $\mathbb{Z}/31\mathbb{Z}$ | $\frac{13}{31},\frac{5}{31},\frac{12}{31},\frac{14}{31}$ | $\frac{44}{31}$
$16$ | $\mathbb{Z}/10\mathbb{Z}$ | $\frac{3}{10},\frac{1}{5},\frac{7}{10},\frac{1}{10}$ | $\frac{17}{10}$
| $\mathbb{Z}/2\mathbb{Z}$ | $\frac{1}{2},\frac{1}{2},0,\frac{1}{2}$ | $\frac{1}{2}$
$17$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{1}{6},\frac{5}{6},\frac{1}{3},\frac{5}{12},\frac{5}{12}$ | $\frac{17}{12}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,\frac{1}{2},0,\frac{1}{2}$ | $0$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}$ | $1$
$18$ | $\mathbb{Z}/44\mathbb{Z}$ | $\frac{13}{44},\frac{5}{44},\frac{27}{44},\frac{31}{44},\frac{25}{44}$ | $\frac{57}{44}$

On the other hand, let $\\{w_{1},\ldots,w_{23-\ell_{k}}\\}$ be the basis of
$E_{k}$ described in Table 9. We can take $\beta\in E_{k}^{\vee}/E_{k}$ with
$\beta=\sum_{k=1}^{23-\ell_{k}}d_{k}w_{k}$ $(d_{k}\in\mathbb{Q}/\mathbb{Z})$
such that $\beta$ generates a direct summand of $\mathcal{G}_{k}$. Here, the
coefficients $d_{1},\ldots,d_{23-\ell_{k}}$ are given in Table 12.

