Families of K3 surfaces and curves of (2,3)-torus type

TL;DR

This paper investigates families of K3 surfaces arising from double covers of the projective plane branched along (2,3)-torus type curves, analyzing their Picard lattices and deformations of singularities.

Contribution

It introduces a detailed study of Picard lattices and dualities, and describes deformations of singularities in these specific K3 surface families.

Findings

Identification of Picard lattice structures

Description of lattice duality phenomena

Analysis of singularity deformations in Gorenstein K3 surfaces

Abstract

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the second part, we describe a deformation of singularities of Gorenstein $K3$ surfaces in these families.

Equations

$$\begin{array}[]{llll}A_{17},\,2A_{8},\,3A_{5},&A_{2}+A_{14},&A_{5}+A_{11},&A_{11}+E_{6},\\ 3E_{6},\,6A_{2},&2A_{2}+A_{11},&A_{2}+A_{5}+A_{8},&3A_{2}+A_{8},\\ 2A_{2}+2A_{5},&4A_{2}+A_{5},&2A_{5}+E_{6},&A_{5}+2E_{6},\\ 2A_{2}+A_{5}+E_{6},&2A_{2}+2E_{6},&A_{2}+A_{8}+E_{6},&4A_{2}+E_{6}.\end{array}$$

$$W^2-F(X,Y,Z)=0,$$

$$U\oplus {\mathrm{Pic}}_{{\Delta }_3^{\ast }}\simeq ({\mathrm{Pic}}_{{\Delta }_3}{)}_{{\Lambda }_{K3}}^{\perp },$$

$$\Delta =\mathrm{Conv}\{v_1,\dots ,v_r\}.$$

$${\Delta }^{\ast }:=\left\{y\in N{\otimes }_{\mathbb{Z}}\mathbb{R}|⟨y,x⟩\ge -1\text{ for all }x\in \Delta \right\}.$$

$$L_D(\Delta ):=\mathrm{Im}r\cap H^2(\tilde{Z},\mathbb{Z}),L_0(\Delta ):=(L_D(\Delta ){)}_{H^2(\tilde{Z},\mathbb{Z})}^{\perp }.$$

$$L_0(\Delta )=\mathrm{coker}r\cap H^2(\tilde{Z},\mathbb{Z}).$$

$$\mathrm{rank}{L}_0(\Delta )=\sum _{\Gamma :\text{edge of }\Delta }{l}^{\ast }(\Gamma )l^{\ast }({\Gamma }^{\ast }).$$

$${\rho }_{\Delta }=\sum _{\Gamma :\text{edges of }\Delta }l({\Gamma }^{\ast })+\mathrm{rank}{L}_0(\Delta )-3.$$

$$D.D=D^2=2l^{\ast }(F)-2.$$

$$D.D^{\prime }=l^{\ast }(\Gamma )+1.$$

$$M:=\left\{(a_0,a_1,a_2,a_3)\in {\mathbb{Z}}^4|a_0+a_1+a_2+3a_3\equiv 0(6)\right\}.$$

$$e_1=(1,0,-1,0),e_2=(0,1,-1,0),e_3=(0,0,-3,1).$$

$$\begin{array}[]{ccc}M&\rightarrow&\mathcal{M}_{6}\\ (a_{0},\,a_{1},\,a_{2},\,a_{3})&\mapsto&X^{a_{0}+1}Y^{a_{1}+1}Z^{a_{2}+1}W^{a_{3}+1},\end{array}$$

$$\begin{array}[]{ccc}W^{2}\leftrightarrow(-1,-1,1),&Y^{6}\leftrightarrow(-1,5,-1),&X^{6}\leftrightarrow(5,-1,-1),\\ Z^{6}\leftrightarrow(-1,-1,-1),&Y^{3}Z^{3}\leftrightarrow(-1,2,-1),&Y^{2}Z^{4}\leftrightarrow(-1,1,-1),\\ X^{4}Z^{2}\leftrightarrow(3,-1,-1),&X^{2}Z^{4}\leftrightarrow(1,-1,-1).\end{array}$$

$${\Delta }_1:=\mathrm{Conv}\left\{(-1,-1,1),(-1,1,-1),(3,-1,-1),(5,-1,-1),(-1,5,-1)\right\},$$

$${\Delta }_2:=\mathrm{Conv}\left\{(-1,-1,1),(-1,2,-1),(3,-1,-1),(5,-1,-1),(-1,5,-1)\right\},$$

