Schiffer variations and the generic Torelli theorem for hypersurfaces.

We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$ , for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call 'Schiffer variations.' We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions. [ABSTRACT FROM AUTHOR]