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On a Family of K3 Surfaces with \(\mathcal{S}_{4}\) Symmetry

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Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Part of the book series: Fields Institute Communications ((FIC,volume 67))

Abstract

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S 4. There are three pairs of three- dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S 4 acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard–Fuchs equation for the third Picard rank 19 family by extending the Griffiths–Dwork technique for computing Picard–Fuchs equations to the case of semi-ample hypersurfaces in toric varieties. The holomorphic solutions to our Picard–Fuchs equation exhibit modularity properties known as “Mirror Moonshine”; we relate these properties to the geometric structure of our family.

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Acknowledgements

The second author was supported by the NSF Grant No. OISE-0965183. The third, fourth, and fifth authors were supported in part by No. DMS-0821725. The authors thank Andrey Novoseltsev for thoughtful discussion of computational techniques and Charles Doran for inspirational conversations.

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Authors and Affiliations

  1. Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA, 91711, USA

    Dagan Karp, Daniel Moore & Dmitri Skjorshammer

  2. Fakultät für Mathematik, Universität Wien, Garnisongasse 3/14, 1090, Wien, Austria

    Jacob Lewis

  3. Department of Mathematics, University of Wisconsin–Eau Claire, 508 Hibbard Humanities Hall, Eau Claire, WI, 54702, USA

    Ursula Whitcher

Authors
  1. Dagan Karp
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  2. Jacob Lewis
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  3. Daniel Moore
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  4. Dmitri Skjorshammer
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  5. Ursula Whitcher
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Correspondence to Ursula Whitcher .

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Editors and Affiliations

  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

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Karp, D., Lewis, J., Moore, D., Skjorshammer, D., Whitcher, U. (2013). On a Family of K3 Surfaces with \(\mathcal{S}_{4}\) Symmetry. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_12

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