Height functions on singular surfaces parameterized by smooth maps $\mathcal{A}$-equivalent to $S_k$, $B_k$, $C_k$ and $F_4$

TL;DR

This paper classifies and analyzes the singularities of height functions on certain singular surfaces in three-dimensional space, using geometric techniques like blowing-ups, and explores their dual surfaces.

Contribution

It provides a detailed description of height function singularities on surfaces with specific singularities, extending geometric analysis methods to these cases.

Findings

Classification of height function singularities for $S_k$, $B_k$, $C_k$, $F_4$ surfaces

Analysis of dual surface singularities

Use of blowing-up techniques for geometric description

Abstract

We describe singularities of height functions on singular surfaces in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to one of $S_k$, $B_k$, $C_k$ and $F_4$ singularities in terms of extended geometric language via finite succession of blowing-ups. We investigate singularities of dual surfaces of such singular surfaces.