Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a family of K3 surfaces with $\mathcal{S}_4$ symmetry   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.

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The categories of this paper are: cs
16
The categories of this paper are: stat
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom