Picard Lattices of Families of K3 Surfaces

Abstract

Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.

Lattice duality for families of K3 surfaces associated to transpose duality

“…Closing RemarksAs is mentioned in[5], almost all mirror symmetric pairs of weight systems in the sense of[2] are also strongly coupled. We would like to study not only the full families of K3 surfaces, but families of K3 surfaces associated to reflexive polytopes ∆ and ∆ ′ obtained in Theorem 5.1.…”

Calabi-Yau manifolds by Borcea-Voisin method

“…3 by ρ. Note that all K3 surfaces in table 2 are all elliptic fibered [25]. The fourth column is the lattice invariants (r, a, δ).…”

“…For K3 cases, Belcastro gave some more improved ways of calculating Pic (K3), finding elliptic fibrations[25]. Her ways are composite ones of using toric varieties and some theorems of algebraic geometry.…”

The Geometry and Moduli of K3 Surfaces

“…A large number of K3 surfaces of this form were found by Reid [55], as toric resolutions of hypersurfaces in weighted projective spaces (the full list is given in [27,Sect. 13.3]); the induced lattice polarizations on them were computed by Belcastro [9]. Further examples of this type have been computed by Rohsiepe [57][58].…”