Isomorphism among Families of Weighted $K3$ Hypersurfaces

Abstract

Some of the 95 families of weighted K3 hypersurfaces have been known to have the isometric lattice polarizations. It is shown that weighted K3 hypersurfaces in such families are to one-to-one correspond by explicitly constructing the monomial birational morphisms among the weighted projective spaces. All the weight systems having the isometric Picard lattices commonly possess an anticanonical sublinear system, being confirmed that the Picard lattice of the sublinear system we obtained is the same as those of the complete linear systems.

Lattice duality for families of K3 surfaces associated to transpose duality

“…In [8], it is proved that the polytopes ∆ (n) for n = 14, 28, 45, and 51 are isomorphic to the polytope ∆ (14) that is the convex hull of vertices (−1, −1, 1), (−1, −1, −1), (6, −1, −1), and (−1, 2, −1) under the above choice of basis. Since the polar dual ∆ (14) * is the convex hull of vertices (1, 0, 0), (0, 1, 0), (0, 0, 1), and (−6, −14, −21), the linear map of R 3 defined by a matrix…”

“…In [8], it is proved that the polytopes ∆ (38) and ∆ (77) , and ∆ (50) , and ∆ (82) are respectively isomorphic to the polytopes…”

“…In [8], it is proved that the polytopes ∆ Since the polar dual polytopes ∆ * 1 and ∆ * 2 of ∆1 and ∆2 are the convex hulls of vertices (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, −2, −3), and (−6, −8, −9), respectively, (0, 1, 0), (0, 0, 1), (2, 0, −1), and (−6, −8, −9), and the linear maps of R 3 determined by a matrix gives isomorphisms from ∆ ′ 1 to ∆ * 1 and from ∆ ′ 2 to ∆ * 2 , the relations ∆ * 1 ≃ ∆ ′ 1 and ∆ * 2 ≃ ∆ ′ 2 hold. No.…”

“…28 and 29. Take a basis {e In [8], it is proved that the polytopes ∆ (27) and ∆ (49) are isomorphic to the polytope ∆ (27,49) with vertices (0, 0, 1), (1, It is straight-forward to see that the relations ∆ *…”

“…32 to 34. Take a basis {e In [8], it is proved that the polytopes ∆ (26) and ∆ (34) are isomorphic to the polytope with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1), and (−4, −5, −10). Since the polar dual polytopes ∆ * 1 and∆ * 1 of ∆1 and∆1 are the convex hulls of vertices (−1, −1, 1), (−1, −1, −1), (4, −1, −1), and (−1, 4, −1), respectively, (−1, −1, −1), (0, 1, 1), (0, −4, 1), and (5, 11, 1), and the lin-…”