Lattice duality for coupling pairs admitting polytope duality with trivial toric contribution

We study a lattice duality among families of $K3$ surfaces associated to coupling pairs that admit polytope duality with trivial toric contribution.


Introduction
Weight systems appear in many interesting spots in algebraic geometry including singularity theory, where singularities have nice properties. We focus on a duality among weight systems called coupling introduced by Ebeling [5], which is for well-posed weight systems associated to simple K3 singularities classified by Yonemura [16]. The coupling duality is in particular admitted by a pair of singularities defined by weightedhomogeneous polynomials f and f ′ as a strange-duality for invertible polynomials introduced by Ebeling and Takahashi in [6]. It is also known that such polynomials f and f ′ in three variables can be projectivised as weighted-homogeneous polynomials F and F ′ as anticanonical divisor of the weighted projective spaces Pa and P b , where the pair (a, b) is coupling among Yonemura's list. Since all the weighted projective spaces with weights being in Yonemura's list are Fano, we obtain subfamilies of K3 surfaces in the space once one finds a reflexive polytope as a subpolytope of the defining polytope of the space. In the author's recent work [12], an existence and duality of such reflexive polytopes are studied and it is concluded that almost all coupling pair extends to a polytope-duality. Once one obtains families of K3 surfaces which already admit several dualities, one may be interested in intrinsic properties of K3 surfaces. We are interested in lattice-duality originally studied by Dolgachev [4]. It is concluded by the author [9,10] that a part of transpose-dual pairs associated to strange duality of bimodal singularities extends to lattice dual, and that some subfamilies of K3 surfaces that are double covering of the projective plane have lattice-dual property as is studied in [11]. In this paper, focusing on polytope-dual pairs associated to coupling, one may pose the following problem.
Problem Determine whether or not the coupling pairs which admit polytope-duality extend to lattice duality of families F∆ and F ∆ ′ in the sense that the relation We give an answer as the main theorem of the article which is presented here : Theorem 3.1 If a coupling pair admits polytope-duality with trivial toric contribution, then, the families of K3 surfaces are lattice dual. Explicite Picard lattices of the families are given in Table 1.
In section 2, we recall the Picard lattice and toric geometry. In section 3, we give a proof of the main theorem. In the last and fourth section, we give a conclusion as the property of the Picard lattices of families that we have obtained.

Preliminery
A lattice is a finitely-generated Z-module with a non-degenerate bilinear form. A K3 surface is a smooth compact complex connected 2dimensional algebraic variety with trivial canonical divisor and irregularity zero. It is known that the second cohomology group with Z-coefficient of a K3 surface S admits a structure of a unimodular lattice of signature (3,19), thus by a classification of lattices, the lattice is in fact isometric to the K3 lattice ΛK3 := U ⊕3 ⊕ E ⊕2 8 , where U is the hyperbolic lattice of rank 2, and E8 is the negative-definite, even unimodular lattice of rank 8. By a standard exact sequence, one gets an inclusion map c1 : H 1 (S, O * S ) → H 2 (S, Z), which makes the Picard group H 1 (S, O * S ) to be a sublattice of H 2 (S, Z). We call the Picard group of S with a lattice structure simply the Picard lattice of S.
Let M be a lattice of rank n, and N := Hom Z (M Z) be the dual lattice of M , with a natural pairing , : N × M → Z with its R-extension denoted by , R . A convex hull of finite-number of points in M ⊗ R is called a polytope, which admits the polar dual polytope ∆ * defined by ∆ * := {y ∈ N ⊗ R | y, x R ≥ −1 for all x ∈ ∆} .
A polytope ∆ is integral if every vertex is in M . An integral polytope ∆ which contains the only lattice point in its interior is reflexive if the polar dual polytope ∆ * is also an integral polytope.
It is observed by [1] that an integral polytope ∆ is reflexive if and only if the resulting projective toric variety P∆ is Fano, in other words, general hypersurfaces that are defined by global anticanonical sections of P∆ are birational to Calabi-Yau.
We only treat with 3-dimensional reflexive polytopes. We call a anticanonical section for hypersurfaces that are defined by global anticanonical sections of P∆ for short. In 3-dimensional case, it is derived by a study of [1], that moreover, singularities in P∆ and in general anticanonical sections Z of P∆ can be simultaneously resolved by a toric resolution called a MPCP-desingularisation, which we denote by P∆ andZ. The natural restriction map is not necessarily surjective in general, and we denote by L0(∆) the rank of the cokernel of the map, which we call the toric contribution, which is known [8] to be given by the formula where the sum runs for all edges in ∆.
Here we recall from [3] that generic anticanonical sections of the Fano 3-fold P∆ admit isomemtric Picard lattices. Thus, we define the Picard lattice of the family F∆ of K3 surfaces in P∆ to be the Picard lattice of the minimal model of any generic anticanonical section of P∆, and denote it by Pic∆.
For a reflexive polytope ∆, one can associate a fan Σ ′ . By definition, lattice points of ∆ * are primitive vector of one-simplices of Σ ′ , and it is clear that the toric varieties P∆ and P Σ ′ coincide. Any divisor D of a generic hypersurface in P∆ is the closure of the torus orbit of a one-simplex v in Σ, in particular, the divisors are called toric divisors. Let F be the face in ∆ that is the polar dual of v. Denote by l(F ) the number of lattice points in the interior of F . The self-intersection number of the divisor D is given by the formula Denote by ∆ (1) the set of all edges in ∆ and l(Γ) be the number of lattice points in the interior of an edge Γ ∈ ∆ (1) . The Picard number ρ(∆) is given by Let e1, e2, e3 be a standard basis for R 3 . Suppose that the fan Σ possesses l one-simplices. The toric divisors D1, . . . , D l admit the linear relations l i=1 vi, ej Di = 0 j = 1, 2, 3.
It is easily seen that the polytope ∆ is of trivial toric contribution if and only if the corresponding fan Σ ′ is simplicial, that is, every triple of one-simplices form a Z-basis of R 3 . Moreover, the restriction of linearlyindependet toric divisors of X = P∆ = P Σ ′ to the anticanonical divisor of X form a basis of the Picard lattice Pic ∆ ′ of the family Denote by M (a 0 ,a 1 ,a 2 ,a 3 ) the lattice consisting of quadruple of integers (i, j, k, l) satisfying an equation a0i + a1j + a2k + a3l = 0 for a weight system (a0, a1, a2, a3; d). There is a one-to-one correspondence between elements in M (a 0 ,a 1 ,a 2 ,a 3 ) and (rational) monomials of degree d by where (W, X, Y, Z) is a coordinate system of the weighted projective space of weight (a0, a1, a2, a3).
We denote by L * , AL, discr L, l(AL), sgn L, qL, and rank L the dual lattice L * := Hom Z (L, Z), the discriminang group L/L * , the discriminant, the minimal number of generators of AL, the signature, the discriminant form, and the rank of a lattice L. It is a standard arithmetic property that if rank L is strictly larger than 5, then, there eists an element representing 0, and if rank L is strictly larger than 12, then, the hyperbolic lattice U is a sublattice of L. We also recall standard properties of lattices from [13] and [14]. A sublattice S of a lattice Λ is called primitive if the quotient lattice Λ/S is torsion-free.
Corollary 2.1 (Corollary 1.6.2 [13]) Let S and T be primitive sublattices of the K3 lattice ΛK3. The lattices S and T are orthogonal in ΛK3 if and only if qS ≃ −qT holds. Corollary 2.2 (Corollary 1.12.3 [13]) Let S be a sublattice with signature (t+, t−) of an even unimodular lattice Λ with signature (l+, l−). The lattice S is a primitive sublattice of Λ if and only if the following three conditions are satisfied.
Remark 1 Note that the K3 lattice ΛK3 is an even unimodular lattice of signature (l+, l−) = (3,19). Thus, l+ − l− = 3 − 19 = −16 ≡ 0 mod 8, and in order to show a lattice S to be a primitive sublattice of ΛK3, it suffices to verify the second and third conditions of Corollary 2.2.
