# Instability and Singularity of Projective Hypersurfaces

## TL;DR

This paper investigates the properties of projective hypersurfaces, demonstrating the invariance of Hesselink stratification and establishing a link between virtual 1-parameter subgroup length and hypersurface multiplicity.

## Contribution

It proves the independence of Hesselink stratification from Plücker coordinates and relates the worst virtual 1-parameter subgroup length to hypersurface multiplicity.

## Key findings

- Hesselink stratification is coordinate-independent.
- A positive correlation exists between virtual subgroup length and multiplicity.
- The results enhance understanding of hypersurface stability and singularity.

## Abstract

In this paper, we will show that the Hesselink stratification of a Hilbert scheme of hypersurfaces is independent of the choice of Pl\"ucker coordinate and there is a positive relation between the length of Hesselink's worst virtual 1-parameter subgroup and multiplicity of a projective hypersurface.