$${\Delta }_3:=\mathrm{Conv}\left\{(-1,-1,1),(-1,1,-1),(1,-1,-1),(5,-1,-1),(-1,5,-1)\right\},$$

$${\Delta }_1^{\ast }=\mathrm{Conv}\left\{(0,0,1),(-1,-1,-3),(0,1,0),(1,2,2),(1,0,0)\right\},$$

$${\Delta }_2^{\ast }=\mathrm{Conv}\left\{(0,0,1),(-1,-1,-3),(0,1,0),(3,4,6),(1,0,0)\right\},$$

$${\Delta }_3^{\ast }=\mathrm{Conv}\left\{(0,0,1),(-1,-1,-3),(0,1,0),(1,1,1),(1,0,0)\right\},$$

$${\Gamma }_1=\mathrm{Conv}\{(-1,-1,1),(-1,1,-1)\}$$

$${\Gamma }_1^{\ast }=\mathrm{Conv}\{(1,0,0),(1,2,2)\}$$

$$\mathrm{rank}{L}_0({\Delta }_1)=l^{\ast }({\Gamma }_1)l^{\ast }({\Gamma }_1^{\ast })=1\cdot 1=1.$$

$${\Gamma }_2=\mathrm{Conv}\{(-1,-1,1),(3,-1,-1)\}$$

$${\Gamma }_2^{\ast }=\mathrm{Conv}\{(3,4,6),(0,1,0)\}$$

$$\mathrm{rank}{L}_0({\Delta }_2)=l^{\ast }({\Gamma }_2)l^{\ast }({\Gamma }_2^{\ast })=1\cdot 2=2.$$

$${\left({\mathrm{Pic}}_{\Delta }\right)}_{{\Lambda }_{K3}}^{\perp }\simeq U\oplus {\mathrm{Pic}}_{{\Delta }^{\ast }}$$

$$U\oplus {\mathrm{Pic}}_{{\Delta }_3^{\ast }}\simeq ({\mathrm{Pic}}_{{\Delta }_3}{)}_{{\Lambda }_{K3}}^{\perp },$$

$$\begin{array}[]{lll}v_{1}=(0,0,1),&v_{2}=(-1,-1,-3),&v_{3}=(0,1,0),\\ v_{4}=(1,2,2),&v_{5}=(1,0,0),&v_{6}=(1,1,1).\end{array}$$

Full text

*Running head. * Families of $K3$ surfaces and curves of $(2,3)$-torus type

1 Introduction

Plane sextic curves of $(2,3)$-torus type, which is defined by a polynomial of the form $F=F_{2}^{3}+F_{3}^{2}$ with polynomials $F_{i}$ of homogeneous degree $i$, that have at most simple singularities, which we simply call curves of $(2,3)$-torus type, are classified by Oka and Pho [OkaPho], and Pho [Pho].

A compact complex connected algebraic surface $S$ is called a Gorenstein $K3$ surface if $S$ has at most simple singularities, has trivial canonical divisor, and the irregularity of $S$ is zero. If a Gorenstein $K3$ surface is non-singular, we simply call it a $K3$* surface*. Note that for any Gorenstein $K3$ surface $S$, there exists a $K3$ surface $\tilde{S}$ such that $S$ and $\tilde{S}$ are birationally equivalent.

Concerning algebraic curves on a $K3$ surface, the study of their Weierstrass semi-group and the existstance of curves that admit a given semi-group are quite authentic and intersting but there are not many known results: see for instance Komeda and Watanabe [KomedaWatanabe]. On the other hand, motivated originally in theoretical physics, many concepts have been discovered related to mirror symmetry for $2$-dimensional Calabi-Yau manifolds such as by Batyrev, Berglund and H$\ddot{\textnormal{u}}$bsch, Dolgachev, Ebeling and Takahashi, and Ebeling and Ploog [BatyrevMirror, BerglundHubsch, DolgachevMirror, EbelingTakahashi11, EbelingPloog].

In this article, we are to focus on families of $K3$ surfaces that are obtained as double covering of the projective plane branching along curves of $(2,3)$-torus type, and their Picard lattices.

Consider the generic non-Galois triple covering $X$ of $\mathbb{P}^{2}$ branching along a curve of $(2,3)$-torus type $B$, and then take the Galois closure $\hat{X}$ of $X$. Moreover, if $Z$ is the minimal model of the double covering $D(X/\mathbb{P}^{2})$ of $\mathbb{P}^{2}$ branching along $B$, then, $\hat{X}$ is also obtained as the cyclic triple covering of $Z$.

