Ursula Whitcher

Short Tops and Semistable Degenerations

One may construct a large class of Calabi–Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi–Yau varieties are fibered by lower-dimensional Calabi–Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classification of reflexive polytopes, there are infinite families of equivalence classes of tops. Tops may be used to describe either fibrations or degenerations of Calabi–Yau varieties. We give a simple combinatorial condition on tops that produces semistable degenerations of K3 surfaces and, when appropriate smoothness conditions are met, semistable degenerations of Calabi–Yau threefolds. Our method is constructive: given a fixed reflexive polytope that will lie on the boundary of the top, we describe an algorithm for constructing tops that yields semistable degenerations of the corresponding hypersurfaces. The properties of each degeneration may be computed directly from the combinatorial structure of the top.

Zeta functions of alternate mirror Calabi–Yau families

We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.

From Polygons to String Theory

Summary We describe special kinds of polygons, called Fano polygons or reflexive polygons, and their higher dimensional generalizations, called reflexive polytopes. Pairs of reflexive polytopes are related by an operation called polar duality. This combinatorial relationship has a deep and surprising connection to string theory: One may use reflexive polytopes to construct “mirror” pairs of geometric spaces called Calabi-Yau manifolds that could represent extra dimensions of the universe. Reflexive polytopes remain a rich source of examples and conjectures in mirror symmetry.

Classifications of Elliptic Fibrations of a Singular K3 Surface

We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to \(\langle 6\rangle \oplus \langle 2\rangle\).

Reflexive Polytopes and Lattice-Polarized K3 Surfaces

We review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for K3 surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of K3 surfaces with high Picard rank.

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 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.