Lemma 2.1 (Lemma 4.3 [14]) There exist primitive embeddings of A1 and A2 into E8 with orthogonal complements being E7 and E6, respectively. We follow the notation of lattices in Bourbaki [2]. Proof. The assertion follows from the proof of [12]. Lemma 3.2 If a coupling pair is in Talbe 1, the toric contribution is trivial.

Main Results
Proof. The assertion follows by case-by-case computation using formula (1) for all polytopes obtained in [12].      Table 1: Lattice duality associated to coupling pairs Remark 2 We present the following data in Table 1. The number(s) in the first column are given in [12]. The second and fifth columns are vertices of polytopes of ∆ ′ and ∆ obtained by [12], and the sets in the same line are polytope-dual. In the third and fourth columns are the Picard lattice of the family F ∆ ′ , resp. F∆, the pair of the rank and the signature of lattices, and the weight systems that are coupling. The latticesL and L ′ are explained in the proof. Other lattices follow notation of [2].
Proof. Take reflexive polytopes ∆ and ∆ ′ as in 1. We explicitely calculate the Picard lattices of the families F∆ and F ∆ ′ . Denote by Σ, respectively Σ ′ the fan associated to polytope ∆ ′ , resoectively ∆. Since the relation ∆ * ≃ ∆ ′ holds, lattice points of ∆ ′ , respectively of ∆ are none other than primitive vectors of one-simplices of Σ, respectively Σ ′ .
and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 12, and D ′ i :=D ′ i |−K X with X := P Σ ′ . It can be easily seen by formulas (3) and (2) that Let L be a lattice generated by divisors one sees that the lattice L ′ is isometric to U ⊕ E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ U ⊕ E7.
and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 12, and D ′ i :=D ′ i |−K X with X := P Σ ′ . It can be easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 12 i=1 . By solving the equation } form a basis for L ′ , with respect to which the intersection matrix of L ′ is U ⊕ E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, It is well-known that lattices U ⊕ A1 ⊕ E8 and U ⊕ E7 are primitive sublattices of the K3 lattice ΛK3. Moreover, by Lemma 2.1, the relation
and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 11, and D ′ i :=D ′ i |−K X with X := P Σ ′ . It can be easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 11 i=1 . By solving the equation one sees that the lattice L ′ is isometric to U ⊕ E6, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, It is well-known that lattices U ⊕ A2 ⊕ E8 and U ⊕ E6 are primitive sublattices of the K3 lattice ΛK3. Moreover, by Lemma 2.1, the relation
Set one-simplices of Σ ′ as follows: and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 13, and It can be easily seen by formulas (3) and (2) that one sees that the lattice L ′ is isometric to U ⊕ A1 ⊕ E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ U ⊕ A1 ⊕ E7. By similar computation, one has Pic∆ ≃ U ⊕ A1 ⊕ E7.
Set one-simplices of Σ ′ as follows: and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 13, and It can be easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors one sees that the lattice L ′ is isometric to U ⊕ A1 ⊕ E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ U ⊕ A1 ⊕ E7. By similar computation, one has Pic ∆ ′ ≃ U ⊕ A1 ⊕ E7. one sees that the lattice L is isometric to U ⊕ A1 ⊕ E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, Pic∆ ≃ U ⊕A1⊕E7.
Set one-simplices of Σ ′ as follows:  (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 13 i=1 . By solving the equation (4), one sees that } form a basis for L ′ . By taking a new basis one sees that the lattice L ′ is isometric to U ⊕A1 ⊕E7, which is hyperbolic and a primitive sublattice of the K3 lattice. Thus, It is well-known that the lattice U ⊕A1 ⊕E7 is a primitive sublattice of the K3 lattice ΛK3. Moreover, by Lemma 2.1, the relation (

Lemma 3.3
If lattices L and L ′ have the signature, the discriminant, and the rank of L and L ′ are respectively (1,9), discr L = discr L ′ = −13, and rank L = rank L ′ = 10, then, the lattices are primitive sublattices of the K3 lattice and U ⊕ L ′ is the orthogonal complement of L.
