Proof of the oval conjecture for proper planar partition surfaces

Abstract

We prove the ‘oval conjecture’ for planar partition functions, which says that the shift plane and the translation plane defined by a planar partition function form an oval pair of planes in the sense that each non-vertical line of one plane defines a topological oval in the projective closure of the other. The proof uses covering space techniques, and we have to assume that the generating function is proper in order to make those techniques available. As an application, we give a natural geometric construction of a homeomorphism between the Cartesian square of the shift line and its tangent bundle.