On a Family of K3 Surfaces with S₄ Symmetry

Student Co-author

HMC Undergraduate

Document Type



Mathematics (HMC)

Publication Date



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.

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