Quotients of the Dwork Pencil

Dwork曲线和镜像五次曲线中的一次曲线

Degree 1 Curves in the Dwork Pencil and the Mirror Quintic

本文给出了五次三重Dwork束中直线的相对Hilbert概型的描述。我们描述了相应的相对希尔伯特计划相关联的五次三重镜像家庭。

Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetr…

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.

Kummer sandwiches and Greene-Plesser construction

In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in $\mathbb{P}^3$, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its ge…

二维和三维中McKay对应之间多余纤维的收缩

Contraction of excess fibres between the McKay correspondences in dimensions two and three

从多面体群和相应的二元多面体群中得到的二维和三维商奇点，承认奇点的自然分解为正则轨道的希尔伯特方案，其在原点上的例外纤维揭示了类似的性质。我们在这两个分辨率之间构造一个态射，正好收缩例外纤维的多余部分。这种结构的动机是研究的一些铅笔的K3-表面所产生的最小分辨率的节点表面的平行分量具有高的对称性。

Geometry of the Wiman-Edge pencil and the Wiman curve

The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $Γ\backslash \Hf$ of the hyperbolic plane $\Hf$, where $Γ<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpret…