2011/03/09 by Dagan Karp, Jacob S. Lewis, Jacob Lewis +8
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Meromorphic and Entire Functions #math.AG
paper · pdf · doi:10.48550/arxiv.1103.1892
13 pages, 2 figures. Updated references. Comments are welcome!
arxiv created 2011/05/31 · arxiv updated 2011/06/02
The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is the symmetric group on four elements. 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 the symmetric group on four elements 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.