Table 12: Coefficients $d_{1},\ldots,d_{23-\ell_{k}}$ and discriminants $q_{E_{k}}(\beta)$ $k$ | Cyclic groups | Coefficients $d_{1},\ldots,d_{23-\ell_{k}}$ | $q_{E_{k}}(\beta)$
---|---|---|---
$6$ | $\mathbb{Z}/4\mathbb{Z}$ | $\frac{1}{2},\frac{1}{4},\frac{1}{2},\frac{1}{4},0,\frac{1}{2},0,\frac{1}{4},\frac{1}{2},\frac{3}{4},\frac{3}{4},\frac{1}{2},\frac{3}{4},0,0,0,0$ | $\frac{1}{4}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,0,0,0,0,0,0,0,0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2}$ | $0$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,\frac{1}{2},\frac{1}{2},0,0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2},\frac{1}{2},0,0,0,0,0,0$ | $1$
$7$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{2}{3},\frac{1}{3},\frac{2}{3},\frac{5}{6},\frac{2}{3},\frac{1}{2},\frac{1}{3},\frac{1}{6},\frac{1}{12},\frac{11}{12},\frac{11}{12},\frac{5}{6},\frac{3}{4},0,\frac{1}{4},\frac{1}{2},\frac{3}{4}$ | $\frac{1}{12}$
$8$ | $\mathbb{Z}/20\mathbb{Z}$ | $\frac{1}{5},\frac{3}{5},\frac{2}{5},\frac{7}{10},\frac{2}{5},\frac{1}{10},\frac{1}{20},\frac{3}{4},\frac{3}{5},\frac{4}{5},\frac{3}{4},\frac{1}{2},\frac{1}{4},\frac{3}{5},\frac{19}{20},\frac{3}{10},\frac{13}{20}$ | $\frac{1}{20}$
$9$ | $\mathbb{Z}/16\mathbb{Z}$ | $\frac{3}{4},\frac{3}{8},\frac{9}{16},\frac{1}{16},\frac{1}{2},\frac{7}{16},\frac{3}{8},\frac{1}{4},\frac{1}{8},\frac{5}{8},\frac{1}{4},\frac{7}{8},\frac{15}{16},0,0,0,0$ | $\frac{1}{16}$
$10$ | $\mathbb{Z}/14\mathbb{Z}$ | $\frac{5}{7},\frac{5}{14},\frac{9}{14},\frac{11}{14},\frac{4}{7},\frac{5}{7},\frac{6}{7},\frac{1}{2},0,\frac{1}{2},0,\frac{6}{7},\frac{5}{7},\frac{4}{7},\frac{3}{7},\frac{2}{7},\frac{1}{7}$ | $\frac{25}{14}$
$11$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{5}{12},\frac{1}{3},\frac{1}{3},\frac{1}{6},\frac{11}{12},\frac{5}{6},\frac{3}{4},\frac{2}{3},\frac{1}{3},0,\frac{2}{3},\frac{1}{3}$ | $\frac{5}{12}$
$12$ | $\mathbb{Z}/18\mathbb{Z}$ | $\frac{1}{3},\frac{2}{3},\frac{1}{3},\frac{1}{18},\frac{1}{9},\frac{5}{6},\frac{5}{9},\frac{5}{18},\frac{2}{9},\frac{1}{9},\frac{1}{3},\frac{2}{3},0,0,0,0,0$ | $\frac{5}{18}$
$13$ | $\mathbb{Z}/14\mathbb{Z}$ | $\frac{3}{7},\frac{5}{7},\frac{13}{14},\frac{5}{14},\frac{2}{7},\frac{9}{14},0,0,\frac{1}{7},0,0,\frac{1}{14},\frac{1}{7},\frac{6}{7},\frac{4}{7},\frac{2}{7}$ | $\frac{2}{7}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,0,0,0,0,0,0,0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2}$ | $0$
$14$ | $\mathbb{Z}/23\mathbb{Z}$ | $\frac{6}{23},\frac{3}{23},\frac{20}{23},\frac{18}{23},\frac{13}{23},\frac{8}{23},\frac{6}{23},\frac{4}{23},\frac{2}{23},\frac{1}{23},\frac{2}{23},\frac{3}{23},\frac{7}{23},\frac{11}{23},\frac{15}{23},\frac{19}{23}$ | $\frac{6}{23}$
$15$ | $\mathbb{Z}/31\mathbb{Z}$ | $\frac{28}{31},\frac{14}{31},\frac{17}{31},\frac{4}{31},\frac{8}{31},\frac{26}{31},\frac{13}{31},\frac{1}{31},\frac{16}{31},\frac{30}{31},\frac{29}{31},\frac{28}{31},\frac{10}{31},\frac{23}{31},\frac{5}{31},\frac{18}{31}$ | $\frac{18}{31}$
$16$ | $\mathbb{Z}/10\mathbb{Z}$ | $0,\frac{1}{2},\frac{1}{2},\frac{3}{10},\frac{3}{5},\frac{2}{5},\frac{1}{5},\frac{1}{10},\frac{1}{5},\frac{3}{10},\frac{2}{5},0,\frac{3}{5},\frac{1}{5},\frac{4}{5},\frac{2}{5}$ | $\frac{3}{10}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,0,0,0,0,0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2}$ | $\frac{3}{2}$
$17$ | $\mathbb{Z}/12\mathbb{Z}$ | $\frac{1}{3},\frac{1}{6},\frac{1}{4},\frac{7}{12},\frac{5}{6},\frac{5}{12},0,0,\frac{5}{12},\frac{5}{12},\frac{1}{2},\frac{3}{4},0,\frac{2}{3},\frac{1}{3}$ | $\frac{7}{12}$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,0,0,0,0,0,0,0,\frac{1}{2},\frac{1}{2},0,\frac{1}{2},0,\frac{1}{2}$ | $0$
| $\mathbb{Z}/2\mathbb{Z}$ | $0,0,0,0,0,0,0,0,\frac{1}{2},\frac{1}{2},0,0,0,0,0$ | $1$
$18$ | $\mathbb{Z}/44\mathbb{Z}$ | $\frac{1}{22},\frac{1}{44},\frac{43}{44},\frac{3}{11},\frac{6}{11},\frac{9}{11},\frac{5}{44},\frac{9}{22},\frac{31}{44},\frac{7}{22},\frac{29}{44},\frac{17}{44},\frac{17}{22},\frac{2}{11},\frac{13}{22}$ | $\frac{31}{44}$

By direct calculations, we can check that

$q_{L_{k}}(\alpha)+q_{E_{k}}(\beta)\equiv 0\quad(\text{mod }2)$

for each direct summand of $\mathcal{G}_{k}$. ∎

######  Theorem 4.1.

For each $k\in\\{6,\ldots,18\\},$ letting $S_{k}$ be a generic member of the
family $\mathscr{F}_{k}$ of (1.10), the Néron-Severi lattice ${\rm NS}(S_{k})$
is isometric to the evident lattice $E_{k}$ of Proposition 2.3.

