  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.
