Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 6

Lines on the Dwork pencil of quintic threefolds

Pages: 1779 – 1836

DOI: https://dx.doi.org/10.4310/ATMP.2012.v16.n6.a4

Authors

Philip Candelas (Mathematical Institute, University of Oxford, United Kingdom)

Bert van Geemen (Dipartimento di Matematica, Università di Milano, Italy)

Xenia de la Ossa (Mathematical Institute, University of Oxford, United Kingdom)

Duco van Straten (Algebraische Geometrie, Johannes Gutenberg-Universität, Mainz, Germany)

Abstract

We present an explicit parameterization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves C~±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, whereas the odd permutations interchange Cφ with Cφ. The ten exceptional curves of dP5 each intersect the Cφ in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the curves Cφ 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 Cφ develops six nodes and may be resolved to a P1. The group A5 acts on this P1 and we describe this action.

Published 28 May 2013