The Alu algebra of hypersurfaces with isolated singularities

The Alu algebra is an algebraic definition of a characteristic cycle of a hypersurface in intersection theory. In this paper, we study the Alu algebra of quasi-homogeneous and locally Eulerian hypersurfaces with only isolated singularities. We prove that the Jacobian ideal of an ane hypersurface 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 with only isolated singularities is of linear type if and only if the ane curve in each ane chart associated to singular points is locally Eulerian. We show that the gradient ideal of Nodal and Cuspidal projective plane curves are of linear type.