On a Family of K3 Surfaces with S₄ Symmetry

Authors

Dagan Karp, Harvey Mudd CollegeFollow
Jacob Lewis, Universitat Wien
Daniel J. Moore '11, Harvey Mudd CollegeFollow
Dmitri Skjorshammer '11, Harvey Mudd CollegeFollow
Ursula Whitcher, University of Wisconsin - Eau Claire

Student Co-author

HMC Undergraduate

Document Type

Article

Department

Mathematics (HMC)

Publication Date

2013

Abstract

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S₄. 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 S₄ 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.

Rights Information

© 2013 Springer

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Terms of Use for work posted in Scholarship@Claremont.

DOI

10.1007/978-1-4614-6403-7_12

Recommended Citation

Karp, Dagan, Jacob Lewis, Daniel Moore, Dmitri Skjorshammer and Ursula Whitcher. "On a family of K3 surfaces with S 4 symmetry." in Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Part II. pp. 367-386. Springer - New York, 2013.