2015 | OriginalPaper | Chapter
Picard Ranks of K3 Surfaces of BHK Type
Published in:
Calabi-Yau Varieties: Arithmetic, Geometry and Physics
Abstract
We give an explicit formula for the Picard ranks of K3 surfaces that have Berglund-Hübsch-Krawitz (BHK) Mirrors over an algebraically closed field, both in characteristic zero and in positive characteristic. These K3 surfaces are those that are certain orbifold quotients of weighted Delsarte surfaces. The proof is an updated classical approach of Shioda using rational maps to relate the transcendental lattice of a Fermat hypersurface of higher degree to that of the K3 surfaces in question. The end result shows that the Picard ranks of a K3 surface of BHK-type and its BHK mirror are intrinsically intertwined. We end with an example of BHK mirror surfaces that, over certain fields, are supersingular.