By this construction, and only in this case, one obtains a cyclic triple covering $\hat{X}\to Z$, where $Z$ is a $K3$ surface, and $\hat{X}$ is either an abelian surface (possibly with singularities), or a Gorenstein $K3$ surface. These two cases are distinguished by an invariant $\delta$ introduced by Ishida and Tokunaga [IshidaTokunaga], in which, the latter case where $\delta=6$ is studied. The former case, in which one has $\delta=9$ is in fact studied in detail by Barth [Barth98] as an analogy of Nikulin’s result [Nikulin75]. It is easily seen by the classification [OkaPho, Pho], that $\delta=9$ occurs if and only if $\mathop{\mathrm{Sing}}\nolimits{B}=9A_{2}$; and together with the fact that the order of the fundamental group of the singularities should be divided by $3$, that $\delta=6$ occurs when $\mathop{\mathrm{Sing}}\nolimits{B}$ is one of the followngs:

[TABLE]

Remark** 1**

The above list of $\mathop{\mathrm{Sing}}\nolimits{B}$ picked up in [Pho] covers all possible cases: indeed, with the aid of Proposition 1.1 [IshidaTokunaga].

The dual curve of a plane smooth cubic curve is a typical example (see [BrieKnor]) of a curve of $(2,3)$-torus type $B$ with nine singularities of type $A_{2}$ (cusps).

Since $\delta=9$ case is well-understood, we focus on the other case in this article.

The defining equation of the Gorenstein $K3$ surface $D(X/\mathbb{P}^{2})$ is given by

[TABLE]

where $F$ is the defining polynomial of the branch curve $B$. Being parameterised by the coefficients of the monomials in $W^{2}-F(X,Y,Z)$, one can construct families of (Gorenstein) $K3$ surfaces. Such a family should be a subfamily of $K3$ surfaces parametrised by the complete anticanonical linear system of the weighted projective space $\mathbb{P}(1,1,1,3)$ with weights $(1,1,1,3)$, which is one of $95$ weights corresponding to simple $K3$ singularities classified by Yonemura [Yonemura].

The aim of this article is to study these families: we first construct polytopes $\Delta_{1},\,\Delta_{2}$, and $\Delta_{3}$ such that the complete anticanonical linear systems of toric Fano $3$-folds associated to them parametrise families $\mathcal{F}_{1},\,\mathcal{F}_{2},\,\mathcal{F}_{3}$ of $K3$ surfaces obtained by a double covering of $\mathbb{P}^{2}$ branching along curves of $(2,3)$-torus type. Then, we shall give the Picard lattice of the families using toric geometry. More precisely, our main theorem is stated as follows:

Theorem 3.1 (1)*  The Picard lattices of the families $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are respectively isometric to $U\oplus\langle-2\rangle\oplus\langle-4\rangle$, and $U\oplus A_{5}$.

(2)  The Picard lattice of the family $\mathcal{F}_{3}$ is isometric to $\langle-2\rangle\oplus\langle 2\rangle$.

(3)  The Picard lattice $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}}$ of the family $\mathcal{F}_{3}$ satisfies the duality*

[TABLE]

*where $\Delta_{3}^{*}$ is the polar dual of $\Delta_{3}$, $U$ is the hyperbolic lattice of rank $2$, and $(L)^{\perp}_{\Lambda_{K3}}$ is the orthogonal complement of a primitive sublattice $L$ in the $K3$ lattice $\Lambda_{K3}$. *

We review fundamental facts of toric geometry necessary in this article in $\S 2$. The main theorem is proced in $\S 3$ after verifying invariants. In $\S 4$, we describe families that contain $K3$ surfaces obtained as double covering of $\mathbb{P}^{2}$ branching along curves of $(2,3)$-torus type.

For a curve $B$ of $(2,3)$-torus type, when $\mathop{\mathrm{Sing}}\nolimits(B)=A$ is the set of singularities of $B$, we call $[A]$ for the curve $B$. If $\mathop{\mathrm{Sing}}\nolimits(B)$ contains more than one singularities, we denote by $A=A^{\prime}+A^{\prime\prime},\,A=2A^{\prime}=A^{\prime}+A^{\prime}$ etc.

