# On the geometry of folded cuspidal edges

## TL;DR

This paper investigates the geometry of cuspidal $S_k$ singularities in 3D, focusing on the cuspidal cross-cap, analyzing its invariants, and classifying its contact with planes to understand its geometric properties.

## Contribution

It introduces a detailed study of the invariants of cuspidal cross-caps and classifies their contact with planes, advancing understanding of their geometric structure.

## Key findings

- Invariants determine the cuspidal cross-cap up to order 5.
- Classification of plane contact reveals geometric properties.
- Relation between singularities of height functions and invariants.

## Abstract

We study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study geometrical invariants associated to $M$ and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.