Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parameterization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves C̃ ± φ, which parameterize the lines. These curves are 125:1 covers of genus six curves C± φ. The C± φ are first presented as curves in P1×P1 that have three nodes. It is natural to blow up P1×P1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P1×P1 in three points is the quintic del Pezzo surface dP5, whose automorphism group is the permutation group S5, which is also a symmetry of the pair of curves C± φ. The subgroup A5, of even permutations, is an automorphism of each curve, where as the odd permutations.