The singularities of type $A_{n}$ is the singularity given locally by $x^{2}+y^{n+1}=0$ for $n\geq 1$, and of type $E_{6}$ is given locally by $x^{3}+y^{4}=0$. In un-necessarily confusing way, we also mean by $A_{n}$ the root lattice of type $A_{n}$. The hyperbolic lattice of rank $2$ is denoted by $U$, and negative definite root lattice of rank $8$ is denoted by $E_{8}$. The $K3$* lattice* is defined by $\Lambda_{K3}:=U^{\oplus 3}\oplus E_{8}^{\oplus 2}$.

*Acknowledgement.  * The author thanks to Professor J.Komeda who gave her an opportunity to study Weierstrass semi-groups, and hopes that the result may enrich the study of that of algebraic curves on $K3$ surfaces. Thanks to Professor C.Hertling for reading through the first draft. * *

2 Setups

A lattice is a finitely-generated $\mathbb{Z}$-module with a non-degenerate bilinear form. Let $M$ be a lattice of rank $3$, and $N:={\rm Hom}_{\mathbb{Z}}(M,\mathbb{Z})$ be its dual.

See Section 3.3 of [FultonToric] for basic facts on toric geometry.

A polytope is a convex hull of finitely-many points, say, $v_{1},\ldots,v_{r}$, in $M\otimes_{\mathbb{Z}}\mathbb{R}$, and is denoted by

[TABLE]

For a polytope $\Delta$, a vertex is a [math]-dimensional face, an edge is a $1$-dimensional face, and a face is a $2$-dimensional face. A polytope is called integral if every vertex is in $M$.

Denote by $\langle\cdot,\cdot\rangle:N\times M\to\mathbb{Z}$ a natural paring, which is the inner product in $\mathbb{R}^{3}$. Let $\Delta$ be a polytope in $M_{\mathbb{R}}$. Define the polar dual polytope of $\Delta$ by

[TABLE]

An integral polytope $\Delta$ is reflexive if $\Delta$ contains the origin in the interior as its only lattice point, and the polar dual $\Delta^{*}$ is also integral.

It is known [BatyrevMirror] that a polytope $\Delta$ is reflexive if and only if the corresponding toric variety $\mathbb{P}_{\!\Delta}$ is Fano, in particular, the complete anticanonical linear system of $\mathbb{P}_{\!\Delta}$ parametrises a family $\mathcal{F}_{\!\Delta}$ of which general sections are Gorenstein $K3$ surfaces. Since every Gorenstein $K3$ surface is birationally equivalent to a unique $K3$ surface, we identify these two surfaces.

Suppose $\Delta$ is a reflexive polytope, and take a general anticanonical section $Z$ of $\mathbb{P}_{\!\Delta}$.

It is also known [BatyrevMirror] that there exists a MPCP desingularization, which is achieved by a simultaneous toric desingularization, and in consequence, we obtain the resulting desingularized varieties $\widetilde{\mathbb{P}_{\!\Delta}}$ and $\widetilde{Z}$ of the ambient space $\mathbb{P}_{\!\Delta}$ and of $Z$, respectively.

Denote by $\widetilde{\mathbb{P}_{\!\Delta}}$ and $\widetilde{Z}$ the resulting desingularized varieties. Define a restriction map $r:H^{1,1}(\widetilde{\mathbb{P}_{\!\Delta}})\to H^{1,1}(\widetilde{Z})$ of Hodge $(1,1)$-parts. Note that the map $r$ is not necessarily surjective. Define

[TABLE]

Then,

[TABLE]

We call the rank of $L_{0}(\Delta)$ the toric correction term.

Denote by $v_{i},\,i=1,\ldots,r$ the vectors starting from the origin and ending at vertices $e_{i}$ of $\Delta$. The vectors $v_{i}$ being as one-simplices, one can construct a fan $\Sigma=\Sigma(\Delta)$ associated to the polytope $\Delta$. It is easily observed that $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta)=0$ if and only if $\Delta$ is simplicial, namely, any three of one-simplices of $\Sigma$ generate $M$, and it is also equivalent that the toric variety $\mathbb{P}_{\!\Delta}$ has at most orbifold singularities.