Proof. Note that the discriminant groups AL, A L ′ of L and L ′ are isomorphic to Z/13Z, and that the minimal number of the generators is l(AL) = l(A L ′ ) = 1. Since the signature of L and L ′ is (t+, t−) = (1,9) and the rank is rank L = rank L ′ = 10, we have one sees that the lattice L is isometric to U ⊕L with some latticeL. By a direct computation, one sees that sgn L = (1,9), discr L = −13, and rank L = 10, thus, discrL = 13 and rankL = 8 hold. In particular, the discriminant group AL of L is isomorphic to Z/13Z, and l(AL) = 1. Set one-simplices of Σ ′ as follows: Let L ′ be a lattice generated by divisors {D ′ i } 13 i=1 . By solving the equation (4), one sees that } form a basis for L ′ . By taking a new basis one sees that the lattice L ′ is isometric to U ⊕L ′ with some latticeL ′ . By a direct computation, one sees that sgn L ′ = (1,9), discr L ′ = −13, and rank L ′ = 10, thus, discrL ′ = 13 and rankL ′ = 8 hold. In particular, the discriminant group A L ′ of L ′ is isomorphic to Z/13Z, and l(A L ′ ) = 1. Case 2 Set one-simplices of Σ as follows: one sees that the lattice L is isometric to U ⊕L with some latticeL. By a direct computation, one sees that sgn L = (1,9), discr L = −13, and rank L = 10, thus, discrL = 13 and rankL = 8 hold. In particular, the discriminant group AL of L is isomorphic to Z/13Z, and l(AL) = 1. Set one-simplices of Σ ′ as follows:  (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 13 i=1 . By solving the equation one sees that the lattice L ′ is isometric to U ⊕L ′ with some latticeL ′ . By a direct computation, one sees that sgn L ′ = (1,9), discr L ′ = −13, and rank L ′ = 10, thus, discrL ′ = 13 and rankL ′ = 8 hold. In particular, the discriminant group A L ′ of L ′ is isomorphic to Z/13Z, and l(A L ′ ) = 1. Case 3 Set one-simplices of Σ as follows: one sees that the lattice L is isometric to U ⊕L with some latticeL. By a direct computation, one sees that sgn L = (1,9), discr L = −13, and rank L = 10, thus, discrL = 13 and rankL = 8 hold. In particular, the discriminant group AL of L is isomorphic to Z/13Z, and l(AL) = 1. Set one-simplices of Σ ′ as follows: and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 13, and D ′ i :=D ′ i |−K X with X := P Σ ′ . It can be easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 13 i=1 . By solving the equation } form a basis for L ′ . By taking a new basis one sees that the lattice L ′ is isometric to U ⊕L ′ with some latticeL ′ . By a direct computation, one sees that sgn L ′ = (1,9), discr L ′ = −13, and rank L ′ = 10, thus, discrL ′ = 13 and rankL ′ = 8 hold. In particular, the discriminant group A L ′ of L ′ is isomorphic to Z/13Z, and l(A L ′ ) = 1. one sees that the lattice L is isometric to U ⊕L with some latticeL. By a direct computation, one sees that sgn L = (1,9), discr L = −13, and rank L = 10, and thus, discrL = 13 and rankL = 8 hold. In particular, the discriminant group AL of L is isomorphic to Z/13Z, and l(AL) = 1.
It is well-known that lattices U and U ⊕ E ⊕2 8 are primitive sublattices of the K3 lattice ΛK3 and it is clear that the relation (Pic∆ Let L ′ be a lattice generated by divisors {D ′ i } 6 i=1 . By solving the equation (4), one sees that { D ′ 1 , D ′ 4 , D ′ 6 } form a basis for L ′ . By taking a new basis {D ′ 1 , D ′ 1 + D ′ 4 , D ′ 6 − D ′ 1 }, one sees that the lattice L ′ is isometric to U ⊕ A1, which is hyperbolic and a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ U ⊕ A1. Note that the discriminant group of Pic ∆ ′ is isomorphic to Z/2Z since 2 is a prime number.