###### Proof.

According to Proposition 1.1, the lattice $L_{k}$ for each
$k\in\\{6,\ldots,18\\}$ is isometric to the Néron-Severi lattice of the $K3$
surface $S_{P_{k}^{\circ}}$. Hence, there exists a primitive embedding
$L_{k}\hookrightarrow L_{K3}$. So, its orthogonal complement $L_{k}^{\perp}$
is a primitive sublattice of $L_{K3}$. Here, recalling Remark 1.1,
$L_{k}^{\perp}$ has a form $U\oplus\widecheck{L}_{k}$. Therefore,
$\widecheck{L}_{k}$ is a primitive sublattice of $L_{K3}$. The invariant of
$\widecheck{L}_{k}$ can be calculated by applying the fact
$q_{\widecheck{L}_{k}}\simeq q_{U\oplus\widecheck{L}_{k}}\simeq
q_{L_{k}^{\perp}}\simeq-q_{L_{k}}$ and Table 11. Then, together with Lemma
4.1, we can see that the invariant of $\widecheck{L}_{k}$ coincides with that
of $E_{k}$ for each $k$. By virtue of Proposition 4.3, there exists a
primitive embedding $E_{k}\hookrightarrow L_{K3}$. Namely, $L_{K3}/E_{k}$ is a
free abelian group.

By the way, $E_{k}$ is a sublattice of ${\rm NS}(S_{k})$. From Theorem 3.1 and
the fact that ${\rm NS}(S_{k})$ is a primitive sublattice of $L_{K3}$,
$L_{K3}/{\rm NS}(S_{k})$ is a free abelian group whose rank is equal to that
of $L_{K3}/E_{k}.$ Therefore, we obtain $E_{k}={\rm NS}(S_{k}).$ ∎

This theorem shows that the primitive closure $\widehat{E}_{k}$ of $E_{k}$,
which is introduced in Section 3, is equal to $E_{k}$ for each
$k\in\\{6,\ldots,18\\}$. From the proof of the theorem, we obtain the
following Corollary.

######  Corollary 4.1.

For each $k\in\\{6,\ldots,18\\},$ let $S_{k}$ be a generic member of the
family $\mathscr{F}_{k}$ of (1.10) and $L_{k}$ be the lattice in Table 1.
Then, the transcendental lattice ${\rm Tr}(S_{k})$ is isometric to $U\oplus
L_{k}$.

######  Remark 4.1.

The lattice structure of $S_{6}$ is determined in [DG] Proposition 5.4. In
[DG], they study $K3$ surfaces, which are coming from the Hessian of a general
cubic surfaces which does not admit a Sylvester form. Such Hessian $K3$
surface is birationally equivalent to our $S_{6}$ (see [Koi2] also).

