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.
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Acknowledgements
The author would like to thank Antonella Grassi, Nathan Priddis, Alessandra Sarti, and Noriko Yui for their conversations on this subject in the context of K3 surfaces. He would like to give special thanks to his advisor, Ron Donagi, for his support, mentoring and conversations during this time period. The author would also like to thank the referee for helpful comments. The author would like to thank the Fields Institute for its hospitality, as portions of this work was done while at its Thematic Program on Calabi-Yau Varieties. This work was done under the support of a National Science Foundation Graduate Research Fellowship.
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Kelly, T.L. (2015). Picard Ranks of K3 Surfaces of BHK Type. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2830-9_2
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DOI: https://doi.org/10.1007/978-1-4939-2830-9_2
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