# Hypersurfaces with linear type singular loci

## TL;DR

This paper establishes criteria for when the Jacobian ideal of a hypersurface with isolated singularities is of linear type, demonstrating that certain plane curves with simple singularities and quartic curves possess this property.

## Contribution

It provides necessary and sufficient conditions for the Jacobian ideal to be of linear type and proves that specific classes of curves have this property.

## Key findings

- Gradient ideal of reduced projective plane curves with ADE singularities is of linear type.
- All reduced projective quartic curves are of gradient linear type.

## Abstract

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type, are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ($\mathrm{ADE}$) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.