Lines on the Dwork Pencil of Quintic Threefolds

TL;DR

This paper explicitly parametrizes lines on the Dwork pencil of quintic threefolds, revealing intricate geometric structures involving genus six curves, del Pezzo surfaces, and symmetries related to the permutation group S_5.

Contribution

It provides an explicit parametrization of lines on the Dwork pencil, connecting genus six curves, del Pezzo surfaces, and symmetry groups, and explores singular cases.

Findings

Genus six curves are 125:1 covers of simpler curves.

The curves are related to the Wiman pencil and del Pezzo surface dP_5.

Symmetries include the permutation group S_5 and its subgroup A_5.

Abstract

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 symmetry of the pair of genus six curves. The subgroup A_5, of even permutations, is an automorphism of each curve, while the odd permutations interchange the two curves. The ten exceptional curves of dP_5 each intersect each of the genus six curves in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the genus six curves 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 genus six curves develop six nodes and may be resolved to a P^1. The group A_5 acts on this P^1 and we describe this action.