Lines on the Dwork pencil of quintic threefolds

Abstract

We present an explicit parameterization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves $\widetilde{C}_{\pm \varphi}$, which parameterize the lines. These curves are 125:1 covers of genus six curves $C_{\pm \varphi }$. The $C_{\pm \varphi}$ are first presented as curves in $\mathbb{P}1 \times \mathbb{P}1$ that have three nodes. It is natural to blow up $\mathbb{P}1 \times \mathbb{P}1$ in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up $\mathbb{P}1 \times \mathbb{P}1$ in three points is the quintic del Pezzo surface $dP_5$, whose automorphism group is the permutation group $S_5$, which is also a symmetry of the pair of curves $C_{\pm \varphi }$. The subgroup $A_5$, of even permutations, is an automorphism of each curve, whereas the odd permutations interchange $C_{\varphi}$ with $C_{- \varphi }$. The ten exceptional curves of $dP_5$ each intersect the $C_{\varphi}$ in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the curves $C_{\varphi}$ are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold, the curve $C_{\varphi}$ develops six nodes and may be resolved to a $\mathbb{P}1$. The group $A_5$ acts on this $\mathbb{P}1$ and we describe this action.