Define the *Picard lattice of the family * $\mathcal{F}_{\!\Delta}$ to be the Picard lattice of $K3$ surfaces that are minimal models of any generic sections in $\mathcal{F}_{\!\Delta}$, which is known to be well-defined [Bruzzo-Grassi]. In other words, the Picard lattice of the family is a lattice generated by the irreducible components of the restrictions to ${-}K_{\!\mathbb{P}_{\!\Delta}}$ of generators, which are torus-action invarant, of the Picard group of the toric $3$-fold $\mathbb{P}_{\!\Delta}$. We denote it by $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta}$ and $\rho_{\!\Delta}$ be its rank.

The toric correction term $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta)$, $\rho_{\!\Delta}$, and intersection numbers $D.D^{\prime}$ for restricted torus-invariant divisors $D,D^{\prime}$ in $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta}$ can be combinatorically computed [Kobayashi].

For a face $\Gamma$ of any dimensional of $\Delta$, denote by $l(\Gamma)$ the number of lattice points in $\Gamma$, and $l^{*}(\Gamma)$ the number of inner lattice points in $\Gamma$.

For an edge $\Gamma$ of $\Delta$, denote by $\Gamma^{*}$ the dual edge of $\Gamma$ in $\Delta^{*}$. The toric correction term is computed by

[TABLE]

The rank $\rho_{\!\Delta}$ is computed by

[TABLE]

A primitive vector $v$ that generates a ray of the fan defining the $3$-fold $\mathbb{P}_{\!\Delta}$ determines a torus-invariant divisor $\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v)}$ on $\mathbb{P}_{\!\Delta}$. By construction, it is equivalent to take a lattice point $v^{*}$ in $\Delta^{*}$. The dual of $v^{*}$ is a face $F$ in $\Delta$. It is well-known that the number of lattice points in the interior of $F$ is equal to the genus of a smooth curve corresponding to the divisor. For $D:=\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v)}|_{{-}K_{\!\mathbb{P}_{\!\Delta}}}$, one has

[TABLE]

Let $D$ and $D^{\prime}$ be the restriction to ${-}K_{\!\mathbb{P}_{\!\Delta}}$ of torus-invariant divisors on $\mathbb{P}_{\!\Delta}$ corresponding to vertices $v$ and $v^{\prime}$, and let $\Gamma$ be the edge of $\Delta$ which connects the vertices $v$ and $v^{\prime}$. The intersection number is thus obtained by

[TABLE]

3 Main Results

Define a lattice $M$ by

[TABLE]

The lattice $M$ is of rank $3$, and one can take a basis $\{e_{1},e_{2},e_{3}\}$ for $M$ over $\mathbb{Z}$, where

[TABLE]

We can associate $M$ with the set of monomials $\mathcal{M}_{6}$ of weighted degree $6$ with weights $(1,1,1,3)$ by

[TABLE]

where the weights of $X,Y,Z,W$ are respectively $1,1,1,3$, and thus there exists a correspondence between lattice points in $M$ and such monomials.

One can embed a polytope of which lattice points are labelled by monomials in $\mathcal{M}_{6}$ into $\mathbb{R}^{3}$ : express elements in $M$ as a linear combination of the chosen basis $\{e_{1},\,e_{2},\,e_{3}\}$ of which the coefficients form a point in $\mathbb{R}^{3}$. Thus, one gets a correspondence between monomials and points in $\mathbb{R}^{3}$ under this choice of a basis, for which some examples are presented in Table 1.

[TABLE]

Define integral polytopes $\Delta_{1},\,\Delta_{2}$, and $\Delta_{3}$ by

[TABLE]

[TABLE]

and

[TABLE]

respectively.

See Figure 1.

Proposition** 3.1**

The polytopes $\Delta_{1},\,\Delta_{2}$, and $\Delta_{3}$ are reflexive.

Proof. It is clear that the origin is the only lattice point contained in the polytopes. By a direct computation, the polar dual polytopes $\Delta_{1}^{*},\,\Delta_{2}^{*}$, and $\Delta_{3}^{*}$ of $\Delta_{1},\,\Delta_{2}$, and $\Delta_{3}$ are as follows:

[TABLE]

[TABLE]

and

[TABLE]

resepctively. Thus $\Delta_{1}^{*},\,\Delta_{2}^{*}$, $\Delta_{3}^{*}$ are integral as well. $\Box$

Proposition** 3.2**

We have $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{1})=1$,  $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{2})=2$, and $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{3})=0$, and $\rho_{\!\Delta_{1}}=4$,  $\rho_{\!\Delta_{2}}=7$,  $\rho_{\!\Delta_{3}}=2$, $\rho_{\!\Delta_{1}^{*}}=17$,  $\rho_{\!\Delta_{2}^{*}}=15$, and $\rho_{\!\Delta_{3}^{*}}=18$. In particular, $\rho_{\!\Delta_{i}}+\rho_{\!\Delta_{i}^{*}}>20$ for $i=1,2$, and $\rho_{\!\Delta_{3}}+\rho_{\!\Delta_{3}^{*}}=20$ hold.

Proof. There exists a lattice point on the edge

[TABLE]

of $\Delta_{1}$, and a lattice point on its dual edge

[TABLE]

of $\Delta_{1}^{*}$. There is no more edge on $\Delta_{1}$ that contributes $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{1})$. Thus, by the formula (1),

[TABLE]

By the formula (2), one has $\rho_{\!\Delta_{1}}=(5+1)+1-3=4$, and $\rho_{\!\Delta_{1}^{*}}=19+1-3=17$. Clearly, $\rho_{\!\Delta_{1}}+\rho_{\!\Delta_{1}^{*}}=4+17=21>20$.

There exists a lattice point on the edge

[TABLE]

of $\Delta_{2}$, and two lattice points on its dual edge

[TABLE]

of $\Delta_{2}^{*}$. There is no more edge on $\Delta_{2}$ that contributes $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{2})$. Thus, by the formula (1),

[TABLE]

By the formula (2), one has $\rho_{\!\Delta_{2}}=5+3+2-3=7$, and $\rho_{\!\Delta_{2}^{*}}=16+2-3=15$. Clearly, $\rho_{\!\Delta_{2}}+\rho_{\!\Delta_{2}^{*}}=7+15=22>20$.

There does not exist an edge on $\Delta_{3}$ that contributes $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{3})$. Thus, by the formula (1), $\mathop{\mathrm{rank}}\nolimits L_{0}(\Delta_{3})=0$. By the formula (2), one has $\rho_{\!\Delta_{3}}=5-3=2$, and $\rho_{\!\Delta_{3}^{*}}=21-3=18$. Clearly, $\rho_{\!\Delta_{3}}+\rho_{\!\Delta_{3}^{*}}=2+18=20$. $\Box$

Denote by $\mathcal{F}_{1},\,\mathcal{F}_{2},\,\mathcal{F}_{3}$, and $\mathcal{F}_{3}^{\prime}$ the families of $K3$ surfaces parametrised by the complete anticanonical linear systems of toric Fano $3$-folds $\mathbb{P}_{\!\Delta_{1}}$,  $\mathbb{P}_{\!\Delta_{2}}$,  $\mathbb{P}_{\!\Delta_{3}}$, and $\mathbb{P}_{\!\Delta_{3}^{*}}$, respectively. Here, $\Delta_{3}^{*}$ is the polar dual polytope of $\Delta_{3}$. Denote by $\mathop{\mathrm{Pic}}\nolimits_{\Delta}$ the Picard lattice of the family of $K3$ surfaces that is associated to a reflexive polytope $\Delta$.

Remark** 2**

We have seen that the sum of Picard numbers $\rho_{\!\Delta_{3}}$ and $\rho_{\!\Delta_{3}^{*}}$ coincides with the rank of the unimodular lattice $U^{\oplus 2}\oplus E_{8}^{\oplus 2}$, and that the rank of $L_{0}(\Delta_{3})$ is [math] means that the toric $3$-fold $\mathbb{P}_{\!\Delta_{3}}$ is simplicial. In [Mase17], it is concluded that if a toric Fano $3$-fold $\mathbb{P}_{\!\Delta}$ is simplicial, then, the family $\mathcal{F}_{\Delta}$ of $K3$ surfaces is lattice dual in the sense that

[TABLE]

holds. Thus, in our situation here, we expect that the family $\mathcal{F}_{3}$ constructed by the toric $3$-fold $\mathbb{P}_{\!\Delta}$ might be lattice dual. Therefore, we have to study the dual $\Delta_{3}^{*}$, or equivalently, the family $\mathcal{F}_{3}^{\prime}$ constructed by the toric $3$-fold $\mathbb{P}_{\!\Delta^{*}}$.

Remark** 3**

Let $\Delta$ be a reflexive polytope. We occasionally identify the complete anticanonical linear system of the toric Fano $3$-fold $\mathbb{P}_{\!\Delta}$, and the family $\mathcal{F}_{\Delta}$. Indeed, a section $f\in|{-}K_{\mathbb{P}_{\!\Delta}}|$ determines a surface $(f=0)$ in $\mathcal{F}_{\Delta}$. Thus, we may also call $(f=0)$ a section as long as there is no confusion.

Theorem** 3.1**

(1)*  The Picard lattices of the families $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are respectively isometric to $U\oplus\langle-2\rangle\oplus\langle-4\rangle$, and $U\oplus A_{5}$.

(2)  The Picard lattice of the family $\mathcal{F}_{3}$ is isometric to $\langle-2\rangle\oplus\langle 2\rangle$.

(3)  The Picard lattice $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}}$ of the family $\mathcal{F}_{3}$ satisfies the duality*

[TABLE]

where $\Delta_{3}^{*}$ is the polar dual of $\Delta_{3}$, $U$ is the hyperbolic lattice of rank $2$, and $(L)^{\perp}_{\Lambda_{K3}}$ is the orthogonal complement of a primitive sublattice $L$ in the $K3$ lattice $\Lambda_{K3}$.

Proof. (1)  We lebel the primitive vectors that generate rays of the fan defining the $3$-fold $\mathbb{P}_{\!\Delta_{1}}$, or equivalently, the lattice points in $\Delta_{1}^{*}$ as follows:

[TABLE]

Let $D_{i}:=\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{i})}|_{{-}K_{\mathbb{P}_{\!\Delta_{1}}}}$ for $i=1,\ldots,5$ be restricted torus-invatiant divisors, and $D_{6},\,D_{7}$ be components of the divisor $\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{6})}|_{{-}K_{\mathbb{P}_{\!\Delta_{1}}}}$. One computes the self-intersection numbers by the formula (3),

[TABLE]

One also has the graph of intersections among these divisors by the formula (4) as in Figure 2.

By solving the linear system

[TABLE]

for $j=1,2,3$, where $\boldsymbol{e}_{j}$ is the $j$-th column of the identity matrix of size $3$, and $(\boldsymbol{x},\boldsymbol{y})$ is the inner product on $\mathbb{R}^{3}$, one obtains a set of linearly-independent divisors $\mathcal{B}=\{D_{4},\,D_{5},\,D_{6},\,D_{7}\}$, the intersection matrix with respect to which is

[TABLE]

By a translation $PA_{\mathcal{B}}{}^{t}P$ with

[TABLE]

one gets a new basis

[TABLE]

with respect to which the intersection matrix is

[TABLE]

Thus, $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{1}}=U\oplus\langle-2\rangle\oplus\langle-4\rangle$, which is clearly primitively embedded into the $K3$ lattice.

We lebel the primitive vectors that generate rays of the fan defining the $3$-fold $\mathbb{P}_{\!\Delta_{2}}$, or equivalently, the lattice points in $\Delta_{2}^{*}$ as follows:

[TABLE]

Let $D_{i}:=\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{i})}|_{{-}K_{\mathbb{P}_{\!\Delta_{2}}}}$ for $i=1,\ldots,6$ be restricted torus-invatiant divisors, and $D_{7},\,D_{7}^{\prime}$ and $D_{8},\,D_{8}^{\prime}$ be components of the divisor $\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{7})}|_{{-}K_{\mathbb{P}_{\!\Delta_{2}}}}$, and $\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{8})}|_{{-}K_{\mathbb{P}_{\!\Delta_{2}}}}$, respectively. One computes the self-intersection numbers by the formula (3),

[TABLE]

One also has the graph of intersections among these divisors by the formula (4) as in Figure 3.

By solving the linear system

[TABLE]

for $j=1,2,3$, where $\boldsymbol{e}_{j}$ is the $j$-th column of the identity matrix of size $3$, and $(\boldsymbol{x},\boldsymbol{y})$ is the inner product on $\mathbb{R}^{3}$, one obtains a set of linearly-independent divisors $\mathcal{B}=\{D_{4},\,D_{5},\,D_{6},\,D_{7},\,D_{7}^{\prime},\,D_{8},\,D_{8}^{\prime}\}$, the intersection matrix with respect to which is

[TABLE]

By a translation $PA_{\mathcal{B}}{}^{t}P$ with

[TABLE]

one gets a new basis

[TABLE]

with respect to which the intersection matrix is

[TABLE]

Thus, $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{2}}=U\oplus A_{5}$, which is clearly primitively embedded into the $K3$ lattice.

