# The Aluffi Algebra of a hypersurface with isolated singularity

## TL;DR

This paper investigates the properties of the Aluffi algebra associated with hypersurfaces having isolated singularities, establishing conditions under which certain ideals are of linear type, with implications for algebraic geometry and intersection theory.

## Contribution

It characterizes when the Jacobian and gradient ideals of hypersurfaces are of linear type, linking this to the locally Eulerian property, especially for isolated singularities.

## Key findings

- Jacobian ideal of affine hypersurface with isolated singularities is of linear type iff locally Eulerian.
- Gradient ideal of a projective hypersurface is of linear type iff the affine curve at singular points is locally Eulerian.
- Gradient ideals of Nodal and Cuspidal plane curves are of linear type.

## Abstract

The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities. We prove that the Jacobian ideal of an affine hypersurfac with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface is of linear type if and only if the corresponding affine curve in the affine chart associated to singular points is locally Eulerian. We prove that the gradient ideal of the Nodal and Cuspidal projective plane curves are of linear type.