Case 2 Set one-simplices of Σ as follows:  (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 6 i=1 . By solving the equation (4), one sees that { D ′ 3 , D ′ 6 , D ′ 5 } form a basis for L ′ , with respect to which the intersection matrix of L ′ is given by one sees that the lattice L ′ is isometric to U ⊕ A1, which is a primitive sublattice of the K3 lattice. Therefore, In cases 1 and 2, we obtain a lattice L ≃ U ⊕L, whereL is a lattice satisfying the assumption of Lemma 3.4. Therefore, L is a primitive sublattice of the K3 lattice, and that Pic∆ = L holds. Since discr Pic∆ = discr (U ⊕ Pic ∆ ′ ) = 2, by Corollary 2.1, the relation (Pic∆) ⊥ Λ K3 ≃ U ⊕Pic ∆ ′ holds. Moreover, by Lemma 2.1, we have Pic∆ one sees that the lattice L is isometric to U ⊕L with some latticeL. By a direct computation, one sees that sgn L = (t+, t−) = (1, 15), discr L = −3, and rank L = 16, and thus, discrL = 3 and rankL = 14 hold. In particulat, the discriminant group AL of L is isomorphic to Z/3Z, and l(AL) = 1. Therefore, one observes that  (3) and (2) that . By solving the equation (4), one sees that { D ′ 4 , D ′ 5 , D ′ 7 , D ′ 6 } form a basis for L ′ . By taking a new basis {D ′ 4 , D ′ 4 + D ′ 5 , −D ′ 4 + D ′ 7 , D ′ 6 }, one sees that the lattice L ′ is isometric to U ⊕ A2, which is a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ U ⊕ A2.
Set one-simplices of Σ ′ in terms of a basis of M (4,5,7,9) ⊗ R and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 5, and D ′ i :=D ′ i |−K X with X := P Σ ′ . It can be easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 5 i=1 . By solving the equation (4), one sees that {D ′ 1 , D ′ 5 } form a basis for L, with respect to which the intersection matrix of L is given by 2 1 1 −2 . One sees that the lattice L ′ is a hyperbolic lattice, that is, of signature (t+, t−) = (1, 1) of rank L ′ = 2 and discr L ′ = −5. In particulat, the discriminant group A L ′ of L ′ is isomorphic to Z/5Z, and l(A L ′ ) = 1. Therefore, one observes that and by Corollary 2.2, L ′ is a primitive sublattice of the K3 lattice. Therefore, Pic ∆ ′ ≃ Z 2 , 2 1 1 −2 .

No. 50
Set one-simplices of Σ ′ in terms of a basis of M (7,8,9,12)  and letD ′ i be the toric divisor determined by the lattice point mi for i = 1, . . . , 4, and D ′ i :=D ′ i |−K X with X := P Σ ′ . One can easily seen by formulas (3) and (2) that Let L ′ be a lattice generated by divisors {D ′ i } 4 i=1 . By solving the equation (4), one sees that {D ′ 1 } form a basis for L ′ . Therefore, Pic ∆ ′ ≃ (Z, 4 ). It is well-known that the lattice (Z, 4 ) is a primitive sublattice of the K3 lattice.

Conclusion
We see in the main theorem that all coupling pairs that are polytope-dual with trivial toric contribution can extend to lattice duality among families of K3 surfaces. Thus, the coupling is partly translated to be the latticeduality. Moreover, all except Nos. 46, 48 and 49, and 50 admit a pair of families of K3 surfaces with generic sections being elliptic: indeed, the Picard lattices Pic∆ and Pic ∆ ′ contain the hyperbolic lattice U of rank 2.
We can conclude that the Picard lattices of the families studied in the article are independent from the choice of reflexive polytopes. In other words, since the choice of a reflexive polytope is that of a way of blow-up of the ambient space, the Picard lattice in the subfamilies is birationally independent.