(2)  We lebel the primitive vectors that generate rays of the fan defining the $3$-fold $\mathbb{P}_{\!\Delta_{3}}$, or equivalently, the lattice points in $\Delta_{3}^{*}$ as follows:

[TABLE]

Let $D_{i}:=\overline{{\rm Orb}(\mathbb{R}_{\geq 0}v_{i})}|_{{-}K_{\mathbb{P}_{\!\Delta_{3}}}}$ for $i=1,\ldots,5$ be restricted torus-invatiant divisors. One computes the self-intersection numbers by the formula (3),

[TABLE]

One also has the graph of intersections among these divisors by the formula (4) as in Figure 4.

By solving the linear system

[TABLE]

for $j=1,2,3$, where $\boldsymbol{e}_{j}$ is the $j$-th column of the identity matrix of size $3$, and $(\boldsymbol{x},\boldsymbol{y})$ is the inner product on $\mathbb{R}^{3}$, one obtains a set of linearly-independent divisors $\mathcal{B}=\{D_{4},\,D_{5}\}$, the intersection matrix with respect to which is

[TABLE]

By a translation $PA_{\mathcal{B}}{}^{t}P$ with

[TABLE]

one gets a new basis

[TABLE]

with respect to which the intersection matrix is

[TABLE]

Thus, $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}}=\langle-2\rangle\oplus\langle 2\rangle$, which is clearly primitively embedded into the $K3$ lattice.

(3)  We lebel the primitive vectors that generate rays of the fan defining the $3$-fold $\mathbb{P}_{\!\Delta_{3}^{*}}$, or equivalently, the lattice points in $\Delta_{3}$ as follows:

[TABLE]

Let $M_{i}:=\overline{{\rm Orb}(\mathbb{R}_{\geq 0}m_{i})}|_{{-}K_{\mathbb{P}_{\!\Delta_{3}^{*}}}}$ for $i=1,\ldots,21$ be restricted torus-invatiant divisors. One computes the self-intersection numbers by the formula (3),

[TABLE]

One also has the graph of intersection numbers among these divisors by the formula (4) as in Figure 5.

By solving the linear system

[TABLE]

for $j=1,2,3$, where $\boldsymbol{e}_{j}$ is the $j$-th column of the identity matrix of size $3$, and $(\boldsymbol{x},\boldsymbol{y})$ is the inner product on $\mathbb{R}^{3}$, one obtains a set of linearly-independent divisors $\mathcal{B}=\{M_{1},\cdots,M_{18}\}$, the intersection matrix with respect to which is

[TABLE]

By a direct computation, we have

[TABLE]

It also immediately follws by considering a map $F:\mathbb{Z}^{19}\to\mathbb{Q}^{15}$ by

[TABLE]

that there exist two distinct elements in the discriminant group of order $2$ that generate the group since all the coefficients of $F[\mathbf{x},2]$ are integral. Thus, the lattice $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}^{*}}$ has the discriminant group $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ that is isometric to the discriminant group of $\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}}=\langle-2\rangle\oplus\langle 2\rangle$. Therefore, by the Lemma below, $U\oplus\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}^{*}}$ is isometric to $\left(\mathop{\mathrm{Pic}}\nolimits_{\!\Delta_{3}}\right)_{\Lambda_{K3}}^{\perp}$.

Lemma** 3.1**** **(Corollary 1.6.2 [Nikulin80])

Let lattices $S$ and $T$ be primitively embedded into the $K3$ lattice. Then $S$ and $T$ are orthogonal to each other in the $K3$ lattice if and only if $q_{S}\simeq-q_{T}$, where $q_{S}$ (resp. $q_{T}$) is the discriminant form of $S$ (resp. $T$).

The assertion is proved. $\Box$

Remark** 4**

Since $\rho_{\!\Delta_{i}}+\rho_{\!\Delta_{i}^{*}}\not=20$, for $i=1,2$ by Proposition 3.2, the families $\mathcal{F}_{1}=\mathcal{F}_{\!\Delta_{1}}$ and $\mathcal{F}_{\!\Delta_{1}^{*}}$, and $\mathcal{F}_{2}=\mathcal{F}_{\!\Delta_{2}}$ and $\mathcal{F}_{\!\Delta_{2}^{*}}$ are not lattice dual in the sense of Dolgachev [DolgachevMirror].

On the other hand, by the statement of part (2) of Theorem 3.1, the families $\mathcal{F}_{3}$ and $\mathcal{F}_{3}^{\prime}$ are lattice dual.