Top

Published in:

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

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter The Geometry and Moduli of K3 Surfaces

next chapter Reflexive Polytopes and Lattice-Polarized K3 Surfaces

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

Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nucl. Phys. B393, 377–391 (1993)CrossRef

Bini, G.: Quotients of hypersurfaces in weighted projective space. Adv. Geom. 11(4), 653–668 (2011)MathSciNetCrossRefMATH

Bruzzo, U., Grassi, A.: Picard group of hypersurfaces in toric 3-folds. Int. J. Math. 23(2), 14 (2012)MathSciNetCrossRef

Chiodo, A., Ruan, Y.: LG/CY correspondence: the state space isomorphism. Adv. Math. 227(6), 2157–2188 (2011)MathSciNetCrossRefMATH

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

Degtyarev, A.: On the Picard group of a Delsarte surface. arxiv: 1307.0382 (2013)

Dimca, A.: Singularities and coverings of weighted complete intersections. J. Reine. Agnew. Math. 366, 184–193 (1986)MathSciNetMATH

Dimca, A., Dimiev, S.: On analytic coverings of weighted projective spaces. Bull. Lond. Math. Soc. 17, 234–238 (1985)MathSciNetCrossRefMATH

10.

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)

11.

Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B 338, 15–37 (1990)MathSciNetCrossRefMATH

12.

Kelly, T.L.: Berglund-Hübsch-Krawitz mirrors via Shioda maps. Adv. Theor. Math. Phys. 17(6), 1425–1449 (2013)MathSciNetCrossRefMATH

13.

Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. arXiv: 0906.0796

14.

Schütt, M.: Picard numbers of quintic surfaces. Proc. Lond. Math. Soc. 110(2), 428–476 (2015)MathSciNetCrossRef

15.

Shioda, T.: The Hodge conjecture for Fermat varieties. Math. Ann. 245(2), 175–184 (1979)MathSciNetCrossRefMATH

16.

Shioda, T.: An explicit algorithm for computing the Picard number of certain algebraic surfaces. Am. J. Math. 108(2), 415–432 (1986)MathSciNetCrossRefMATH

17.

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)

Title: Picard Ranks of K3 Surfaces of BHK Type
Author: Tyler L. Kelly
Publisher: Springer New York
Book: Calabi-Yau Varieties: Arithmetic, Geometry and Physics
Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4939-2830-9_2

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner