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K3 and Enriques Surfaces Read chapter Read first chapter

2013 | OriginalPaper | Chapter

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

Authors : Dagan Karp, Jacob Lewis, Daniel Moore, Dmitri Skjorshammer, Ursula Whitcher

Published in: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Publisher: Springer New York

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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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previous chapter Fourier–Mukai Partners and Polarised Surfaces
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Metadata
Title
On a Family of K3 Surfaces with Symmetry
Authors
Dagan Karp
Jacob Lewis
Daniel Moore
Dmitri Skjorshammer
Ursula Whitcher
Copyright Year
2013
Publisher
Springer New York
Book
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Print ISBN: 978-1-4614-6402-0

Electronic ISBN: 978-1-4614-6403-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-6403-7_12

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