A generalization of Griffiths theorem on rational integrals, II

We show that the Hodge and pole order filtrations are globally different for sufficiently general singular projective hypersurfaces in case the degree is 3 or 4 assuming the dimension of the projective space is at least 5 or 3 respectively. We then study an algebraic formula for the global Hodge filtration in the ordinary double point case conjectured by the third named author. This is more explicit and easier to calculate than the previous one in this case. We prove a variant of it under the assumption that the image of the singular points by the $e$-fold Veronese embedding consists of linearly independent points, where $e$ is determined only by the dimension and the degree. In particular, the original conjecture is true in case the above condition is satisfied for $e=1$.