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 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.