Picard ranks of \(K3\) surfaces of BHK type. (English) Zbl 1329.14077
Laza, Radu (ed.) et al., Calabi-Yau varieties: arithmetic, geometry and physics. Lecture notes on concentrated graduate courses, Toronto, Canada, July 1 – December 31, 2013. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer (ISBN 978-1-4939-2829-3/hbk; 978-1-4939-2830-9/ebook). Fields Institute Monographs 34, 45-63 (2015).
Summary: 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.
For the entire collection see [Zbl 1329.14008].
For the entire collection see [Zbl 1329.14008].
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
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