##  5 Mordell-Weil groups for our elliptic fibrations

Let $\pi:S\rightarrow C$ be a Jacobian elliptic fibration. We assume that
$\pi$ has singular fibres. We let ${\rm MW}(\pi,O)$ denote the Mordell-Weil
group of sections of $\pi$. For all $P\in{\rm MW}(\pi,O)$ and $v\in C$, we
have $(P\cdot\pi^{-1}(v))=1$. Note that the section $P$ intersects an
irreducible component with multiplicity $1$ of every fibre $\pi^{-1}(v)$. Set
$R=\\{v\in\mathbb{C}|\pi^{-1}(v){\rm\enspace is\enspace a\enspace
singular\enspace fibre\enspace of\enspace}\pi\\}.$ For all $v\in R$, we have
an expression
$\pi^{-1}(v)=\Theta_{v,0}+\sum_{j=1}^{m_{v}-1}\mu_{v,j}\Theta_{v,j},$ where
$m_{v}$ is the number of irreducible components of $\pi^{-1}(v)$,
$\Theta_{v,j}\hskip 2.84526pt(j=0,\cdots,m_{v}-1)$ are irreducible components
with multiplicity $\mu_{v,j}$ of $\pi^{-1}(v)$ and $\Theta_{v,0}$ is the
component such that $\Theta_{v,0}\cap O\not=\phi$. Let $F$ be a general fibre
of $\pi$. The lattice $T=\langle F,O,\Theta_{v,j}|v\in R,1\leq j\leq
m_{v}-1\rangle_{\mathbb{Z}}\subset{\rm NS}(S)$ is called the trivial lattice
for $\pi$. We set $\hat{T}=(T\otimes_{\mathbb{Z}}\mathbb{Q})\cap{\rm NS}(S)$
for the trivial lattice $T$.

######  Proposition 5.1.

([Sh]) (1) There is an isomorphism ${\rm MW}(\pi,O)\simeq{\rm NS}(S)/T$ of
groups.

(2) The rank of ${\rm MW}(\pi,O)$ is equal to ${\rm rank}({\rm
NS}(S))-2-\sum_{v\in R}(m_{v}-1).$

(3) Let ${\rm MW}(\pi,O)_{tor}$ be the torsion part of ${\rm MW}(\pi,O)$.
Then, ${\rm MW}(\pi,O)_{tor}\simeq\hat{T}/T.$

######  Theorem 5.1.

Let $\pi_{k}$ $(k\in\\{6,\ldots,18\\})$ be the elliptic fibration given in
Proposition 2.1. Then, the rank of the Mordell-Weil group ${\rm
MW}(\pi_{k},O)$ is equal to $1$. Also, for each $k$, the free part of ${\rm
MW}(\pi_{k},O)$ is generated by the non-trivial section $Q$ in Table 7.
Moreover, if $k\in\\{6,9,13,17\\}$, the section $O^{\prime}$ in Table 7
generates the torsion part of ${\rm MW}(\pi_{k},O)$.

###### Proof.

By applying Proposition 5.1 (2) to our Jacobian elliptic fibration $\pi_{k}$
for each $k$ whose singular fibres are illustrated in Table 8, we can see that
${\rm rank}({\rm MW}(\pi_{k},O))=1.$ Now, $Q\not\in{\rm MW}(\pi_{k},O)_{tor}.$
Also, according to Theorem 4.1, together with Proposition 2.3, we obtain a
basis of ${\rm NS}(S_{k})$ for each $k$ consisting of $Q$, $O^{\prime}\in{\rm
MW}(\pi_{k},O)_{tor}$ and appropriate members of $T$. Hence, by virtue of
Proposition 5.1 again, the section $Q$ must generates the free part of ${\rm
MW}(\pi_{k},O)$, which is isomorphic to $\mathbb{Z}$. ∎

Thus, we prove Theorem 5.1 on the basis of Theorem 4.1. We remark that the
first author gives another proof of Theorem 4.1 and Theorem 5.1 in [Mat1] and
[Mat2]. In fact, he first proves Theorem 5.1 by applying detailed techniques
of Mordell-Weil lattices (see [Sh]) to our elliptic fibrations $\pi_{k}$.
Then, he shows that Theorem 4.1 follows from Theorem 5.1. The argument in the
present paper is simpler than the proof of them.

## Acknowledgment

This second author is supported by JSPS Grant-in-Aid for Scientific Research
(22K03226, 18K13383) and MEXT LEADER.

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Tomonao Matsumura
Core Concept Technologies Inc.,
11F DaiyaGate Ikebukuro,
1-16-15 Minamiikebukuro, Toshima-ku, Tokyo
171-0022, Japan
(E-mail<EMAIL_ADDRESS>
Atsuhira Nagano
Faculty of Mathematics and Physics,
Institute of Science and Engineering,
Kanazawa University,
Kakuma, Kanazawa, Ishikawa
920-1192, Japan
(E-mail<EMAIL_ADDRESS>