  We show how to recover a general hypersurface in $\mathbb{P}^n$ of
sufficiently large degree $d$ dividing $n+1$, from its finite order variation
of Hodge structure. We also analyze the two other series of cases not covered
by Donagi's generic Torelli theorem. Combined with Donagi's theorem, this shows
that the generic Torelli theorem for hypersurfaces holds with finitely many
exceptions.
