  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 $e=1$.
