Fast loops on semi-weighted homogeneous hypersurface singularities

TL;DR

This paper investigates the existence of fast loops in semi-weighted homogeneous hypersurface singularities and establishes conditions under which these singularities possess a metrical conical structure, based on their weights.

Contribution

It demonstrates the existence of specific fast loops in semi-weighted homogeneous hypersurface singularities and characterizes when these singularities have a metrical conical structure.

Findings

Existence of (1 + w_2/w_3)-fast loops in semi-weighted homogeneous hypersurface singularities.

Metrical conical structure occurs only when the two lowest weights are equal.

Provides conditions linking weights to geometric properties of singularities.

Abstract

We show the existence of ($1+\frac{w_2}{w_3}$)-fast loops on semi-weighted homogeneous hypersurface singularities with weights $w_1\geq w_2>w_3$. In particular we show that semi-weighted homogeneous hypersurface singularities have metrical conical structure only if its two low weights are equal.