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2015 | OriginalPaper | Chapter

Picard Ranks of K3 Surfaces of BHK Type

Author: Tyler L. Kelly

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

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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.
Literature
1.
go back to reference Artebani, M., Boissière, S., Sarti, A.: The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces. J. Math. Pure. Appl. 102, 758–781 (2014) CrossRefMATH Artebani, M., Boissière, S., Sarti, A.: The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces. J. Math. Pure. Appl. 102, 758–781 (2014) CrossRefMATH
2.
go back to reference Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nucl. Phys. B393, 377–391 (1993) CrossRef Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nucl. Phys. B393, 377–391 (1993) CrossRef
6.
go back to reference Comparin, P., Lyons, C., Priddis, N., Suggs, R.: The mirror symmetry of K3 surfaces with non-symplectic automorphisms of prime order. Adv. Theor. Math. Phys. 18(6), 1335–1368 (2014) MathSciNetCrossRef Comparin, P., Lyons, C., Priddis, N., Suggs, R.: The mirror symmetry of K3 surfaces with non-symplectic automorphisms of prime order. Adv. Theor. Math. Phys. 18(6), 1335–1368 (2014) MathSciNetCrossRef
7.
go back to reference Degtyarev, A.: On the Picard group of a Delsarte surface. arxiv: 1307.0382 (2013) Degtyarev, A.: On the Picard group of a Delsarte surface. arxiv: 1307.0382 (2013)
8.
go back to reference Dimca, A.: Singularities and coverings of weighted complete intersections. J. Reine. Agnew. Math. 366, 184–193 (1986) MathSciNetMATH Dimca, A.: Singularities and coverings of weighted complete intersections. J. Reine. Agnew. Math. 366, 184–193 (1986) MathSciNetMATH
10.
go back to reference Goto, Y.: K3 surfaces with symplectic group actions. In: Calabi-Yau Varieties and Mirror Symmetry. Fields Institute Communications, vol. 38, pp. 167–182. American Mathematical Society, Providence (2003) Goto, Y.: K3 surfaces with symplectic group actions. In: Calabi-Yau Varieties and Mirror Symmetry. Fields Institute Communications, vol. 38, pp. 167–182. American Mathematical Society, Providence (2003)
13.
go back to reference Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. arXiv: 0906.0796 Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. arXiv: 0906.0796
16.
go back to reference Shioda, T.: An explicit algorithm for computing the Picard number of certain algebraic surfaces. Am. J. Math. 108(2), 415–432 (1986) MathSciNetCrossRefMATH Shioda, T.: An explicit algorithm for computing the Picard number of certain algebraic surfaces. Am. J. Math. 108(2), 415–432 (1986) MathSciNetCrossRefMATH
17.
go back to reference Tate, J.: Algebraic cycles and poles of zeta functions. In: Arithmetical Algebraic Geometry (Proceedings Conference Purdue University, 1963), pp. 93–110. Harper and Row, New York (1965) Tate, J.: Algebraic cycles and poles of zeta functions. In: Arithmetical Algebraic Geometry (Proceedings Conference Purdue University, 1963), pp. 93–110. Harper and Row, New York (1965)
Metadata
Title
Picard Ranks of K3 Surfaces of BHK Type
Author
Tyler L. Kelly
Copyright Year
2015
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_2

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