Height functions on a cuspidal edge with a distinguished singular curve

Abstract

We classify submersions on a cuspidal edge $M$ in ${\mathbb{R}}^3$ with a distinguished singular curve $S$ in it. We use this classification to study the contact of the pair $(M,S)$ with planes, which is measured by the singularities of the height functions on $(M,S)$. The discriminants of the versal deformations of the submersions obtained are also described.