# Irreducibility of analytic arc-sections of hypersurface singularities

## TL;DR

This paper investigates the conditions under which hypersurface singularities admit irreducible arc-sections, providing examples, an algorithm for checking their existence, and demonstrating that reducible arc-sections always exist.

## Contribution

It introduces an algorithm to determine the existence of irreducible arc-sections in hypersurface singularities and provides examples where such sections do not exist.

## Key findings

- Some hypersurface singularities have no irreducible arc-sections.
- Reducible arc-sections always exist for these singularities.
- An algorithm can determine the existence of irreducible arc-sections.

## Abstract

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no irreducible arc-sections exist, and show that reducible ones always exist. Moreover, we give an algorithm to check if a given projection allows irreducible arc-sections, and find them if they exist.