The local cohomology of the jacobian ring

We study the 0-th local cohomology module of the jacobian ring of a singular reduced complex projective hypersurface X, by relating it to the sheaf of logarithmic vector field along X. We investigate the analogies between the local cohomology and the well known properties of the jacobian ring of a nonsingular hypersurface. In particular we study self-duality, Hodge theoretic and Torelli type questions.