Polytope duality for families of $K3$ surfaces and coupling

We study a relation between coupling introduced by Ebeling and the polytope duality among families of $K3$ surfaces.

projectivisations F and F ′ of f and f ′ in the weighted projective spaces P(a) and P(b), respectively, such that they are polytope dual in the sense that they satisfy the following conditions: Here ∆F and ∆ F ′ are Newton polytopes of F and F ′ , respectively, and ∆a and ∆ b are polytopes that define the weighted projective spaces P(a) and P(b).
In section 2, we recall the definitions concerning the weighted projective spaces and the strange duality. In section 3, we recall the definition of coupling. In section 4, we explain the polytope duality after recalling necessary notions of toric geometry.
Acknowledgement. The author thanks to Professor Wolfgang Ebeling for his suggestion of the study and discussions.

Preliminary
A K3 surface is a compact complex 2-dimensional non-singular algebraic variety with trivial canonical bundle and irregularity zero.
By [4], the anticanonical sheaf of P(a) is isomorphic to O P(a) (−d). All weight systems that give simple K3 hypersurface singularities are classified by Yonemura [10]. Namely, if a weight a is in Yonemura's list, general anticanonical sections of P(a) are birational to K3 surfaces. Thus, one can consider families of K3 surfaces.
For a polynomial f in three variables, a polynomial F in the weighted projective space P(a) is called a projectivisation of f if there exists a linear form l in P(a) such that f = F | l=0 holds. In this case, the form l is called a section of f for F .
A weighted magic square C for the weight systems w and w ′ is a square matrix of size 3 that satisfies relations The pair of weight systems (w, The pair of weight systems (w, w ′ ) is strongly coupled if it is coupled and the weighted magic square C has entries zero in every column and row.
Define an anticanonical section F of weight system (a0, a1, a2, a3; d) so that l is the section of f for F , where l is a linear form defined by Note that the choice of variables is different from the original Ebeling's paper [5]. And then define a polynomial F ′ so that t A F ′ = AF holds. Note that F ′ is a projectivisation of f ′ , and an anticanonical section in the weighted projective space of weight system (b0, b1, b2, b3; h).

Duality of polytopes
Let M be a lattice of rank 3, and N be its dual lattice Hom Z (M, Z) that is again of rank 3. A polytope is a convex hull of finite number of points in M ⊗ R. If vertices of a polytope ∆ are v1, . . . , vr, we denot it by ∆ = Conv{v1, . . . , vr}.
We call a polytope integral if all the vertices of the polytope are in M . For a polytope ∆, define the polar dual polytope ∆ * by where , is a natural pairing N × M → Z, and , R is the extension to R-coefficients. Let ∆ be an integral polytope that contains the origin in its interior as the only lattice point. The polytope ∆ is reflexive if the polar dual ∆ * is also an integral polytope.
Recall an interesting property of reflexive polytopes related to K3 surfaces due to Batyrev [1]: [1] Denote by P∆ the toric 3-fold associated to an integral polytope ∆. The following conditions are equivalent.
In particular, the weighted projective space P(a) with weight system a = (a0, a1, a2, a3) in Yonemura's list is a toric Fano 3-fold determined by a reflexive polytope ∆ (n) in the R-extension of the lattice where the weight system a is assigned No. n in Yonemura's list. The anticanonical sections are weighted homogeneous polynomial of degree d := a0 + a1 + a2 + a3, thus, there is a one-to-one correspondence between a lattice point (i, j, k, l) in Mn and a rational monomial W i+1 X j+1 Y k+1 Z l+1 . In this way, once a Z-basis is taken for Mn, we identify lattice points in ∆ (n) and monomials of weighted degree d.

Main Result
In this section, we prove the main theorem.   Table 1, the reflexive polytopes ∆ and ∆ ′ are given as a set of monomials that are vertices of them. If there are more than one pairs, they are separated by a dotted line and polytopes in the same row give the polytope duality.
Proof. Take polynomials F and F ′ that are respectively anticanonical sections of the weighted projective spaces P(a) and P(b) as in Table 1 in each case.
Recall that a pair of reflexive polytopes ∆ and The strategy of the proof is that in each case, after taking a basis of the lattice Mn, we observe if the Newton polytope ∆F of the polynomial F is reflexive by a direct computation. If the polytope is not reflexive, then, we search a reflexive polytope ∆ satisfying inclusions ∆F ⊂ ∆ ⊂ ∆ (n) of polytopes. The analogous observations should be made for F ′ . Once one gets a candidate reflexive polytope ∆ and ∆ ′ , we then study whether they satisfy a relation ∆ * ≃ ∆ ′ .
The assertion is proved case by case.
In order that∆ should be reflexive, one has a = b = c = −1, that is,∆ should be ∆ (14) .

No. 15-No. 18
We claim that there exists a unique polytope-dual pair for Nos. 17, 18, and that two pairs for Nos. 15, 16. Take a basis {e

No. 20
We claim that there exist two polytope-dual pairs. Take a basis {e

No. 22-No. 24
We claim that there exist a unique polytope-dual pair for Nos. 22 and 23, and that none for No. 24. Take a basis {e gives an isomorphism from ∆2 to ∆ * 1 , the relation ∆ * 1 ≃ ∆2 holds. In the polytope ∆ (71) , the vertex (−1, −1, −1) is adjacent to three other vertices: vertex (−1, −1, 1) with an edge e1, vertex (−1, 2, −1) with an edge e2, and vertex (4, −1, −1) with an edge e3. On the edges e1, e2, e3, respectively, there are one, two, and four inner lattice points. Thus, the polar dual polytope ∆ (71) * must contain a triangle as a two-dimensional face that is adjacent to other two-dimensional faces with inner lattice points one, two, and four. However, it is easily observed that there is no such a configuration in the polytope ∆ (71) * . Thus, ∆ (71) is not self-dual, and there does not exist a polytope-dual pair.

No. 26
We claim that there exist four polytope-dual pairs. Take a basis {e

No. 30
We claim that there exist two polytope-dual pairs. Take a basis {e

No. 31
We claim that there exist four polytope-dual pairs. Take a basis {e

No. 32-No. 34
We claim that there exist two polytope-dual pairs for Nos. 32 to 34. Take a basis {e

No. 46-No. 47
We claim that there exists a unique polytope-dual pair for Nos. 46 and 47. Take a basis {e